Decision problems with quantum black boxes

Abstract

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.

Keywords:

• unambiguous discrimination
• optimum discrimination
• operator comparison
• generalized measurements

Acknowledgements

DKLO acknowledges the support of the Scottish Universities Physics Alliance (SUPA) and the Quantum Information Scotland Network (QUISCO). MH acknowledges support from the Royal Society of Edinburgh and from the SUPA Distinguished Visitor programme. SMB thanks the Royal Society and the Wolfson Foundation for their generous support.