Information processing in generalized probabilistic theories

By Jonathan Barrett. 2005

765 citations recorded by the source corpus.

I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the nonunique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distribution is generic), and a no-cloning theorem. Two particular theories are then investigated in detail, for the sake of comparison with the classical and quantum cases. One of these includes states that can give rise to arbitrary nonsignaling correlations, including the superquantum correlations that have become known in the literature as nonlocal machines or Popescu-Rohrlich boxes. By investigating these correlations in the context of a theory with well-defined dynamics, I hope to make further progress with a question raised by Popescu and Rohrlich, which is why does quantum theory not allow these strongly nonlocal correlations? The existence of such correlations forces much of the dynamics in this theory to be, in a certain sense, classical, with consequences for teleportation, cryptography, and computation. I also investigate another theory in which all states are local. Finally, I raise the question of what further axiom(s) could be added to the framework in order to identify quantum theory uniquely, and hypothesize that quantum theory is optimal for computation.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/d5dfa9ddf84734d1f759ed5a0ff209d10b99736e

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Citations in the corpus

Beyond Bell's theorem: correlation scenarios (2012, background): “he same property not only holds for a pair of subsets of V, but for any number of pairwise not connected subsets. Problem3.4. For every standard Bell scenario, there is a general probabilistic theory [Bar07] which reproduces all no-signaling correlations in that scenario2. Is this also true that for”
Quantum resource theories (2018, background): “D.5, the induced resource theory here is a resource theory of nonlocality, and it involves the conversion of one bipartite channel p(a, b|x, y) (or “box”) into another p(a′, b′|x′, y′) by LOSR superchannels (Barrett, 2007; Gallego et al., 2012; Jones and Masanes, 2005; de Vicente, 2014).”
Information causality as a physical principle (2009, background): “Only a few discrepancies have been noticed: some no-signaling theories would lead to an implausible simplification of distributed computational tasks[17–20] and would exhibit very limited dynamics[21].”
Convex Spaces I: Definition and Examples (2009, background): “he information states has the mathematical structure of convex combinations, which correspond to statistical superpositions of ensembles. This is the framework known as general probabilistic theories [Bar06], where the set of information states is taken to be a convex subset of a vector space. Howev”
Reformulating and Reconstructing Quantum Theory (2011, background): “Recent treatments and developments of this framework can be found in [36], [6], [5], [14, 13], [40, 42, 41, 44].”
Theory-independent limits on correlations from generalized Bayesian networks (2014, background): “What limits the correlations in a general probabilistic theory? In the Bell scenario, a general probabilistic theory is limited only by the no-signalling principle (see for example [4]).”
A derivation of quantum theory from physical requirements (2010, background): “The redundant component ψ in (1) allows to extend T to a linear map T : R → R [3, 6].”
The sheaf-theoretic structure of non-locality and contextuality (2011, neutral): “This characteristic is shared to some extent by other foundational approaches, such as generalized probabilistic theories [49], but these formalisms are still rather closer to that of quantum mechanics, and indeed have been directly suggested by it.”
Higher-order interference and single-system postulates characterizing quantum theory (2014, background): “For example, we could postulate the existence of a tradeoff between information gained in a measurement and disturbance to the measured state [70], or the existence of at least one state that has two distinct convex decompositions into pure states, or the existence of interference; the existence of ”
A Combinatorial Approach to Nonlocality and Contextuality (2012, background): “This notation is supposed to suggest the reading ‘general probabilistic’ in the sense of general probabilistic theories [12].”