Compatibility of quantum instruments

Abstract

Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{$d$-compatibility} for measurements, channels, and instruments whose parent passes a $d$-dimensional quantum system and unlimited classical information to the post-processing. We formulate $d$-compatibility as a common linear factorization of operator spaces and operator systems and show that $d$-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify $d$-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a $d$-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are $d$-incompatible in certain regimes of $d$. We also link $d$-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.