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We consider the problem of detecting the true quantum state among r possibleones, based on measurements performed on n of copies of a finite dimensionalquantum system. It is known that the exponent for the rate of decrease of theaveraged error probability cannot exceed the multiple quantum Chernoff bound(MQCB) defined as the worst case (smallest) quantum Chernoff distance betweenany possible pair of the r states. This error exponent is attainable for r purestates, but for the general case of mixed states only attainability up to afactor 1/3 is known. Here we show that the MQCB is attainable for mixed statesif there is a pair which is closer in quantum Chernoff distance than 1/6 timesthe distance between all other pairs.

Attainment of the multiple quantum Chernoff bound for certain ensembles of mixed states