Symmetric distinguishability as a quantum resource

Abstract

We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\mathrm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.

arXiv comment59 pages main text + 25 pages of appendices, 4 figures

References in corpus (22)

• Concentrating Partial Entanglement by Local Operations
• Quantum Resource Theories
• Operational Resource Theory of Coherence
• The operational meaning of min- and max-entropy
• The Resource Theory of Quantum States Out of Thermal Equilibrium
• The Quantum Chernoff Bound
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• Reversible transformations from pure to mixed states, and the unique measure of information
• Decoherence in Solid State Qubits
• The Chernoff lower bound for symmetric quantum hypothesis testing
• The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states
• Resource theory of asymmetric distinguishability for quantum channels
• Discriminating quantum states: the multiple Chernoff distance
• Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination
• An information-theoretic treatment of quantum dichotomies
• Reverse Test and Characterization of Quantum Relative Entropy
• The Conditional Uncertainty Principle
• How many copies are needed for state discrimination?
• The resource theory under conditioned thermal operations
• Symmetric distinguishability as a quantum resource
• Jointly constrained semidefinite bilinear programming with an application to Dobrushin curves

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• Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems
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• Inevitability of knowing less than nothing
• Quantum Coherence and Distinguishability as Complementary Resources: A Resource-Theoretic Perspective from Wave-Particle Duality
• Quantum distinguishability measures: projectors vs. states maximization
• Interpolating between symmetric and asymmetric hypothesis testing