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Quantum entanglement

Quantum entanglement is a fundamental quantum mechanical phenomenon in which two or more particles become correlated such that the quantum state of each particle cannot be described independently, even when separated by large distances, leading to instantaneous correlations upon measurement that defy classical intuitions of locality.^[1] However, these correlations do not allow for faster-than-light communication, as the no-signaling principle ensures that no information can be transmitted instantaneously.^[1] Quantum entanglement, often described by Albert Einstein as "spooky action at a distance" in his skeptical critique of quantum non-locality, arises when particles interact and their wave functions combine into a single entangled state, where measuring a property of one particle immediately reveals the corresponding property of the other, regardless of the separation between them.^[2] The concept was first highlighted in the 1935 Einstein-Podolsky-Rosen (EPR) thought experiment, which questioned the completeness of quantum mechanics by arguing that such correlations implied either faster-than-light influences or incomplete physical descriptions.^[3] For instance, in the EPR thought experiment, two entangled particles have opposite spins, so measuring one as 'up' instantly determines the other as 'down.'^[4] In 1964, physicist John Bell formulated Bell's theorem, which provided a testable prediction distinguishing quantum entanglement from local hidden variable theories; subsequent experiments, starting with those by John Clauser and Alain Aspect in the 1970s and 1980s, confirmed quantum predictions by violating Bell inequalities, solidifying entanglement as a real and non-local feature of nature. This work was recognized by the 2022 Nobel Prize in Physics awarded to Clauser, Aspect, and Anton Zeilinger.^[5]^[6]^[7] Entanglement exhibits key characteristics such as maximal violation of classical correlations for certain particle pairs like photons or electrons, and it preserves coherence only under specific conditions, making it fragile to environmental decoherence.^[8] Examples include entangled photon pairs produced via spontaneous parametric down-conversion, where polarization measurements on one photon correlate perfectly with the other.^[9] Quantum entanglement underpins modern quantum technologies, serving as a resource for quantum computing, where it enables operations like quantum gates on qubits to achieve exponential speedup over classical systems, and quantum communication, including protocols for secure key distribution via quantum key distribution (QKD) and quantum teleportation of information without physical transfer.^[8] Recent advances include the first entanglement of individual molecules in 2023, opening pathways for scalable quantum networks, and NASA's SEAQUE experiment, deployed in 2024, which tests entanglement sources and self-healing technologies in space environments to assess persistence against radiation.^[10]^[11] These developments highlight entanglement's role not only in foundational physics but also in emerging applications like quantum sensing and simulation of complex systems.^[12]

History

Origins and early concepts

The development of quantum mechanics in the 1920s brought to light fundamental issues concerning the description of physical systems, setting the stage for the concept of entanglement. At the 1927 Solvay Conference on Electrons and Photons, Albert Einstein raised doubts about the completeness of quantum mechanics, arguing that its probabilistic nature might overlook underlying deterministic elements of reality. Niels Bohr, defending the Copenhagen interpretation, countered that the theory's statistical predictions were inherently complete for observable phenomena, though this sparked ongoing debates about the role of measurement and the wave function in determining system states. These exchanges, involving key figures like Werner Heisenberg and Max Born, underscored tensions between classical separability and quantum interconnectedness, without yet explicitly addressing correlated distant systems. Early mathematical frameworks for quantum mechanics incorporated descriptions of composite systems, enabling the representation of states that defied independent characterization. Werner Heisenberg's matrix mechanics, initially formulated in 1925 and extended to multi-particle interactions by the late 1920s, treated joint systems through non-commuting operators, implicitly allowing for correlated behaviors in atomic and molecular contexts. Similarly, Paul Dirac's 1927 transformation theory and his 1930 book The Principles of Quantum Mechanics formalized composite systems using the direct product of Hilbert spaces, where the total wave function could not always factorize into separate functions for each subsystem, laying the groundwork for inseparable quantum descriptions. These developments prioritized computational tools for spectroscopy and atomic structure over interpretive implications.^[13] The term "entanglement" was coined by Erwin Schrödinger in 1935 to encapsulate the peculiar inseparability of quantum states in composite systems following interaction. In his seminal paper "Discussion of Probability Relations between Separated Systems," Schrödinger described how two particles, after temporary interaction, possess a joint wave function—such as one correlating their positions and momenta—that cannot be expressed as a product of individual states, even if spatially separated. He emphasized: "When two different systems enter into temporary physical interaction... the two systems are no longer independent," highlighting the non-local correlations this implies. This formulation played a central role in the Copenhagen interpretation debates between Bohr and Einstein, particularly regarding wave function collapse during measurement, as it illustrated how observing one subsystem instantaneously affects the description of the other, challenging classical notions of locality and independence.

EPR paradox and debates

In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen published a seminal paper challenging the completeness of quantum mechanics through a thought experiment involving two entangled particles.^[14] They proposed criteria for physical reality, stating that if, without disturbing a system, the value of a physical quantity can be predicted with certainty, then there exists an element of physical reality corresponding to that quantity.^[14] Applying this to quantum mechanics, they considered two particles in an entangled state where measuring the position of one precisely determines the position of the other, regardless of distance, while a similar perfect correlation holds for momenta.^[14] Einstein, Podolsky, and Rosen argued that since quantum mechanics allows simultaneous predictions of both position and momentum for the distant particle—outcomes that would violate the Heisenberg uncertainty principle if the particles had definite values independently—the theory must be incomplete, requiring hidden variables to describe underlying physical realities.^[14] This setup highlighted what they termed "spooky action at a distance," suggesting that quantum mechanics fails to provide a local, realistic description of such systems.^[14] The paper, received by Physical Review on March 25, 1935, and published on May 15, 1935, ignited immediate philosophical debates on the foundations of quantum theory.^[14] Niels Bohr responded in a paper published in Physical Review on October 1, 1935, defending the completeness of quantum mechanics through his principle of complementarity. Bohr contended that the EPR criteria for reality presupposed an independent, classical notion of measurement inapplicable to quantum systems, where the act of measurement on one particle defines the context for the other's properties, rendering them non-simultaneously measurable due to wave-particle duality. He emphasized that quantum descriptions do not attribute definite values to unmeasured observables, avoiding any paradox by rejecting the assumption of undisturbed predetermination. Erwin Schrödinger, in a contemporaneous paper published in the Proceedings of the Cambridge Philosophical Society in October 1935, introduced the term "entanglement" to characterize the inseparability of composite quantum systems described by the EPR scenario. He described entanglement as a situation where the state vector of the whole cannot be factored into individual states for the parts, leading to correlations stronger than classical ones, and viewed it as the characteristic trait of quantum mechanics that EPR had illuminated. These exchanges in 1935 marked the onset of enduring debates on quantum reality, which persisted unresolved until the development of Bell's theorem in the 1960s.

Bell's theorem and resolution

In 1964, John Stewart Bell formulated a theorem that addressed the Einstein-Podolsky-Rosen (EPR) paradox by deriving mathematical inequalities that must hold under the assumption of local hidden variable theories.^[15] These theories posit that quantum outcomes are determined by local variables carried by particles, independent of distant measurements, thereby bounding the possible correlations between entangled particles' measurement results.^[15] Bell's work provided a quantitative criterion to test whether quantum mechanics could be completed by such local realism, transforming the EPR debate from philosophical speculation into an experimentally verifiable prediction.^[15] Bell's theorem emerged from his engagement with earlier interpretations of quantum mechanics, particularly David Bohm's 1952 pilot-wave theory and a 1957 formulation of the EPR thought experiment by Bohm and Yakir Aharonov using spin-entangled particles.^[15] In his seminal paper, "On the Einstein-Podolsky-Rosen Paradox," published in Physics Physique Fizika, Bell explicitly referenced this Bohm-Aharonov example to frame the EPR argument in terms of spin correlations for two spin-1/2 particles in a singlet state.^[15] The theorem's development in 1964 marked a pivotal theoretical advance, but its inequalities were initially abstract; subsequent refinements in the late 1960s made them more amenable to laboratory testing. In 1969, John Clauser, Michael Horne, Abner Shimony, and Richard Holt derived the Clauser-Horne-Shimony-Holt (CHSH) inequality, a specific form of Bell's inequality that relates joint measurement probabilities without requiring prior knowledge of single-particle outcomes, thus facilitating experimental implementation.^[16] Bell's theorem resolved the EPR paradox by demonstrating that quantum mechanics predicts correlations exceeding the bounds set by local hidden variable theories, thereby ruling out local realism as a complete description of nature and affirming the intrinsic nonlocality of quantum entanglement.^[15] Specifically, for entangled particles, quantum predictions for certain measurement angles violate Bell's inequalities by up to $2\sqrt{2} in the CHSH form, whereas local hidden variables limit violations to 2, providing clear evidence that no such local theory can reproduce all quantum results.^[16] This outcome supported the EPR critics' view that quantum mechanics necessitates "spooky action at a distance" or a rejection of locality, without invoking hidden variables to restore determinism.^[15] Key advancements in testing Bell's theorem involved experimental proposals and implementations by prominent physicists. In 1972, John Clauser, collaborating with Stuart Freedman, proposed and executed the first experimental test using calcium atoms to produce entangled photon pairs, observing a violation of the CHSH inequality consistent with quantum predictions and inconsistent with local realism.^[17] Building on this, Alain Aspect's 1982 experiments employed time-varying polarizers on entangled photons to address potential signaling loopholes, achieving a violation of Bell's inequalities by more than five standard deviations and further solidifying the theorem's empirical support.^[18] These efforts established Bell's framework as the cornerstone for resolving foundational debates on quantum nonlocality.

Conceptual Foundations

Definition and basic meaning

Quantum entanglement is a fundamental phenomenon in quantum mechanics where the quantum state of a composite system consisting of two or more particles cannot be described as a product of individual states for each particle, even if the particles are separated by arbitrary distances in space.^[19] This inseparability leads to correlations between the particles' properties that are stronger and more intricate than those possible in classical physics. The term "entanglement" was coined by Erwin Schrödinger to characterize this peculiar interconnectedness of separated quantum systems.^[19] To understand entanglement, it is essential to recall key principles of quantum mechanics: superposition and measurement. Superposition allows a quantum system to exist in multiple states simultaneously, represented by a wave function that is a linear combination of basis states; for example, a single particle's spin can be in a superposition of "up" and "down" orientations. Upon measurement, however, the wave function collapses probabilistically to one definite outcome, with the probability determined by the coefficients in the superposition. These concepts underpin how entangled states behave, as the joint wave function of the system governs the outcomes rather than independent descriptions. A basic illustration of entanglement involves two spin-1/2 particles prepared in a singlet state, where their total spin angular momentum is zero. In this configuration, a measurement of the spin of one particle along any axis immediately determines the spin of the other particle to be opposite, with perfect anticorrelation, irrespective of the distance separating them.^[3] This correlation arises without any prior exchange of information between the particles after their creation. This idea was vividly exemplified in the 1935 EPR thought experiment, which questioned the completeness of quantum mechanics based on such remote correlations.^[3] Unlike classical correlations—such as two gloves, one left and one right, packed in separate boxes, where finding one reveals the other's type through pre-shared information—quantum entanglement does not rely on local hidden variables or classical signaling. Instead, the correlations stem from the non-separable, holistic nature of the joint quantum wave function, which treats the particles as a single entity until measurement.^[19] This distinction highlights entanglement's inherently quantum character, defying intuitive classical separability.

