$\alpha$-Logarithmic negativity

Abstract

The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory, due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the $\kappa$-entanglement of a bipartite state was shown to be the first entanglement measure that is both easily computable and has a precise information-theoretic meaning, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose [Wang and Wilde, Phys. Rev. Lett. 125(4):040502, July 2020]. In this paper, we provide a non-trivial link between these two entanglement measures, by showing that they are the extremes of an ordered family of $\alpha$-logarithmic negativity entanglement measures, each of which is identified by a parameter $\alpha \in [1,\infty ]$. In this family, the original logarithmic negativity is recovered as the smallest with $\alpha =1$, and the $\kappa$-entanglement is recovered as the largest with $\alpha =\infty$. We prove that the $\alpha$-logarithmic negativity satisfies the following properties: entanglement monotone, normalization, faithfulness, and subadditivity. We also prove that it is neither convex nor monogamous. Finally, we define the $\alpha$-logarithmic negativity of a quantum channel as a generalization of the notion for quantum states, and we show how to generalize many of the concepts to arbitrary resource theories.

arXiv commentv3: 15 pages, accepted for publication in Physical Review A

References in corpus (59)

• Quantum entanglement
• Mixed State Entanglement and Quantum Error Correction
• Separability Criterion for Density Matrices
• A computable measure of entanglement
• Separability of Mixed States: Necessary and Sufficient Conditions
• Distributed Entanglement
• The logarithmic negativity: A full entanglement monotone that is not convex
• Entanglement Measures and Purification Procedures
• Separability criterion and inseparable mixed states with positive partial transposition
• Quantum Resource Theories
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Quantum Nonlocality without Entanglement
• On quantum Renyi entropies: a new generalization and some properties
• Monogamy of entanglement and other correlations
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• From Classical to Quantum Shannon Theory
• Entanglement negativity in quantum field theory
• The general structure of quantum resource theories
• The asymptotic entanglement cost of preparing a quantum state
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• An improved bound on distillable entanglement
• The entanglement cost under operations preserving the positivity of partial transpose
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Quantifying Entanglement with Witness Operators
• Is Entanglement Monogamous?
• General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks
• Finite temperature entanglement negativity in conformal field theory
• Counterexample to an additivity conjecture for output purity of quantum channels
• On the quantum Renyi relative entropies and related capacity formulas
• Operational Advantage of Quantum Resources in Subchannel Discrimination
• Entanglement negativity and topological order
• One-shot coherence distillation
• Operational Resource Theory of Quantum Channels
• Negativity and topological order in the toric code
• The squashed entanglement of a quantum channel
• Comments on Entanglement Negativity in Holographic Field Theories
• Convex geometry of quantum resource quantification
• Monotonicity of the Quantum Relative Entropy Under Positive Maps
• Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States
• On the power of PPT-preserving and non-signalling codes
• Holographic Quantum Entanglement Negativity
• Entanglement Cost of Quantum Channels
• Strong converse rates for quantum communication
• Quantifying resource in catalytic resource theory
• Cost of quantum entanglement simplified
• Non-asymptotic entanglement distillation
• Improved Semidefinite Programming Upper Bound on Distillable Entanglement
• Binegativity and geometry of entangled states in two qubits
• Entanglement purification via separable superoperators
• Semidefinite programming converse bounds for quantum communication
• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces
• Resource theories of quantum channels and the universal role of resource erasure
• Inequalities for quantum divergences and the Audenaert-Datta conjecture
• Measurement-Device-Independent Approach to Entanglement Measures
• Resource theories of knowledge
• Additivity of minimal entropy output for a class of covariant channels
• Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose

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