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Fortschritte der Physik

Optimum Discrimination of Non-orthogonal Quantum States: An Optical Realization

János A. Bergou, Mark S. Hillery, Yuqing Sun

Published Oct 1, 2001• 7 citations
Publisher

Abstract

Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.

DOI
10.1002/1521-3978(200110)49:10/11<915::aid-prop915>3.0.co;2-r

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Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.
Unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here we consider a variant of that problem; rather than discriminating among all of the different states, we shall discriminate between two subsets of them. In particular, for the case of 3 states, we find the optimal strategy to distinguish one from the other two. This procedure has a higher probability of success than distinguishing all 3 states. We show how to construct generalized interferometers that provide an implementation of the optimal strategy.
It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.
Abstract Unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a new variant of that problem. Instead of discriminating among all of the N different states, we now ask for less. We want to unambiguously assign the state to one of two complementary subsets of the set of N given non‐orthogonal quantum states, each occurring with given a priori probabilities. We refer to the special case when one subset contains only one state and the other contains the remaining N‐1 states as unambiguous quantum state filtering. We present an optimal analytical solution for the special case of N=3, and discuss the optimal strategy to unambiguously distinguish |ψ1〉 from the set {|ψ2〉,|ψ3〉}. For unambiguous filtering the subsets need not be linearly independent. We briefly discuss how to construct generalized interferometers (multiports) which provide a fully linear optical implementation of the optimal strategy.
Unambiguously distinguishing between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In principle, the problem can be solved by mapping the set of nonorthogonal quantum states onto a set of orthogonal ones, which then can be distinguished without error. Such nonunitary transformations can be performed conditionally on quantum systems; a unitary transformation is carried out on a larger system of which the system of interest is a subsytem, a measurement is performed, and if the proper result is obtained the desired nonunitary transformation has been performed on the subsystem. We show how to construct generalized interferometers (multiports), which when combined with measurements on some of the output ports, implement nonunitary transformations of this type. The input states are single-photon states in which the photon is divided among several modes. A number of explicit examples of distinguishing among three nonorthogonal states are discussed, and the networks that optimally distinguish among these states are presented.
In quantum information and quantum computing, the carrier of information is some quantum system and information is encoded in its state. After processing the state in the quantum processor, the information has to be read out. Clearly, this task is equivalent to determining the final state of the system. We begin by briefly reviewing various possible state discrimination strategies that are optimal with respect to some reasonable criteria and report on recent advances in the unambiguous discrimination of mixed quantum states. This strategy has been successfully applied to devise a class of novel probabilistic quantum algorithms and has been demonstrated experimentally, using a linear optical implementation via generalized interferometers. In the second part we present a scheme for communication via completely unknown quantum states. In this context we discuss programmable quantum state discriminators that are universal, i.e. perform optimally on average, independently of the actual states used for the communication scheme. We conclude with a discussion of possible experimental implementations of the proposed device.
We present the results of generalized measurements of optical polarization designed to provide one of three or four distinct outcomes. This has allowed us to discriminate between non-orthogonal polarization states with an error probability that is close to the minimum allowed by quantum theory. The sets of states we have prepared and measured are (i) two non-orthogonal states of linear polarisation, (ii) the trine, consisting of three states of linear polarization separated by 60º and, (iii) the tetrad, consisting of four polarization states (two linear and two elliptical) arranged in a tetrahedron on the Poincaré sphere. This has enabled us to realize the generalized measurements required for (i) optimal unambiguous discrimination between two non-orthogonal polarization states, (ii) discrimination between the three trine states and four tetrad states with minimum probability of error and, (iii) maximizing the mutual information associated with our trine and tetrad measurements and showing that it exceeds the value assocated with the best possible von Neumann measurement. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.