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arXiv:quant-ph/0012131 · Physical Review A

Optimum unambiguous discrimination between linearly independent nonorthogonal quantum states and its optical realization

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

Published Jul 1, 2001• 80 citations
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Abstract

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.

DOI
10.1103/physreva.64.022311

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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.
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.
Quantum mechanics forbids deterministic discrimination among nonorthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore, we present the first realization of unambiguous discrimination between a pure state and a nonorthogonal 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.
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.
We show how to construct devices which implement non-unitary transformations on quantum systems with a certain probability of success. The first transforms two specified non-orthogonal states to orthogonal ones while the second can be used to enhance the entanglement of quantum states. The states themselves are composed of qubits which are realized as photons in the dual-rail representation, and these photons serve as the inputs to a six-port. One of the outputs is measured, and if a null result is obtained, the desired non-unitary transformation has been carried out. The transformation which takes non-orthogonal to orthogonal states is an optical realization of an optimal quantum measurement for distinguishing the two non-orthogonal states.