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arXiv:1110.5482 · Physical Review Letters

Deriving Quantum Theory from Its Local Structure and Reversibility

Gonzalo de la Torre, Lluís Masanes, Anthony J. Short, Markus P. Müller

Published Aug 30, 2012• 54 citations
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Abstract

We investigate the class of physical theories with the same local structure as quantum theory but potentially different global structure. It has previously been shown that any bipartite correlations generated by such a theory can be simulated in quantum theory but that this does not hold for tripartite correlations. Here we explore whether imposing an additional constraint on this space of theories-that of dynamical reversibility-will allow us to recover the global quantum structure. In the particular case in which the local systems are identical qubits, we show that any theory admitting at least one continuous reversible interaction must be identical to quantum theory.

DOI
10.1103/physrevlett.109.090403

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We investigate the class of physical theories with the same local structure as quantum theory but potentially different global structure. It has previously been shown that any bipartite correlations generated by such a theory can be simulated in quantum theory but that this does not hold for tripartite correlations. Here we explore whether imposing an additional constraint on this space of theories-that of dynamical reversibility-will allow us to recover the global quantum structure. In the particular case in which the local systems are identical qubits, we show that any theory admitting at least one continuous reversible interaction must be identical to quantum theory.
The richness of quantum theory’s reversible dynamics is one of its unique operational characteristics, with recent results suggesting deep links between the theory’s reversible dynamics, its local state space and the degree of non-locality it permits. We explore the delicate interplay between these features, demonstrating that reversibility places strong constraints on both the local and global state space. Firstly, we show that all reversible dynamics are trivial (composed of local transformations and permutations of subsytems) in maximally non-local theories whose local state spaces satisfy a dichotomy criterion; this applies to a range of operational models that have previously been studied, such as d -dimensional ‘hyperballs’ and almost all regular polytope systems. By separately deriving a similar result for odd-sided polygons, we show that classical systems are the only regular polytope state spaces whose maximally non-local composites allow for non-trivial reversible dynamics. Secondly, we show that non-trivial reversible dynamics do exist in maximally non-local theories whose state spaces are reducible into two or more smaller spaces. We conjecture that this is a necessary condition for the existence of such dynamics, but that reversible entanglement generation remains impossible even in this scenario.
In order to better understand the structure of quantum theory, or speculate about theories that may supersede it, it can be helpful to consider alternative physical theories. 'Boxworld' describes one such theory, in which all non-signaling correlations are achievable. In a limited class of multipartite Boxworld systems—wherein all subsystems are identical and all measurements have the same number of outcomes—it has been demonstrated that the set of reversible dynamics is 'trivial', generated solely by local relabellings of measurement choices and outcomes, and permutations of subsystems. We develop the convex formalism of Boxworld to give an alternative proof of this result, then extend this proof to all multipartite Boxworld systems, and discuss the potential relevance to other theories. These results lend further support to the idea that the rich reversible dynamics in quantum theory may be the key to understanding its structure and its informational capabilities.
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We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this we impose two very natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories interacting dynamics is impossible. This result reveals that "existence of entanglement" is the requirement with minimal logical content which singles out quantum theory from our family of theories.
We consider a very natural generalization of quantum theory by letting the dimension of the Bloch ball be not necessarily three. We analyze bipartite state spaces where each of the components has a d-dimensional Euclidean ball as state space. In addition to this, we impose two very natural assumptions: the continuity and reversibility of dynamics and the possibility of characterizing bipartite states by local measurements. We classify all these bipartite state spaces and prove that, except for the quantum two-qubit state space, none of them contains entangled states. Equivalently, in any of these non-quantum theories, interacting dynamics is impossible. This result reveals that “existence of entanglement” is the requirement with minimal logical content which singles out quantum theory from our family of theories.
Correlation self-testing of quantum theory involves identifying a task or set of tasks whose optimal performance can be achieved only by theories that can realise the same set of correlations as quantum theory in every causal structure. Following this approach, previous work has ruled out various classes of generalised probabilistic theories whose joint state spaces have a certain regularity in the sense of a (discrete) rotation symmetry of the bipartite state spaces. Here we consider theories whose bipartite state spaces lack this regularity. We form them by taking the convex hull of all the local states and a finite number of non-local states. We show that a criterion of compositional consistency is needed in such theories: for a measurement effect to be valid, there must exist at least one measurement that it is part of. This goes beyond previous consistency criteria and corresponds to a strengthening of the no-restriction hypothesis. We show that quantum theory outperforms these theories in a task called the adaptive CHSH game, which shows that they can be ruled out experimentally. We further show a connection between compositional consistency and Tsirelson's bound.