Generalized quantum Chernoff bound

TL;DR

This paper extends the quantum Chernoff bound to multiple sets of quantum states, providing a generalized error exponent for quantum hypothesis testing, and introduces an optimal state-agnostic test with broad applicability.

Contribution

It establishes a generalized quantum Chernoff bound for multiple quantum state sets and characterizes an optimal state-agnostic discrimination test.

Findings

The optimal error exponent is given by the regularized quantum Chernoff divergence.

Discriminating between sets is no harder than between their worst-case elements.

The maximum overlap with free states equals the quantum Chernoff divergence between sets.

Abstract

We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.