State-independent error-disturbance trade-off for measurement operators

In general, classical measurement statistics of a quantum measurement is disturbed by performing an additional incompatible quantum measurement beforehand. Using this observation, we introduce a state-independent definition of disturbance by relating it to the distinguishability problem between two classical statistical distributions – one resulting from a single quantum measurement and the other from a succession of two quantum measurements. Interestingly, we find an error-disturbance trade-off relation for any measurements in two-dimensional Hilbert space and for measurements with mutually unbiased bases in any finite-dimensional Hilbert space. This relation shows that error should be reduced to zero in order to minimize the sum of error and disturbance. We conjecture that a similar trade-off relation with a slightly relaxed definition of error can be generalized to any measurements in an arbitrary finite-dimensional Hilbert space.