| relative entropy of entanglement | |
|---|---|
| Name | Relative entropy of entanglement |
| Field | Quantum information theory |
| Introduced | 1990s |
| Users | Quantum information science researchers |
relative entropy of entanglement
The relative entropy of entanglement is an entanglement measure quantifying how distinguishable a given quantum state is from the set of separable states using the quantum relative entropy as a distance-like functional. It plays a central role in resource-theoretic treatments of entanglement (quantum) and is widely used to bound rates in tasks such as entanglement distillation and quantum communication.
The relative entropy of entanglement E_R(ρ) for a density operator ρ on a composite Hilbert space H_A ⊗ H_B is defined as the minimal quantum relative entropy S(ρ||σ) between ρ and any separable state σ: E_R(ρ) = inf_{σ ∈ S} S(ρ || σ), where S(ρ||σ) = Tr[ρ (log ρ − log σ)] and S denotes the convex set of separable states. The optimization is performed over mixed states in S and the logarithm is the matrix logarithm with respect to the von Neumann entropy formalism. The definition generalizes naturally to multipartite systems and to other convex sets such as the set of PPT states (positive partial transpose) when alternative resource restrictions are considered.
E_R is nonnegative, vanishes exactly on separable states, and is invariant under local unitary operations induced by elements of the unitary groups U(H_A) and U(H_B). It is nonincreasing under LOCC (local operations and classical communication) and thus qualifies as an entanglement monotone in the sense used by Vlatko Vedral and collaborators. E_R is subadditive in general, and satisfies bounds relating it to the entanglement of formation and distillable entanglement E_D, providing upper bounds on E_D via the quantum relative entropy's contractivity under completely positive trace-preserving maps. In operational terms, E_R quantifies the optimal asymptotic distinguishability between an entangled resource and separable noise in hypothesis testing scenarios related to quantum hypothesis testing and the one-shot resource theory framework.
Computing E_R requires a convex optimization over separable states, which is generally NP-hard because of the difficulty of characterizing separability; complexity results connect this to the separability problem and complexity classes studied in computational complexity theory. For low-dimensional systems closed-form expressions exist for special families: e.g., for two-qubit Bell-diagonal states and isotropic states (studied by Bennett et al. and Horodecki family), or for pure bipartite states where E_R reduces to the Schmidt decomposition-based entropy of the reduced state (the entanglement entropy). Semidefinite programming relaxations using the Peres–Horodecki criterion (PPT test) and hierarchies like the Doherty–Parrilo–Spedalieri (DPS) method yield computable upper and lower bounds. Numerical approaches often employ convex optimization toolkits and methods from matrix analysis.
E_R is intimately related to several entanglement quantifiers. It upper-bounds the distillable entanglement and is upper-bounded by the entanglement cost under certain asymptotic settings. The measure connects to the relative entropy functionals used in resource theories, such as the relative entropy of coherence and the relative entropy of magic. For pure states, E_R equals the entropy of entanglement (the von Neumann entropy of the reduced density matrix). Comparisons with entanglement monotones like concurrence, negativity, and squashed entanglement often clarify ordering of mixed states, and differences in additivity properties highlight distinct operational meanings; for instance, squashed entanglement satisfies strong subadditivity properties mediated by conditional mutual information while E_R does not always.
E_R is used to derive bounds for protocols in quantum communication, including capacities of quantum channels and upper bounds on secret key rates in quantum cryptography. In entanglement manipulation, E_R provides constraints on convertibility under asymptotic LOCC and serves as a resource monotone in the resource theory of entanglement. It appears in analyses of entanglement-assisted communication and in studies of many-body physics as a diagnostic for quantum phase transitions when applied to reduced density matrices in lattice models studied at institutions such as CERN, Institute for Quantum Computing, and research groups at MIT and Caltech. Connections to thermodynamics are explored through links to relative entropy as a measure of irreversible work and free energy differences in quantum thermodynamic protocols.
Several extensions generalize the relative entropy of entanglement by replacing the separable set S with other free sets, producing measures like the relative entropy of nonlocality, the relative entropy of coherence, and the relative entropy with respect to PPT or k-extendible states. One-shot variants and smoothed versions relate E_R to single-shot information quantities and smooth min/max entropies studied by Renner and others. Operational generalizations include hypothesis-testing relative entropy variants that yield strong converse bounds and finite-blocklength refinements in the style of quantum Stein's lemma. Research frontiers investigate additivity conjectures, regularized versions E_R^∞, and connections to holographic entanglement entropy in the context of the AdS/CFT correspondence and high-energy theory groups at Perimeter Institute and Institute for Advanced Study.