Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)

TL;DR

This paper extends Kirkwood-Dirac representations to a fully compositional framework for quantum theory, establishing their properties and linking real, nonnegative representations to noncontextuality, with implications for quantum foundations.

Contribution

It introduces a fully compositional extension of Kirkwood-Dirac representations for all quantum theory elements and connects their properties to noncontextuality principles.

Findings

Extended Kirkwood-Dirac representations to channels and measurements.

Proved functoriality, linearity, and quasistochasticity of the extension.

Linked real, nonnegative representations to generalized noncontextuality.

Abstract

Kirkwood-Dirac representations of quantum states are increasingly finding use in many areas within quantum theory. Usually, representations of this sort are only applied to provide a representation of quantum states (as complex functions over some set). We show how standard Kirkwood-Dirac representations can be extended to a fully compositional representation of all of quantum theory (including channels, measurements and so on), and prove that this extension satisfies the essential features of functoriality (namely, that the representation commutes with composition of channels), linearity, and quasistochasticity. Interestingly, the representation of a POVM element is uniquely picked out to be the collection of weak values for it relative to the bases defining the representation. We then prove that if one can find any Kirkwood-Dirac representation that is everywhere real and nonnegative for a given experimental scenario or fragment of quantum theory, then the scenario or fragment is consistent with the principle of generalized noncontextuality, a key notion of classicality in quantum foundations. We also show that the converse does not hold: even if one verifies that all Kirkwood-Dirac representations (as defined herein) of an experiment require negativity or imaginarity, one cannot generally conclude that the experiment witnesses contextuality.