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IEEE International Symposium on Information Theory (ISIT)

Guesswork with Quantum Side Information: Optimal Strategies and Aspects of Security

Eric P. Hanson, Vishal Katariya, Nilanjana Datta, Mark M. Wilde

Published Jun 1, 2020• 0 citations
Publisher

Abstract

What is the minimum number of guesses needed on average to correctly guess a realization of a random variable? The answer to this question led to the introduction of the notion 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 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, and we prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.

DOI
10.1109/isit44484.2020.9174107

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What is the minimum number of guesses needed on average to correctly guess a realization of a random variable? The answer to this question led to the introduction of the notion 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 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, and we prove a continuity result that certifies the security of slightly imperfect key states when the guesswork is used as the security criterion.
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 calledguessworkby 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.
Guesswork is an information-theoretic quantity which can be seen as an alternate security criterion to entropy. Recent work has established the theoretical framework for guesswork in the presence of quantum side information, which we extend both theoretically and experimentally. We consider guesswork when the side information consists of the BB84 states and their higher-dimensional generalizations. With this side information, we compute the guesswork for two different scenarios for each dimension. We then performed a proof-of-principle experiment using Laguerre-Gauss modes to experimentally compute the guesswork for higher-dimensional generalizations of the BB84 states. We find that our experimental results agree closely with our theoretical predictions. This work shows that guesswork can be a viable security criterion in cryptographic tasks and is experimentally accessible in a number of optical setups.
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.
The guesswork of a classical-quantum channel quantifies the cost incurred in guessing the state transmitted by the channel when only one state can be queried at a time, maximized over any classical pre-processing and minimized over any quantum post-processing. For arbitrary-dimensional covariant classical-quantum channels, we prove the invariance of the optimal pre-processing and the covariance of the optimal ost-processing. In the qubit case, we compute the optimal guesswork for the class of so-called highly symmetric informationally complete classical-quantum channels.
The guesswork quantifies the minimum cost incurred in guessing the state of an ensemble, when only one state can be queried at a time. In the classical case, it is well known that the optimal strategy trivially consists of querying the states in their non-increasing order of posterior probability. In the quantum case, on the other hand, the most general strategy to obtain the optimal ordering in which to perform the queries consist of a quantum measurement. Here, we solve such an optimization problem by deriving the quantum measurement attaining the guesswork for a broad class of ensembles and cost functions.