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Quantum superposition

Quantum superposition is a core principle of quantum mechanics in which a quantum system, such as an electron or photon, can exist simultaneously in multiple mutually exclusive states—described mathematically as a linear combination of those states via the wave function—until a measurement is performed, causing the system to collapse into a single definite state.^[1] This phenomenon arises from the linearity of the Schrödinger equation, which governs the evolution of quantum systems and permits such overlapping probability waves.^[2] The concept of superposition originated in the development of wave mechanics by Erwin Schrödinger in 1926, building on Louis de Broglie's 1924 hypothesis of wave-particle duality, where particles like electrons exhibit wave-like behavior.^[3] Schrödinger's formulation provided a mathematical framework for quantum states as superpositions, resolving earlier inconsistencies in atomic models and proving equivalent to Werner Heisenberg's matrix mechanics.^[3] This breakthrough, published in a series of papers in 1926, laid the foundation for modern quantum theory and earned Schrödinger the 1933 Nobel Prize in Physics (shared with Paul Dirac).^[3] Key demonstrations of superposition include the double-slit experiment, where particles like electrons fired at a barrier with two slits interfere with themselves, producing a pattern consistent with each particle traversing both paths in superposition until detection.^[2] Another iconic illustration is Schrödinger's cat thought experiment from 1935, which posits a cat in a sealed box linked to a quantum event (e.g., radioactive decay) as being in a superposition of alive and dead states until observed, highlighting the counterintuitive scale of quantum effects.^[1] These examples underscore that superposition is not mere ignorance of state but a fundamental property, where the probability amplitudes of different states can interfere constructively or destructively.^[2] Superposition has profound implications for technology, particularly in quantum computing, where qubits can represent both 0 and 1 simultaneously, enabling exponential parallelism for solving complex problems intractable for classical computers.^[1] It also underpins phenomena like quantum tunneling and entanglement, driving advances in fields from cryptography to materials science, though challenges like decoherence—where environmental interactions collapse superpositions—remain central to ongoing research.^[4]

Foundations

Superposition principle

The superposition principle is a foundational postulate of quantum mechanics, stating that any quantum system can be described by a state that is a linear combination of multiple possible states, allowing the system to exist simultaneously in those states until a measurement forces it to collapse into one.^[5] This principle arises from the mathematical structure of quantum theory, where states are vectors in a Hilbert space, and any normalized linear superposition of valid states remains a valid quantum state due to the space's linearity.^[5] In contrast to classical systems, quantum superposition enables coherent interference between the component states, leading to phenomena impossible in classical physics, such as wave-like probability distributions.^[6] Unlike classical "superpositions," which are merely probabilistic mixtures—such as a coin flip where the outcome is either heads or tails with certain probabilities but never both at once—quantum superposition involves definite amplitudes that can constructively or destructively interfere, producing observable effects like enhanced or suppressed probabilities.^[7] For instance, a classical wave superposition, like combining sound tones, adds amplitudes linearly without the discrete state scaling and interference probabilities seen in quantum systems.^[8] The superposition principle was introduced by Erwin Schrödinger in 1926 through his formulation of wave mechanics, where he proposed that quantum systems behave as waves governed by a linear partial differential equation now known as the Schrödinger equation. This equation, i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, describes the time evolution of the system's wave function \psi, ensuring that superpositions of solutions remain solutions due to the equation's linearity, thus underpinning the wave-like nature of quantum superposition.^[9]

Historical development

The concept of quantum superposition emerged from early explorations of wave-particle duality in the 1920s. In 1924, Louis de Broglie proposed that particles, such as electrons, exhibit wave-like properties, introducing the idea of matter waves with a wavelength given by \lambda = h / p, where h is Planck's constant and p is momentum.^[10] This hypothesis laid the groundwork for duality, suggesting that quantum entities could manifest both particle and wave characteristics, which inherently implied the possibility of coherent superpositions of states.^[11] In 1925, Werner Heisenberg developed matrix mechanics, the first mathematical formulation of quantum mechanics, which implicitly incorporated superposition through the use of non-commuting observables represented by infinite matrices.^[12] These matrices described transitions between quantum states, allowing for linear combinations of basis states without direct reference to waves, yet the formalism required systems to exist in superposed configurations to account for interference-like effects in observables.^[13] This approach marked a shift from classical determinism, emphasizing probabilistic outcomes that superposition would later explain. Erwin Schrödinger's wave mechanics, introduced in 1926, explicitly formalized superposition by representing quantum states as wave functions \psi, which could be linear combinations of basis functions satisfying the Schrödinger equation.^[14] Building on de Broglie's ideas, Schrödinger demonstrated that the hydrogen atom's energy levels arise from superposed standing waves, providing a intuitive picture of how quantum systems occupy multiple states simultaneously until measurement.^[15] This wave-based framework made superposition a central tenet, contrasting with Heisenberg's abstract matrices but equivalent in predictive power. Paul Dirac's transformation theory in 1927 unified matrix and wave mechanics, formalizing superpositions through a general framework of linear vector spaces.^[16] Dirac showed that quantum states are elements in a linear vector space, where superpositions are arbitrary linear combinations of basis states, enabling transformations between representations and solidifying superposition as a universal principle across formulations.^[16] A pivotal challenge to superposition's implications came in 1935 with the Einstein-Podolsky-Rosen (EPR) paradox, where Albert Einstein, Boris Podolsky, and Nathan Rosen argued that entangled superpositions, such as those in a two-particle system with correlated positions and momenta, implied "spooky action at a distance" violating locality and realism.^[17] They contended that quantum mechanics must be incomplete, as measuring one particle's state instantaneously determines the other's, seemingly without physical influence.^[18] This critique sparked intense debates on the reality of superposed states. By the late 1930s, superposition gained widespread acceptance within the Copenhagen interpretation, championed by Niels Bohr and Heisenberg, which resolved EPR-like issues through complementarity and the role of measurement in collapsing superpositions.^[19] Bohr emphasized that quantum phenomena, including superposition, are context-dependent on the experimental apparatus, integrating the observer into the theory without requiring hidden variables.^[20] This pragmatic framework dominated quantum theory, framing superposition as an essential, if counterintuitive, feature verified by subsequent developments.

