Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. - quantum lattice systems and more

Abstract

Quantum Stein's Lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the knowledge that it is one of two specific states ($\rho$ or $\sigma$). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, $n$) identical copies of the state (${\rho }^{\otimes n}$ or ${\sigma }^{\otimes n}$) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability ${\alpha }_n$ of erroneously inferring the state to be $\sigma$, the probability ${\beta }_n$ of erroneously inferring the state to be $\rho$ decays exponentially in $n$, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. setting independently by Tomamichel and Hayashi, and Li. We extend this result to settings beyond i.i.d.. Examples of these include Gibbs states of quantum spin systems (with finite-range, translation-invariant interactions) at high temperatures.

34 pages and 2 figures. Version 4: a new proposition, Proposition 1, (and its proof) added and the proof of Equation (3.4) corrected using it

References in corpus (10)

• The Quantum Chernoff Bound
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Error exponents in hypothesis testing for correlated states on a spin chain
• Large deviations and Chernoff bound for certain correlated states on a spin chain
• Hypothesis testing for Gaussian states on bosonic lattices
• Typical support and Sanov large deviations of correlated states
• Fermionic Quasi-free States and Maps in Information Theory
• Asymptotic distinguishability measures for shift-invariant quasi-free states of fermionic lattice systems

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• Applications of position-based coding to classical communication over quantum channels
• Upper bounds on secret key agreement over lossy thermal bosonic channels
• Entanglement-assisted private communication over quantum broadcast channels
• Union bound for quantum information processing
• Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems
• Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Quantum hypothesis testing in many-body systems
• An invitation to the sample complexity of quantum hypothesis testing
• Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities
• Classical capacities of memoryless but not identical quantum channels
• Asymptotic Equipartition Theorems in von Neumann algebras