# Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories

- 207 Downloads
- 8 Citations

## Abstract

In any probabilistic theory, we say that a bipartite state *ω* on a composite system *AB* *steers* its marginal state *ω* ^{ B } if, for any decomposition of *ω* ^{ B } as a mixture *ω* ^{ B }=∑_{ i } *p* _{ i } *β* _{ i } of states *β* _{ i } on *B*, there exists an observable {*a* _{ i }} on *A* such that the conditional states \(\omega_{B|a_{i}}\) are exactly the states *β* _{ i }. This is always so for pure bipartite states in quantum mechanics, a fact first observed by Schrödinger in 1935. Here, we show that, for *weakly* self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system *A* is steered by some bipartite state of a composite *AA* consisting of two copies of *A*, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical.

## Keywords

Probabilistic theories Non-signaling states Steering Self-duality## Notes

### Acknowledgements

This research was supported by the United States Government through grant OUR-0754079 from the National Science Foundation. It was also supported by 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. H. Barnum and A. Wilce also wish to thank: Samson Abramsky and Bob Coecke for extending them the hospitality of the Oxford University Computing Laboratory during November of 2009, when parts of this paper were written; C. Martin Edwards for referring us to the work of H. Hanche-Olsen; the Foundational Questions Institute and the University of Cambridge’s DAMTP for sponsoring the workshop Operational Probabilistic Theories as Foils to Quantum Theory, Cambridge (UK), July 2007, where some of these ideas were initially presented and their development into the present paper begun. Further work was done at other workshops and conferences: we thank Renato Renner, Oscar Dahlsten, and the Pauli Center for Theoretical Studies and the initiative “Quantum Science and Technology” at ETH Zürich, for sponsoring the workshop “Information Primitives and Laws of Physics”, March, 2008; Jeffrey Bub, Robert Rynasiewicz, and the University of Maryland, College Park and Georgetown University for organizing and sponsoring New Directions in the Foundations of Physics, May 2008; Hans Briegel, Bob Coecke, and the other organizers and the European network QICS, the EPSRC, and the IQOQI of the Austrian Academy of Sciences for the workshop “Foundational Structures for Quantum Information and Computation”, September 2008.

