Construction of the least informative observable conserved by a given quantum instrument

For a quantum measurement process described by a quantum instrument $\mathcal{I}$ and a system observable corresponding to a positive-operator valued measure (POVM) $E ,$ $\mathcal{I}$ is said to conserve the information of $E$ if the joint successive measurement of $\mathcal{I}$ followed by $E$ is equivalent to a single measurement of $E .$ We show that for any quantum instrument $\mathcal{I}$ we can construct a POVM conserved by $\mathcal{I}$. Intuitively the construction gives the infinite joint successive measurement of $\mathcal{I}.$ We also show that the constructed POVM is the least informative observable among POVMs conserved by $\mathcal{I}$, i.e. the constructed POVM can be realized by a classical post-processing of any POVM conserved by $\mathcal{I} .$ As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.