Mathematical Formalism

Wave function representation

In quantum mechanics, the wave function \psi(x, t) describes the state of a single particle in position space as a function of position x and time t, serving as a complex-valued function that encodes the probability amplitudes for the particle's location. This complex nature arises from the need to account for interference effects inherent in quantum systems, where the real and imaginary parts contribute to the phase relationships between different spatial components.^[21] Quantum superposition in the wave function representation manifests as a continuous linear combination of basis wave functions, expressed as \psi(x, t) = \int c_k \phi_k(x, t) \, dk, where \phi_k(x, t) are orthonormal basis functions (such as plane waves labeled by wave number k) and the coefficients c_k are complex amplitudes determining the contribution of each basis component.^[21] This integral form allows the wave function to spread over multiple positions simultaneously, reflecting the particle's delocalized nature before measurement.^[22] The time evolution of the superposed wave function is governed by the time-dependent Schrödinger equation, i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, where \hat{H} is the Hamiltonian operator and \hbar is the reduced Planck's constant. Due to the linearity of this equation, if \psi_1 and \psi_2 are solutions, then any superposition c_1 \psi_1 + c_2 \psi_2 also evolves as a solution, preserving the superposition structure over time without introducing nonlinear distortions. For physical interpretability, the wave function must satisfy the normalization condition \int_{-\infty}^{\infty} |\psi(x, t)|^2 \, dx = 1, ensuring that the total probability of finding the particle somewhere in space is unity, as per the probabilistic interpretation of |\psi(x, t)|^2 as the probability density.^[23] A representative example of superposition in position space is the initial state of a free particle formed by two Gaussian wave packets centered at different positions, \psi(x, 0) = N \left[ e^{-(x - x_1)^2 / (2 \sigma^2)} + e^{-(x - x_2)^2 / (2 \sigma^2)} \right], where N is a normalization constant, \sigma is the width, and x_1 \neq x_2.^[24] Under free evolution, each packet spreads and propagates with its group velocity, but their coherent overlap leads to interference patterns in the probability density |\psi(x, t)|^2, demonstrating the non-classical superposition effects.^[24]

Basis states and linear combinations

In quantum mechanics, the state of a physical system is represented by a vector |\psi\rangle in a complex separable Hilbert space, which provides the mathematical structure for describing quantum states and their evolution. This formulation allows for an abstract treatment independent of specific representations, capturing the essential features of quantum systems through linear algebra.^[25]^[26] Quantum superposition arises when the state vector is expressed as a linear combination of orthonormal basis states: |\psi\rangle = \sum_i c_i |\phi_i\rangle, where the |\phi_i\rangle form a complete orthonormal basis satisfying \langle \phi_i | \phi_j \rangle = \delta_{ij}, and the complex coefficients c_i obey the normalization condition \sum_i |c_i|^2 = 1 to ensure the total probability is unity. This structure generalizes the superposition principle to discrete bases, such as the energy eigenstates of a bound system like the hydrogen atom, where |\psi\rangle = \sum_n c_n |n\rangle and the |n\rangle are eigenstates of the Hamiltonian with discrete eigenvalues E_n. In contrast, for continuous spectra, such as free particles, the sum becomes an integral over a continuous basis.^[26] The linearity of quantum mechanical evolution, governed by the time-dependent Schrödinger equation or unitary operators, preserves superpositions: if |\psi(0)\rangle = \sum_i c_i |\phi_i\rangle, then at later time t, |\psi(t)\rangle = \sum_i c_i e^{-i E_i t / \hbar} |\phi_i\rangle (in the energy basis), maintaining the form of a linear combination. This property holds because the evolution operator is linear, ensuring that superposed states evolve as superpositions of the evolved basis states.^[26] For multi-particle systems, the Hilbert space is constructed as the tensor product of the individual particle spaces, allowing superpositions to extend naturally: the total state is |\Psi\rangle \in \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \cdots \otimes \mathcal{H}_N, which can be a superposition such as |\Psi\rangle = \sum_{i_1, \dots, i_N} c_{i_1 \dots i_N} |i_1\rangle_1 \otimes \cdots \otimes |i_N\rangle_N, with normalization \sum_{i_1, \dots, i_N} |c_{i_1 \dots i_N}|^2 = 1. This framework enables the description of entangled states and collective quantum behaviors in composite systems.^[27]

