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On the conditions for discrimination between quantum states with minimum error

Barnett, S.M. and Croke, S. (2009) On the conditions for discrimination between quantum states with minimum error. Journal of Physics A: Mathematical and Theoretical, 42 (6). pp. 1-4. ISSN 0305-4470

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Abstract

In quantum communications a transmitting party, Alice, selects from among a set of agreed quantum states to prepare a quantum system for transmission to the receiving party, Bob. Both the set of possible states, {ρˆi}, and the associated probabilities for selection, {pi}, are known to Bob but not, of course, the selected state. His task is to determine as well as he can which state was prepared and he does this by choosing a measurement to perform. If the states are not mutually orthogonal then there is no measurement that will reveal the selected state with certainty. The strategy he chooses will depend on the use for which the information is intended and there exist many figures of merit for Bob's measurement [1, 2]. Among these the simplest is the minimum probability of error or, equivalently, the maximum probability for correctly identifying the state. Necessary and sufficient conditions for realizing a minimum error measurement are known [3-6], but it has proven to be easier to prove sufficiency than necessity. This letter presents an appealingly simple proof that the conditions are necessary.