Asymptotic quantification of entanglement with a single copy

Abstract

Despite the central importance of quantum entanglement in quantum technologies, the understanding of the optimal ways to exploit it is still beyond our reach, and even measuring entanglement in an operationally meaningful way is prohibitively difficult. Here we study two fundamental tasks in the processing of entanglement: entanglement testing, which is a quantum state discrimination problem concerned with entanglement detection in the many-copy regime, and entanglement distillation, concerned with purifying entanglement from noisy entangled states. We introduce a way of benchmarking the performance of distillation that focuses on the best achievable error rather than its yield in the asymptotic limit. When the underlying set of operations used for entanglement distillation is the axiomatic class of non-entangling operations, we show that the two figures of merit for entanglement testing and distillation coincide. We solve both problems by proving a generalised quantum Sanov's theorem, enabling the exact evaluation of asymptotic error rates of composite quantum hypothesis testing. We show in particular that the asymptotic figure of merit is given by the reverse relative entropy of entanglement, a single-letter quantity that can be evaluated using only a single copy of a quantum state -- a distinct feature among measures of entanglement that quantify the optimal performance of information-theoretic tasks.

22+23 pages. v3: accepted version; changes limited to presentation

References in corpus (40)

• Quantum entanglement
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Bell nonlocality
• Quantifying Entanglement
• Quantum Resource Theories
• Distillation of secret key and entanglement from quantum states
• Superadditivity of communication capacity using entangled inputs
• The Physics and Mathematics of the Second Law of Thermodynamics
• From Classical to Quantum Shannon Theory
• Entanglement Measures under Symmetry
• Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
• The general structure of quantum resource theories
• The asymptotic entanglement cost of preparing a quantum state
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• One-Shot Classical-Quantum Capacity and Hypothesis Testing
• The entanglement cost under operations preserving the positivity of partial transpose
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Entanglement Theory and the Second Law of Thermodynamics
• The quantum capacity of channels with arbitrarily correlated noise
• A Generalization of Quantum Stein's Lemma
• Are the laws of entanglement theory thermodynamical?
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• No-Go Theorems for Quantum Resource Purification
• Entangled Mixed States and Local Purification
• A reversible theory of entanglement and its relation to the second law
• Relative Entropy of Entanglement and Restricted Measurements
• On Variational Expressions for Quantum Relative Entropies
• A quantum version of Sanov's theorem
• Remarks on entanglement measures and non-local state distinguishability
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• On Composite Quantum Hypothesis Testing
• Equivalence of Additivity Questions in Quantum Information Theory
• Hypothesis Testing on Invariant Subspaces of the Symmetric Group, Part I - Quantum Sanov's Theorem and Arbitrarily Varying Sources
• A minimax approach to one-shot entropy inequalities
• Reversibility of quantum resources through probabilistic protocols
• A solution of the generalised quantum Stein's lemma
• Distillable entanglement under dually non-entangling operations
• Black Box Work Extraction and Composite Hypothesis Testing
• Computable entanglement cost under positive partial transpose operations