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arXiv:1508.03491 · New Journal of Physics

Reversibility and the structure of the local state space

Sabri W. Al-Safi, Jonathan G. Richens

Published Dec 4, 2015• 15 citations
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Abstract

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.

DOI
10.1088/1367-2630/17/12/123001

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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.
A remarkable feature of quantum theory is nonlocality (Bell inequality violations). However, quantum correlations are not maximally nonlocal, and it is natural to ask whether there are compelling reasons for rejecting theories in which stronger violations are possible. To shed light on this question, we consider post-quantum theories in which maximally nonlocal states (nonlocal boxes) occur. We show that reversible transformations in such theories are trivial: they consist solely of local operations and permutations of systems. In particular, no correlations can be created; nonlocal boxes cannot be prepared from product states and classical computers can efficiently simulate all such processes.
The outcomes of measurements on entangled quantum systems can be nonlocally correlated. However, while it is easy to write down toy theories allowing arbitrary nonlocal correlations, those allowed in quantum mechanics are limited. Quantum correlations cannot, for example, violate a principle known as macroscopic locality, which implies that they cannot violate Tsirelson's bound. This paper shows that there is a connection between the strength of nonlocal correlations in a physical theory and the structure of the state spaces of individual systems. This is illustrated by a family of models in which local state spaces are regular polygons, where a natural analogue of a maximally entangled state of two systems exists. We characterize the nonlocal correlations obtainable from such states. The family allows us to study the transition between classical, quantum and super-quantum correlations by varying only the local state space. We show that the strength of nonlocal correlations—in particular whether the maximally entangled state violates Tsirelson's bound or not—depends crucially on a simple geometric property of the local state space, known as strong self-duality. This result is seen to be a special case of a general theorem, which states that a broad class of entangled states in probabilistic theories—including, by extension, all bipartite classical and quantum states—cannot violate macroscopic locality. Finally, our results show that models exist that are locally almost indistinguishable from quantum mechanics, but can nevertheless generate maximally nonlocal correlations.
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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.
This thesis explores various perspectives on quantum phenomena, and how our understanding of these phenomena is informed by the study of general probabilistic theories. Particular attention is given to quantum nonlocality, and its interaction with areas of physical and mathematical interest such as entropy, reversible dynamics, information-based games and the idea of negative probability. We begin with a review of non-signaling distributions and convex operational theories, including “black box” descriptions of experiments and the mathematics of convex vector spaces. In Chapter 3 we derive various classical and quantum-like quasiprobabilistic representations of arbitrary non-signaling distributions. Previously, results in which the density operator is allowed to become non-positive [1] have proved useful in derivations of quantum theory from physical requirements [2]; we derive a dual result in which the measurement operators instead are allowed to become non-positive, and show that the generation of any non-signaling distribution is possible using a fixed separable state with negligible correlation. We also derive two distinct “quasi-local” models of non-signaling correlations. Chapter 4 investigates non-local games, in particular the game known as Information Causality. By analysing the probability of success in this game, we prove the conjectured tightness of a bound given in [3] concerning how well entanglement allows us to perform the task of random access coding, and introduce a quadratic bias bound which seems to capture a great deal of information about the set of quantum-achievable correlations. By reformulating Information Causality in terms of entropies, we find that a sensible measure of entropy precludes many general probabilistic theories whose non-locality is stronger than that of quantum theory. Chapter 5 explores the role that reversible transitivity (the principle that any two pure states are joined by a reversible transformation) plays as a characteristic feature of quantum theory. It has previously been shown that in Boxworld, the theory allowing for the full set of non-signaling correlations, any reversible transformation on a restricted class of composite systems is merely a composition of relabellings of measurement choices and outcomes, and permutations of subsystems [4]. We develop a tabular description of Boxworld states and effects first introduced in [5], and use this to extend this reversibility result to any composite Boxworld system in which none of the subsystems are classical.
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.