Notation and transformations

In quantum mechanics, the Dirac notation, also known as bra-ket notation, offers a concise framework for expressing superpositions of states. A general quantum state is represented as a ket vector |ψ⟩ in a Hilbert space, and its superposition over a complete orthonormal basis {|i⟩} is written as

|\psi\rangle = \sum_i c_i |i\rangle,

where the complex coefficients c_i satisfy the normalization condition ⟨ψ|ψ⟩ = 1, ensuring that the sum of probabilities |c_i|^2 equals unity. This notation abstracts away from specific representations, emphasizing the linear combination structure inherent to superposition.^[28]^[26] The representation of a superposition depends on the chosen basis, but the underlying state remains invariant. To transform the coefficients from one orthonormal basis {|i⟩} to another {|j⟩}, a unitary matrix U is employed, where the basis states transform as |j⟩ = ∑i U{ji} |i⟩. The new coefficients then become c'_j = ∑i U{ji} c_i, preserving the normalization due to the unitarity condition U^\dagger U = I. This change of basis highlights the abstract nature of quantum states, as the same |ψ⟩ admits different expansions without altering its physical content.^[26] A prominent example of such a basis transformation occurs between position and momentum representations. The momentum-space wavefunction ψ(p), which gives the coefficients in the momentum basis {|p⟩}, is obtained via the Fourier transform of the position-space wavefunction ψ(x):

\psi(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-i p x / \hbar} \, dx.

This unitary transformation underscores the duality between conjugate observables in quantum mechanics.^[26] Superpositions are invariant under unitary transformations, which map states within the Hilbert space while preserving inner products and thus probabilities in a fixed basis. For instance, the time evolution of a state under a time-independent Hamiltonian H is governed by the unitary operator U(t) = e^{-i H t / \hbar}, yielding |ψ(t)⟩ = U(t) |ψ(0)⟩, where the initial superposition evolves coherently without collapse. While the superposition itself is basis-independent—meaning |ψ⟩ exists independently of any expansion—the probabilities |⟨j|ψ⟩|^2 for outcomes in a measurement basis {|j⟩} explicitly depend on that basis, reflecting the observer's choice.^[26]

Physical Implications

Measurement outcomes

In quantum mechanics, when a system is in a superposition of states, a measurement of an observable corresponding to a complete set of orthonormal basis states \{ |\phi_i\rangle \} yields one of the eigenvalues associated with those basis states, with the pre-measurement state described by the superposition principle as |\psi\rangle = \sum_i c_i |\phi_i\rangle.^[29] The probability of obtaining the outcome corresponding to the basis state |\phi_i\rangle is given by the Born rule, which states that this probability is |\langle \phi_i | \psi \rangle|^2 = |c_i|^2. This rule, introduced by Max Born, provides the probabilistic interpretation essential for connecting quantum superpositions to empirical outcomes.^[29] Upon measurement yielding the outcome i, the wave function collapses instantaneously to the state |\phi_i\rangle, according to the projection postulate formalized by John von Neumann. This collapse is non-unitary and resets the system to an eigenstate of the measured observable. For an observable represented by a Hermitian operator \hat{O} with eigenvalues o_i and eigenvectors |\phi_i\rangle, the expectation value of the measurement outcome over many identical preparations of the state |\psi\rangle is \langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle = \sum_i |c_i|^2 o_i, as derived in the Dirac formalism. This average aligns with the statistical predictions of the Born rule.^[30] A representative example is a spin-1/2 particle prepared in the superposition |\psi\rangle = c_+ |+\rangle + c_- |-\rangle of spin-up and spin-down states along the z-axis, where |+\rangle and |-\rangle are eigenstates of the spin operator \hat{S}_z with eigenvalues +\hbar/2 and -\hbar/2, respectively. Measuring \hat{S}_z will yield +\hbar/2 with probability |c_+|^2 or -\hbar/2 with probability |c_-|^2, followed by collapse to the corresponding eigenstate.^[30] The irreversibility of this measurement process, where the superposition is lost and cannot be restored unitarily without additional information, fundamentally distinguishes quantum measurements from reversible classical dynamics.

Interference effects

In quantum mechanics, interference effects arise from the superposition of quantum states, where the probability amplitudes—complex numbers associated with each possible path or state—combine constructively or destructively to produce observable patterns that cannot be explained by classical probabilities alone.^[31] Unlike classical waves, where interference depends on intensity addition, quantum interference stems from the addition of amplitudes before squaring to obtain probabilities, leading to non-additive outcomes such as enhanced or suppressed detection probabilities at specific locations.^[31] The complex nature of these amplitudes introduces phase differences that critically determine the interference pattern. Each coefficient c_i in the superposition |\psi\rangle = \sum_i c_i |i\rangle can be expressed as c_i = |c_i| e^{i \theta_i}, where |c_i| is the magnitude and \theta_i is the phase; relative phases \theta_i - \theta_j dictate whether amplitudes add in phase (constructive interference, maximizing probability) or out of phase (destructive interference, minimizing it). This phase sensitivity manifests as oscillatory fringes in interference experiments, underscoring the wave-like behavior inherent to superpositions. A prominent demonstration occurs in the Mach-Zehnder interferometer, where a single photon enters a beam splitter, creating a superposition of traversing two paths, and recombines at a second beam splitter to produce interference fringes at the detectors.^[32] Depending on the path length difference, the photon's detection probability at one output can approach 100% (constructive) or 0% (destructive), illustrating self-interference without multiple particles.^[32] Environmental interactions cause decoherence, which suppresses these interference effects by entangling the quantum system with the surroundings, effectively randomizing phases and converting the coherent superposition into an incoherent mixture. In the Mach-Zehnder setup, even weak coupling to the environment—such as air molecules or thermal vibrations—can wash out fringes, transitioning the system toward classical behavior. The visibility of interference fringes provides a quantitative measure of superposition coherence, defined as V = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}, where I_{\max} and I_{\min} are the maximum and minimum intensities in the pattern; full coherence yields V = 1, while decoherence reduces V toward 0, directly quantifying the preservation of quantum superposition.^[33]

