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Quantum Information Processing

Quantum decision theory—minimax approach

Andrzej Łuczak

Published Sep 23, 2025• 0 citations
Publisher

Abstract

We show a connection between the minimax and Bayes approaches in quantum decision theory in a general setting of normal states on a von Neumann algebra. In particular, the quantum minimax theorem is proven in a fairly general situation, and it is shown that every minimax strategy is Bayes for some a priori distribution on the set of states—a so-called least favourable prior . Minimax strategies with constant risk are investigated in some detail. It turns out that in dimension greater than two such a strategy can be Bayes for a non-uniform a priori distribution which, at the same time, is a least favourable prior.

DOI
10.1007/s11128-025-04924-1

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We show a connection between the minimax and Bayes approaches in quantum decision theory in a general setting of normal states on a von Neumann algebra. In particular, the quantum minimax theorem is proven in a fairly general situation, and it is shown that every minimax strategy is Bayes for some a priori distribution on the set of states—a so-called least favourable prior . Minimax strategies with constant risk are investigated in some detail. It turns out that in dimension greater than two such a strategy can be Bayes for a non-uniform a priori distribution which, at the same time, is a least favourable prior.
Recently, many fundamental and important results in statistical decision theory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples. In the present paper, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics.
The necessary and sufficient conditions for the minimax strategy of a finite number of decisions in quantum signal detection are derived in the case that the average Bayes cost is used as a quality function. This is a simple extension of the result of Hirota and Ikehara [1]. Applying the necessary and sufficient conditions, a closed-form expression of the minimax strategy for binary pure state signal is shown. Furthermore, the minimax strategy for a discrimination problem of the quasi-Bell states is considered. In this case, the square-root measurement consisting of the quasi-Bell states yields the minimax strategy.
We consider the application of the minimax strategy to discrimination between quantum states with a certain fraction of inconclusive results. An inconclusive minimax solution is defined as the pair of a collection of prior probabilities and a quantum measurement that achieves the minimax probability of a detection error subject to the constraint that the highest probability of an inconclusive result does not exceed a given value. We first show a necessary and sufficient condition for an inconclusive minimax solution, and derive that the problem of obtaining an inconclusive minimax measurement can be expressed as a semidefinite programming problem. We then present a condition under which there exists a solution such that at least one of the prior probabilities is a zero. We finally show that for any state set with a certain symmetry, an inconclusive minimax solution exists having the same symmetry.
We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.
After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.