Information processing in generalized probabilistic theories

Abstract

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 non-unique 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 non-signalling correlations, including the super-quantum 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 uniquely to identify quantum theory, and hypothesize that quantum theory is optimal for computation.

The Communication Complexity of Non-signaling Distributions

“…We give a simple proof for the case of binary outcomes but this carries over to the general case. This was shown independently of us, on a few occasions in different communities [RF81,FR81,KRF87,Wil92,Bar07]. Proof.…”

Reconstruction of Quantum Theory

“…They do not question the validity of quantum mechanics and do not compete with it in explaining empirical phenomena. Methodologically, as this is routinely emphasized in the opening paragraphs of articles introducing toy models, e.g., in [3], toy models focus on important physics principles, which are upheld while other possibilities are modified. This way the consequences of such and such principle are investigated, independently of other principles.…”

“…We learn that the continuous state space, the existence of a Bell theorem, or contextuality, all of which are absent from this toy model, go unconnected with the appearance of the phenomena that are reproduced. Analogously, the toy model known as non-local, or PR, boxes [3,46] allows non-local correlations that are strictly stronger than those allowed by quantum mechanics, while it only slightly modifies the quantum mechanical state space. One then sees that non-locality is not an exclusively quantum feature and, further, that the amount of non-locality in quantum theory is smaller than in some other theories.…”

“…Recently, a new type of information-theoretic reconstructions appeared, intentionally not aimed at deriving the whole quantum theoretic structure [1,2,3,22,46,50,52]. Christened pejoratively by their own authors, these "toy models", "fantasy quantum mechanics," or "quantum mechanics lite" employ a methodology of reconstruction that has been overlooked by the previous generations of researchers: it is now claimed as helpful to reconstruct, not the full version but only a certain part of quantum theory.…”

Limited Holism and Real-Vector-Space Quantum Theory

“…We employ here some elements of the convex probabilities framework that has been developed by various authors [7,8,9,1,10,11,12,13,14,15,16]. The state of a system A, in general, can be represented by a list of probabilities associated with a fiducial set of measurement outcomes labeled k A ∈ Ω A ,…”