Paradoxes of entanglement

Quantum entanglement gives rise to several counterintuitive paradoxes that challenge classical intuitions about locality and causality in physics. One of the most famous is the Einstein-Podolsky-Rosen (EPR) paradox, proposed in 1935, which highlighted the apparent "spooky action at a distance" in entangled systems. Einstein coined the phrase "spooky action at a distance" to describe the instantaneous correlations between distant entangled particles, suggesting that measuring one particle's property seemed to influence the other's without any mediating signal, violating the principle of locality in special relativity. This paradox arises because, in an entangled pair of particles, the quantum state is such that the outcome of a measurement on one particle determines the state of the other, regardless of the distance separating them, as if information travels faster than light. A classic thought experiment illustrating this involves two entangled particles with opposite spins in a singlet state, separated by a vast distance. If an observer measures the spin of the first particle along a particular axis and finds it "up," the second particle's spin must instantaneously be "down" along the same axis, even if the particles are light-years apart. This setup appears to contradict relativity's prohibition on faster-than-light influences, as the measurement outcome on the distant particle seems predetermined or affected immediately upon the first measurement. Einstein, Podolsky, and Rosen argued that this implied quantum mechanics was incomplete, requiring hidden variables to explain the correlations without nonlocality. The resolution to these paradoxes lies in the no-signaling theorem of quantum mechanics, which demonstrates that while correlations are instantaneous, no usable information can be transmitted faster than light, thus preserving causality and relativity. Entanglement allows perfect correlations but not control over the distant outcome; the measurement result on the first particle is random, and the observer at the second site learns nothing new until classical communication arrives. This ensures that no superluminal signaling occurs, resolving the apparent violation. Common misconceptions about entanglement often stem from these paradoxes, particularly the erroneous belief that it enables faster-than-light communication. In reality, to verify the correlation and extract any meaning, the observers must compare results via a classical channel limited by the speed of light, preventing any practical FTL messaging. Such misunderstandings persist in popular media but are firmly ruled out by the foundational principles of quantum theory.

Challenges to local realism

Local realism is a foundational concept in classical physics, asserting that physical properties of a system exist objectively and independently of any measurement performed on it, and that any influences between spatially separated systems propagate at or below the speed of light, preserving locality.^[20] This view combines realism—the idea that systems possess definite attributes prior to observation—with locality, ensuring no faster-than-light causal connections.^[20] In their 1935 paper, Einstein, Podolsky, and Rosen (EPR) invoked local realism to critique quantum mechanics, arguing through a thought experiment involving entangled particles that the theory must be incomplete because it allows measurements on one particle to instantaneously determine properties of a distant one without local influence. They contended that under local realism, the distant particle must possess predetermined values for incompatible observables like position and momentum, which quantum mechanics cannot simultaneously assign definite values to.^[20] Quantum entanglement directly challenges local realism by generating correlations between distant particles that exceed what any local hidden variable theory—hypothesized by EPR to complete quantum mechanics—could produce.^[15] In his 1964 theorem, John Bell formalized this conflict, deriving inequalities that local realistic models must satisfy; quantum predictions for entangled states, however, violate these bounds, implying that either locality or realism (or both) fails.^[15] Entanglement thus reveals "stronger than classical" correlations, where outcomes on one subsystem appear to depend nonlocally on the other's measurement choice, undermining the independent reality of separated systems.^[20] Subsequent experiments, such as those confirming Bell inequality violations, provide empirical evidence that quantum mechanics aligns with these nonlocal correlations rather than local realism.^[21] The philosophical implications of entanglement's assault on local realism have prompted a reevaluation of objective reality in quantum theory, favoring non-realist interpretations that dispense with absolute, measurement-independent properties.^[22] For instance, QBism (Quantum Bayesianism) treats quantum states as subjective credences or personal beliefs of an agent about measurement outcomes, rendering entanglement correlations as epistemic relations among an agent's experiences rather than ontological nonlocal influences, thereby restoring locality at the cost of realism.^[23] Similarly, relational quantum mechanics posits that physical states and outcomes are relative to interacting systems, eliminating the need for a unique, observer-independent reality and resolving entanglement paradoxes by viewing correlations as perspectival.^[22] These approaches shift the focus from an absolute physical world to relational or informational structures, accommodating quantum predictions without hidden variables or superluminal signaling.^[23] Historically, the EPR paradox ignited intense debates in the mid-20th century, with Niels Bohr countering that quantum mechanics' wholeness precludes independent subsystem realities, emphasizing complementary descriptions over local realism.^[20] These exchanges, spanning Solvay conferences and responses like Bohr's 1935 reply, highlighted tensions between classical intuitions and quantum formalism, with Einstein persisting in his advocacy for hidden variables to preserve realism.^[21] Bell's theorem marked a turning point, enabling experimental tests that, by the 1980s, confirmed quantum mechanics' predictions and led to widespread acceptance of entanglement's "weirdness" as a fundamental feature, not a theoretical flaw.^[21] This resolution integrated nonlocal correlations into mainstream physics, influencing fields from quantum information to foundational philosophy.^[20]

Quantum Nonlocality

Bell inequalities

Bell inequalities constitute a class of mathematical constraints derived from the assumptions of locality and realism, bounding the possible correlations between measurement outcomes on spatially separated entangled particles. These inequalities enable quantitative tests to determine whether observed quantum correlations can be reproduced by local hidden variable theories, which posit that particle properties are predetermined and measurements are independent given the separation. In the context of quantum entanglement, violations of these inequalities demonstrate that quantum mechanics cannot be explained by such classical models without abandoning locality or realism. A canonical example is the Clauser-Horne-Shimony-Holt (CHSH) inequality, formulated in 1969 for bipartite systems involving two particles and two possible measurement settings per observer.^[16] Consider two parties, Alice and Bob, each performing one of two dichotomic measurements (yielding outcomes \pm 1) on their respective subsystems: Alice chooses between observables A and A', while Bob chooses between B and B'. The CHSH correlator is defined as

S = \langle AB \rangle + \langle AB' \rangle + \langle A' B \rangle - \langle A' B' \rangle,

where \langle \cdot \rangle denotes the expectation value of the product of outcomes. Under local realism, the inequality asserts |S| \leq 2.^[16] The derivation of the CHSH inequality proceeds from the premises of locality—outcomes depend only on local settings and shared hidden variables \lambda—and realism—outcomes are predetermined for all settings. Let the outcomes be functions a(x, \lambda) for Alice's setting x \in \{A, A'\} and b(y, \lambda) for Bob's y \in \{B, B'\}, with hidden variable distribution \rho(\lambda) \geq 0 normalized to 1. The correlations are then \langle AB \rangle = \int a(A, \lambda) b(B, \lambda) \rho(\lambda) \, d\lambda, and similarly for the others. For fixed \lambda, the combination a(A, \lambda)b(B, \lambda) + a(A, \lambda)b(B', \lambda) + a(A', \lambda)b(B, \lambda) - a(A', \lambda)b(B', \lambda) equals $2 [a(A, \lambda) (b(B, \lambda) + b(B', \lambda))/2 + a(A', \lambda) (b(B, \lambda) - b(B', \lambda))/2 ] \leq 2, since each term in brackets is at most 1 in absolute value given the \pm 1 outcomes. Integrating over \lambda yields |S| \leq 2.^[24] In quantum mechanics, the expectation values are computed as \langle XY \rangle = \mathrm{Tr}(\rho \, X \otimes Y), where \rho is the joint density operator and X, Y are the measurement operators. For a maximally entangled two-qubit state, such as the Bell state |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), appropriate choices of Pauli-like observables (e.g., rotated spin projections) yield |S| = 2\sqrt{2} \approx 2.828, violating the local realistic bound. This maximum value, known as the Tsirelson bound, represents the supremum of quantum correlations for this inequality and holds for any quantum system.^[25] Understanding these inequalities presupposes familiarity with basic quantum measurement theory, particularly the probabilistic nature of outcomes for projective measurements on entangled states. For dichotomic observables with eigenvalues \pm 1, the expectation value simplifies to \langle O \rangle = P(+1) - P(-1), where P(\pm 1) are the outcome probabilities derived from the state's projection onto the eigenspaces. The CHSH inequality emerged as a refinement of John Bell's 1964 theorem, adapting it for direct experimental implementation with photon polarization or spin measurements.^[16]

No-signaling principle

The no-signaling principle, also known as the no-communication theorem, states that quantum entanglement allows for nonlocal correlations between distant subsystems without permitting one party to transmit information to another faster than light. This ensures that the marginal probability distribution for outcomes on one subsystem remains independent of the measurement choice performed on the distant subsystem.^[26]^[27] Mathematically, this principle arises from the structure of quantum mechanics, where the reduced density matrix \rho_A for subsystem A is obtained by tracing over the degrees of freedom of the complementary subsystem B: \rho_A = \Tr_B (\rho_{AB}). For an entangled state \rho_{AB}, the reduced density matrix \rho_A is independent of any local operation or measurement basis chosen on B, ensuring that the local statistics observed by party A are unaffected by distant actions.^[27]^[26] A representative example is the Bell singlet state |\psi^-\rangle = \frac{1}{\sqrt{2}} (|01\rangle - |10\rangle), where the reduced density matrix for either qubit is the maximally mixed state \rho_A = \frac{1}{2} I, yielding uniform local outcome probabilities (50% for each basis state) regardless of the remote measurement basis chosen.^[27] These properties imply that while entanglement exhibits nonlocality through correlated measurement outcomes, it does not violate causality or special relativity, as no controllable information transfer is possible without classical communication.^[26]

