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RESEARCH ARTICLE Decision problems with quantum black boxes

Mark S. Hillery, Erika Andersson, Stephen Mark Barnett, Daniel K. L. Oi

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

(September 23, 2011)We examine how to distinguish between unitary operators, when the exact form of thepossible operators is not known. Instead we are supplied with “programs” in the form ofunitary transforms, which can be used as references for identifying the unknown unitarytransform. All unitary transforms should be used as few times as possible. This situationis analoguous to programmable state discrimination. One difference, however, is that thequantum state to which we apply the unitary transforms may be entangled, leading to aricher variety of possible strategies. By suitable selection of an input state and generalizedmeasurement of the output state, both unambiguous and minimum-error discrimination canbe achieved. Pairwise comparison of operators, comparing each transform to be identifiedwith a program transform, is often a useful strategy. There are, however, situations in whichmore complicated strategies perform better. This is the case especially when the number ofallowed applications of program operations is different from the number of the transforms tobe identified.Keywords:unambiguous discrimination; optimum discrimination; operator comparison;generalized measurements

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We examine how to distinguish between unitary operators, when the exact form of the possible operators is not known. Instead we are supplied with ‘programs’ in the form of unitary transforms, which can be used as references for identifying the unknown unitary transform. All unitary transforms should be used as few times as possible. This situation is analogous to programmable state discrimination. One difference, however, is that the quantum state to which we apply the unitary transforms may be entangled, leading to a richer variety of possible strategies. By suitable selection of an input state and generalized measurement of the output state, both unambiguous and minimum-error discrimination can be achieved. Pairwise comparison of operators, comparing each transform to be identified with a program transform, is often a useful strategy. There are, however, situations in which more complicated strategies perform better. This is the case especially when the number of allowed applications of program operations is different from the number of the transforms to be identified.

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Unambiguous discrimination between quantum operations

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We address the problem of unambiguous discrimination between a given set of quantum operations. The necessary and sufficient condition for them to be unambiguously distinguishable is derived in the cases of single use and multiple uses respectively. For the latter case we explicitly construct the input states and corresponding measurements that accomplish the task. It is found that the introduction of entanglement can improve the discrimination. The advantage of our strategy over minimum-error discrimination is also discussed.

