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W state

The W state is a fundamental entangled quantum state involving three qubits, mathematically defined as |W\rangle = \frac{1}{\sqrt{3}} (|001\rangle + |010\rangle + |100\rangle), which exemplifies one of two inequivalent classes of genuine tripartite entanglement.^[1] Unlike the GHZ state, |GHZ\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle), the W state exhibits remarkable robustness: tracing out any single qubit leaves the remaining two in a mixed state with maximal bipartite entanglement, specifically \rho_{AB} = \frac{2}{3} |\Psi^+\rangle\langle \Psi^+| + \frac{1}{3} |00\rangle\langle 00|, where |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle).^[1] This property maximizes both the average and minimum bipartite entanglement across all possible one-qubit losses, making it \overline{E(W)} = \frac{4}{9}, in contrast to the GHZ state's complete loss of entanglement under similar conditions.^[1] The W state's entanglement cannot be transformed into the GHZ class (or vice versa) via local operations and classical communication with nonzero probability, highlighting their distinct SLOCC (stochastic local operations and classical communication) classes.^[1] This inequivalence underscores the richness of multipartite entanglement, where the W state prioritizes distributed pairwise correlations over the fully correlated structure of GHZ.^[1] Generalizations to N qubits extend this robustness, defining |W_N\rangle = \frac{1}{\sqrt{N}} \sum_{i=1}^N |00\dots 1_i \dots 00\rangle, a symmetric state that preserves entanglement even after losing N-2 qubits.^[1] In quantum information science, W states play a crucial role due to their resilience against decoherence and particle loss, enabling applications such as quantum teleportation schemes that outperform GHZ-based protocols in noisy environments.^[2] They facilitate multipartite quantum secret sharing, where information is split among parties such that reconstruction requires cooperation, leveraging the state's partial traceability properties.^[3] Experimental realizations have advanced in photonic, atomic, and superconducting systems, with recent breakthroughs including the first collective entangled measurement of photonic W states in 2025, enhancing prospects for scalable quantum networks.^[4]

Definition and Fundamentals

Definition

The W state is a fundamental tripartite entangled quantum state defined for three qubits, serving as one of the two canonical representatives of genuine three-qubit entanglement, alongside the GHZ state. It embodies a symmetric configuration in which exactly one qubit is excited to the |1⟩ state, while the other two remain in the |0⟩ ground state, capturing a form of multipartite correlation that is inequivalent to other entanglement classes under local operations.^[1]^[5] This state plays a key role in illustrating multipartite entanglement, where the quantum dependencies among the three qubits are inherently collective and cannot be reduced to simpler bipartite links. A distinctive feature is its robustness: upon measurement of any single qubit, the resulting state of the remaining two qubits often preserves entanglement, highlighting a resilient structure not shared by all multipartite states.^[1]^[5] The W state was introduced in 2000 by Wolfgang Dür, Guifré Vidal, and J. Ignacio Cirac as part of their seminal classification of three-qubit entanglement into inequivalent classes, distinguishing it from biseparable and other fully separable forms.^[1]^[5]

Mathematical Representation

The three-qubit W state is mathematically represented in the computational basis as the equal superposition of all single-excitation states:
|W\rangle = \frac{1}{\sqrt{3}} \left( |100\rangle + |010\rangle + |001\rangle \right),
where |0\rangle and |1\rangle denote the standard qubit basis states, and the notation |abc\rangle represents the tensor product state of three qubits with amplitudes a, b, c \in \{0,1\}.^[5] This form captures a symmetric distribution of the single excitation across the three qubits, distinguishing it as a canonical representative of tripartite entanglement.^[5] The normalization factor $1/\sqrt{3} ensures that the state has unit norm, as required for pure quantum states. To verify this, compute the inner product \langle W | W \rangle: the basis states |100\rangle, |010\rangle, and |001\rangle are mutually orthogonal, so their cross terms vanish, yielding \langle W | W \rangle = \frac{1}{3} (1 + 1 + 1) = 1. Without the factor, the unnormalized superposition would have norm \sqrt{3}, confirming the necessity of division by \sqrt{3} for proper normalization.^[5] This representation is unique up to local unitary transformations for states in the W equivalence class under stochastic local operations and classical communication (SLOCC). Specifically, any three-qubit state equivalent to |W\rangle can be transformed into this standard form via invertible local operators, and the state is invariant under local phase shifts, preserving its entanglement structure.^[5]

Key Properties

Entanglement Structure

The entanglement in the W state is distributed in a way that ensures genuine tripartite entanglement while exhibiting significant bipartite correlations across subsystems. For any partition into two qubits (A and B) with the third qubit (C) traced out, the reduced density matrix takes the form

