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

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.

20 pages

References in corpus (3)

• Sharp and fuzzy observables on effect algebras
• A Kolmogorov Extension Theorem for POVMs
• Classicality condition on a system's observable in a quantum measurement and relative-entropy conservation law

Cited by in corpus (3)

• Minimal sufficient positive-operator valued measure on a separable Hilbert space
• Saturation of repeated quantum measurements
• Order preserving maps on quantum measurements