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arXiv:0712.1616 · Physical Review A

Unambiguous comparison of ensembles of quantum states

Michal Sedlák, Mário Ziman, Vladimír Bužek, Mark S. Hillery

Published Apr 3, 2008• 20 citations
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Abstract

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.

DOI
10.1103/physreva.77.042304

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We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states ∣\ensuremathψ1⟩\ensuremath⨂k{|{\ensuremath{\psi}}_{1}⟩}^{\ensuremath{\bigotimes}k}∣\ensuremathψ1​⟩\ensuremath⨂k and ∣\ensuremathψ2⟩\ensuremath⨂l{|{\ensuremath{\psi}}_{2}⟩}^{\ensuremath{\bigotimes}l}∣\ensuremathψ2​⟩\ensuremath⨂l. We consider two cases: (1) The two unknown states ∣\ensuremathψ1⟩|{\ensuremath{\psi}}_{1}⟩∣\ensuremathψ1​⟩ and ∣\ensuremathψ2⟩|{\ensuremath{\psi}}_{2}⟩∣\ensuremathψ2​⟩ 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.
Quantum theory of unambiguous measurements In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.
We consider an unambiguous identification of an unknown coherent state with one of two unknown coherent reference states. Specifically, we consider two modes of an electromagnetic field prepared in unknown coherent states $\ensuremath{\mid}{\ensuremath{\alpha}}_{1}⟩$ and $\ensuremath{\mid}{\ensuremath{\alpha}}_{2}⟩$, respectively. The third mode is prepared either in the state $\ensuremath{\mid}{\ensuremath{\alpha}}_{1}⟩$ or in the state $\ensuremath{\mid}{\ensuremath{\alpha}}_{2}⟩$. The task is to identify (unambiguously) which of the two modes are in the same state. We present a scheme consisting of three beam splitters capable to perform this task. Although we do not prove the optimality, we show that the performance of the proposed setup is better than the generalization of the optimal measurement known for a finite-dimensional case. We show that a single beam splitter is capable to perform an unambiguous quantum state comparison for coherent states optimally. Finally, we propose an experimental setup consisting of $2N\ensuremath{-}1$ beam splitters for unambiguous identification among NNN unknown coherent states. This setup can be considered as a search in a quantum database. The elements of the database are unknown coherent states encoded in different modes of an electromagnetic field. The task is to specify the two modes that are excited in the same, though unknown, coherent state.
We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of NNN pure states, and find an optimal solution for some parameter range when N=3N=3N=3. In both cases we use the reduction method for the corresponding problem of mixed-state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for N=2N=2N=2. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.
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 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.