On the conditions for discrimination between quantum states with minimum error
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چکیده
Submitted to: J. Phys. A: Math. Gen. 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, {ρ̂i}, 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 realising a minimum error measurement are known [3, 4, 5, 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. A minimum error measurement will, in general, be a generalised measurement described not by projectors but rather by a probability operator measure (POM) [5], also referred to as a positive operator valued measure [7]. The probability that a generalised measurement gives the result j is
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تاریخ انتشار 2008