Interpretations

Quantum superposition lies at the heart of various interpretations of quantum mechanics, each offering distinct perspectives on whether superpositions represent objective physical states, subjective knowledge, or something else entirely. These interpretations do not alter the predictive power of quantum theory but differ in their ontological commitments regarding the "reality" of superposition prior to measurement. While the mathematical formalism remains unchanged, philosophical debates center on the wave function's role and the nature of measurement outcomes. The Copenhagen interpretation, developed primarily by Niels Bohr and Werner Heisenberg in the 1920s, views quantum superposition as a description of incomplete knowledge about a system before measurement, rather than an objective feature of reality. In this framework, the wave function encodes probabilities for potential outcomes, and superposition reflects the limitations of classical concepts in describing quantum phenomena; upon measurement, the superposition "collapses" to a definite state corresponding to the observed result. Bohr emphasized complementarity, arguing that superposition has no independent empirical meaning outside the context of a specific experimental setup, where classical apparatus descriptions are essential for interpreting results.^[34] In contrast, the many-worlds interpretation, proposed by Hugh Everett in 1957, treats superposition as a fully objective and universal aspect of reality, with no collapse occurring during measurement. Instead, the entire universe evolves linearly according to the Schrödinger equation, and a measurement on a superposed system causes the universal wave function to branch into multiple, non-interacting parallel worlds, each realizing one possible outcome of the superposition. Observers in different branches experience different results, preserving the superposition across the multiverse without invoking subjective collapse. This approach eliminates the measurement problem by positing that all outcomes in a superposition occur, albeit in separate realities.^[35] Bohmian mechanics, also known as pilot-wave theory and introduced by David Bohm in 1952, interprets superposition through a deterministic framework involving hidden particle positions guided by the wave function. Here, the wave function is a real physical field that pilots particles along definite trajectories, even when the system is in a superposition; the superposition influences these trajectories via an interference pattern in the guiding field, producing quantum effects without requiring collapse. Particles always have well-defined positions as the hidden variables, resolving the apparent indefiniteness of superposition into a complete, nonlocal description of motion.^[36] Quantum Bayesianism (QBism), developed in the early 2000s by Christopher Fuchs and others, regards superposition as encoding an agent's subjective degrees of belief about future measurement outcomes, rather than an objective state of the world. The wave function serves as a personal probabilistic tool, akin to Bayesian updating in statistics, where superposition represents updated expectations based on prior information; there is no collapse, as quantum states are inherently epistemic and observer-dependent. This view emphasizes that quantum mechanics is a calculus for personal predictions, with superposition lacking ontological status beyond the agent's perspective.^[37] Despite these diverse views, there remains no consensus among physicists on the nature of superposition's reality, as interpretations primarily address foundational questions without affecting empirical predictions. A 2025 survey of over 1,100 physicists revealed deep divisions: only 31% affirmed that an unobserved particle in superposition traverses multiple paths, 14% disagreed, and 48% deemed the question meaningless, underscoring ongoing debates on whether superposition is physically real, epistemic, or otherwise.^[38]^[39]

Experimental Verification

Early demonstrations

The foundational demonstrations of quantum superposition emerged in the early 20th century, building on classical wave optics experiments that hinted at wave-like behavior in light and extending these principles to particles. Thomas Young's double-slit experiment in 1801 served as a crucial precursor, where light passing through two closely spaced slits produced an interference pattern on a screen, indicating that light exists in a coherent wave state capable of superposition across multiple paths.^[40] This observation, initially interpreted through classical wave theory, laid the groundwork for understanding quantum superposition by illustrating how waves from different paths interfere constructively and destructively, a phenomenon later applied to quantum particles.^[40] In 1922, Otto Stern and Walther Gerlach conducted their seminal experiment, directing a beam of silver atoms through an inhomogeneous magnetic field, which split the beam into two discrete components corresponding to the atom's spin angular momentum aligned either parallel or antiparallel to the field. This spatial separation demonstrated the quantization of spin, providing direct evidence for the existence of superposition in the spin degree of freedom: prior to measurement, the atoms' spins were in a superposition of the two possible states, as classical theory would predict a continuous distribution rather than discrete spots.^[41] The experiment confirmed the directional quantization of angular momentum, a cornerstone of quantum mechanics that underpins spin superposition. The wave nature of matter, essential for particle superposition, was experimentally verified in 1927 by Clinton Davisson and Lester Germer, who observed diffraction patterns when electrons were scattered off a nickel crystal surface, producing intensity maxima consistent with wave interference.^[42] This electron diffraction matched the predictions of Louis de Broglie's hypothesis that particles possess wave properties with wavelength \lambda = h/p, where h is Planck's constant and p is momentum, implying that individual electrons traverse the crystal in a superposition of multiple scattering paths, leading to observable interference.^[42] The results extended Young's interference effects from light to electrons, solidifying wave-particle duality as the basis for quantum superposition in matter.^[42] Although occurring in the 1970s, neutron interferometry rooted its design in these early quantum ideas, with the first successful demonstration in 1974 using a silicon perfect-crystal interferometer to split and recombine a neutron beam, producing clear interference fringes that confirmed the superposition of neutron matter waves over macroscopic distances of several centimeters. In this setup, neutrons entering the interferometer were placed in a superposition of two spatially separated paths, which interfered upon recombination, directly verifying de Broglie wave coherence for neutral particles and reinforcing the principles established by prior electron and atom experiments. These early demonstrations collectively established wave-particle duality as the underpinning of quantum superposition, showing that particles maintain coherent superpositions until measurement collapses the state.^[43]

