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arXiv:0901.3206 · Physical Review A

Unambiguous identification of coherent states. II. Multiple resources

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

Published Jun 9, 2009• 10 citations
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Abstract

We analyze unambiguous identification of coherent states of an electromagnetic field. In particular, we study possible generalizations of an optical setup proposed in Sedl\'ak et al. [Phys. Rev. A 76, 022326 (2007)]. We show how the unambiguous identification of coherent states can be performed in a general case when multiple copies of unknown and reference states are available. Under the condition that the experimental setup consists only of linear optical elements and photodetectors, we prove the optimality of the setup. We also investigate whether reference states after the measurement can be ``recovered'' and further used for subsequent unambiguous identification tasks. We show that in spite of the fact that the recovered reference states are disturbed by measurements, they can be repeatedly used for unambiguous identifications. We analyze the influence of particular type of noise in the preparation of the unknown and the reference coherent states on the performance of our unambiguous-identification setup.

DOI
10.1103/physreva.79.062305

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We analyze unambiguous identification of coherent states of an electromagnetic field. In particular, we study possible generalizations of an optical setup proposed in Sedl'ak et al. [Phys. Rev. A 76, 022326 (2007)]. We show how the unambiguous identification of coherent states can be performed in a general case when multiple copies of unknown and reference states are available. Under the condition that the experimental setup consists only of linear optical elements and photodetectors, we prove the optimality of the setup. We also investigate whether reference states after the measurement can be ``recovered'' and further used for subsequent unambiguous identification tasks. We show that in spite of the fact that the recovered reference states are disturbed by measurements, they can be repeatedly used for unambiguous identifications. We analyze the influence of particular type of noise in the preparation of the unknown and the reference coherent states on the performance of our unambiguous-identification setup.
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.
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.
An optical implementation of the recently proposed unambiguous identification of coherent states is presented. Our system works as a programmable discriminator between two, in general nonorthogonal, weak coherent states. The principle of operation lies in the interference of three light beams--two program states and one unknown coherent state which can be equal to either one of the two program states. The experiment is based on fiber optics. Its results confirm theoretical predictions, and the experimental setup can be straightforwardly extended for higher numbers of program states.
Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are a prime candidate as information carriers for optical communications. A large body of literature exists on quantum-limited parameter estimation and discrimination for coherent states. However, very little is known about practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and establish a theory of unambiguous discrimination of coherent states, with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, un-excited auxiliary input modes, and on-off photon detection. Our results indicate that these currently-available optical components are near optimal for unambiguous discrimination of multiple coherent states in a constellation.
Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.
We discuss several methods for unambiguous state discrimination of N symmetric coherent states using linear optics and photodetectors. One type of measurement is shown to be optimal in the limit of small photon numbers for any N. For the special case of N=4N=4N=4 this measurement can be fruitfully used by the receiving end (Bob) in an implementation of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol using faint laser pulses. In particular, if Bob detects only a single photon the procedure is equivalent to the standard measurement that he would have to perform in a single-photon implementation of BB84, if he detects two photons Bob will unambiguously know the bit sent to him in 50% of the cases without having to exchange basis information, and if three photons are detected, Bob will know unambiguously which quantum state was sent.