Locally Tomographic Shadows (Extended Abstract)

TL;DR

This paper introduces a framework for constructing locally tomographic probabilistic theories from monoidal categories, highlighting how global states can become indistinguishable locally, with detailed analysis in the context of real quantum theory.

Contribution

It develops a method to derive locally tomographic shadows of monoidal probabilistic theories, clarifying local indistinguishability phenomena and process restrictions, especially in real quantum theory.

Findings

Locally tomographic shadows capture local indistinguishability of states.

The construction restricts processes to respect local indistinguishability.

Application to real quantum theory illustrates the framework.

Abstract

Given a monoidal probabilistic theory -- a symmetric monoidal category $\mathcal{C}$ of systems and processes, together with a functor $\mathbf{V}$ assigning concrete probabilistic models to objects of $\mathcal{C}$ -- we construct a locally tomographic probabilistic theory LT$(\mathcal{C},\mathbf{V})$ -- the locally tomographic shadow of $(\mathcal{C},\mathbf{V})$ -- describing phenomena observable by local agents controlling systems in $\mathcal{C}$, and able to pool information about joint measurements made on those systems. Some globally distinct states become locally indistinguishable in LT$(\mathcal{C},\mathbf{V})$, and we restrict the set of processes to those that respect this indistinguishability. This construction is investigated in some detail for real quantum theory.