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arXiv:1010.2937 · Physical Review A

Optimal unambiguous comparison of two unknown squeezed vacua

Stefano Olivares, Michal Sedlák, Peter Rapčan, Matteo G. A. Paris, Vladimír Bužek

Published Jan 24, 2011• 4 citations
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Abstract

We propose a scheme for the unambiguous state comparison (USC) of two unknown squeezed vacuum states of the electromagnetic field. Our setup is based on linear optical elements and photon-number detectors, and it achieves optimal USC in an ideal case of unit quantum efficiency. In realistic conditions, i.e., for nonunit quantum efficiency of photodetectors, we evaluate the probability of getting an ambiguous result as well as the reliability of the scheme, thus showing its robustness in comparison to previous proposals.

DOI
10.1103/physreva.83.012313

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We propose a scheme for the unambiguous state comparison (USC) of two unknown squeezed vacuum states of the electromagnetic field. Our setup is based on linear optical elements and photon-number detectors, and it achieves optimal USC in an ideal case of unit quantum efficiency. In realistic conditions, i.e., for nonunit quantum efficiency of photodetectors, we evaluate the probability of getting an ambiguous result as well as the reliability of the scheme, thus showing its robustness in comparison to previous proposals.
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We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states ${|{\ensuremath{\psi}}_{1}⟩}^{\ensuremath{\bigotimes}k}$ and ${|{\ensuremath{\psi}}_{2}⟩}^{\ensuremath{\bigotimes}l}$. We consider two cases: (1) The two unknown states $|{\ensuremath{\psi}}_{1}⟩$ and $|{\ensuremath{\psi}}_{2}⟩$ are arbitrary states of qudits. (2) Alternatively, they are coherent states of a harmonic oscillator. For the case of coherent states we propose a simple experimental realization of the optimal ``comparison'' machine composed of a finite number of beam splitters and a single photodetector.
In this paper, we propose an experimental scheme for unambiguous quantum state comparison assisted by linear optical manipulations, twin-photons produced from parametric down-conversion, and postselection from the coincidence measurement. In this scheme the preparation of the general two mixed qubit states with arbitrary prior probabilities and the realization of the optimal POVMs for unambiguous quantum state comparison are presented. This proposal is feasible by current experimental technology, and may be used in single-qubit quantum flngerprinting.
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The goal of comparison is to reveal the difference of compared objects as fast and reliably as possible. In this paper we formulate and investigate the unambiguous comparison of unknown quantum measurements represented by nondegenerate sharp positive operator valued measures. We distinguish between measurement devices with a priori labeled and unlabeled outcomes. In both cases we can unambiguously conclude only that the measurements are different. For the labeled case it is sufficient to use each unknown measurement only once and the average conditional success probability decreases with the Hilbert space dimension as 1/d1/d1/d. If the outcomes of the apparatuses are not labeled, then the problem is more complicated. We analyze the case of two-dimensional Hilbert space. In this case single shot comparison is impossible and each measurement device must be used (at least) twice. The optimal test state in the two-shot scenario gives the average conditional success probability 4/9. Interestingly, the optimal experiment detects unambiguously the difference with nonvanishing probability for any pair of observables.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.