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arXiv:2308.16494 · Electronic Proceedings in Theoretical Computer Science

Locally Tomographic Shadows (Extended Abstract)

Howard Barnum, Matthew A. Graydon, Alexander Wilce

Published Aug 23, 2023• 0 citations
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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.

DOI
10.4204/eptcs.384.3

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Given a monoidal probabilistic theory -- a symmetric monoidal category C\mathcal{C}C of systems and processes, together with a functor V\mathbf{V}V assigning concrete probabilistic models to objects of C\mathcal{C}C -- we construct a locally tomographic probabilistic theory LT(C,V)(\mathcal{C},\mathbf{V})(C,V) -- the locally tomographic shadow of (C,V)(\mathcal{C},\mathbf{V})(C,V) -- describing phenomena observable by local agents controlling systems in C\mathcal{C}C, and able to pool information about joint measurements made on those systems. Some globally distinct states become locally indistinguishable in LT(C,V)(\mathcal{C},\mathbf{V})(C,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.
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.
Entanglement is a central and subtle feature of quantum theory, whose structure and operational behavior can change dramatically when additional physical constraints, such as symmetries or superselection rules, are imposed. Such constraints can give rise to striking and counter-intuitive phenomena, including local broadcasting of entangled states and failures of entanglement monogamy. These effects naturally arise in tomographically nonlocal theories (like real quantum theory, twirled worlds, or fermionic quantum theory), where composite systems possess holistic degrees of freedom that are inaccessible to local measurements. In this work, we study entanglement in such theories within the framework of generalized probabilistic theories. We show that the failure of tomographic locality leads to two qualitatively distinct forms of entanglement, which we term tomographically-local\textit{tomographically-local}tomographically-local entanglement and tomographically-nonlocal\textit{tomographically-nonlocal}tomographically-nonlocal entanglement. We analyze the operational consequences of this distinction, proving that tomographically-nonlocal entanglement is useless for Bell nonlocality, steering, and teleportation, but sufficient for dense coding and perfectly secure data hiding. This framework clarifies the origin of several previously puzzling features of entanglement that arise when tomographic locality fails, as can happen even in quantum theory when one considers fermions or fundamental superselection rules.
This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u from V_o to 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V_o(C) as consisting of concrete operational models. In this case, if A and B are objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and V_o(B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V_o a (strict) monoidal functor.
We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category $\Prob$ of probabilistic models (test spaces with designated state spaces) and morphisms thereof. Restricted to any subcategory of OUS monoidal with respect to a positive, normalized, bilinear composition rule, our functor is also monoidal. This shows that the convex-operational approach to physical theories can be subsumed by the test-space approach, without resort to ``generalized test spaces''. A second construction, equipping a probabilistic model with tests representing ``weighted coins'', also sheds light on the nature of unsharp observables.
We give a shadow tomography algorithm that uses O(log⁡(m)log⁡(m/δ)/ε2)O(\log(m) \log(m/δ) / \varepsilon^2)O(log(m)log(m/δ)/ε2) copies of the input state. Specifically, given a quantum state ρρρ and observables O1,…,OmO_1,\ldots,O_mO1​,…,Om​, the algorithm outputs estimates of tr(Oiρ)\mathrm{tr}(O_iρ)tr(Oi​ρ) for all i∈[m]i\in[m]i∈[m] that are all accurate to within ±ε\pm\varepsilon±ε, except with probability at most δδδ. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on O(log⁡(m)/ε2)O(\log(m)/\varepsilon^2)O(log(m)/ε2) copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining log⁡(m)\log(m)log(m) factor separating the complexity of the quantum problem from its classical counterpart.