\rho_{AB} = \Tr_C (|W\rangle\langle W|) = \frac{2}{3} |\psi\rangle\langle\psi| + \frac{1}{3} |00\rangle\langle 00|,

where |\psi\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle) is a maximally entangled Bell state.^[6] This structure reveals that the two qubits share maximal entanglement within the singly excited subspace, diluted by a separable vacuum component, leading to a concurrence of C = \frac{2}{3} that quantifies the pairwise entanglement strength. The tripartite nature of the entanglement is evident in the state's inseparability across any bipartition: the W state cannot be decomposed into a product of states involving fewer than all three qubits, as confirmed by the non-vanishing concurrence C = \frac{2\sqrt{2}}{3} for every 1-versus-2 qubit cut. This distinguishes it as a prototype of genuine multipartite entanglement within its SLOCC class. Unlike the GHZ state, the W state's entanglement spectrum for single-qubit reductions is asymmetric, with the reduced density matrix \rho_A = \Tr_{BC} (|W\rangle\langle W|) = \frac{2}{3} |0\rangle\langle 0| + \frac{1}{3} |1\rangle\langle 1| having eigenvalues \frac{2}{3} and \frac{1}{3}.^[6] This unequal spectrum underscores the state's biased excitation distribution, contributing to its unique multipartite architecture.

Symmetry and Stability

The W state possesses full permutation invariance, remaining unchanged under any exchange of its three qubits, and thereby occupies the totally symmetric subspace within the three-qubit Hilbert space.^[1] This symmetry distinguishes it as a representative of the W-class entanglement, where the state's structure ensures equivalence under stochastic local operations and classical communication (SLOCC) transformations that preserve the permutation group.^[5] A direct consequence of this permutation symmetry is the state's notable stability against qubit loss. Tracing out any single qubit from the W state yields a reduced two-qubit density matrix of the form

\rho = \frac{2}{3} |\Psi^+\rangle\langle \Psi^+| + \frac{1}{3} |00\rangle\langle 00|,

where |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle) is a maximally entangled Bell state; the entanglement in the remaining pair is thus preserved within the entangled subspace, up to a normalization factor of \frac{2}{3}.^[1] This residual bipartite entanglement maximizes both the average and minimum pairwise entanglement across all three-qubit states, highlighting the W state's robustness to such partial decoherence.^[1] Furthermore, the W state exhibits resistance to phase damping decoherence, decaying more slowly than the GHZ state due to its singly excited, distributed structure. Under local phase damping channels modeled by Lindblad master equations, the fidelity of multipartite entanglement for the W state follows a decay rate governed by single-qubit dephasing (rate \gamma), whereas the GHZ state's tripartite coherence erodes at a rate scaling with $3\gamma, leading to faster loss of global entanglement.^[7] This differential sensitivity underscores the W state's suitability for noisy environments, as confirmed by analyses of residual negativity and concurrence under such dynamics.

Comparisons with Other States

Versus GHZ State

The W state and the Greenberger–Horne–Zeilinger (GHZ) state represent two distinct forms of tripartite entanglement in three-qubit systems, differing fundamentally in their structural composition. The GHZ state is given by |\text{GHZ}\rangle = \frac{1}{\sqrt{2}} (|000\rangle + |111\rangle), which features a balanced superposition of the all-zero and all-one basis states, encoding a highly correlated, phase-sensitive entanglement across all three qubits.^[1] In contrast, the W state, |W\rangle = \frac{1}{\sqrt{3}} (|001\rangle + |010\rangle + |100\rangle), distributes a single excitation symmetrically among the qubits, resulting in a more decentralized structure where entanglement is shared in a permutation-invariant manner without relying on global phase coherence.^[1] This structural disparity leads to the GHZ state having a minimal tensor product decomposition into two terms, while the W state requires three, highlighting their inequivalent entanglement geometries.^[1] A key behavioral difference arises in their response to qubit measurement or loss, underscoring the W state's greater robustness. For the GHZ state, measuring or tracing out one qubit causes the remaining two-qubit reduced density matrix to become fully separable and unentangled, effectively destroying the multipartite entanglement entirely.^[1] Conversely, the W state preserves significant bipartite entanglement upon such an operation; tracing out one qubit yields a mixed state with a two-thirds fidelity to the maximally entangled Bell state |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle), ensuring that the residual system remains entangled.^[1] This fragility of the GHZ state stems from its reliance on perfect correlation among all qubits, whereas the W state's distributed excitation allows it to maintain connectivity even after partial decoherence.^[1] Under stochastic local operations and classical communication (SLOCC), the W and GHZ states belong to two inequivalent entanglement classes for three qubits, meaning no local operations with invertible matrices can transform one into the other.^[1] The W class is characterized by its decentralized nature, where entanglement is more evenly distributed and less vulnerable to local perturbations, distinguishing it from the GHZ class's centralized, all-or-nothing correlation.^[1] This classification underscores the W state's utility in scenarios requiring resilience, while the GHZ state excels in applications demanding maximal tripartite sensitivity.^[1]