Interpretations of nonlocality

Quantum nonlocality, as evidenced by violations of Bell inequalities in entangled systems, challenges classical intuitions of locality and has prompted diverse interpretive frameworks within quantum mechanics to reconcile these correlations without invoking signaling faster than light.^[28] These interpretations seek to explain how distant entangled particles exhibit perfect correlations upon measurement, while preserving the no-signaling principle. In the many-worlds interpretation (MWI), proposed by Hugh Everett and further developed by others, nonlocality is resolved through the branching of the universal wavefunction into parallel worlds upon interaction, eliminating the need for wavefunction collapse. Entangled particles in a singlet state, for instance, lead to a superposition that splits the universe into branches where observers in different worlds record complementary outcomes, such as opposite spins, ensuring correlations without any action at a distance across a single shared reality.^[28] This approach treats both the quantum system and the measuring apparatus quantum mechanically, so the apparent nonlocality arises from the multiverse structure rather than instantaneous influences, aligning with empirical Bell violations through local branching dynamics.^[29] Bohmian mechanics, or the de Broglie-Bohm pilot-wave theory, accommodates entanglement's nonlocality via an explicitly nonlocal guiding equation that determines particle trajectories.^[30] In this deterministic framework, particles possess definite positions at all times, guided by a pilot wave derived from the universal wavefunction, which encompasses the entire configuration space of the system.^[31] For entangled particles, a measurement on one instantly affects the pilot wave, influencing the distant particle's trajectory through the holistic wavefunction, producing the observed correlations without collapse or randomness in outcomes.^[29] This nonlocality is "gross" in the sense that it permeates the theory's dynamics but remains consistent with quantum predictions, as the pilot wave's influence cannot transmit usable information superluminally. Relational quantum mechanics (RQM), formulated by Carlo Rovelli, addresses nonlocality by positing that physical states and properties are inherently relational, defined only relative to interacting systems rather than absolutely.^[32] In the context of entanglement, such as the EPR paradox, the correlated outcomes are not nonlocal influences but emerge from the relative information exchanged between subsystems during measurement interactions.^[33] For example, when two entangled particles are measured by different observers, each observer's state description is valid only with respect to their local interaction, avoiding any need for absolute simultaneity or distant causation; the apparent nonlocality dissolves because reality is observer-relative, with no privileged global state. This perspective aligns with quantum formalism by treating all systems equivalently, resolving paradoxes through relational facts without hidden variables or branching. Among physicists, there is no consensus on a preferred interpretation of quantum nonlocality, as each viable framework—MWI, Bohmian mechanics, RQM, and others—reproduces the same empirical predictions for entanglement while offering distinct ontological resolutions.^[34] A 2025 Nature survey of over 1,100 physicists confirmed sharp divisions in preferred interpretations of quantum mechanics, with no single view holding majority support.^[35] However, all accepted interpretations accommodate the observed nonlocal correlations, emphasizing that nonlocality does not imply superluminal signaling but rather underscores the holistic nature of quantum systems.^[36]

Mathematical Description

Pure bipartite states

In quantum mechanics, a bipartite system consists of two subsystems, labeled A and B, whose combined state resides in the tensor product of their individual Hilbert spaces, \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B.^[37] This structure allows the total Hilbert space to encompass all possible joint states, including those where the subsystems are correlated beyond classical expectations. For pure states, the wavefunction |\psi\rangle \in \mathcal{H} describes the entire system without classical uncertainty, enabling precise analysis of quantum correlations.^[37] The Schmidt decomposition provides a canonical representation for any pure bipartite state, expressing it as |\psi\rangle = \sum_i \lambda_i |i_A\rangle |i_B\rangle, where \{\lambda_i\} are non-negative real Schmidt coefficients satisfying \sum_i \lambda_i^2 = 1, and \{|i_A\rangle\}, \{|i_B\rangle\} form orthonormal bases for \mathcal{H}_A and \mathcal{H}_B, respectively. This decomposition arises from the singular value decomposition of the coefficient matrix in the state vector, revealing the inherent correlations between subsystems. A pure bipartite state is entangled if and only if more than one Schmidt coefficient is non-zero, indicating that the state cannot be written as a product |\psi\rangle = |\phi_A\rangle \otimes |\chi_B\rangle. For two-qubit systems, a common measure of entanglement for pure states is the concurrence C(|\psi\rangle) = \sqrt{2(1 - \mathrm{Tr}(\rho_A^2))}, where \rho_A = \mathrm{Tr}_B(|\psi\rangle\langle\psi|) is the reduced density operator of subsystem A. This quantity ranges from 0 for separable states to 1 for maximally entangled states, quantifying the degree of inseparability based on the purity of the reduced state. A prominent example of maximally entangled pure bipartite states are the Bell states, which form an orthonormal basis for the two-qubit Hilbert space and exhibit perfect correlations. These states are:

\begin{align} |\Phi^+\rangle &= \frac{1}{\sqrt{2}} \left( |00\rangle + |11\rangle \right), \\ |\Phi^-\rangle &= \frac{1}{\sqrt{2}} \left( |00\rangle - |11\rangle \right), \\ |\Psi^+\rangle &= \frac{1}{\sqrt{2}} \left( |01\rangle + |10\rangle \right), \\ |\Psi^-\rangle &= \frac{1}{\sqrt{2}} \left( |01\rangle - |10\rangle \right), \end{align}

where |0\rangle and |1\rangle denote the computational basis states. In the Schmidt decomposition, each Bell state has two equal coefficients \lambda_0 = \lambda_1 = 1/\sqrt{2}, confirming their maximal entanglement with concurrence C = 1.

Mixed states and density matrices

In quantum mechanics, mixed states of composite systems are described by density operators \rho that represent statistical ensembles of pure states. For a bipartite system AB, the reduced density operator for subsystem A is obtained by performing a partial trace over B: \rho_A = \operatorname{Tr}_B(\rho). This operation captures the local description of A while accounting for correlations with B. A mixed state \rho is separable (unentangled) if it can be decomposed as a convex combination of product states, \rho = \sum_i p_i \rho_A^{(i)} \otimes \rho_B^{(i)}, where p_i \geq 0, \sum_i p_i = 1, and each \rho_A^{(i)}, \rho_B^{(i)} are density operators. In this case, the reduced density operator \rho_A is itself a convex combination (hence decomposable) of the local \rho_A^{(i)}. Conversely, if the total state is a pure product state, \rho_A is pure; however, for general mixed separable states, \rho_A is typically mixed but decomposable in the above sense. Detecting entanglement in mixed states is more challenging than in pure states, where a mixed reduced density operator \rho_A (with \operatorname{Tr}(\rho_A^2) < 1) directly indicates entanglement. For mixed states, no simple local purity test suffices, as separable mixtures can yield mixed \rho_A. Instead, operational criteria are required to distinguish entangled mixtures from separable ones. A key necessary condition for separability, proposed by Peres, is that the partial transpose \rho^{T_B} (transposing the matrix elements in the B basis while keeping A fixed) must have non-negative eigenvalues. Horodecki et al. later proved this positivity of the partial transpose (PPT) criterion is also sufficient for separability in systems of dimensions $2 \times 2 and $2 \times 3, making it a complete test for low-dimensional bipartite mixed states. Violations of PPT, manifesting as negative eigenvalues, confirm entanglement. This criterion outperforms Bell inequalities for mixed states, as it directly probes the density operator structure without assuming local measurements.^[38]^[39] Werner states provide a canonical example illustrating mixed-state entanglement and the PPT criterion. For two qubits, a Werner state is \rho_p = p |\psi^-\rangle\langle\psi^-| + (1-p) \frac{I}{4}, where |\psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle) is the singlet state, I is the $4 \times 4 identity, and p \in [0,1]. These states are invariant under simultaneous unitary rotations U \otimes U and represent mixtures of the maximally entangled singlet with white noise. The reduced density operators are maximally mixed (\rho_A = \rho_B = I/2) for all p, so local tests cannot detect entanglement. However, the partial transpose \rho_p^{T_B} has eigenvalues \frac{1-p}{4} (threefold degenerate) and \frac{1-3p}{4}. For p > \frac{1}{3}, the latter is negative, violating PPT and confirming entanglement; for p \leq \frac{1}{3}, \rho_p is separable. This threshold highlights how noise can mask entanglement in mixed states, with Werner states achieving the maximum noise tolerance for bound entanglement in higher dimensions.^[40]

Entanglement measures

Entanglement measures quantify the degree of quantum correlations in a bipartite quantum state that cannot be explained by classical means, serving as essential tools for assessing resources in quantum information processing. These measures must satisfy key axioms, including monotonicity under local operations and classical communication (LOCC), meaning they do not increase on average when parties perform local quantum operations assisted by classical messages, ensuring they reflect genuine entanglement that cannot be created or enhanced by such means.^[41] This property distinguishes valid entanglement quantifiers from other correlation measures, as LOCC preserves separability but cannot generate entanglement from separable states.^[42] For pure bipartite states, the standard entanglement measure is the entropy of entanglement, defined using the von Neumann entropy of the reduced density matrix of one subsystem. The reduced density matrix \rho_A for subsystem A is obtained by tracing out the degrees of freedom of subsystem B. The entanglement entropy is then given by

E(|\psi\rangle_{AB}) = S(\rho_A) = -\operatorname{Tr}(\rho_A \log_2 \rho_A),

where |\psi\rangle_{AB} is the pure state of the composite system AB, and the logarithm is base 2 to express the measure in ebits (entanglement bits). This quantity equals the entropy of \rho_B due to the purity of the overall state, and it vanishes if and only if the state is a product state.^[41] The measure captures the maximum number of ebits that can be distilled from the state via LOCC, making it operationally significant. For mixed bipartite states, the entanglement of formation extends this concept by considering the minimum resources required to prepare the state through mixing pure entangled states. It is defined as

E_F(\rho_{AB}) = \min_{\{p_i, |\psi_i\rangle\}} \sum_i p_i E(|\psi_i\rangle_{AB}),

where the minimum is taken over all ensemble decompositions \rho_{AB} = \sum_i p_i |\psi_i\rangle\langle\psi_i|_{AB} of the mixed state \rho_{AB}, and E(|\psi_i\rangle) is the entanglement entropy of each pure state. This measure is convex and LOCC-monotone, providing a lower bound on the entanglement cost of state preparation. For two-qubit systems, an explicit formula exists in terms of the concurrence, facilitating computations.^[43] The squashed entanglement offers another LOCC-monotone measure, particularly valued for its additivity and monogamy properties, which bound how entanglement is shared among multiple parties. It is defined as half the infimum of the conditional quantum mutual information over all possible extensions of the bipartite state:

E_{\text{sq}}(\rho_{AB}) = \frac{1}{2} \inf_{\rho_{ABE}} I(A:B|E)_{\rho_{ABE}},

where the infimum is over all tripartite extensions \rho_{ABE} such that \operatorname{Tr}_E(\rho_{ABE}) = \rho_{AB}, and I(A:B|E) = S(AE) + S(BE) - S(ABE) - S(E) is the conditional mutual information with S denoting the von Neumann entropy. This measure is zero for separable states and provides a tight bound on distillable entanglement in certain scenarios.^[44] Its additivity under tensor products ensures it behaves well in asymptotic resource theories.^[41]

Multipartite and Advanced Entanglement

Multipartite systems

Quantum entanglement extends beyond bipartite systems to multipartite scenarios involving three or more parties, where the correlations cannot be explained by pairwise interactions alone. In such systems, a state is considered fully multipartite entangled if it remains entangled across every possible bipartition of the subsystems, meaning no subset of parties can be separated from the rest without destroying the overall quantum correlations. This generalization reveals a richer structure of entanglement, with multiple inequivalent classes that cannot be interconverted via local operations. A paradigmatic example of fully multipartite entanglement is the Greenberger-Horne-Zeilinger (GHZ) state for three qubits, defined as

\frac{1}{\sqrt{2}} \left( |000\rangle + |111\rangle \right),

which exhibits perfect correlations under specific joint measurements, violating local realism in a manner stronger than Bell inequalities. This state, introduced to highlight contradictions in local hidden variable theories, is fully entangled across all bipartitions, such as A|BC or AB|C. In contrast, the W state for three qubits,

\frac{1}{\sqrt{3}} \left( |100\rangle + |010\rangle + |001\rangle \right),

represents a distinct entanglement class that is inequivalent to the GHZ state. Unlike the GHZ state, the W state remains entangled even after the loss of one qubit, making it more robust against decoherence, though it shows different correlation properties under local measurements. These two classes illustrate that multipartite entanglement for three qubits falls into two inequivalent families under stochastic local operations and classical communication (SLOCC). Multipartite systems allow for entanglement analysis across various cuts or partitions, differing from bipartite cases where only one division exists. For an N-party system, entanglement can be assessed by bipartitioning into groups of sizes k and N-k (with 1 ≤ k ≤ ⌊N/2⌋), revealing partial separability if entanglement vanishes across some cut while persisting across others. Bipartite measures, such as concurrence, can quantify entanglement for individual cuts but fail to capture the global multipartite structure comprehensively. Classifying multipartite entanglement under SLOCC—invertible local operations with nonzero success probability followed by classical communication—poses significant challenges due to the exponential growth in inequivalent classes with increasing particle number. For three qubits, only two SLOCC classes exist (GHZ and W types), but for four qubits, at least nine distinct classes emerge, complicating the identification of entanglement types and their transformations. This classification is crucial for understanding resource equivalence in multipartite quantum information tasks.