## References

- 1.Araki, H.: On a characterization of the state space of quantum mechanics. Commun. Math. Phys.
**75**, 1–24 (1980) MathSciNetADSCrossRefzbMATHGoogle Scholar - 2.Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Cloning and broadcasting in generic probabilistic models (2006). arXiv:quant-ph/0611295
- 3.Barnum, H., Barrett, J., Leifer, M., Wilce, A.: A general no-cloning theorem. Phys. Rev. Lett.
**99**, 240501 (2007). arXiv:0707.0620 ADSCrossRefGoogle Scholar - 4.Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Teleportation in general probabilistic theories. In: Abramsky, S., Milsove, M. (eds.) Mathematical Foundations of Information Flow. Proceedings of Symposia in Applied Mathematics, vol. 71. Am. Math. Soc., Providence (2012). arXiv:0805.3553 Google Scholar
- 5.Barnum, H., Barrett, J., Clark, L., Leifer, M., Spekkens, R., Stepanik, N., Wilce, A., Wilke, R.: Entropy and information causality in non-signaling theories. New J. Phys.
**12**, 038824 (2010). arXiv:0909.5075 MathSciNetCrossRefGoogle Scholar - 6.Barnum, H., Dahlsten, O., Leifer, M., Toner, B.: Nonclassicality without entanglement enables bit committment. In: IEEE Information Theory Workshop, Porto (2008). arXiv:0803.1264 Google Scholar
- 7.Barnum, H., Wilce, A.: Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci.
**270**, 3–15 (2011). arXiv:0908.2352 CrossRefGoogle Scholar - 8.Barnum, H., Wilce, A.: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory (2009). arXiv:0908.2354
- 9.Barnum, H., Wilce, A.: Local tomography and the Jordan structure of quantum theory (2012). arXiv:1202.4513
- 10.Barrett, J.: Information processing in general probabilistic theories. Phys. Rev. A
**75**, 032304 (2007). arXiv:quant-ph/0508211 ADSCrossRefGoogle Scholar - 11.Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Phys. Rev. A
**81**, 062348 (2010). arXiv:0908.1583 ADSCrossRefGoogle Scholar - 12.Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers Systems and Signal Processing, Bangalore India, pp. 175–179 (1984) Google Scholar
- 13.Dakic, B., Brukner, C.: Quantum theory and beyond: is entanglement special? (2009). arXiv:0911.0695
- 14.Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys.
**17**, 239–260 (1970) MathSciNetADSCrossRefzbMATHGoogle Scholar - 15.Edwards, C.M.: The operational approach to quantum probability I. Commun. Math. Phys.
**17**, 207–230 (1971) Google Scholar - 16.Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, London (1994) zbMATHGoogle Scholar
- 17.Foulis, D.J., Randall, C.H.: Empirical logic and tensor products. In: Neumann, H. (ed.) Interpretations and Foundations of Quantum Mechanics. Wissenschaftsverlag, Mannheim (1981) Google Scholar
- 18.Hanche-Olsen, H.: In: Jordan Algebras with Tensor Products Are
*C*^{∗}-Algebras. Springer Lecture Notes in Mathematics, vol. 1132 (1985) Google Scholar - 19.Hadjisaavas, N.: Properties of mixtures of non-orthogonal quantum states. Lett. Math. Phys.
**5**, 327–332 (1981) MathSciNetADSCrossRefGoogle Scholar - 20.Hardy, L.: Quantum theory from five reasonable axioms. J. Phys. A
**40**, 3081 (2007). arXiv:quant-ph/0101012. See also, Hardy, L.: A framework for probabilistic theories with non-fixed causal structure. J. Phys. A.**40**, 3081 (2007) MathSciNetADSCrossRefzbMATHGoogle Scholar - 21.Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A
**183**, 14–18 (1993) MathSciNetADSCrossRefGoogle Scholar - 22.Kläy, M.: Einstein-Podolski-Rosen Experiments: the structure of the sample space. Found. Phys. Lett.
**1**, 205–244 (1988) MathSciNetCrossRefGoogle Scholar - 23.Kläy, M., Randall, C.H., Foulis, D.J.: Tensor products and probability weights. Int. J. Theor. Phys.
**26**, 199–219 (1987) CrossRefzbMATHGoogle Scholar - 24.Koecher, M.: Die geoodätischen von positivitaätsbereichen. Math. Ann.
**135**, 192–202 (1958) MathSciNetCrossRefzbMATHGoogle Scholar - 25.Koecher, M.: Jordan Algebras and Their Applications Univ. Minnesota Lecture Notes, Minneapolis (1962) Reprinted as Krieger, A., Walcher, S.: The Minnesota Notes on Jordan Algebras and their Applications. Lecture Notes in Mathematics, vol. 1710. Springer, Berlin (1999) zbMATHGoogle Scholar
- 26.Lo, H.K., Chau, H.F.: Is quantum bit commitment really possible? Phys. Rev. Lett.
**78**, 3410–3413 (1997) ADSCrossRefGoogle Scholar - 27.Mackey, G.: Mathematical Foundations of Quantum Mechanics. Addison-Wesley, Reading (1963) zbMATHGoogle Scholar
- 28.Mayers, D.: Unconditionally secure quantum bit commitment is impossible. Phys. Rev. Lett.
**78**, 3414 (1997). arXiv:quant-ph/9605044 ADSCrossRefGoogle Scholar - 29.Pawlowski, M., Paterek, T., Kazlikowski, D., Scarani, V., Winter, A., Żukowski, M., et al.: A new physical principle: information causality. Nature
**461**, 1101–1104 (2009). arXiv:0905.2292 ADSCrossRefGoogle Scholar - 30.Satake, I.: Algebraic Structures of Symmetric Domains. Publications of the Mathematical Society of Japan, vol. 14. Iwanami Shoten/Princeton University Press, Tokyo/Princeton (1980) zbMATHGoogle Scholar
- 31.Schrödinger, E.: Probability relations between separated systems. Proc. Camb. Philos. Soc.
**32**, 446–452 (1936) ADSCrossRefGoogle Scholar - 32.Vinberg, E.B.: Homogeneous cones. Dokl. Akad. Nauk SSSR
**141**, 270–273 (1960). English transl., Sov. Math. Dokl.**2**, 1416–1619 (1961) MathSciNetGoogle Scholar - 33.Vinberg, E.B.: The theory of convex homogeneous cones. Tr. Mosk. Mat. Obŝ.
**12**, 303–358 (1963). English transl., Trans. Mosc. Math. Soc. 340–403 (1963) MathSciNetGoogle Scholar - 34.Wilce, A.: Tensor products in generalized measure theory. Int. J. Theor. Phys.
**31**, 1915 (1992) MathSciNetCrossRefzbMATHGoogle Scholar - 35.Wilce, A.: Four and a half axioms for quantum mechanics. In: Ben-Menahem, Y., Hemmo, M. (eds.) Probability in Physics. Springer, Berlin (2012) Google Scholar