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arXiv:quant-ph/0504048 · Physical Review A

Minimax quantum-state discrimination

Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi, Jonas Kahn

Published Sep 8, 2005• 50 citations
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Abstract

We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.

DOI
10.1103/physreva.72.032310

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We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.
In this paper, we consider the quantum unambiguous discrimination problem of a set of linearly independent pure states. With a constructive procedure, we establish a necessary and sufficient condition for feasible POVM measurements for unambiguous quantum state discrimination. We develop methods to calculate the optimal discrimination efficiencies for minimizing the inconclusive probability and the optimal discrimination efficiencies for the minimax criterion. Finally, we prove a necessary and sufficient condition to have a feasible POVM measurement which achieves the minimum inconclusive probability criterion and the minimax criterion simultaneously and show that geometrically uniform states with equal a priori probabilities meet this condition.
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 study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.