Modern precision tests

Modern precision tests of quantum superposition have advanced significantly since the late 20th century, focusing on verifying superposition in increasingly complex and macroscopic systems while addressing challenges like decoherence and experimental loopholes. These experiments not only confirm the foundational principles of quantum mechanics but also probe the boundaries between quantum and classical regimes. Key developments include refined Bell inequality tests that demonstrate superposition through entanglement, matter-wave interference with massive molecules, prolonged coherence in solid-state qubits, and superposition states in mechanical oscillators via optomechanics. These tests collectively push the limits of superposition verification, revealing how environmental interactions can suppress quantum behavior in larger systems. Bell test experiments provide rigorous confirmation of quantum superposition in entangled systems by violating local realism, a direct consequence of superposition and nonlocality. In 1982, Alain Aspect and colleagues conducted pioneering photon-based tests that violated Bell's inequalities, demonstrating correlations incompatible with classical local hidden variables and thus affirming superposition in entangled pairs. Subsequent loophole-free tests in 2015 eliminated detection, locality, and fair-sampling loopholes, achieving violations with a CHSH value of S = 2.42 \pm 0.20 using entangled electron spins separated by 1.3 km, providing the strongest evidence to date for superposition-driven entanglement without classical explanations. Matter-wave interference experiments with large organic molecules have extended superposition demonstrations to macroscopic scales, showing de Broglie wave behavior in objects far beyond atomic dimensions. Markus Arndt's group first observed interference fringes from C_{60} fullerene molecules (mass ~720 u, 60 atoms) in 1999 using a grating interferometer, confirming self-interference consistent with quantum superposition. Building on this, experiments progressed to more complex molecules; by 2019, the same team achieved high-contrast interference with phthalocyanine tetramers and derivatives containing up to ~2000 atoms and masses exceeding 25 kDa, verifying superposition in thermal beams at room temperature with visibility up to 40% despite decoherence from internal vibrations. In solid-state systems, superconducting qubits have enabled precise control and measurement of superposition states with extended coherence times, testing limits against environmental noise. During the 2010s and early 2020s, advances in circuit design and materials reduced decoherence, allowing transmon and fluxonium qubits to maintain superposition for up to 1.48 ms in Ramsey experiments, a tenfold improvement over prior benchmarks and enabling multi-gate operations without loss of quantum information. These durations approach the millisecond regime, highlighting superposition robustness in mesoscopic electrical circuits cooled to millikelvin temperatures. Optomechanical systems couple light to mechanical oscillators, facilitating the creation and detection of superposition states in vibrational modes of massive objects. In the 2020s, experiments have generated nonclassical mechanical states, such as Schrödinger cat-like superpositions, by parametrically driving cavity modes to entangle photons with phonons in silicon nitride membranes or levitated nanoparticles. For instance, a 2022 demonstration prepared displaced coherent states in a 10-MHz oscillator, achieving fidelities >90% to ideal cat states and verifying superposition via homodyne detection of quadrature squeezing beyond classical limits. These precision tests also explore decoherence boundaries, with recent 2023–2025 advances demonstrating room-temperature superpositions in molecular and nanoparticle systems. Arndt's ongoing work with thermal molecular beams continues to push atomic limits at ambient conditions, while ETH Zurich researchers in 2025 levitated nano-glass spheres (mass ~10^8 u) in optical traps, achieving 92% quantum state purity without cryogenic cooling by suppressing thermal phonons via feedback, thus extending macroscopic superposition viability and informing quantum-to-classical transitions.

