Probability in Physics pp 281-298 | Cite as

# Four and a Half Axioms for Finite-Dimensional Quantum Probability

## Abstract

It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus *per se*. However, from this angle, the rather special *C* ^{*}-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis of which this apparatus can be reconstructed. In this paper, I explore one route to such a derivation of *finite-dimensional* quantum mechanics, by means of a set of strong, but probabilistically intelligible, axioms. Stated very informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of (basic, discrete) measurements as orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space – in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.

## Keywords

Real Hilbert Space Jordan Algebra Hermitian Operator Test Space Classical Probability Theory## Notes

### Acknowledgement

I wish to thank Howard Barnum for reading and commenting on an earlier draft of this paper, and, more especially, for introducing me to the papers of Koecher and Vinberg, on which the present exercise depends. Thanks also to C. M. Edwards for pointing out the paper [21] of Hanche-Olsen.

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