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(September 23, 2011)We examine how to distinguish between unitary operators, when the exact form of thepossible operators is not known. Instead we are supplied with “programs” in the form ofunitary transforms, which can be used as references for identifying the unknown unitarytransform. All unitary transforms should be used as few times as possible. This situationis analoguous to programmable state discrimination. One difference, however, is that thequantum state to which we apply the unitary transforms may be entangled, leading to aricher variety of possible strategies. By suitable selection of an input state and generalizedmeasurement of the output state, both unambiguous and minimum-error discrimination canbe achieved. Pairwise comparison of operators, comparing each transform to be identifiedwith a program transform, is often a useful strategy. There are, however, situations in whichmore complicated strategies perform better. This is the case especially when the number ofallowed applications of program operations is different from the number of the transforms tobe identified.Keywords:unambiguous discrimination; optimum discrimination; operator comparison;generalized measurements
We examine how to distinguish between unitary operators, when the exact form of the possible operators is not known. Instead we are supplied with ‘programs’ in the form of unitary transforms, which can be used as references for identifying the unknown unitary transform. All unitary transforms should be used as few times as possible. This situation is analogous to programmable state discrimination. One difference, however, is that the quantum state to which we apply the unitary transforms may be entangled, leading to a richer variety of possible strategies. By suitable selection of an input state and generalized measurement of the output state, both unambiguous and minimum-error discrimination can be achieved. Pairwise comparison of operators, comparing each transform to be identified with a program transform, is often a useful strategy. There are, however, situations in which more complicated strategies perform better. This is the case especially when the number of allowed applications of program operations is different from the number of the transforms to be identified.
How to discriminate or compare two unitary transformations that are completely unknown? We first examine the unambiguous discrimination of two unknown unitary transformations; we show that the results are the same as those found for the programmable discrimination of two unknown quantum states. Next we consider the minimum-error comparison of two unknown unitary transformations; the results are obtained in the general case where the prior probabilities are different. Last we study the unambiguous discrimination of two unknown unitary transformations in the case where multiple copies of data are available.
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.
Quantum state discrimination is one of the most fundamental tasks in quantum information since it describes the read-out process of classical information from quantum states in quantum communications.In quantum state discrimination, a given test state is guaranteed to be identical to one of a set of candidate quantum states.The task is to determine the identity of the test state by performing an appropriate quantum measurement based on the classical descriptions of the candidate states.Quantum state discrimination also has several connections to the principles of quantum mechanics such as the no-cloning theorem and the no-signaling principle.One of fundamental properties of quantum mechanics is that the exact classical description of unknown quantum states cannot be retrieved from a single copy of the physical state.Quantum state discrimination in the case that the candidate states are given as two unknown physical states (reference states) but their classical descriptions are not given, has been investigated.The two reference states are labeled one and two, respectively.The task is to determine which of the two reference states is equivalent to the test state, whereas the descriptions of the states are not of interest.We call such a task as equivalence determination of quantum states.Discrimination of quantum operations has been also investigated and it has been shown that there is an intrinsic difference between discrimination of quantum state and discrimination of quantum operations.We analyze a unitary operation version of the equivalence determination task by using the concept of higher-order quantum computation to investigate properties of discrimination of other quantum objects.We consider three black-boxes (test box and reference boxes) implementing unknown unitary operations given as physical systems.The test box is guaranteed to implement one of the two unitary operations given by the reference boxes.Equivalence determination of unitary operations is a task to determine the reference box implementing the same unitary operation as the one by the test box.We assume that the unitary operations of the reference boxes are randomly sampled from the Haar measure of SU(2).We first consider the case that each of three black-boxes can be used only once both in parallel and ordered uses of the black-boxes.We show that the optimal success probability for equivalence determination is 7/8 by obtaining analytical solutions of the corresponding semidefinite programmings (SDPs) both in the parallel and ordered strategies.The optimal success probability can be achieved even without using one of the reference boxes.We also showed that entanglement of an initial state across the systems on which the reference boxes act and the test box acts is necessary to achieve the optimal success probability.We then consider the case that the multiple uses of the reference boxes are allowed.We consider the parallel strategies up to four uses of each of the reference boxes and obtain the optimal success probabilities by numerically solving the semidefinite programmings.We numerically show that the ordered strategies give improvement over the parallel strategies and the optimal success probability varies depending on the order of the black-boxes when one reference box is used once and the other reference box is used twice.This result indicates that an appropriate order of the black-boxes under the ordered strategies is necessary to obtain improvement over the parallel strategies. Chapter 1 Introduction Notations• H i , K j : Hilbert spaces labeled with i and j.• L(H): The set of linear operators on H.• L(H, K): The set of linear operators from H to K.• S(H) : The set of positive semidefinite linear operation of unit trace on H • Tr H : The partial trace over H • U † : The conjugate transpose of U • {Π i }: The positive operator-valued measure • { Π i }: The quantum tester • δ ij : The Kronecker delta, equal to 1 if i = j and 0 otherwise • dµ(•) : The Haar measure, the uniform distribution of the space of unitary operations • |I⟩⟩ : Unnormalized maximally entangled vector defined as |I⟩⟩ = i |i⟩|i⟩ • M : A quantum channel • M : Choi operator of a quantum channel M• U j : The irreducible subspace of the total angular momentum j.
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.
We address the problem of unambiguous discrimination between a given set of quantum operations. The necessary and sufficient condition for them to be unambiguously distinguishable is derived in the cases of single use and multiple uses respectively. For the latter case we explicitly construct the input states and corresponding measurements that accomplish the task. It is found that the introduction of entanglement can improve the discrimination. The advantage of our strategy over minimum-error discrimination is also discussed.