On the error exponents of binary state discrimination with composite hypotheses

Abstract

The trade-off between the two types of errors in binary state discrimination may be quantified in the asymptotics by various error exponents. In the case of simple i.i.d. hypotheses, each of these exponents is equal to a divergence (pseudo-distance) of the two states. In the case of composite hypotheses, represented by sets of states $R,S$, one always has the inequality $\mathrm{e}(R\Vert S)\le \mathrm{E}(R\Vert S)$, where $\mathrm{e}$ is the exponent, $\mathrm{E}$ is the corresponding divergence, and the question is whether equality holds. The relation between the composite exponents and the worst pairwise exponents may be influenced by a number of factors: the type of exponents considered; whether the problem is classical or quantum; the cardinality and the geometric properties of the sets representing the hypotheses; and, on top of the above, possibly whether the underlying Hilbert space is finite- or infinite-dimensional. Our main contribution in this paper is clarifying this landscape considerably: We exhibit explicit examples for hitherto unstudied cases where the above inequality fails to hold with equality, while we also prove equality for various general classes of state discrimination problems. In particular, we show that equality may fail for any of the error exponents even in the classical case, if the system is allowed to be infinite-dimensional, and the alternative hypothesis contains countably infinitely many states. Moreover, we show that in the quantum case strict inequality is the generic behavior in the sense that, starting from any pair of non-commuting density operators of any dimension, and for any of the exponents, it is possible to construct an example with a simple null-hypothesis and an alternative hypothesis consisting of only two states, such that strict inequality holds for the given exponent.

arXiv comment50 pages. v2: Considerably expanded, also includes the discussion of the strong converse exponent. Comments are welcome

References in corpus (30)

• On quantum Renyi entropies: a new generalization and some properties
• The Quantum Chernoff Bound
• 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
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Monotonicity of a relative Rényi entropy
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• On the quantum Renyi relative entropies and related capacity formulas
• An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
• A Generalization of Quantum Stein's Lemma
• Strong converse exponent for classical-quantum channel coding
• A quantum version of Sanov's theorem
• Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics
• Discriminating quantum states: the multiple Chernoff distance
• Error exponents in hypothesis testing for correlated states on a spin chain
• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces
• Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination
• On Composite Quantum Hypothesis Testing
• Coding theorems for compound problems via quantum Rényi divergences
• Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences
• Hypothesis Testing on Invariant Subspaces of the Symmetric Group, Part I - Quantum Sanov's Theorem and Arbitrarily Varying Sources
• Quantum hypothesis testing with group symmetry
• Rényi relative entropies and noncommutative $L_p$-spaces II
• Two approaches to obtain the strong converse exponent of quantum hypothesis testing for general sequences of quantum states
• Asymptotic relative submajorization of multiple-state boxes
• On a conjecture regarding quantum hypothesis testing

Cited by in corpus (5)

• 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
• Postselected quantum hypothesis testing
• Monotonic multi-state quantum $f$-divergences
• Asymptotic relative submajorization of multiple-state boxes