Bound and distillable entanglement

In quantum information theory, distillable entanglement quantifies the maximum number of Einstein-Podolsky-Rosen (EPR) pairs, or ebits, that can be extracted from many identical copies of a given entangled quantum state using local operations and classical communication (LOCC). This asymptotic yield represents the usable entanglement resource available for tasks like quantum teleportation or dense coding, where the protocol achieves a rate approaching the distillable entanglement in the limit of infinitely many copies. A key challenge arises with bound entanglement, which refers to entangled states that possess non-zero entanglement yet yield zero distillable entanglement under LOCC. These states are characterized by having a positive partial transpose (PPT), meaning their partial transpose with respect to any subsystem has non-negative eigenvalues, yet they remain inseparable.^[45] Unlike free states in the resource theory of entanglement, bound entangled states cannot be transformed into pure EPR pairs, rendering their entanglement "locked" and unusable for distillation, though they may assist in other quantum protocols.^[46] Examples of bound entanglement include states derived from unextendible product bases (UPBs), where the uniform mixture over the orthogonal complement to a UPB in a multipartite system is PPT entangled but undistillable. In particular, for a tripartite 2×2×2 system, such states exhibit no bipartite entanglement across any cut while being globally bound entangled.^[45] Distillation protocols like hashing and breeding, which operate on multiple copies to project onto subspaces with high fidelity to EPR pairs, fail for these bound states, as the yield asymptotically approaches zero. Within the resource theory of entanglement, where LOCC acts as free operations, the entanglement cost of a state is the minimum number of ebits required to prepare it asymptotically via LOCC, providing a complementary measure to distillable entanglement. This cost highlights irreversibility in entanglement manipulation, as bound entangled states demand ebits for creation but return none upon distillation, underscoring the theory's foundational asymmetry.^[46]

Entanglement in quantum fields

In quantum field theory (QFT), entanglement arises inherently from the structure of the vacuum state and the locality of field operators, extending the notion of quantum correlations beyond finite-dimensional systems to infinite degrees of freedom. Unlike in non-relativistic quantum mechanics, where entanglement is typically analyzed for isolated particles or qubits, QFT describes fields propagating continuously in spacetime, leading to observer-dependent and spatially extended entanglement. This framework reveals how the vacuum, appearing empty to inertial observers, encodes pervasive quantum correlations that manifest differently under Lorentz transformations or acceleration. The Unruh effect exemplifies observer-dependent entanglement in the vacuum. A uniformly accelerated observer perceives the Minkowski vacuum as a thermal state at temperature T = \frac{a}{2\pi}, where a is the proper acceleration, due to the mixing of positive and negative frequency modes in the Rindler coordinates. This thermal perception arises from entanglement between left- and right-moving modes across the Rindler horizon, such that the vacuum state for inertial observers appears entangled when traced over one wedge of spacetime for the accelerated observer. This entanglement degrades correlations between field modes as observed by non-inertial detectors, highlighting how acceleration entangles the vacuum in a way that simulates particle creation. The Reeh-Schlieder theorem further underscores the global nature of local operations in entangled QFT vacua. It states that, for any bounded spacetime region V, the algebra of local operators \mathcal{A}(V) acting on the vacuum |\Omega\rangle densely spans the full Hilbert space, meaning \mathcal{A}(V) |\Omega\rangle is cyclic and can approximate any global state. Consequently, local field excitations generate states with entanglement across the entire spacetime, implying that the vacuum is fundamentally non-separable with respect to any spatial bipartition. This theorem demonstrates that entanglement in QFT is unavoidable and spans all scales, as even operators confined to a small region entangle the system globally without violating causality.^[47] Entanglement entropy in QFT quantifies these correlations, particularly in ground states, where it follows an area-law scaling. For a subsystem defined by a spatial region A with boundary area \mathcal{A}, the von Neumann entropy S_A = -\operatorname{Tr}(\rho_A \log \rho_A), with \rho_A the reduced density matrix, behaves as S_A \sim \mathcal{A}/\epsilon^{d-2} in d-dimensional spacetime, where \epsilon is a UV cutoff, rather than scaling with the volume. This area law emerges from the short-range correlations in gapped systems or the conformal structure in critical theories, reflecting how entanglement is concentrated near boundaries in relativistic vacua. Multipartite aspects, such as multi-region entropies, can extend this but remain tied to local field dynamics.^[48]^[49] These concepts link to the black hole information paradox through Hawking radiation, where particle-antiparticle pairs entangle across the event horizon, with one member falling in and the other escaping. The paradox arises because the radiation appears thermal and entangled in a way that purifies the interior state, yet unitarity demands information preservation, challenging naive semiclassical evaporation. Entanglement entropy calculations, including the area-law analogy to black hole entropy S_{BH} = \frac{A}{4G}, suggest resolutions via quantum corrections that maintain information in correlations, as explored in firewall proposals and replica wormhole geometries.

Generation Methods

Laboratory creation techniques

Laboratory creation of quantum entanglement relies on controlled interactions between quantum systems, such as photons, atoms, or ions, to produce correlated states like Bell states. These methods enable the generation of entangled pairs or larger systems on demand, forming the foundation for quantum information experiments. One of the most widely used techniques is spontaneous parametric down-conversion (SPDC), where a high-energy pump photon interacts with a nonlinear crystal, such as beta-barium borate (BBO), splitting into two lower-energy photons whose properties—such as polarization, momentum, or frequency—are entangled due to conservation laws. This process, first demonstrated for entanglement in the 1980s, produces photon pairs at rates up to millions per second with crystals pumped by lasers in the visible or near-infrared range, achieving high-fidelity entanglement suitable for quantum key distribution and teleportation.^[50] Atomic cascades provide another early method, involving the excitation of atoms or ions to a high-energy state, followed by sequential decays that emit two photons in correlated polarizations or directions. Pioneered in the 1960s with calcium atoms, this technique generates entangled photon pairs from the atomic transitions, though it suffers from lower efficiency due to isotropic emission and requires precise timing to collect the photons. Modern implementations use quantum dots or Rydberg atoms to improve directionality and fidelity.^[51] In ion traps, entanglement is created by confining charged atoms in electromagnetic fields and applying laser pulses to induce controlled interactions, such as the Mølmer-Sørensen gate, which entangles the internal spin states of multiple ions through shared vibrational modes. This approach, developed in the 1990s, yields near-perfect entanglement fidelities exceeding 99% for up to 20 ions, making it ideal for quantum computing demonstrations.^[52] Cavity quantum electrodynamics (QED) exploits the strong coupling between an atom and the quantized electromagnetic field inside an optical cavity to generate entanglement via resonant interactions, such as the Jaynes-Cummings model dynamics. Atoms or artificial qubits placed in high-finesse cavities emit or absorb photons in a way that correlates their states, enabling deterministic entanglement of distant systems through cavity-mediated photon exchange. Early experiments in the 2000s achieved two-atom entanglement with fidelities around 80%, with recent advances pushing toward scalable networks.^[53] A recent advancement in 2023 demonstrated on-demand entanglement of individual molecules using optical tweezers at Princeton University, where laser-cooled calcium monofluoride molecules were trapped in a reconfigurable array and brought into proximity to interact via electric dipole forces, producing a two-qubit entangling gate with fidelity over 80%. This technique extends entanglement to more complex molecular systems, opening pathways for quantum simulation of chemical processes. In May 2025, researchers demonstrated deterministic entanglement generation via elastic collisions between ultracold atoms, enabling high-fidelity two-qubit gates without probabilistic photon emission, advancing fault-tolerant quantum computing.^[54]^[55]

Natural and emergent entanglement

Quantum entanglement arises naturally in various physical processes, independent of laboratory interventions. A prominent example occurs in the decay of the neutral pion (π⁰), which predominantly decays into two photons via the electromagnetic interaction π⁰ → γγ. Due to conservation of angular momentum, the pion's spin-zero state requires the two photons to have opposite helicities in its rest frame, resulting in a maximally entangled Bell state in their polarization degrees of freedom.^[56]^[57] This entanglement persists as the photons propagate, demonstrating non-local correlations inherent to the decay process. In cosmic contexts, entanglement emerges theoretically from the early universe and extreme gravitational environments. Big Bang cosmology implies that particles produced in the initial expansion are highly entangled due to the universe's origin from a low-entropy, highly coherent quantum state.^[58] Similarly, the formation of black holes generates entanglement between infalling matter and outgoing Hawking radiation, where particle-antiparticle pairs created near the event horizon become entangled, with one partner escaping as thermal radiation while the other falls in.^[59] These scenarios highlight entanglement as a fundamental feature of cosmological evolution, though direct observation remains challenging. Entanglement also emerges spontaneously in many-body quantum systems, where collective interactions lead to correlated states without external pairing. In one-dimensional spin chains, such as the Heisenberg model, ground-state entanglement structures develop through nearest-neighbor antiferromagnetic couplings, manifesting as area-law scaling of entanglement entropy and self-similar patterns in the entanglement network.^[60] In superconductors described by Bardeen-Cooper-Schrieffer (BCS) theory, electrons form Cooper pairs in a superposition of momentum states, yielding spin-singlet entanglement between paired fermions that underlies the macroscopic coherence and zero-resistance state.^[61] These emergent correlations drive phase transitions and collective phenomena in condensed matter. High-energy collisions at particle accelerators reveal entanglement within hadronic structure. Observations from proton-proton interactions at the Large Hadron Collider (LHC) and electron-proton scattering at HERA indicate that quarks and gluons inside protons are entangled over sub-femtometer scales, influencing parton distribution functions and leading to high-entropy configurations post-collision.^[62] Specifically, analyses of LHC data show quantum entanglement in top-quark pairs produced via gluon fusion, confirming non-local spin correlations at TeV energies.^[63] Additionally, the quantum chromodynamics vacuum in quantum field theory exhibits area-law entanglement, contributing to the overall entangled nature of particle interactions.^[58]

