| entangled state | |
|---|---|
| Name | Entangled state |
| Type | Quantum state |
| Fields | Quantum mechanics; Quantum information science |
| Introduced | 1935 |
| Introduced by | Einstein, Podolsky, Rosen; formalized by Schrödinger |
entangled state
An entangled state is a quantum state of two or more quantum systems in which the joint state cannot be factored into a product of individual subsystem states. Entanglement is a nonclassical correlation that underlies fundamental tests of local realism and enables tasks in quantum information theory such as quantum teleportation and quantum cryptography. Its study links foundational questions in physics to applications in quantum computing and quantum technologies.
An entangled state is formally defined within the Hilbert space formalism of quantum mechanics: for composite systems with Hilbert space H = H_A ⊗ H_B, a pure state |ψ⟩ is entangled if it cannot be written as |ψ⟩ = |φ⟩_A ⊗ |χ⟩_B. For mixed states represented by a density operator ρ_AB, separability is characterized by the existence of a convex decomposition ρ_AB = Σ_i p_i ρ_A^i ⊗ ρ_B^i; absence of such a decomposition denotes entanglement. Mathematical tools include the Schmidt decomposition for pure bipartite states and the partial trace to obtain reduced density matrices. The formalism generalizes to multipartite systems (H = ⊗_i H_i), where entanglement structure can be richer, and to continuous-variable systems described by Gaussian state formalism and Wigner function techniques.
Quantifying entanglement employs operational and axiomatic measures. For bipartite pure states, the von Neumann entropy S(ρ_A) of the reduced state (entropy of entanglement) is canonical. Mixed-state measures include entanglement of formation, concurrence (Wootters), negativity and logarithmic negativity, and relative entropy of entanglement. Criteria for detecting entanglement include the Peres–Horodecki criterion (positive partial transpose, PPT) and entanglement witnesses constructed from observables. For continuous-variable systems, the Simon criterion and Duan criterion give necessary and sufficient conditions for Gaussian entanglement. Resource-theoretic approaches formalize monotones and free operations, such as local operations and classical communication (LOCC), and link measures to conversion rates like distillable entanglement and entanglement cost.
Canonical bipartite examples include the two-qubit Bell states (EPR pairs), e.g., |Φ+⟩ = (|00⟩+|11⟩)/√2, and Werner states that interpolate between entangled and mixed separable forms. Multipartite classes include the GHZ state and the W state, which exhibit distinct entanglement persistences and convertibility under stochastic LOCC (SLOCC). Continuous-variable examples include two-mode squeezed vacuum states produced by optical parametric amplification and entangled coherent states. Cluster states and graph states are structured multipartite entangled resources for measurement-based quantum computation and error-correcting codes such as surface code implementations rely on entangled stabilizer states.
Entanglement generation spans many physical platforms. In quantum optics, entangled photon pairs are commonly produced by spontaneous parametric down-conversion (SPDC) in nonlinear crystals or by four-wave mixing in optical fibers. In atomic systems, cold atom ensembles and Bose–Einstein condensates can be entangled via controlled collisions or spin-squeezing protocols. Solid-state approaches include entanglement of superconducting qubits via microwave resonators in circuit quantum electrodynamics (cQED), and entanglement between quantum dot spins via exchange interactions. Ion-trap quantum processors use entangling gates such as the Mølmer–Sørensen gate. Hybrid schemes link disparate systems using entanglement swapping and quantum interfaces, for example between photons and ensembles via quantum memories developed at institutions like ICFO and NIST.
Entangled states are resources for numerous quantum information tasks. They enable quantum teleportation of unknown quantum states (Bennett et al.), form the basis of entanglement-based quantum key distribution such as the E91 scheme, and are central to entanglement-assisted quantum error correction and superdense coding. Entanglement boosts metrological sensitivity in quantum metrology (e.g., using spin-squeezed states to reach beyond the standard quantum limit). In quantum computing, entanglement is necessary for universal quantum computation and serves as a diagnostic of quantum advantage in algorithms implemented on platforms by companies like IBM and Google.
Key experiments demonstrating entanglement include early photon polarization tests by Clauser and Freedman, Bell test violations by Aspect's group, and loophole-free Bell tests by teams at Delft, NIST, and Vienna that closed detection and locality loopholes. Platforms showing scalable entanglement include trapped-ion chains at UCSB and IQIM, superconducting qubit arrays realized by Google's Sycamore and IBM quantum processors, photonic networks at University of Bristol and Caltech, and neutral-atom arrays via optical tweezers developed at Harvard and MPQ (Max Planck Quantum Optics). Satellite-based entanglement distribution has been demonstrated by the Micius satellite mission.
Entanglement is fragile to decoherence from environmental coupling, modeled by quantum channels and noise processes such as amplitude damping and dephasing. Decoherence limits entanglement lifetime and motivates error correction and dynamical decoupling techniques. Monogamy relations (e.g., Coffman–Kundu–Wootters inequality) constrain how entanglement can be shared among parties and underpin security proofs in quantum cryptography. Resource-theoretic frameworks treat entanglement as a resource with free operations (LOCC) and quantify conversion rates, distillation, and dilution; central theoretical contributions include work by the Horodecki family and operational links to thermodynamics of quantum systems. Categories: Category:Quantum information theory Category:Quantum states