Minimum guesswork discrimination between quantum states

Abstract

Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy.

Guesswork of a Quantum Ensemble

“…The guesswork of a quantum ensemble quantifies the minimum number of guesses needed in average to correctly guess the state of the ensemble, when only one state can be queried at a time. Here, we derived analytical solutions subject to a finite set of conditions, including analytical solutions for any qubit ensemble with uniform probability distribution, thus disproving the conjecture [13] that analytical solutions only exist for binary and symmetric ensembles. As explicit examples, we computed the guesswork for any qubit regular polygonal and polyhedral ensemble, thus proving a conjecture [14] on the guesswork of the square qubit ensemble.…”

“…Our main result, Theorem 1, provides an analytical solution subject to a finite set of conditions. In particular, Corollary 1 provides the analytical solution for any qubit ensemble with uniform probability distribution, thus disproving the conjecture [13] that analytical solutions exist only for binary and symmetric ensembles. As explicit examples, in Corollaries 2 and 3 we explicitly compute the minimum guesswork of any qubit regular polygonal and polyhedral ensebles, respectively.…”

“…The guesswork has been extensively studied for classical ensembles [2]- [12], but only very recently tackled for quantum ensembles [13]- [15]. While previous works focused on the derivation of entropic bounds, our aim is instead the derivation of analytical solutions.…”

Guesswork of a quantum ensemble

“…In the classical case, the guesswork has been extensively studied [2]- [11]. However, the quantum case considered here has been tackled only recently in References [12], [13], whose main contribution is the derivation of entropic bounds.…”

Guesswork with Quantum Side Information

“…In this paper, we consider a natural generalization of the above guessing problem to the case in which the classical side information is replaced by quantum side information. This generalization was first considered in [13]. In this case, the guesser (say, Bob) holds a quantum system B, instead of a classical random variable (or equivalently, a classical system) Y .…”