Scalable sources for applications

Solid-state systems such as semiconductor quantum dots and nitrogen-vacancy (NV) centers in diamond have emerged as promising platforms for generating scalable entanglement due to their ability to produce on-demand entangled photon pairs with high purity and integration potential into photonic circuits. Quantum dots, particularly those based on GaAs or InAs, enable the biexciton-exciton cascade process to emit polarization-entangled photons, achieving fidelities exceeding 90% and indistinguishability greater than 95% under strain tuning or cavity enhancement.^[64] For instance, droplet-etched GaAs quantum dots have demonstrated wavelength-tunable entangled photon sources with fidelities exceeding 90% through combined AC Stark and quantum-confined Stark effects. Similarly, NV centers facilitate spin-photon entanglement via optical spin initialization and readout, supporting hybrid architectures where nuclear spins provide long coherence times for error-corrected entanglement distribution. Recent integrations of NV centers into nanophotonic platforms have yielded entanglement rates in the kHz range with fidelities above 80%, enabling scalable quantum repeaters. In October 2025, multimode quantum entanglement was achieved via dissipation engineering in optical systems, allowing simultaneous entanglement across multiple degrees of freedom for enhanced quantum information capacity.^[65] Fiber-optic and satellite-based systems address the distribution of entanglement over long distances, forming the backbone of quantum networks by mitigating losses in transmission. In fiber networks, polarization-entangled photons have been distributed over 96 km in submarine cables with fidelities maintained above 80% using active compensation for birefringence and low-loss telecom wavelengths around 1550 nm. Hybrid satellite-fiber architectures further extend this to global scales, combining ground-fiber links with medium Earth orbit satellites to achieve entanglement distribution rates of several pairs per second over thousands of kilometers, as demonstrated in protocols integrating free-space and guided-wave channels. Satellite platforms, such as those employing down-conversion sources on orbit, have realized entanglement over 1200 km with Bell inequality violations confirming non-locality, paving the way for intercontinental quantum key distribution. A notable 2025 advancement involves the demonstration of entanglement in the total angular momentum of near-field photons confined in nanoscale structures, reported by researchers at the Technion-Israel Institute of Technology. This form of entanglement, observed between photons' spin and orbital angular momentum components in nanophotonic waveguides, achieves non-classical correlations with violation of classical bounds by over 5 standard deviations, offering potential for compact, on-chip quantum information processing without polarization degree-of-freedom limitations. Despite these progresses, key challenges persist in achieving practical scalability, including maintaining high entanglement fidelity against environmental noise, boosting generation and distribution rates to meet network demands, and mitigating decoherence from interactions like phonons or atmospheric turbulence. Fidelity degradation often limits effective rates to below 1 Hz over metropolitan scales, necessitating advanced error correction and purification protocols. Decoherence times in solid-state sources, typically milliseconds for NV spins but shorter for quantum dot excitons, require cryogenic operation or dynamical decoupling to extend coherence, while distribution losses in fibers and free space demand heralded schemes to herald successful entanglement events.

Detection and Characterization

Entanglement witnesses

Entanglement witnesses provide an operational method to detect the presence of quantum entanglement in a multipartite quantum state described by a density matrix \rho. These are Hermitian operators W such that the expectation value \Tr(W \rho) \geq 0 for all separable states \rho, while \Tr(W \rho) < 0 for at least some entangled states. This property allows witnesses to serve as binary classifiers for entanglement, distinguishing entangled states from the convex set of separable ones without requiring complete knowledge of the state. The concept was formalized through the connection between positive maps and block-positive operators, where a witness corresponds to an operator that is positive on product states but not necessarily on all separable states.^[66] One standard construction of entanglement witnesses leverages the partial transpose operation, a linear map that transposes the density matrix with respect to one subsystem. If the partial transpose \rho^\Gamma of a state \rho exhibits negative eigenvalues, indicating entanglement via the Peres-Horodecki criterion, a witness can be built by taking the partial transpose of the projector onto the subspace spanned by the corresponding negative eigenvectors. Specifically, let P_- be the projector onto the eigenspace of \rho^\Gamma with negative eigenvalues; then W = P_-^\Gamma satisfies \Tr(W \sigma) = \Tr(P_- \sigma^\Gamma) \geq 0 for any separable \sigma (since \sigma^\Gamma \geq 0), but \Tr(W \rho) < 0 due to the negative contributions. This method is particularly effective for detecting bound entanglement in higher-dimensional systems where the partial transpose alone is inconclusive.^[66] In continuous-variable systems, such as those involving Gaussian states, entanglement witnesses are often constructed from the covariance matrix V of quadrature observables. The covariance matrix encodes second-order correlations, and criteria like the Duan-Simon inseparability condition derive witnesses by checking if V + i \Omega \geq 0, where \Omega is the symplectic form; violation implies entanglement. A witness operator can then be formulated as W = V + i \Omega - \epsilon I for small \epsilon > 0, ensuring positivity on separable Gaussian states while detecting entangled ones through negative expectation values. This approach is advantageous for experimental setups with optical or mechanical modes.^[66] For discrete-variable systems like two qubits, projector-based witnesses are commonly used for maximally entangled Bell states. Consider the Bell state |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle); a suitable witness is W = \frac{1}{2} I - |\Phi^+\rangle\langle\Phi^+|, which is non-negative on separable states but yields \Tr(W |\Phi^+\rangle\langle\Phi^+|) = -\frac{1}{2} < 0. Similar constructions apply to other Bell states by projecting onto the orthogonal complement adjusted for the maximally entangled subspace. These examples highlight the versatility of witnesses in targeted detection.^[66] A key advantage of entanglement witnesses over full quantum state tomography is the reduced number of measurements required. Tomography demands exponential resources in system size to reconstruct the entire density matrix, whereas evaluating \Tr(W \rho) typically involves only a few local observables, making witnesses scalable for larger systems and practical in noisy experimental environments. This efficiency has enabled entanglement verification in photonic, atomic, and superconducting platforms.^[66]

Quantum state tomography

Quantum state tomography (QST) is a comprehensive method for reconstructing the full density matrix of a quantum system from experimental measurements, enabling the detailed verification of entanglement by revealing all correlations and coherences in the state. Unlike preliminary detection techniques such as entanglement witnesses, which offer only partial information, QST provides a complete characterization essential for benchmarking quantum devices and protocols involving entangled particles. This approach is particularly crucial for multipartite systems where entanglement structure must be precisely quantified to assess its utility in applications like quantum computing. The reconstruction process begins with preparing an ensemble of identical copies of the unknown quantum state and performing projective measurements in a tomographically complete set of bases, ensuring that the data spans the entire Hilbert space. For qubit-based entangled states, measurements are typically conducted in the eigenbases of the Pauli operators—\sigma_x, \sigma_y, and \sigma_z—for each qubit, either individually or in tensor products for multi-qubit systems, to obtain the necessary expectation values \langle \sigma_i \otimes \sigma_j \cdots \rangle. These projections yield the coefficients in the Pauli basis expansion of the density matrix \rho = \frac{1}{2^n} \sum_{k} r_k P_k, where P_k are the Pauli strings and n is the number of qubits. The collected statistics are then analyzed using maximum likelihood estimation, which maximizes the likelihood function L(\rho) = \prod_m p_m^{N_m} (with p_m the predicted probabilities and N_m the observed counts for outcome m) under the constraints of positivity and unit trace, yielding the most probable physical state consistent with the data. This iterative method, detailed by James et al.,^[67] robustly handles experimental noise and incomplete datasets. A primary challenge in QST for entangled systems is the curse of dimensionality, where the parameter space grows as d^2 - 1 (with d = 2^n for n qubits), demanding exponentially many measurements—on the order of O(4^n)—to achieve reliable reconstruction, which becomes infeasible beyond a handful of qubits due to resource limitations and statistical errors. Adaptive strategies and compressed sensing can mitigate this scaling for low-rank or structured states, but full tomography remains computationally intensive even with modern optimizations. In applications to entanglement, the reconstructed density matrix allows computation of the fidelity F(\rho, |\psi\rangle\langle\psi|) = \langle\psi| \rho |\psi\rangle to an ideal entangled state |\psi\rangle, such as a Bell state, providing a direct measure of entanglement quality; for example, two-qubit experiments have demonstrated fidelities exceeding 0.99, confirming high-purity entanglement suitable for quantum information tasks.

Bell test protocols

Bell test protocols begin with the generation of entangled particle pairs, such as polarization-entangled photons produced via spontaneous parametric down-conversion or electron spins in nitrogen-vacancy centers in diamond. These pairs are distributed to two spatially separated observers, Alice and Bob, who independently and randomly select measurement settings on their respective particles. For photonic implementations, Alice measures the polarization in bases at 0° (horizontal/vertical) and 45° (diagonal/antidiagonal), while Bob measures at 22.5° and 67.5° to optimize for the Clauser-Horne-Shimony-Holt (CHSH) inequality. The outcomes are recorded as binary results (±1), and correlations are computed across many trials to evaluate the CHSH parameter. To achieve conclusive evidence against local realism, protocols must address key experimental loopholes, particularly the locality loophole—ensured by space-like separation of measurement events to prevent light-speed signaling—and the detection (fair-sampling) loophole, mitigated by detection efficiencies exceeding approximately 66.7% for optimal CHSH angles. In 2015, researchers at Delft University demonstrated a loophole-free test using entangled electron spins separated by 1.3 km, with measurements performed within 4.3 μs to enforce locality; they reported a CHSH value of S = 2.42 ± 0.20, violating the classical bound of |S| ≤ 2 with a p-value of 0.039 under local realist models.^[68] Concurrently, the NIST group conducted a photonic loophole-free Bell test with superconducting nanowire single-photon detectors achieving 90% efficiency and space-like separation over 184 m, yielding S = 2.427 ± 0.020 and a significance exceeding 11 standard deviations.^[69] The CHSH parameter quantifies nonlocality through S = |⟨AB⟩ + ⟨A′B⟩ + ⟨AB′⟩ − ⟨A′B′⟩|, where ⟨⋅⟩ denotes expectation values of joint measurement outcomes for settings A, A′ (Alice) and B, B′ (Bob); quantum predictions allow S up to 2√2 ≈ 2.828 for maximally entangled states, with violations (S > 2) indicating incompatibility with local hidden variables. Statistical significance is evaluated via the deviation from the classical threshold, often using p-values from the observed distribution or equivalent sigma levels, accounting for finite trial numbers and noise; for instance, the Delft experiment's modest p-value improved to >5σ upon reanalysis with no-signaling constraints.^[70] In the 2020s, continuous-variable (CV) Bell test protocols have gained prominence, employing homodyne or heterodyne detection of field quadratures (position- and momentum-like observables) on entangled optical modes, such as Gaussian states from parametric down-conversion. These tests extend to infinite-dimensional Hilbert spaces, potentially enhancing violation magnitudes, but require careful handling of vacuum noise and inefficiency. A 2023 experiment demonstrated CV nonlocality via phase-space Bell inequalities on squeezed-state entanglement, achieving violations that certify quantum steering and dimension witnesses without full tomography.^[71] In 2025, a Bell inequality violation was demonstrated using gate-defined quantum dots, achieving 97.17% Bell state fidelity without readout error correction.^[72] Advances in integrated optics and high-efficiency detectors are paving the way for loophole-free CV implementations, complementing discrete-variable tests in quantum network applications.

