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.
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.
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.
(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 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.
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.
The discrimination of any pair of unknown quantum states is performed by devices processing three parts of inputs: copies of the pair of unknown states we want to discriminate are, respectively, stored in two program systems and copies of data, which is guaranteed to be one of the unknown states, in a third system. We study the efficiency of such programmable devices with the inputs prepared with n and m copies of unknown qubits used as programs and data, respectively. By finding a symmetry in the average inputs, we apply the Jordan-basis method to derive their optimal unambiguous discrimination and minimum-error discrimination schemes. The dependence of the optimal solutions on the a priori probabilities of the mean input states is also demonstrated.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.