A Generalization of Quantum Stein's Lemma

Abstract

We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d.. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.

30 pages. (see posting by M. Piani arXiv:0904.2705 for a different proof of the strict positiveness of the regularized relative entropy of entanglement on every entangled state). published version.

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• Quantum Resource Theories
• The Resource Theory of Stabilizer Computation
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• Faithful Squashed Entanglement
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• Comparison of Quantum Channels by Superchannels
• Operational Resource Theory of Quantum Channels
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• How to quantify a dynamical quantum resource
• Experimental bound entanglement in a four-photon state
• Relative Entropy Bounds on Quantum, Private and Repeater Capacities
• Distilling entanglement from arbitrary resources
• Fundamental limitations on distillation of quantum channel resources
• Entanglement cost in practical scenarios
• Catalysis of entanglement and other quantum resources
• A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels
• Discriminating quantum states: the multiple Chernoff distance
• Relative entropy and squashed entanglement
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• Applications of position-based coding to classical communication over quantum channels
• Energy-constrained discrimination of unitaries, quantum speed limits and a Gaussian Solovay-Kitaev theorem
• Framework for resource quantification in infinite-dimensional general probabilistic theories
• Classical communication over a quantum interference channel
• The first law of general quantum resource theories
• Quantum de Finetti theorem under fully-one-way adaptive measurements
• On Composite Quantum Hypothesis Testing
• One-shot entanglement distillation beyond local operations and classical communication
• A quasipolynomial-time algorithm for the quantum separability problem
• Testing the quantum nature of gravity without entanglement
• Quantum state discrimination bounds for finite sample size
• General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures
• Disentanglement Cost of Quantum States
• Hypothesis Testing on Invariant Subspaces of the Symmetric Group, Part I - Quantum Sanov's Theorem and Arbitrarily Varying Sources
• General theory of environment-assisted entanglement distillation
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• Quantum data hiding with continuous variable systems
• One-shot assisted concentration of coherence
• Reversibility of quantum resources through probabilistic protocols
• Operational meaning of quantum measures of recovery
• Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems
• Epsilon-smooth measure of coherence
• A solution of the generalised quantum Stein's lemma
• Distillable entanglement under dually non-entangling operations
• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Postselected quantum hypothesis testing
• Monotonic multi-state quantum $f$-divergences
• On the error exponents of binary state discrimination with composite hypotheses
• Evidence procedure for efficient quantum state tomography
• Simple Bounds for One-shot Pure-State Distillation in General Resource Theories
• Asymptotically Consistent Measures of General Quantum Resources: Discord, Non-Markovianity, and Non-Gaussianity
• Exact entanglement cost of quantum states and channels under PPT-preserving operations
• Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols
• Entanglement monogamy via multivariate trace inequalities
• Local hypothesis testing between a pure bipartite state and the white noise state
• Inferring the Gibbs state of a small quantum system
• A fermionic de Finetti theorem
• Continuity of entropies via integral representations
• Second Law of Entanglement Manipulation with Entanglement Battery
• Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma
• Lower Bounds for Quantum Parameter Estimation
• Convergence of macrostates under reproducible processes
• Asymptotic relative submajorization of multiple-state boxes
• Black Box Work Extraction and Composite Hypothesis Testing
• Continuity of the relative entropy of resource
• Asymptotic quantification of entanglement with a single copy
• Assessing thermalization and estimating the Hamiltonian with output data only
• Towards the ultimate limits of quantum channel discrimination and quantum communication
• Universal tester for multiple independence testing and classical-quantum arbitrarily varying multiple access channel
• De Finetti Theorems for Braided Parafermions
• Appearance of Gibbs states in quantum-state tomography
• Quantum hypothesis testing between qubit states with parity
• Multipartite entanglement theory with entanglement-nonincreasing operations
• A de Finetti theorem for quantum causal structures
• Quantum geometric tensor determines the pure-state i.i.d. conversion rate in the resource theory of asymmetry for any compact Lie group
• Tight relations and equivalences between smooth relative entropies