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Fortschritte der Physik

Quantum Limited State Discrimination

Stephen Mark Barnett

Published Oct 1, 2001• 9 citations
Publisher

Abstract

Quantum theory restricts our ability to determine the state of a physical system. This is true even if we know for certain that it was prepared in one of a known set of possible states. I describe two types of optimal strategy for state discrimination. These are (i) state discrimination with minimum probability of error and (ii) unambiguous or error-free state discrimination.

DOI
10.1002/1521-3978(200110)49:10/11<909::aid-prop909>3.0.co;2-f

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Quantum theory restricts our ability to determine the state of a physical system. This is true even if we know for certain that it was prepared in one of a known set of possible states. I describe two types of optimal strategy for state discrimination. These are (i) state discrimination with minimum probability of error and (ii) unambiguous or error-free state discrimination.
There are fundamental limits to the accuracy with which one can determine the state of a quantum system. I give an overview of the main approaches to quantum state discrimination. Several strategies exist. In quantum hypothesis testing, a quantum system is prepared in a member of a known, finite set of states, and the aim is to guess which one with the minimum probability of error. Error free discrimination is also sometimes possible, if we allow for the possibility of obtaining inconclusive results. If no prior information about the state is provided, then it is impractical to try to determine it exactly, and it must be estimated instead. In addition to reviewing these various strategies, I describe connections between state discrimination, the manipulation of quantum entanglement and quantum cloning. Recent experimental work is also discussed.
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.
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 present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.
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.
Abstract Consider a quantum system prepared in a state chosen from a known set, the aim of quantum state discrimination is to perform a measurement on the system which can determine that a subset of the possible preparation procedures have been prepared. However, the aim of quantum state exclusion is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures have not taken place. For classical states, probabilities of error in discrimination and exclusion are always equal. For quantum states, we point out that probabilities of error in discrimination and exclusion are still equal in two-state ensembles, however they are no longer equal in multiple-state ensembles. More specifically, the error-probability of discrimination is not less than the error-probability of exclusion. We also show that the error-probability of exclusion is continuous which is similar with the error-probability of discrimination.