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

TL;DR

This paper introduces a method to construct the least informative observable conserved by a quantum instrument, providing insights into the structure of quantum measurements and their informational properties.

Contribution

The authors present a construction of the minimal observable conserved by any quantum instrument, extending understanding of measurement invariance in quantum systems.

Findings

Constructed the least informative conserved POVM for any quantum instrument.

Explicitly evaluated infinite successive measurement POVMs for photon counting.

Demonstrated the classical post-processing relation among conserved POVMs.

Abstract

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.