Multipartite Entanglement Metrics

The multipartite entanglement of the W state is quantified using bipartite measures applied to its reductions and multipartite invariants that capture genuine multi-party correlations. The concurrence, originally defined for bipartite systems, can be applied to reduced density matrices obtained by tracing out qubits from the W state to assess the entanglement structure. Tracing out any single qubit from the three-qubit W state yields a two-qubit reduced density matrix with concurrence C = 2/3.^[8] This value reflects the distributed bipartite entanglement inherent to the W state's structure, where the excitation is shared equally among the qubits. In comparison, the GHZ state has a reduced two-qubit concurrence of 0 after tracing out one qubit, though its concurrence across a one-qubit-versus-two-qubit bipartition is 1.^[8] The Coffman-Kundu-Wootters (CKW) inequality relates these bipartite concurrences (from two-qubit reductions) to the genuine tripartite entanglement via the three-tangle τ through the relation τ = C_{A|BC}^2 - C_{AB}^2 - C_{AC}^2 ≥ 0 (and cyclic permutations), where C_{A|BC} is the concurrence for the one-vs-two bipartition A|BC.^[8] The three-tangle τ, a SL(2,ℂ)-invariant measure of genuine tripartite entanglement, vanishes for the W state (τ(W) = 0), indicating an absence of the phase-sensitive three-way correlations present in the GHZ state (τ(GHZ) = 1).^[8] Despite this, the W state's entanglement persists in its bipartite reductions; for the W state, with τ(W) = 0, the CKW relation implies that the squared concurrence for each one-vs-two bipartition C_{A|BC}^2 = 8/9 equals the sum of the squared bipartite concurrences C_{AB}^2 + C_{AC}^2 = 4/9 + 4/9, meaning there is no residual genuine tripartite entanglement, and the total sum of the three pairwise squared concurrences is 4/3.^[8] This distributed nature makes the W state more robust to local losses than the GHZ state, as the remaining subsystems retain entanglement even after one qubit is removed. Other metrics further illustrate the W state's "weaker" yet broadly distributed multipartite entanglement. The geometric measure of entanglement, defined as E_G = 1 - \max |\langle \phi | \psi \rangle|^2 over all product states |\phi\rangle, yields E_G(W) \approx 0.556 for the three-qubit W state, corresponding to a maximum overlap of 4/9 with the closest product state.^[9] The negativity N, computed as (||\rho^{T_B}||_1 - 1)/2 for a bipartition, is approximately 0.5 for the W state across a one-versus-two cut, underscoring its multipartite character while being lower than the maximal value of 0.5 achieved exactly by the GHZ state in the same bipartition. These measures collectively demonstrate how the W state's entanglement, though lacking the concentrated tripartite strength of the GHZ state, provides a more even distribution suitable for certain quantum protocols.

Generalizations

To N Qubits

The generalized W state for an arbitrary number N of qubits is defined as

|W_N\rangle = \frac{1}{\sqrt{N}} \sum_{k=1}^N |00\dots 1_k \dots 00\rangle,

where the subscript k denotes the position of the single |1\rangle excitation in the otherwise all-zero basis state.^[10] This form extends the three-qubit W state by symmetrically distributing a single excitation equally across all N qubits, maintaining the state's permutation symmetry under qubit exchanges.^[5] A key feature of |W_N\rangle is the scaling of its bipartite entanglement. The bipartite concurrence across the partition of one qubit versus the remaining N-1 qubits, quantified as the square root of the one-tangle \tau_1 = 4 \det(\rho_A) where \rho_A is the reduced density matrix of the single qubit, decreases as O(1/\sqrt{N}). Specifically, \det(\rho_A) = (N-1)/N^2, yielding \tau_1 = 4(N-1)/N^2 \approx 4/N for large N, so the concurrence C = \sqrt{\tau_1} \approx 2/\sqrt{N}.^[1] Despite this dilution of bipartite entanglement with increasing N, the state remains genuinely multipartite entangled for all N > 2, as it cannot be expressed as a product state across any bipartition and exhibits non-zero entanglement across multiple cuts. Under stochastic local operations and classical communication (SLOCC), the generalized W state |W_N\rangle for N > 3 belongs to a distinct equivalence class from the GHZ state |GHZ_N\rangle = (|0\rangle^{\otimes N} + |1\rangle^{\otimes N})/\sqrt{2}. This separation arises because no invertible local operations can interconvert states between these classes, reflecting fundamentally different entanglement structures: the W class features distributed single-particle excitations, while the GHZ class relies on global phase coherence.^[11] This distinction enables unique applications of |W_N\rangle in larger quantum systems, such as robust multipartite protocols where delocalized entanglement provides resilience to local losses.

