The Uniqueness Theorem for Entanglement Measures

Abstract

We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyse properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entanglement.

22 pages, LaTeX, version accepted by J. Math. Phys

References in corpus (2)

• A uniqueness theorem for entanglement measures
• The asymptotic relative entropy of entanglement

Cited by in corpus (113)

• Quantum entanglement
• The logarithmic negativity: A full entanglement monotone that is not convex
• Quantum Resource Theories
• Local versus non-local information in quantum information theory: formalism and phenomena
• Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
• Aspects of generic entanglement
• Further results on the cross norm criterion for separability
• Open-System Dynamics of Entanglement
• Some Properties of the Computable Cross Norm Criterion for Separability
• Faithful Squashed Entanglement
• Gaussian Entanglement of Formation
• Genuine quantum correlations in quantum many-body systems: a review of recent progress
• Information as a resource in distributed quantum systems
• Generalizations of entanglement based on coherent states and convex sets
• Large violation of Bell inequalities with low entanglement
• General paradigm for distilling classical key from quantum states
• The maximally entangled set of multipartite quantum states
• Entanglement of multiparty stabilizer, symmetric, and antisymmetric states
• Quantifying entanglement in two-mode Gaussian states
• Almost all multipartite qubit quantum states have trivial stabilizer
• Entangled pure state transformations via local operations assisted by finitely many rounds of classical communication
• On asymptotic continuity of functions of quantum states
• Semidefinite programming relaxations for quantum correlations
• Entanglement manipulation of multipartite pure states with finite rounds of classical communication
• No second law of entanglement manipulation after all
• Axiomatic and operational connections between the $l_1$-norm of coherence and negativity
• Distribution of bipartite entanglement for random pure states
• Entanglement of the Antisymmetric State
• Entanglement of Purification in Many Body Systems and Symmetry Breaking
• Quantum and Classical Correlations Inside the Entanglement Wedge
• Remarks on entanglement measures and non-local state distinguishability
• Are all maximally entangled states pure?
• Quantifying Entanglement Production of Quantum Operations
• Gravity entanglement, quantum reference systems, degrees of freedom
• The first law of general quantum resource theories
• Distillation protocols: Output entanglement and local mutual information
• The maximally entangled set of 4-qubit states
• Entanglement cost of implementing controlled-unitary operations
• Multipartite entanglement outperforming bipartite entanglement under limited quantum system sizes
• Rates of asymptotic entanglement transformations for bipartite mixed states: Maximally entangled states are not special
• General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures
• Multipartite entanglement measures: a review
• Continuity of the quantum Fisher information
• On extremal quantum states of composite systems with fixed marginals
• Flag Additivity in Quantum Resource Theories
• Maximally entangled mixed states for qubit-qutrit systems
• Asymptotic state transformations of continuous variable resources
• Entangled bases with fixed Schmidt number
• Unextendible entangled bases with fixed Schmidt number
• Channel kets, entangled states, and the location of quantum information
• Dual entanglement measures based on no local cloning and no local deleting
• On the irreversibility of entanglement distillation
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• Upper bound on the secret key rate distillable from effective quantum correlations with imperfect detectors
• Simple test for quantum channel capacity
• Ancilla dimension in quantum channel discrimination
• The Min-entropy as a Resource for One-Shot Private State Transfer, Quantum Masking and State Transition
• Measure-independent anomaly of nonlocality
• Entanglement enhancement induced by noise in inhomogeneously monitored systems
• Entanglement convertibility for infinite dimensional pure bipartite states
• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Sensitivity of entanglement measures in bipartite pure quantum states
• Information-theoretical formulation of anyonic entanglement
• No-go theorem for entanglement distillation using catalysis
• Strict entanglement monotonicity under local operations and classical communication
• Towards an Entanglement Measure for Mixed States in CFTs Based on Relative Entropy
• Remarks on the equivalence of full additivity and monotonicity for the entanglement cost
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Delocalization power of global unitary operations on quantum information
• Local Transformations of Multiple Multipartite States
• State transformations within entanglement classes containing permutation-symmetric states
• Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols
• Local hypothesis testing between a pure bipartite state and the white noise state
• Entropy of quantum states
• Optimized Correlation Measures in Holography
• Entanglement enhancement and postselection for two atoms interacting with thermal light
• Continuity of entropies via integral representations
• Second Law of Entanglement Manipulation with Entanglement Battery
• Introduction to quantum entanglement in many-body systems
• Smooth Metric Adjusted Skew Information Rates
• Hierarchy of quantum operations in manipulating coherence and entanglement
• Accessible bounds for general quantum resources
• Complementarity of Entanglement and Interference
• Quantum entanglement correlations in double quark PDFs
• Entanglement theory with limited computational resources
• A measure of quantum correlations that lies approximately between entanglement and discord
• Volume of the set of LOCC-convertible quantum states
• Entanglement cost for infinite-dimensional physical systems
• Continuity of robustness measures in quantum resource theories
• Bipartite Bound Entanglement
• Computable entanglement cost under positive partial transpose operations
• A generalization of Schmidt number for multipartite states
• A new entanglement measure based on the total concurrence
• Non-Gaussian Entangled States
• Counting atypical black hole microstates from entanglement wedges
• Characterization of entanglement transformation via group representation theory
• On construction of finite averaging sets for $SL(2,\mathbb{C})$ via its Cartan decomposition
• Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication
• Relative entanglement entropy for widely separated regions in curved spacetime
• Local channels preserving maximal entanglement or Schmidt number
• Geometric Entanglement in Valance-Bond-Solid state
• Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs
• Separability of Hermitian Tensors and PSD Decompositions
• Identifying families of multipartite states with non-trivial local entanglement transformations
• Complementary Relationships between Entanglement and Measurement
• Dynamics of Correlations and Entanglement Generation in Electron-Molecule Inelastic Scattering
• Entanglement behavior and localization properties in monitored fermion systems
• Approximate and ensemble local entanglement transformations for multipartite states
• Highly Nonclassical Quantum States and Environment Induced Decoherence
• Entanglement Cost for Steering Assemblages
• Superactivation of Bell nonlocality in pure anyonic states
• Some Remarks on the Entanglement Number
• Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group