Applications

Quantum information protocols

Quantum information protocols exploit quantum entanglement to perform tasks that surpass classical communication limits, enabling secure and efficient transfer of quantum states over distances. These protocols typically rely on shared entangled resources, such as Bell states, combined with classical communication channels to achieve outcomes impossible with classical means alone. Seminal developments in this area, starting from the early 1990s, have laid the foundation for practical quantum networks by demonstrating how entanglement facilitates state transfer without physical transport of the quantum system itself. One foundational protocol is quantum teleportation, which allows the transfer of an unknown quantum state from a sender (Alice) to a receiver (Bob) using a shared entangled pair and a classical channel. Proposed by Bennett et al. in 1993, the protocol begins with Alice and Bob sharing a maximally entangled Bell state, such as \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle). Alice performs a joint Bell-basis measurement on her qubit to teleport and one half of the entangled pair, yielding two classical bits that she sends to Bob. Bob then applies a conditional Pauli operation based on these bits to reconstruct the original state on his qubit. This process destroys the state on Alice's side while faithfully reproducing it on Bob's, consuming one ebit (unit of entanglement) and two classical bits per teleported qubit. The protocol's security stems from the no-cloning theorem, ensuring the state cannot be intercepted without detection. Experimental realizations have achieved fidelities exceeding 90% over optical fibers spanning hundreds of kilometers. Superdense coding, introduced by Bennett and Wiesner in 1992, leverages entanglement to enhance classical information transmission efficiency. In this protocol, Alice and Bob share a Bell state, enabling Alice to encode two classical bits into a single qubit by applying one of four unitary operations (identity, Pauli-X, Pauli-Z, or both) to her half of the entangled pair before sending it to Bob. Bob measures the received qubit jointly with his entangled partner in the Bell basis, directly decoding the two bits with perfect fidelity. This doubles the classical channel capacity compared to sending an unentangled qubit, which conveys only one bit. The protocol requires prior entanglement distribution, often via methods like spontaneous parametric down-conversion, and has been experimentally verified with photonic systems achieving near-unity efficiency. Entanglement swapping extends the range of quantum correlations by linking two independent entangled pairs through a joint measurement. First proposed by Żukowski et al. in 1993, the protocol involves two parties, Alice and Bob, each sharing an entangled pair with a central station, Charlie. Charlie performs a Bell-state measurement on his two qubits, projecting them into an entangled state and thereby entangling Alice's and Bob's distant qubits despite no direct interaction. The resulting swapped entanglement can be used for further protocols like teleportation over longer distances. This technique is crucial for quantum repeaters, mitigating decoherence in extended networks, and has been demonstrated experimentally with fidelities above 80% using trapped ions and photons. Recent advances in multi-party protocols have focused on W states, symmetric entangled states like the three-qubit W state \frac{1}{\sqrt{3}} (|001\rangle + |010\rangle + |100\rangle), which offer robustness against particle loss compared to GHZ states. In 2025, researchers at Kyoto University developed an entangled measurement scheme for direct, high-fidelity identification of photonic W states, achieving a measurement discrimination fidelity of 0.871 ± 0.039 (87.1%) in distinguishing the W state. This method uses collective measurements on multiple modes to project the system onto the W subspace, enabling efficient verification for multi-party quantum tasks such as multipartite teleportation and secret sharing. The protocol's scalability supports applications in distributed quantum computing and sensing, with experimental implementation on a three-photon system via linear optics and single-photon detectors.^[73]

Quantum computing resources

In quantum computing, entanglement serves as a fundamental resource for implementing multi-qubit operations, particularly through two-qubit gates like the controlled-NOT (CNOT) gate, which generates bipartite entanglement between qubits. When applied to a separable state such as the product of a single-qubit superposition and a basis state, the CNOT gate transforms it into a maximally entangled Bell state, enabling the creation of quantum correlations essential for universal quantum computation. This entangling capability, combined with single-qubit rotations, forms a universal gate set for quantum circuits.^[74] Cluster states represent a highly entangled resource specifically tailored for measurement-based quantum computation (MBQC), an alternative paradigm to the circuit model where computation proceeds via adaptive single-qubit measurements on a pre-prepared entangled state rather than direct gate applications. In MBQC, a large-scale cluster state—a graph-like multipartite entangled state where qubits are connected via controlled-phase gates—encodes the logical quantum information, and measurements in the X-Y plane drive the computation while projecting the state onto the desired output. This approach leverages the entanglement structure to perform arbitrary quantum algorithms fault-tolerantly, with the cluster's graph topology determining the computational power; for instance, a 2D cluster state suffices for universal computation. The concept was introduced as the "one-way quantum computer," highlighting how entanglement distribution in the cluster replaces sequential gate operations.^[75] Recent experimental advances have demonstrated entanglement of logical qubits, which encode quantum information redundantly across multiple physical qubits to enable error correction and fault tolerance. In 2024, Microsoft and Quantinuum achieved a milestone by creating and entangling 12 logical qubits using a qubit-virtualization system on Quantinuum's H2 trapped-ion processor, encoding logical qubits using the tesseract code, with four logical qubits protected in 16 physical qubits and achieving logical error rates an order of magnitude lower than physical qubits. This demonstration involved preparing high-fidelity graph states across the logical qubits, showcasing scalable entanglement for practical quantum algorithms while suppressing errors through repeated syndrome measurements. Such entangled logical qubits pave the way for reliable, large-scale quantum computing by mitigating decoherence in noisy intermediate-scale quantum devices.^[76] Entangled states are crucial for quantum simulation of many-body physics, where classical computers struggle with the exponential complexity of strongly correlated systems. By preparing specific entangled configurations, such as those mimicking spin chains or Hubbard models, quantum processors can efficiently simulate ground states and dynamics of materials exhibiting phenomena like high-temperature superconductivity or quantum phase transitions. For example, entanglement measures like concurrence or negativity quantify correlations in these simulations, revealing scaling laws that align with theoretical predictions for 1D and higher-dimensional systems. This capability exploits the natural entanglement in quantum hardware to model intractable problems, with distillable entanglement providing a metric for the extractable pure entanglement resource from mixed many-body states.^[77]

Fundamental physics probes

Quantum entanglement serves as a powerful tool for probing foundational aspects of physics, particularly in testing the compatibility of quantum mechanics with relativity and exploring connections to quantum gravity. In Bell tests designed to close the locality loophole, entangled particles are measured at spacelike separations to ensure that no causal influence can propagate between the measurement sites faster than light, thereby invoking special relativity's prohibition on superluminal signaling.^[78] A seminal loophole-free Bell test in 2015 using entangled electron spins separated by 1.3 kilometers demonstrated a violation of the Clauser-Horne-Shimony-Holt inequality by 2.42 standard deviations, confirming quantum nonlocality while respecting relativistic causality and ruling out local hidden variable theories.^[78] Subsequent experiments, such as a 2023 superconducting circuit-based test achieving a CHSH violation exceeding the classical bound by more than 22 standard deviations over a 30-meter separation, further reinforced these findings by minimizing readout times to enforce strict locality.^[79] In the quest for quantum gravity, entanglement plays a central role in the AdS/CFT correspondence, a conjectured duality between quantum gravity in anti-de Sitter (AdS) space and a conformal field theory (CFT) on its boundary, suggesting that spacetime geometry emerges from quantum entanglement. The Ryu-Takayanagi formula posits that the entanglement entropy of a boundary region in the CFT equals one-fourth the area of the minimal surface in the bulk AdS geometry homologous to that region, providing a holographic prescription that links quantum information measures to gravitational structures. This relation implies that changes in entanglement, such as during quantum quenches, correspond to dynamical evolutions in bulk geometry, offering hints toward a unified theory where gravity arises from entangled quantum degrees of freedom.^[80] A 2025 theoretical study proposed that entanglement entropy directly contributes to spacetime curvature by introducing an "informational stress-energy tensor" into Einstein's field equations, modifying gravitational dynamics in regimes involving strong quantum correlations.^[81] Authored by Florian Neukart, this work derives perturbative corrections to Newton's gravitational constant that scale with energy density and entanglement strength, predicting subtle effects on black hole entropy and early-universe inflation, though these remain below current observational thresholds.^[81] Such modifications suggest entanglement as a fundamental ingredient in reconciling quantum mechanics and general relativity, potentially resolvable only near the Planck scale. In particle physics, entanglement has been observed in top quark-antiquark pairs produced at the Large Hadron Collider (LHC), enabling probes of quantum coherence at high energies and testing standard model predictions.^[63] The ATLAS collaboration reported in 2024 the first high-energy observation of spin entanglement in top quark pairs from proton-proton collisions at 13 TeV, using dilepton decay channels to measure an entanglement witness that deviated from classical expectations by more than 5 standard deviations, confirming quantum correlations persisting despite the quarks' short lifetimes of about 5 × 10^{-25} seconds.^[63] Complementarily, the CMS experiment observed similar entanglement in 2024 with a significance of 5.1 standard deviations, analyzing over 7 million candidate events to quantify spin correlations, which could reveal beyond-standard-model physics if deviations appear in future higher-luminosity runs.^[82] These measurements establish entanglement as a viable tool for precision tests in high-energy environments, bridging quantum information concepts with particle phenomenology.

Experimental Developments

Early demonstrations

The first experimental confirmation of quantum entanglement came in 1972 through a test of Bell's inequality using polarized photons emitted from an atomic cascade in calcium atoms. Stuart Freedman and John Clauser measured correlations between the photons' polarizations, demonstrating a violation of the inequality by approximately 5%, consistent with quantum mechanical predictions and ruling out local hidden-variable theories for this system.^[17] This experiment, while leaving some loopholes such as the detection efficiency, marked the initial empirical support for entanglement's non-local correlations as predicted by Bell's theorem.^[5] A significant advance occurred in 1982 with Alain Aspect's experiment, which addressed concerns about the locality loophole by implementing rapidly switching polarizers to choose measurement bases after the photons were emitted, ensuring the choice was independent of the particles' separation. Using entangled photon pairs from calcium atoms separated by 12 meters, the setup violated Bell's inequality by more than 5 standard deviations, providing stronger evidence for quantum entanglement while partially closing the locality loophole through dynamic basis selection.^[18] A key milestone in 1995 when researchers at NIST demonstrated the first entanglement between the internal states of two trapped 9Be+ ions, using their shared motional state as an intermediary for the coupling, laying the groundwork for quantum logic operations with atoms.^[83] By preparing states like Schrödinger cat superpositions, the experiment highlighted entanglement's role in atomic quantum information processing, achieving fidelities that confirmed quantum coherence over multiple operations. In 1998, Anton Zeilinger's group achieved the first demonstration of entanglement swapping using photons from parametric down-conversion, entangling two initially independent photon pairs without direct interaction between the final particles. By performing a joint Bell-state measurement on one photon from each pair, the remaining photons became entangled, with a fidelity of about 0.6 to the expected maximally entangled state, verifying the protocol's potential for quantum repeaters and networks.^[84] This experiment extended entanglement's reach beyond direct pairwise correlations, confirming theoretical predictions for scalable quantum communication.

