Quantum state discrimination bounds for finite sample size

TL;DR

This paper derives finite-sample bounds for quantum state discrimination errors, addressing practical scenarios where only limited copies of quantum states are available, extending asymptotic results to finite cases.

Contribution

It introduces finite-size bounds on various quantum state discrimination error probabilities, bridging the gap between theoretical asymptotic results and practical applications.

Findings

Finite bounds for Stein errors, Chernoff errors, and Hoeffding errors.

Extension of asymptotic quantum discrimination results to finite samples.

Improved understanding of error rates in realistic quantum measurement scenarios.

Abstract

In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of two given and completely known states, rho or sigma. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking rho for sigma, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between rho and sigma (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors and the mixed error probabilities related to the Chernoff and the Hoeffding errors.