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arXiv:quant-ph/0112051 · Physical Review A

Optimum unambiguous discrimination between subsets of nonorthogonal quantum states

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

Published Sep 23, 2002• 72 citations
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Abstract

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.

DOI
10.1103/physreva.66.032315

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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, ∣\ensuremathψ1〉,∣\ensuremathψ2〉,∣\ensuremathψ3〉,{|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},∣\ensuremathψ1​〉,∣\ensuremathψ2​〉,∣\ensuremathψ3​〉, we show that the optimal strategy to distinguish ∣\ensuremathψ1〉|{\ensuremath{\psi}}_{1}〉∣\ensuremathψ1​〉 from the set ∣\ensuremathψ2〉,∣\ensuremathψ3〉{|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}∣\ensuremathψ2​〉,∣\ensuremathψ3​〉 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.
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.
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.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
With the quantum state filtering as a known example, it has been shown that the unambiguous discrimination between subsets of nonorthogonal quantum states is possible with a certain possibility of success. In present work, besides reconsidering the known case of quantum state filtering, we also ask what happens if parts of the states are shared by both the subsets or if there are more than two subsets. The procedure, which is designed to find the optimal way for quantum set discrimination, contains two steps. For each case, we at first construct a one-photon interferometer (OPI) to realize the unambiguous discrimination in an enlarged Hilbert space according to the Neumark's theorem. The awards from the OPI are double: an optical experiment and the operators in terms of positive operator-valued measures. Then, with the operators from its corresponding OPI and the given a priori probability for each state, we are able to find out the optimal operators which can discriminate the subsets in an optimal way. It is shown that the OPIs, which have been discussed in present work, can be constructed by taking the quantum state filtering as their elementary process. This fact indicates that the problem of quantum set discrimination can be reduced to the known problem of quantum state filtering.
We propose an experimental setup for discriminating four linearly independent nonorthogonal symmetric quantum states. The setup is based on linear optics only and can be configured to implement both optimal unambiguous state discrimination [Chefles and Barnett, Phys. Lett. A 250, 223 (1998)] and minimum error discrimination. In both cases, the setup is characterized by an optimal success probability. The experimental setup can be generalized to the case of discrimination among NNN linearly nonorthogonal symmetric quantum states. We also study the discrimination between two incoherent superpositions of symmetric states. In this case, the setup also achieves an optimal success probability in the case of unambiguous discrimination as well as minimum error discrimination.