Long-distance and macroscopic cases

One landmark achievement in long-distance quantum entanglement distribution occurred in 2017 with the Micius satellite, which successfully delivered entangled photon pairs to two ground stations in China separated by 1203 kilometers, achieving a Bell inequality violation with a parameter of S = 2.37 \pm 0.09, thus confirming the preservation of entanglement despite atmospheric turbulence and vast separation.^[85] This experiment utilized spontaneous parametric down-conversion in a compact source aboard the satellite to generate polarization-entangled photons at 810 nm, which were then transmitted via free-space optical links to telescopes at the receiving sites.^[85] Such demonstrations highlight the potential for satellite-based quantum communication networks, overcoming fiber-optic limitations in distance. In macroscopic systems, entanglement has been observed between collective vibrational modes of millimeter-scale objects, as exemplified by a 2011 experiment entangling the motional states of two spatially separated diamond crystals at room temperature.^[86] Researchers employed a Sagnac interferometer configuration with continuous-wave lasers to couple the Raman-scattered phonons in the diamonds' vibrational spectra around 40 THz, verifying entanglement through a violation of the continuous-variable Bell inequality with a value of -0.61 \pm 0.04.^[86] This work demonstrated that quantum correlations can persist in systems containing approximately $10^{17} atoms, bridging microscopic quantum effects with larger-scale mechanical oscillators. Entanglement in high-energy particle physics represents another frontier for macroscopic cases, with the ATLAS and CMS collaborations reporting its observation in top quark-antitop quark pair production at the Large Hadron Collider in 2023.^[87]^[88] Using proton-proton collisions at 13 TeV center-of-mass energy, ATLAS measured spin entanglement via dilepton decay channels, achieving a concurrence of $0.183 \pm 0.013 (stat.) \pm 0.023 (syst.), while CMS confirmed similar correlations in the same dataset, marking the highest-energy entanglement detection to date and probing quantum effects in systems with combined particle masses exceeding 346 GeV/c².^[87]^[88] These results stem from quantum chromodynamics predictions and underscore entanglement's role in validating standard model processes at extreme scales. A primary challenge in both long-distance and macroscopic entanglement is decoherence, arising from environmental interactions that rapidly degrade quantum coherence in extended systems, such as photon absorption in free space or thermal phonons coupling to vibrational modes.^[89] For instance, in satellite experiments, atmospheric seeing and pointing errors introduce loss rates up to 50 dB over 1000 km, necessitating error correction and quantum repeaters for practical scalability.^[90] In macroscopic mechanical systems, coupling to numerous environmental degrees of freedom limits entanglement lifetimes to microseconds, though cryogenic cooling and isolation techniques have extended coherence in diamond-based setups.^[86] Addressing these issues remains crucial for advancing applications in quantum sensing and networks.

Recent advances (2020s)

In 2023, researchers at Princeton University achieved the first entanglement of individual molecules using optical tweezers to trap and manipulate calcium monofluoride molecules at ultracold temperatures. By positioning pairs of molecules approximately 3 micrometers apart and applying microwave fields to couple their rotational states, the team generated entangled states with fidelities exceeding 0.7, marking a milestone for molecular quantum platforms that could enable scalable quantum simulation and computing.^[54] Advancing entanglement in nanostructured systems, a 2025 experiment at the Technion – Israel Institute of Technology demonstrated a novel form of entanglement involving the total angular momentum of photons confined in nanoscale waveguides. This near-field entanglement, observed between two photons with non-classical correlations in their combined spin and orbital angular momentum, achieved violation of classical bounds through observation of non-classical correlations in TAM, opening pathways for compact quantum devices integrated into photonic chips.^[91] Theoretical and experimental progress in 2025 linked classical gravity to quantum entanglement, showing that even non-quantum descriptions of gravity can induce entanglement between massive particles. In a Nature study, researchers proposed and analyzed a scenario where two particles in superposition experience differential gravitational fields, generating observable entanglement quantified by a phase parameter θ of up to 0.1, challenging assumptions about the necessity of quantized gravity for such effects and suggesting testable predictions with optomechanical systems.^[92] A breakthrough in multi-particle entanglement detection occurred in 2025, when scientists at Kyoto University developed a method for direct, high-fidelity measurement of W states involving three entangled photons. Using a collective quantum measurement protocol with adaptive optics, the team achieved a success probability of over 80% in identifying the symmetric W state, resilient to single-photon loss, which advances applications in quantum networks and error-corrected quantum information processing.^[93] In 2024, NASA's SEAQUE experiment aboard the International Space Station demonstrated persistent entanglement of photons over orbital distances, confirming robustness in space environments for quantum networks.^[94]