Applications

Quantum computing

In quantum computing, the fundamental unit of information is the qubit, which unlike a classical bit, can exist in a superposition of states described by the wave function |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, where \alpha and \beta are complex amplitudes satisfying |\alpha|^2 + |\beta|^2 = 1. This allows a single qubit to represent both 0 and 1 simultaneously, with the probabilities of measuring each outcome given by the squared magnitudes of the coefficients. For n qubits, the system can occupy a superposition spanning all $2^n possible basis states, enabling the representation of an exponentially large computational space in a linear number of qubits. This property underpins the potential for quantum computers to process vast amounts of information in parallel. Quantum gates manipulate these superpositions to perform computations. For instance, the Hadamard gate H, a single-qubit unitary operator with matrix representation

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix},

transforms the basis state |0\rangle into the equal superposition \frac{|0\rangle + |1\rangle}{\sqrt{2}}, and similarly for |1\rangle. Algorithms exploit this parallelism; Grover's search algorithm, for example, uses amplitude amplification on a superposition of database entries to achieve a quadratic speedup over classical exhaustive search, querying an unsorted database of N items in O(\sqrt{N}) steps.^[44] A prominent application is Shor's algorithm, which factors large integers by creating a superposition over exponentially many periods in the modular exponentiation step, enabling efficient identification of prime factors through quantum Fourier transform—a task intractable for classical computers on large scales.^[45] However, maintaining these superpositions is challenging due to decoherence, where environmental interactions cause qubits to lose their quantum coherence over time. In superconducting qubit hardware, energy relaxation times T_1 and dephasing times T_2 have reached up to several milliseconds in leading systems as of 2025, though typical values remain in the hundreds of microseconds to low milliseconds range, limiting the depth of computations before errors accumulate.^[46] Despite this, superposition has demonstrated computational speedups over classical systems; Google's 2019 quantum supremacy experiment using a 53-qubit processor performed a random circuit sampling task in 200 seconds that would take a supercomputer 10,000 years, leveraging superposition for exponential parallelism.^[47] More recent demonstrations, such as Google's 2025 quantum processor achieving a 13,000× speedup over classical supercomputers in physics simulations, continue to showcase superposition's role in practical advantages.^[48] Similar advantages have been shown in IBM's systems, where superposition enables tasks like variational quantum eigensolver simulations unattainable classically at scale.^[49] Measurement ultimately collapses the superposition to a classical outcome, providing the computational result.

Quantum sensing and metrology

Quantum superposition enables enhanced precision in sensing and metrology by allowing quantum systems to probe physical parameters with reduced uncertainty, surpassing classical limits through coherent superpositions of states. In quantum sensing, superpositions of quantum states, such as spin or photonic modes, are manipulated to encode information about external fields or forces, leveraging interference effects to amplify signals while suppressing noise. This approach is foundational to devices that achieve sensitivities unattainable with classical methods, where measurement precision is constrained by the standard quantum limit (SQL).^[50] Squeezed states represent a key application of superposition in quantum metrology, where the quantum uncertainty in one observable, such as position or phase, is reduced below the vacuum level at the expense of increased uncertainty in the conjugate variable, in accordance with the Heisenberg uncertainty principle. These states are generated via nonlinear optical processes, like parametric down-conversion, creating superpositions of photon number states that correlate fluctuations between quadratures. In sensing, squeezed vacuum or light fields are injected into interferometers or resonators to minimize shot noise, enabling sub-SQL precision for detecting weak signals such as magnetic fields or gravitational perturbations. For instance, frequency-dependent squeezing tailors the noise reduction across spectral bands, optimizing broadband sensitivity.^[51]^[52] Atomic clocks exemplify the use of superposition for time and frequency metrology, employing coherent superpositions of hyperfine ground states in atoms or ions to achieve exceptional stability. In optical lattice clocks at NIST, aluminum ions are prepared in superpositions via quantum logic spectroscopy, allowing interrogation of narrow-linewidth transitions with fractional frequency uncertainties of 5.5 × 10^{-19} as demonstrated in 2025.^[53] This precision, demonstrated in comparisons between independent clocks, stems from the superposition-enhanced Ramsey interferometry, which mitigates decoherence and scales sensitivity with probe time. Such clocks support applications in relativistic geodesy and fundamental constant tests.^[54] Nitrogen-vacancy (NV) centers in diamond utilize electron spin superpositions for nanoscale magnetometry, where the spin-1 ground state is initialized into a coherent superposition of m_s = 0 and m_s = \pm 1 levels using microwave pulses. This superposition evolves under Zeeman splitting induced by external magnetic fields, enabling detection via optically detected magnetic resonance (ODMR) with sensitivities reaching sub-nanotesla (sub-nT) levels at room temperature. Ensemble NV centers in bulk diamonds have achieved broadband sensitivities of approximately 1 nT/\sqrt{\mathrm{Hz}}, while single NV sensors offer nanometer spatial resolution for biomedical imaging and material characterization. Recent optimizations, including dynamical decoupling, extend coherence times to milliseconds, enhancing signal-to-noise ratios.^[55]^[56] Interferometric sensors, such as those in gravitational wave detectors, exploit superpositions of light fields to measure minute displacements. In LIGO, squeezed vacuum states—superpositions reducing amplitude quadrature noise—are injected into the interferometer's antisymmetric port, suppressing quantum radiation pressure and shot noise to below the SQL. This has improved strain sensitivity by up to 3 dB in the 35–75 Hz band, facilitating the detection of binary neutron star mergers and other astrophysical events. The technique relies on the superposition's ability to redistribute quantum noise, achieving broadband enhancements without sacrificing dynamic range.^[57]^[58] The Heisenberg limit (HL) in quantum metrology arises from entangled superpositions of N probes, yielding precision scaling as $1/N, compared to the classical SQL of $1/\sqrt{N}. This fundamental bound is approached using multipartite entangled states, such as NOON states or spin-squeezed ensembles, where collective superposition enhances phase accumulation. In practice, entangled photonic or atomic superpositions in distributed sensor networks have neared HL performance, with recent demonstrations achieving $1.1 \times 10^{-18} fractional precision in spin-squeezed clocks. Advances in portable quantum sensors, including chip-scale NV magnetometers and squeezed-light atomic devices, are extending HL scaling to field-deployable systems for navigation and environmental monitoring as of 2025.^[59]^[60]^[61]