In Different Quantum Systems

The W state has been generalized to qudit systems, where each particle possesses d levels instead of the binary levels of qubits. In these higher-dimensional setups, the qudit W state for d parties is defined as |W_d\rangle = \frac{1}{\sqrt{d}} \sum_{i=1}^d |e_i 0 \cdots 0\rangle_{\text{sym}}, where |e_i\rangle denotes the i-th excited state, the sum runs over permutations to ensure placement in the symmetric subspace, and the normalization accounts for the equally weighted single excitations.^[12] This construction extends the permutation invariance of the qubit W state while leveraging the increased Hilbert space dimensionality.^[12] The enhanced entanglement capacity arises from the ability to encode more quantum information per qudit, enabling stronger multipartite correlations that are robust against certain forms of noise and support applications in high-dimensional quantum protocols.^[12] In continuous-variable (CV) quantum systems, exact W states are not directly realizable due to the infinite-dimensional nature of the Hilbert space, but close analogues have been developed using Gaussian states. A prominent example is the symmetric three-mode Gaussian state, characterized by a covariance matrix that maximizes tripartite residual entanglement and pairwise correlations, serving as the CV counterpart to the three-party W state.^[13] These states are constructed through Gaussian operations on squeezed vacuum inputs, preserving the promiscuous entanglement sharing property of the original W state.^[13] Approximations to W-like entanglement in CV systems can also involve non-Gaussian resources, such as photon-number superpositions in optical modes, often derived from truncated cluster states or NOON-like configurations to mimic the single-excitation symmetry.^[14] Bosonic realizations of W-like states exploit the indistinguishable nature of bosons to maintain permutation symmetry across modes. In photonic systems, the W state manifests as a multi-mode Fock state with a single photon delocalized symmetrically, |W\rangle = \frac{1}{\sqrt{N}} \sum_{i=1}^N |1_i 0 \cdots 0\rangle, generated via linear optics and postselection from parametric down-conversion sources. This bosonic encoding preserves the state's robustness to single-particle loss, a key feature inherited from the qubit version. In trapped-ion setups, analogous W-like states appear in the collective phonon modes, where a single vibrational quantum is distributed symmetrically across ion chains using sideband interactions and motional coupling. These phonon-based states retain the permutation symmetry through the shared harmonic potential, facilitating entanglement in the bosonic degree of freedom without relying on internal spin states.

Applications

Quantum Computing Protocols

W states serve as entangled resources in multipartite quantum teleportation protocols, particularly for three-party schemes where a single qubit state is transferred among Alice, Bob, and Charlie. In such protocols, the three parties share a W state, and through local operations and classical communication, the sender (Alice) can teleport an unknown qubit to the receivers (Bob and Charlie) with shared entanglement. This approach leverages the permutation symmetry of the W state to enable joint reconstruction at the receivers.^[15] Compared to GHZ states, W-state-based teleportation achieves higher average fidelity in the presence of noise, owing to the W state's robustness against single-qubit decoherence or loss, where the remaining parties retain bipartite entanglement unlike the fully separable GHZ under similar conditions. For instance, under white noise, the average fidelity for W-state teleportation exceeds that of GHZ in certain parameter regimes, making it preferable for fault-tolerant implementations. This stability stems from the W state's structure, which preserves multipartite entanglement better under local noise than the more fragile GHZ.^[15] In quantum error correction, W states play a role in stabilizer-based encodings for detecting phase errors, particularly in photonic implementations compatible with cluster-state architectures. By encoding logical qubits into W states via linear optics and post-selection, phase (dephasing) noise can be mapped to detectable heralding failures without active feedback, enabling passive error detection in measurement-based schemes. This method suppresses dephasing effects, which are prevalent in optical systems, and integrates with fault-tolerant protocols by flagging errors during computation.^[16] For measurement-based quantum computation (MBQC), W states can serve as resources in fusion-based protocols to generate larger graph states required for one-way quantum computers. This approach exploits the W state's multipartite entanglement to efficiently expand resource states while maintaining fault tolerance through error detection in fusion steps.^[17]

