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International Journal of Theoretical Physics

Partition a Quantum Pure-state Set into Unambiguously Discriminable Subsets

Jianting Wang, Jianyu Wang, Zhongqian Fu

Published Nov 18, 2014• 1 citation
Publisher
DOI
10.1007/s10773-014-2406-0

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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, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ 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.
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.