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