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Heisenberg's Breakthrough - American Institute of Physics
The first page of Heisenberg's break-through paper on quantum mechanics, published in the Zeitschrift für Physik, 33 (1925), 879-893, received 29 July 1925.
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Understanding Heisenberg's 'Magical' Paper of July 1925 - arXiv
Apr 1, 2004 · In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which ended the period of `the Old Quantum Theory' and ushered in the new era of ...
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[PDF] 1926-Schrodinger.pdf
The paper gives an account of the author's work on a new form of quantum theory. §1. The Hamiltonian analogy between mechanics and optics. §2. The analogy is to ...
[15]
An Undulatory Theory of the Mechanics of Atoms and Molecules
The paper gives an account of the author's work on a new form of quantum theory. §1. The Hamiltonian analogy between mechanics and optics.
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[PDF] The Quantum Theory of the Electron - UCSD Math
Oct 26, 2013 · By P. A. M. DIRAC, St. John's College, Cambridge. (Communicated by R. H. Fowler, F.R.S.-Received January 2, 1928.) The new quantum mechanics, ...
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The quantum theory of the electron - Journals
The quantum theory of the electron. Paul Adrien Maurice Dirac.
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[PDF] Can Quantum-Mechanical Description of Physical Reality Be
A. EINSTEIN, B. PODOLSKY AND N. ROSEN, Institute for Advanced Study, Princeton, New Jersey. (Received March 25, 1935). In a complete theory there is an ...
[19]
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 ...
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Copenhagen Interpretation of Quantum Mechanics
May 3, 2002 · The Copenhagen interpretation was the first general attempt to understand the world of atoms as this is represented by quantum mechanics.The Interpretation of the... · The Measurement Problem · Complementarity...
[21]
Triumph of the Copenhagen Interpretation - Heisenberg Web Exhibit
Heisenberg formulated the uncertainty principle in February 1927 while employed as a lecturer in Bohr's Institute for Theoretical Physics at the University of ...
[22]
[PDF] Meaning of the wave function - arXiv
The wave function is the most fundamental concept of quantum mechanics. It was first introduced into the theory by analogy (Schrödinger 1926); the behavior of ...
[23]
[PDF] A trajectory-based understanding of quantum interference - arXiv
Oct 1, 2008 · The wave function describing the evolution of these two wave packets is a superposition which will display interference when they “collide” ...
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[PDF] Zur Quantenmechanik der Stoßvorgänge - psiquadrat
(Eingegangen am 25. Juni 1926.) Durch eine Untersuchung der S~oflvorg~nge wird die Auffassung entwickelt, daft die Quantenmechanik in der S chrSdingerschen Form ...
[25]
[PDF] arXiv:1906.08526v3 [quant-ph] 22 Aug 2020
Aug 22, 2020 · Two examples are considered, the free evolution of one and two Gaussian wave packets as well as the time evolution under a constant field.
[26]
Mathematical Foundations of Quantum Mechanics
### Book Description
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[PDF] PRINCIPLES QUANTUM MECHANICS
The general principle of superposition of quantum mechanics applies to ... now use form a more general space than a Hilbert space. We tan now see that ...
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[PDF] MULTIPARTICLE STATES AND TENSOR PRODUCTS
Dec 14, 2013 · In this section, we develop the tools needed to describe a system that contains more than one particle. Most of the required ideas appear ...
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A new notation for quantum mechanics - Cambridge University Press
A NEW NOTATION FOR QUANTUM MECHANICS. BY P. A. M. DIRAC. Received 29 April 1939. In mathematical theories the question of notation, while not of primary impor-.
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Zur Quantenmechanik der Stoßvorgänge | Zeitschrift für Physik A ...
Download PDF · Zeitschrift für Physik. Zur Quantenmechanik ... Cite this article. Born, M. Zur Quantenmechanik der Stoßvorgänge. Z. Physik 37, 863–867 (1926).
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The Principals Of Quantum Mechanics : Dirac. P.a.m - Internet Archive
Oct 11, 2020 · The Principals Of Quantum Mechanics. by: Dirac. P.a.m ... PDF WITH TEXT download · download 1 file · SINGLE PAGE PROCESSED JP2 ZIP ...
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The Feynman Lectures on Physics Vol. III Ch. 3: Probability Amplitudes
Equation (3.7) says in effect that the particle has wavelike properties, the amplitude propagating as a wave with a wave number equal to the momentum divided by ...
[33]
Experimental observation of simultaneous wave and particle ...
Apr 30, 2015 · If the second BS is present, the photon travels through both beams of the Mach-Zehnder interferometer and interference fringes can be observed, ...
[34]
Duality of quantum coherence and path distinguishability
Jul 22, 2015 · For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known ...
[35]
How Bohr's Copenhagen interpretation is realist and solves ... - arXiv
