Guesswork with Quantum Side Information

Abstract

What is the minimum number of guesses needed on average to guess a realization of a random variable correctly? The answer to this question led to the introduction of a quantity called guesswork by Massey in 1994, which can be viewed as an alternate security criterion to entropy. In this paper, we consider the guesswork in the presence of quantum side information, and show that a general sequential guessing strategy is equivalent to performing a single quantum measurement and choosing a guessing strategy based on the outcome. We use this result to deduce entropic one-shot and asymptotic bounds on the guesswork in the presence of quantum side information, and to formulate a semi-definite program (SDP) to calculate the quantity. We evaluate the guesswork for a simple example involving the BB84 states, both numerically and analytically, and we prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.

Classical computation of quantum guesswork

“…The cost function is represented by the average number of queries needed to correctly guess the state of the ensemble, and is therefore referred to as the quantum guesswork [1]- [14]. Notice that, if multiple states could be queried at a time, the corresponding cost function would instead be the entropy [13] of the ensemble.…”

“…If the answer is on the negative, one will proceed asking about the second state appearing in the output tuple, and so on, with the goal of correctly guessing with the minimum number of queries. Notice, in particular, that a strategy employing sequential uses of quantum instruments, each producing one state to be queried, reduces [13] to the present strategy by considering the composition of the instruments and discarding the final state.…”

Guesswork of a quantum ensemble

“…Optimality is usually measured in terms of Bob's expected probability of correctly guessing m [1]. Here, we focus on an alternative figure of merit, namely, the expected number of guesses Bob has to do before correctly guessing m. Such a loss function is known as the minimum guesswork [2]- [13] associated to the particular encoding. In the classical case, the guesswork has been extensively studied [2]- [11].…”

“…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.…”