Quantum Communication Schemes

W states have been employed in quantum secret sharing (QSS) protocols to securely distribute cryptographic keys among multiple parties. In a typical three-party QSS scheme using the symmetric W state, the dealer encodes the secret key into the quantum state and distributes the entangled qubits to Alice, Bob, and Charlie. Any two parties can collaborate to reconstruct the full secret by performing joint measurements, as the residual bipartite entanglement between their qubits allows faithful recovery, while a single party possesses only partial information insufficient for reconstruction. This threshold access structure exploits the unique symmetry and robustness of W states against single-qubit loss, ensuring security against eavesdropping through the no-cloning theorem and entanglement verification.^[18]^[19] Multipartite dense coding protocols leverage W states to enhance classical information transmission over quantum channels among multiple users. In a three-party setup, one sender can encode classical messages into a shared W state by applying local operations on their qubit before sending it to the receivers. The receivers then perform collective measurements to decode the message, achieving higher efficiency than unentangled channels. This advantage stems from the genuine tripartite entanglement in W states, which allows for superadditive coding gains.^[20]^[21] Entanglement swapping with W states facilitates the extension of multipartite entanglement over long distances, a key component in quantum repeater architectures. In a repeater protocol, short-distance W states are generated between adjacent nodes; local Bell-state measurements on intermediate qubits from adjacent W states then swap the entanglement, probabilistically creating a longer-range W state between end nodes. For three-qubit W states, this process preserves the multipartite nature while mitigating decoherence losses, with success probabilities optimized through error correction on the swapped links. Such schemes are particularly suited for quantum networks requiring distributed tripartite entanglement, as W states maintain connectivity even after single-qubit failures.^[22]^[23]

Experimental Developments

Early Realizations

The first experimental realization of a photonic W state was reported in 2004 by Eibl et al., who generated a three-photon polarization-entangled W state through postselection from multiple spontaneous parametric down-conversion events in a single-mode optical parametric oscillator pumped by a mode-locked titanium-sapphire laser. This approach produced an approximate W state with a measured fidelity of 0.71 ± 0.04 relative to the ideal state, verified via quantum state tomography.^[24] In parallel, trapped-ion systems enabled high-fidelity implementations of W states. In 2005, Häffner et al. created multipartite W states involving up to eight ^{40}Ca^{+} ions confined in a linear Paul trap, using a sequence of individually addressed laser pulses for state initialization and global bichromatic laser fields to induce collective spin flips that entangle the ions. For the three-ion W state, they achieved a fidelity exceeding 0.9, as confirmed by full state tomography, demonstrating scalability while maintaining multipartite entanglement.^[25] These early experiments highlighted significant challenges in generating W states. Photonic realizations suffered from low success probabilities, often below 10^{-6}, due to the probabilistic nature of postselection from rare higher-order emissions in spontaneous parametric down-conversion processes. Additionally, both platforms contended with decoherence: in ion traps, environmental interactions limited coherence times to milliseconds, while photonic setups were affected by losses and imperfect mode matching, reducing overall state purity.^[24]^[25]

Recent Measurements and Advances

In September 2025, researchers at Kyoto University demonstrated the first experimental entangled measurement of three-qubit photonic W states, utilizing linear optics and photon number-resolving detectors to exploit the cyclic shift symmetry via a three-mode discrete Fourier transform circuit. This breakthrough achieved a measurement discrimination fidelity of 0.871 ± 0.039, marking a significant step in directly identifying W-state entanglement without full state tomography.^[4]^[26] The technique addresses long-standing challenges in W-state detection by enabling unambiguous discrimination among W states and their permutations, which has implications for quantum teleportation protocols and scalable quantum networks, as it allows for efficient identification of multipartite entanglement in photonic systems.^[4]^[27] In May 2025, a study introduced and experimentally verified the first multi-qubit quantum energy teleportation protocol leveraging W-state entanglement, implemented on superconducting quantum processors for three- and four-qubit systems, demonstrating robust energy transfer without direct interaction between sender and receiver.^[28] Concurrent 2025 investigations highlighted the robustness of W states in noisy intermediate-scale quantum (NISQ) devices, with superconducting circuit experiments achieving high-fidelity generation of up to nine-qubit W states, underscoring their resilience to decoherence compared to GHZ states in practical quantum hardware.^[28]^[29]

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