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arXiv:0807.4576 · Physical Review A

One-photon interferometer for realizing optimal unambiguous discrimination among quantum subsets

Xiaohua Wu, Shaolan Yu, Tao Zhou

Published May 5, 2009• 12 citations

Abstract

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.

DOI: 10.1103/physreva.79.052302

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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.

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.

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.

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

N

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.

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.

Quantum-state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs. The simplest case, when the

N

given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum-state filtering. We derived previously the optimal strategy for the case of

N

nonorthogonal states, ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$, for distinguishing $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. In a previous paper [Phys. Rev. Lett. 90, 257901 (2003)], we sketched an appplication of the results to probabilistic quantum algorithms. Here we fill the gaps and give the complete derivation of the probabilstic quantum algorithm that can optimally distinguish between two classes of Boolean functions, that of the balanced functions and that of the biased functions. The algorithm is probabilistic, it fails sometimes but when it does it lets us know that it did. Our approach can be considered as a generalization of the Deutsch-Jozsa algorithm that was developed for the discrimination of balanced and constant Boolean functions.