References

[1]
Quantum Entanglement and Information
Aug 13, 2001 · Quantum entanglement is a physical resource, like energy, associated with the peculiar nonclassical correlations that are possible between separated quantum ...
[2]
What Is the Spooky Science of Quantum Entanglement?
Quantum entanglement is the theory that particles of the same origin, which were once connected, always stay connected, even through time and space.Missing: key | Show results with:key
[3]
[PDF] Can Quantum-Mechanical Description of Physical Reality Be
In quantum mechanics it is usually assumed that the wave function does contain a complete description of the physical reality of the system. (5) in the state to ...
[4]
Bell's Theorem - Stanford Encyclopedia of Philosophy
Jul 21, 2004 · Moreover, Bell's theorem reveals that the entanglement-based correlations predicted by quantum mechanics are strikingly different from the ...Introduction · Proof of a Theorem of Bell's Type · Some Variants of Bell's Theorem
[5]
Proving that Quantum Entanglement is Real - Caltech
Sep 20, 2022 · Bell showed that quantum entanglement is, in fact, incompatible with EPR's notion of locality and causality.
[6]
What Is Entanglement and Why Is It Important?
When two particles, such as a pair of photons or electrons, become entangled, they remain connected even when separated by vast distances.
[7]
5 Concepts Can Help You Understand Quantum Mechanics and ...
Feb 5, 2025 · Quantum entanglement occurs when the quantum states of two or more particles become strongly correlated. This means the state of one particle ...
[8]
Physicists 'entangle' individual molecules for the first time, bringing ...
Dec 7, 2023 · The latter concept, entanglement, is a major cornerstone of quantum mechanics, and occurs when two particles become inextricably linked with ...
[9]
New Form of Quantum Entanglement Gives Insight into Nuclei
Sep 21, 2023 · Quantum entanglement is when two particles have a special relationship to each other. They don't and can't act independently. Instead, each ...Missing: definition key facts
[10]
On the theory of quantum mechanics | Proceedings of the Royal ...
The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the ...
[11]
Can Quantum-Mechanical Description of Physical Reality Be ...
Feb 18, 2025 · Einstein and his coauthors claimed to show that quantum mechanics led to logical contradictions. The objections exposed the theory's strangest ...
[12]
[PDF] ON THE EINSTEIN PODOLSKY ROSEN PARADOX*
THE paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented ...
[13]
Proposed Experiment to Test Local Hidden-Variable Theories
Proposed Experiment to Test Local Hidden-Variable Theories. John F. Clauser*. Michael A. Horne · Abner Shimony · Richard A. Holt.
[14]
Experimental Test of Local Hidden-Variable Theories | Phys. Rev. Lett.
Nobel Prize: Quantum Entanglement Unveiled ... The 2022 Nobel Prize in Physics honors research on the foundations of quantum mechanics, which opened up the ...Missing: proposal | Show results with:proposal
[15]
Experimental Test of Bell's Inequalities Using Time-Varying Analyzers
The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.
[16]
[PDF] Discussion of Probability Relations between Separated Systems
Oct 24, 2008 · Schrödinger (1935). Discussion of Probability Relations between Separated Systems. Mathematical Proceedings of the Cambridge Philosophical ...
[17]
Einstein-Podolsky-Rosen argument in quantum theory
Oct 31, 2017 · The paper features a striking case where two quantum systems interact in such a way as to link both their spatial coordinates in a certain direction and also ...Can Quantum Mechanical... · Einstein's versions of the... · Development of EPR
[18]
This Month in Physics History
### Summary of EPR Paradox, Debates, and Acceptance of Quantum Entanglement
[19]
Relational Quantum Mechanics - Stanford Encyclopedia of Philosophy
Feb 4, 2002 · This leads to a stronger version of realism that QBism, and to the emphasis on the relational aspect of all variables.Main Ideas · Quantum superposition: can a... · Related Issues · General Comments
[20]
Quantum-Bayesian and Pragmatist Views of Quantum Theory
Dec 8, 2016 · QBists argue that from this point of view quantum theory faces no conceptual problems associated with measurement or non-locality. While QBism ...
[21]
[PDF] CHSH Inequality on a single probability space - arXiv
Jan 9, 2017 · 4 Derivation of the CHSH inequality. We now explore the derivation of the CHSH inequality (1) given above. The expression makes use of the ...
[22]
Quantum generalizations of Bell's inequality
Even though quantum correlations violate Bell's inequality, they satisfy weaker inequalities of a similar type. Some particular inequalities of this ki.
[23]
The relativistic causality versus no-signaling paradigm for multi-party ...
Apr 12, 2019 · The quantum nonlocal correlations are known to be fully compatible with the no-signaling principle, i.e., the space-like separated parties ...
[24]
[PDF] arXiv:1904.08381v2 [quant-ph] 12 Dec 2019
Dec 12, 2019 · Consequently, no signaling can be proven by showing that the reduced density matrix of Bob is unchanged whatever operation Alice does. In the ...
[25]
[PDF] Does Quantum Nonlocality Exist? Bell's Theorem and the Many ...
Quantum nonlocality may be an artifact of the assumption that observers obey the laws of classical mechanics, while observed systems obey quantum mechanics.
[26]
[PDF] Nonlocality in Bell's theorem, in Bohm's theory, and in Many ... - arXiv
The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new ...
[27]
https://arxiv.org/pdf/1904.08381
[28]
https://arxiv.org/pdf/quant-ph/0003146
[29]
[quant-ph/9609002] Relational Quantum Mechanics - arXiv
Aug 31, 1996 · I consider a reformulation of quantum mechanics in terms of information theory. All systems are assumed to be equivalent, there is no observer-observed ...Missing: entanglement nonlocality
[30]
[quant-ph/0604064] Relational EPR - arXiv
Apr 10, 2006 · We argue that these correlations do not entail any form of 'non-locality', when viewed in the context of this interpretation. The abandonment of ...
[31]
Many-Worlds: Why Is It Not the Consensus? - MDPI
Bell's inequality and its violation arguably show that any theory which reproduces the predictions of quantum theory has to be nonlocal, if we assume that the ...
[32]
Quantum Nonlocality - PMC - NIH
Apr 29, 2019 · Quantum nonlocality, in single-world interpretations, is actions at a distance, like entangled particles where measurement on one changes the ...
[33]
[PDF] Lecture Notes for Ph219/CS219: Quantum Information Chapter 2
A qubit is a quantum system described by a two-dimensional Hilbert space, whose state can take any value of the form eq.(2.15). We can perform a measurement ...
[34]
[quant-ph/9604005] Separability Criterion for Density Matrices - arXiv
Apr 8, 1996 · Abstract: A quantum system consisting of two subsystems is separable if its density matrix can be written as \rho=\sum_A w_A\ ...
[35]
Separability Criterion for Density Matrices | Phys. Rev. Lett.
Aug 19, 1996 · In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of ρ, has only non-negative ...
[36]
Separability of Mixed States: Necessary and Sufficient Conditions
View a PDF of the paper titled Separability of Mixed States: Necessary and Sufficient Conditions, by Michal Horodecki and 1 other authors.
[37]
Quantum states with Einstein-Podolsky-Rosen correlations admitting ...
Oct 1, 1989 · Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Reinhard F. Werner. Dublin Institute for Advanced ...
[38]
Quantum entanglement | Rev. Mod. Phys.
Jun 17, 2009 · This article reviews basic aspects of entanglement including its characterization, detection, distillation, and quantification.
[39]
[quant-ph/0702225] Quantum entanglement - arXiv
Feb 26, 2007 · This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying.
[40]
Entanglement of Formation of an Arbitrary State of Two Qubits
Mar 9, 1998 · The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.Missing: original | Show results with:original
[41]
An additive entanglement measure | Journal of Mathematical Physics
Mar 1, 2004 · Matthias Christandl, Andreas Winter; “Squashed entanglement”: An additive entanglement measure. J. Math. Phys. 1 March 2004; 45 (3): 829–840 ...
[42]
Unextendible Product Bases and Bound Entanglement - arXiv
Aug 18, 1998 · We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We ...
[43]
[PDF] DISTILLATION AND BOUND ENTANGLEMENT 1. Introduction
The aim of this present paper is to present an overview of that part of the quantum information theory which concerns entanglement distillation together with ...
[44]
Invited article on entanglement properties of quantum field theory
Oct 23, 2018 · The Reeh-Schlieder theorem says that, in quantum field theory, if A V and A V ′ are the algebras of operators supported in complementary regions ...
[45]
Colloquium: Area laws for the entanglement entropy | Rev. Mod. Phys.
Feb 4, 2010 · In this paper, we discuss the study of entanglement entropy primarily (i) from the viewpoint of quantum information theory, (ii) with an ...
[46]
[hep-th/0405152] Entanglement Entropy and Quantum Field Theory
May 18, 2004 · Abstract: We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann ...Missing: law seminal
[47]
[PDF] Spontaneous parametric down-conversion - arXiv
Spontaneous Parametric Down-Conversion (SPDC) is a non-linear optical process where a photon splits into two lower-energy photons.
[48]
Quantum entanglement of optical photons: the first experiment, 1964 ...
Aug 21, 2024 · This experiment would be the first attempt to count and analyze single optical photons and pairs of photons emitted in an atomic cascade.
[49]
Entanglement of Trapped-Ion Qubits Separated by 230 Meters
We report on an elementary quantum network of two atomic ions separated by 230 m. The ions are trapped in different buildings and connected with 520(2) m of ...Abstract · Article Text · ACKNOWLEDGMENTSMissing: creation | Show results with:creation
[50]
Generation of entangled states in cavity QED | Phys. Rev. A
We propose a scheme to generate four Bell states and a four-atom entangled cluster state in a thermal cavity.
[51]
On-demand entanglement of molecules in a reconfigurable optical ...
Dec 7, 2023 · In this work, we realized on-demand entanglement between individual laser-cooled molecules trapped in a reconfigurable optical tweezer array ( ...
[52]
Einstein’s entanglement – CERN Courier
### Summary of Quantum Entanglement from Neutral Pion Decay to Two Photons
[53]
Flavor Patterns of Fundamental Particles from Quantum Entanglement?
### Summary of Quantum Entanglement in Neutral Pion Decay to Two Photons
[54]
[1205.1584] Everything is Entangled - arXiv
May 8, 2012 · Title:Everything is Entangled ... Abstract:We show that big bang cosmology implies a high degree of entanglement of particles in the universe. In ...
[55]
Quantum entanglement produced in the formation of a black hole
Jul 16, 2010 · This entanglement provides quantum information resources between the modes in the asymptotic future (thermal Hawking radiation) and those which ...
[56]
Emergent entanglement structures and self-similarity in quantum ...
Jul 14, 2020 · We introduce an experimentally accessible network representation for many-body quantum states based on entanglement between all pairs of its ...Missing: superconductors | Show results with:superconductors
[57]
Entanglement of electron spins in superconductors | Phys. Rev. B
Apr 29, 2005 · We investigate entanglement of two electron spins forming Cooper pairs in an s -wave superconductor. The two-electron space-spin density ...Abstract · Article Text
[58]
Scientists Map Out Quantum Entanglement in Protons
Dec 2, 2024 · The results reveal that quarks and gluons, the fundamental building blocks that make up a proton's structure, are subject to so-called quantum entanglement.
[59]
Observation of quantum entanglement with top quarks at the ATLAS ...
Sep 18, 2024 · Here we report the highest-energy observation of entanglement, in top–antitop quark events produced at the Large Hadron Collider, using a proton–proton ...
[60]
Quantum entanglement
**Summary of Entanglement Witnesses from https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.81.865:**
[61]
Loophole-free Bell inequality violation using electron spins ... - Nature
Oct 21, 2015 · Here we report a Bell experiment that is free of any such additional assumption and thus directly tests the principles underlying Bell's inequality.
[62]
Significant-loophole-free test of Bell's theorem with entangled photons
Here we report a Bell test that closes the most significant of these loopholes simultaneously. Using a well-optimized source of entangled photons.
[63]
[2209.00702] Optimal statistical analyses of Bell experiments - arXiv
Sep 1, 2022 · The methodology is illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST ...
[64]
Certifying quantum state and dimension via phase-space Bell tests ...
May 1, 2023 · We investigate generalized Bell tests in phase space for continuous variables that can be used to certify quantum states and witness the dimension of a given ...Missing: review | Show results with:review
[65]
Entangled measurement for W states | Science Advances
Sep 12, 2025 · Vidal, J. I. Cirac, Three qubits can be entangled in two inequivalent ways. ... Dür, Quantum repeater for W states. PRX Quantum 4, 040323 (2023).
[66]
[quant-ph/9503016] Elementary gates for quantum computation - arXiv
Mar 23, 1995 · We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates.
[67]
Measurement-based quantum computation with cluster states - arXiv
Jan 13, 2003 · A scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states.
[68]
[2409.04628] Demonstration of quantum computation and error ...
Sep 6, 2024 · Using the tesseract code on Quantinuum's trapped-ion quantum computers, we prepare high-fidelity encoded graph states on up to 12 logical qubits ...
[69]
Entanglement in many-body systems | Rev. Mod. Phys.
May 6, 2008 · ... von Neumann entropy as a measure for the quantum entanglement has been analyzed by. Li et al. (2001) and. Paškauskas and You (2001) . It is ...
[70]
Significant-Loophole-Free Test of Bell's Theorem with Entangled ...
Dec 16, 2015 · Bell's theorem states that this worldview is incompatible with the predictions of quantum mechanics, as is expressed in Bell's inequalities.
[71]
Loophole-free Bell inequality violation with superconducting circuits
May 10, 2023 · In a Bell test4, two distinct parties A and B each hold one part of an entangled quantum system, for example, one of two qubits. Each party ...
[72]
Holographic Derivation of Entanglement Entropy from AdS/CFT - arXiv
Feb 28, 2006 · A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence.
[73]
Quantum entanglement contributions to gravitational dynamics
These results suggest that entanglement entropy plays a direct role in gravitational dynamics, influencing the curvature of spacetime and contributing to the ...
[74]
[2406.03976] Observation of quantum entanglement in top quark ...
Jun 6, 2024 · Entanglement is an intrinsic property of quantum mechanics and is predicted to be exhibited in the particles produced at the Large Hadron ...
[75]
Experimental issues in coherent quantum-state manipulation of ...
Abstract. Methods for, and limitations to, the generation of entangled states of trapped atomic ions are examined. As much as possible, state manipulations ...
[76]
Entangling Photons That Never Interacted | Phys. Rev. Lett.
This demonstrates that quantum entanglement requires the entangled particles neither to come from a common source nor to have interacted in the past. In our ...
[77]
Satellite-based entanglement distribution over 1200 kilometers
Jun 16, 2017 · Here we demonstrate satellite-based distribution of entangled photon pairs to two locations separated by 1203 kilometers on Earth.
[78]
Entangling Macroscopic Diamonds at Room Temperature - Science
Dec 2, 2011 · We generated motional entanglement between vibrational states of two spatially separated, millimeter-sized diamonds at room temperature.
[79]
[2311.07288] Observation of quantum entanglement in top-quark ...
Nov 13, 2023 · Here we report the highest-energy observation of entanglement, in top-antitop quark events produced at the Large Hadron Collider, using a proton-proton ...
[80]
[PDF] Probing entanglement in top quark production with the CMS detector
Mar 27, 2024 · We focus on a measurement of entanglement in top quark pairs, which provides a new probe for understanding top quark physics in the larger ...
[81]
Decoherence, the measurement problem, and interpretations of ...
Feb 23, 2005 · The key idea promoted by decoherence is the insight that realistic quantum systems are never isolated, but are immersed in the surrounding ...
[82]
Progress in satellite quantum key distribution - Nature
Aug 9, 2017 · The Micius satellite collects its entangled photons into single mode fibres. These are sensitive to vibrations and after launch introduced ...
[83]
Near-field photon entanglement in total angular momentum - PubMed
Apr 2, 2025 · Photons can carry angular momentum, which is conventionally attributed to two constituents-spin angular momentum (SAM), ...Missing: nanoscale | Show results with:nanoscale
[84]
Classical theories of gravity produce entanglement - Nature
Oct 22, 2025 · Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity. ... Quantum Gravity 42, ...Feynman's Experiment · Quantum Gravity · Discussion
[85]
Measuring the quantum W state: Seeing a trio of entangled photons ...
Sep 12, 2025 · It could also lead to new quantum communication protocols, the transfer of multi-photon quantum entangled states, and new methods for ...
[86]
Quantum Entanglement and Information
Stanford Encyclopedia of Philosophy entry on quantum entanglement, covering the no-signaling principle.
[87]
Chapter 4: Quantum Entanglement
Notes from John Preskill's quantum computation course at Caltech, providing an example of entangled spins in the EPR context.