Post-processing minimal joint observables

Abstract

A finite set of quantum observables (positive operator valued measures) is called compatible if these observables are marginals of a some observable, called a joint observable of them. For a given set of compatible observables, their joint observable is in general not unique and it is desirable to take a minimal joint observable in the post-processing order since a less informative observable disturbs less the system. We address the question of the minimality of finite-outcome joint observables and prove that any joint observable is lower bounded by a minimal joint observable in the post-processing order. We also give characterizations of the minimality of a joint observable that can be checked by finitestep algorithms and apply them to the case of non-commuting dichotomic qubit observables.

The compatibility dimension of quantum measurements

“…Consider now an isometry V : C r → C d and note that the operators Ãi = V * |i i|V and Bi = V * |u i u i |V have rank at most one. Compatibility of unit rank POVMs is essentially the same as equality, up to permutation of effect operators and summing together collinear effects [Kur15,HK19]. We have thus the following lower bound; we conjecture that the bound is tight for generic, non-degenerate unitary matrices.…”

A resource theory of quantum measurements

“…That is, our free transformations are a subset of classical processing operations on POVMs, sometimes known as post-processing. Classical processing on POVMs has been widely studied [20][21][22][23][24][25][26][27][28][29][30][31], often in the context of measurement compatibility [7,8,21,32], but not in the context of a resource theory with a tensor product structure. These works often focus on simulating POVMs using projection-valued measurements (or some standard set of POVMs).…”

A Resource Theory of Quantum Measurements

“…That is, our free transformations are a subset of classical processing operations on POVMs, sometimes known as post-processing. Classical processing on POVMs has been widely studied [18][19][20][21][22][23][24][25][26][27][28], often in the context of measurement compatibility [7,8,19,29], but not in the context of a resource theory with a tensor product structure. This work often focuses on simulating POVMs using projection-valued measurements (or some standard set of POVMs).…”