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It must be noticed that in the present paper we are focusing only on the statistical description of the measurement process—not on the change of state due to the measurement, nor on its realization scheme—and for this reason we identify the measurement covariance with the covariance of the POVM. The state change due to the measurement is described by the so-called "instrument" (Ref. 11), and the measuring scheme is given in terms of a unitary interaction with an ancilla and an orthogonal measurement on it. Clearly, the POVM covariance does not imply the covariance of all of its instruments, and the covarance of an instrument does not imply the covariance of all of the measurement realizations of the instrument.
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