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Unambiguous discrimination between quantum operations

Guoming Wang, Mingsheng Ying

Published Dec 18, 2005• 2 citations
OpenAlex

Abstract

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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Unambiguous discrimination among quantum operations

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We address the problem of unambiguous discrimination among 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 also found that the introduction of entanglement can improve the discrimination.

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RESEARCH ARTICLE Decision problems with quantum black boxes

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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

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Computer Engineering and Applications Journal•Jan 1, 2009•0 citations
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.
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.
The discrimination of two mixed quantum states is a fundamental task in quantum state estimation and quantum information theory. In quantum state discrimination a quantum system is assumed to be in one of two possible – in general mixed – non-orthogonal quantum states. The discrimination then consists of a measurement strategy that allows to decide in which state the system was before the measurement. In unambiguous state discrimination the aim is to make this decision without errors, but it is allowed to give an inconclusive answer. Especially interesting are measurement strategies that minimize the probability of an inconclusive answer. A starting point for the analysis of this optimization problem was a result by Eldar et al. [Phys. Rev. A 69, 062318 (2004)], which provides non-operational necessary and sufficient conditions for a given measurement strategy to be optimal. These conditions are reconsidered and simplified in such a way that they become operational. The simplified conditions are the basis for further central results: It is shown that the optimal measurement strategy is unique, a statement that is e.g. of importance for the complexity analysis of optimal measurement devices. The optimal measurement strategy is derived for the case, where one of the possible input states has at most rank two, which was an open problem for many years. Furthermore, using the optimality criterion it is shown that there always exists a threshold probability for each state, such that below this probability it is optimal to exclude this state from the discrimination strategy. If the two states subject to discrimination can be brought to a diagonal structure with (2×2)-dimensional blocks, then the unambiguous discrimination of these states can be reduced to the unambiguous discrimination of pure states. A criterion is presented that allows to identify the presence of such a structure for two self-adjoint operators. This criterion consists of the evaluation of three commutators and allows an explicit construction of the (2×2)-dimensional blocks. As an important application of unambiguous state discrimination, unambiguous state comparison, i.e., the question whether two states are identical or not, is generalized and optimal measurements for this problem are constructed. If for a certain family of states, a physical device maps the input state to an output state, such that a second device can be built that yields back the original input state, such a map is called reversible on this family. With respect to state discrimination, such reversible maps are particularly interesting, if the output states are pure. A complete characterization of all families that allow such a reversible and purifying map is provided. If the states are mapped to pure states, but the map itself is not reversible, upper and lower bounds are analyzed for the “deviation from perfect faithfulness”, a quantity which measures the deviation from a reversible mapping.
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.
(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