Jul 29, 2023 · Bohr's interpretation is argued to be realist and solves the measurement problem, unlike textbook quantum mechanics, and eliminates the problem ...Missing: superposition | Show results with:superposition
[36]
"Relative State" Formulation of Quantum Mechanics
### Summary of Everett's 1957 Paper on Many-Worlds Interpretation
[37]
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[38]
QBism, the Perimeter of Quantum Bayesianism
### Summary of QBism's View on Quantum Superposition as Subjective Probabilities
[39]
Physicists Divided on What Quantum Mechanics Says about Reality
Aug 8, 2025 · A survey of more than 1,000 physicists finds deep disagreements in what quantum theories mean in the real world.
[40]
https://www.aps.org/publications/apsnews/200805/physicshistory.cfm
[41]
Thomas Young and the Nature of Light - American Physical Society
In November 1801 Young presented his paper, titled “On the theory of light and color” to the Royal Society. In that lecture, he described interference of light ...
[42]
How the Stern–Gerlach experiment made physicists believe in ...
Nov 1, 2022 · Their paper presenting the results, published in December 1922 (Zeit. Phys. 9 349), provided the first experimental evidence for space ...
[43]
Diffraction of Electrons by a Crystal of Nickel | Phys. Rev.
Feb 3, 2025 · Quantum Milestones, 1927: Electrons Act Like Waves. Published 3 February, 2025. Davisson and Germer showed that electrons scatter from a ...
[44]
The neutron interferometer as a device for illustrating the strange ...
Oct 1, 1983 · The neutron interferometer is a unique instrument that allows one to construct a neutron wave packet of macroscopic size, divide it into two ...Missing: paper | Show results with:paper
[45]
A fast quantum mechanical algorithm for database search - arXiv
Nov 19, 1996 · The quantum algorithm can find a phone number in O(sqrt(N)) steps, using quantum superposition to examine multiple names simultaneously.
[46]
[quant-ph/9508027] Polynomial-Time Algorithms for Prime ... - arXiv
Aug 30, 1995 · This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer.
[47]
Quantum error correction below the surface code threshold - Nature
Dec 9, 2024 · The qubits have a mean coherence time T1 of 68 μs and T2,CPMG of 89 μs, which we attribute to improved fabrication techniques, participation ...
[48]
Quantum supremacy using a programmable superconducting ...
Oct 23, 2019 · A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018). Article CAS ADS MathSciNet Google ...Missing: superposition speedup
[49]
USC researchers show exponential quantum scaling speedup - IBM
Jul 16, 2025 · An exponential scaling speedup means that as the problem size increases owing to the addition of variables, the gap between quantum and ...
[50]
Quantum sensing | Rev. Mod. Phys. - Physical Review Link Manager
Jul 25, 2017 · This review provides an introduction to the basic principles, methods, and concepts of quantum sensing from the viewpoint of the interested experimentalist.Article Text · Examples of Quantum Sensors · The Quantum Sensing Protocol
[51]
Quantum Sensing with Squeezed Light | ACS Photonics
May 13, 2019 · This Perspective reviews the dramatic advances made in the use of squeezed light for sub-shot-noise quantum sensing in recent years and highlights emerging ...
[52]
Squeezing metrology: a unified framework - Quantum Journal
Jul 9, 2020 · Here we focus instead on quantum squeezing and provide a unified framework for metrology with squeezing, showing that, similarly, one can generally attain a ...
[53]
An 27Al+ quantum-logic clock with systematic uncertainty below 10-18
Jul 15, 2019 · We describe an optical atomic clock based on quantum-logic spectroscopy of the 1S0 <- -> 3P0 transition in 27Al+ with a systematic uncertainty ...
[54]
An atomic clock with 10-18 instability | NIST
Sep 13, 2013 · A measurement comparing these systems demonstrates an unprecedented atomic clock instability of 1.6x10 -18 after only 7 hours of averaging.
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Sensitivity optimization for NV-diamond magnetometry
Mar 31, 2020 · This review analyzes present and proposed approaches to enhance the sensitivity of broadband ensemble-NV-diamond magnetometers.
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Nitrogen-vacancy centers in diamond for nanoscale magnetic ...
Nov 4, 2019 · The nitrogen-vacancy (NV) center is a point defect in diamond with unique properties for use in ultra-sensitive, high-resolution magnetometry.
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Squeezing the quantum noise of a gravitational-wave detector ...
Sep 19, 2024 · We found that the upgrade reduces quantum noise below the SQL by a maximum of three decibels between 35 and 75 hertz while achieving a broadband sensitivity ...
[58]
LIGO Surpasses the Quantum Limit | LIGO Lab | Caltech
Oct 23, 2023 · Since 2019, LIGO's twin detectors have been squeezing light in such a way as to improve their sensitivity to the upper frequency range of ...
[59]
Achieving the Heisenberg limit in quantum metrology using ... - Nature
Jan 8, 2018 · According to HL, the scaling of precision with t can be no better than 1/t; equivalently, precision scales no better than 1/N with the total ...
[60]
Clock precision beyond the Standard Quantum Limit at 10^-18 level
May 7, 2025 · Here we demonstrate clock performance beyond the SQL, achieving a fractional frequency precision of 1.1 \times 10^{-18} for a single spin-squeezed clock.