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arXiv:quant-ph/0702245 · Journal of Physics A Mathematical and Theoretical

Unambiguous discrimination among oracle operators

Anthony Chefles, Akira Kitagawa, Masahiro Takeoka, Masahide Sasaki, Jason Twamley

Published Aug 1, 2007• 40 citations
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Abstract

We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show that, except in certain trivial cases, unambiguous discrimination among all standard oracle operators corresponding to integer functions with fixed domain and range is impossible. However, we find that it is possible to unambiguously discriminate among the Grover oracle operators corresponding to an arbitrarily large unsorted database. The unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions, which possess a strong form of classical indistinguishability, is analysed. We prove that these operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a Boolean domain and with arbitrary fixed range. Sets of such functions on a larger domain can have unambiguously distinguishable standard oracle operators, and we provide a complete analysis of the simplest case, that of four functions. We also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls and investigate an intriguing unitary superoperator transformation between standard and entanglement-assisted minimal oracle operators.

DOI
10.1088/1751-8113/40/33/016

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We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show that, except in certain trivial cases, unambiguous discrimination among all standard oracle operators corresponding to integer functions with fixed domain and range is impossible. However, we find that it is possible to unambiguously discriminate among the Grover oracle operators corresponding to an arbitrarily large unsorted database. The unambiguous distinguishability of standard oracle operators corresponding to totally indistinguishable functions, which possess a strong form of classical indistinguishability, is analysed. We prove that these operators are not unambiguously distinguishable for any finite set of totally indistinguishable functions on a Boolean domain and with arbitrary fixed range. Sets of such functions on a larger domain can have unambiguously distinguishable standard oracle operators, and we provide a complete analysis of the simplest case, that of four functions. We also examine the possibility of unambiguous oracle operator discrimination with multiple parallel calls and investigate an intriguing unitary superoperator transformation between standard and entanglement-assisted minimal oracle operators.
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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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.