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derivations (or reconstructions") of nite-dimensional quantum mechanics [1]. These rest of some strong assumptions. In particular, most assume the state of a composite system is entirely determined by the joint probabilities it assigns to the outcomes of measurements on the component systems. This condition, often called local tomography, is satis ed by neither real nor quaternionic quantum theory. If we are interested in the possibilities for theories more general than orthodox quantum mechanics, we might wish to relax this constraint.

In this talk I will discuss a weaker system of assumptions, involving correla-

tions between a probabilistic system and an isomorphic conjugate system, that leads to a representation of such systems in terms of euclidean Jordan algebras [2]. These have a well-known classi cation as direct sums of real, complex or quaternionic quantum systems, possibly the exceptional Jordan algebra, and spin factors. The last are a form of bit", characterized by a family of two-valued observables, parametrized by antipodal vectors in a sphere of arbitrary dimension. Orthodox quantum mechanics can be singled out by imposing local tomography, plus the existence of a qubit as additional axioms [3]. However, there is a natural way to form non-signaling, but generally non-locally tomo-graphic, composites of systems based on special euclidean Jordan algebras (that is, excluding the exceptional one). This yields a probabilistic theory strictly, but not wildly, more general than orthodox nite-dimensional QM; one that elegantly uni es real, complex and quaternionic quantum theory, has a simple operational basis, and allows for a spectrum of bits more general than permitted in orthodox quantum theory. Parts of this talk reflect ongoing joint work with Howard Barnum, Matthew Graydon and Cozmin Ududec.

References:

[1] G. Chiribella, M. D'Ariano and P. Perinotti, Phys. Rev. A 84, (2011), 012311-

012350; B. Dakic and C. Brukner, in H. Halvorson (ed.), Deep Beauty, Cambridge,

2011; P. Goyal, Phys. Rev. A 78 (2008), 052120-052146; L. Hardy, arXiv:quant-

ph/0101012 (2001); L. Masanes and M. Mueller, New J. Phys. 13 (2011) ; J. Rau,

Annals of Physics 324 (2009) 2622{2637;

[2] A. Wilce, Conjugates, correlation and quantum mechanics, arXiv:1206.2897 (2012)

[3] H. Barnum and A. Wilce, Local tomography and the Jordan structure of quantum

theory, arXiv:1202.4513, (2012)]]>

Motivated by a histories outlook, we propose the following condition: there exists a positive semi-definite matrix in which the indices run over all possible experimental outcomes, from which the experimental probabilities arise as “marginals” in a similar way. This is a much simpler condition than the usual statement of the existence of a quantum model for the probabilities, and suggests an underlying connection with Bell’s derivation of his bound on local correlations. I will outline existing proofs that this condition places strong bounds on correlations consistent with QM, and ask whether it could completely characterise quantum non-locality

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