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arXiv:0809.4884 · Physical Review A

Optimum unambiguous identification of d unknown pure qudit states

Ulrike Herzog, János A. Bergou

Published Sep 17, 2008• 18 citations
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Abstract

We address the problem of unambiguously identifying the state of a probe qudit with the state of one of $d$ reference qudits. The $d$ reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the $d$ unknown reference states. We derive the optimum measurement strategy that maximizes the success probability of unambiguous identification and find that the optimum strategy is a generalized measurement. We give both the measurement operators and the optimum success probability explicitly. Technically, the problem we solve amounts to the optimum unambiguous discrimination of $d$ known mixed quantum states.

DOI
10.1103/physreva.78.032320

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We address the problem of unambiguously identifying the state of a probe qudit with the state of one of ddd reference qudits. The ddd reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the ddd unknown reference states. We derive the optimum measurement strategy that maximizes the success probability of unambiguous identification and find that the optimum strategy is a generalized measurement. We give both the measurement operators and the optimum success probability explicitly. Technically, the problem we solve amounts to the optimum unambiguous discrimination of ddd known mixed quantum states.
We consider the problem of optimally identifying the state of a probe qudit, prepared with given prior probability in a pure state belonging to a finite set of possible states which together span a DDD-dimensional subspace of the ddd-dimensional Hilbert space the qudit is defined in. It is assumed that we do not know some or all of the states in the set, but for each unknown state we are given a reference qudit into which this state is encoded. We show that from the measurement for optimal state identification with d=Dd=Dd=D one can readily determine the optimal figure of merit for qudits with d>Dd>Dd>D, without solving a new optimization problem. This result is applied to the minimum-error identification and to the optimal unambiguous identification of two qudit states with $d\ensuremath{\ge}2$, where either one or both of the states are unknown, and also to the optimal unambiguous identification of NNN equiprobable linearly independent unknown pure qudit states with $d\ensuremath{\ge}N$. In all cases the optimal figure of merit, averaged over the unknown states, increases with growing dimensionality ddd of the qudits.
Given nnn linearly independent pure states and their prior probabilities, we study the optimum unambiguous state discrimination problem. We derive the conditions for the optimum measurement strategy to achieve the maximum average success probability and establish two sets of equations that must be satisfied by the optimum solution in different situations. We also provide the detailed steps to find the optimum measurement strategy. The method and results we obtain are given a geometrical illustration with a numerical example. Furthermore, using these equations, we derive a formula which shows a clear analytical relation between the optimum solution and the nnn states to be discriminated. We also solve a generalized equal-probability measurement problem analytically. Finally, as another application of our result, the unambiguous discrimination problem of three pure states is studied in detail and analytical solutions are obtained for some interesting cases.
We study how to unambiguously identify a given quantum pure state with one of the two reference pure states when no classical knowledge on the reference states is given but a certain number of copies of each reference quantum state are presented. By unambiguous identification, we mean that we are not allowed to make a mistake but our measurement can produce an inconclusive result. Assuming the two reference states are independently distributed over the whole pure state space in a unitary invariant way, we determine the optimal mean success probability for an arbitrary number of copies of the reference states and a general dimension of the state space. It is explicitly shown that the obtained optimal mean success probability asymptotically approaches that of the unambiguous discrimination as the number of the copies of the reference states increases.
We consider the problem of determining the state of an unknown quantum sequence without error. The elements of the given sequence are drawn with equal probability from a known set of linearly independent pure quantum states with the property that their mutual inner products are all real and equal. This problem can be posed as an instance of unambiguous state discrimination where the states correspond to that of all possible sequences having the same length as the given one. We calculate the optimum probability by solving the optimality conditions of a semidefinite program. The optimum value is achievable by measuring individual members of the sequence, and no collective measurement is necessary.
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.
In this paper, we present a general upper bound on the success probability for unambiguous discrimination among arbitrary mmm mixed quantum states with given a priori probability. We further analyze how this upper bound can be achievable by presenting a sufficient condition related to it, and we compare this upper bound with three other upper bounds. Moreover, for the issue of the unambiguous identification of mmm unknown multicopy qudit pure states, we also get a bound.