# Local Tomography and the Jordan Structure of Quantum Theory

## Abstract

Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.

## Keywords

General probabilistic theories Local tomography Jordan algebras## Notes

### Acknowledgments

We thank C. M. Edwards for drawing our attention to Hanche-Olsen’s paper. Part of this work was done while the authors were guests of the Oxford University Computing Laboratory, whose hospitality is also gratefully acknowledged. H. B. thanks the Foundational Questions Institute (FQXi) for travel support for the visit. Additional work was done at the Perimeter Institute for Theoretical Physics; work at Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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