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Psi-Epistemic Theories: The Role of Symmetry · Around

2013

Psi-Epistemic Theories: The Role of Symmetry

Aaronson, Scott, Bouland, Adam, Chua, Lynn et al.

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Formalizing an old desire of Einstein, "psi-epistemic theories" try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states." Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out, by proving no-go theorems analogous to the Bell Inequality.

In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant psi-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic psi-epistemic theory must be "measurement contextual" in dimensions 3 and higher.
Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under tensor product." In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that satisfy two reasonable conditions: (1) symmetry under unitary transformations, and (2) "maximum nontriviality" (meaning that the probability distributions corresponding to any two non-orthogonal states overlap).
This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries.

Built on

S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. Mech. 17
1967
Earlier work this paper cites.
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics,
1987
Earlier work this paper cites.
T. Rudolph, “Ontological models for quantum mechanics and the Kochen-Specker theorem,” (2006), arXiv: quant-ph/0608120v1
2006
Earlier work this paper cites.
N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” (2007), arXiv: quant-ph/0709.4266v1
2007
Earlier work this paper cites.

Similar

N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states”, Found. Phys. 40, 125 (2010), arXiv:0706.2661
Original
2010
Cited alongside, same era.
M. F. Pusey, J. Barrett and T. Rudolph, “On the reality of the quantum state,” Nature Physics 8
2012
Cited alongside, same era.
O. J. E. Maroney, “How statistical are quantum states?”� (2012), arXiv:1207.6906
Original
2012
Cited alongside, same era.
M. S. Leifer and O. J. E. Maroney, “Maximally epistemic interpretations of the quantum state and contextuality” (2012), arXiv:1208.5132v1
Original
2012
Cited alongside, same era.
http://mathoverflow.net/questions/95537/psi-epistemic-theories-in-3-or-more-dimensions
Cited in the paper.
http://mathoverflow.net/questions/108507/a-kakeya-like-problem-must-a-union-of-annuli-fill-the-plane
Cited in the paper.

Then

L. Hardy, “Are quantum states real?” (2012), arXiv:1205.1439
Original
2012
Later among the works it cites.
M. K. Patra, S. Pironio and S. Massar. “No-go theorems for ψ \psi -epistemic models based on a continuity assumption” (2012), arXiv:1211.1179v2
Original
2012
Later among the works it cites.
P. G. Lewis, D. Jennings, J. Barrett and T. Rudolph, “The quantum state can be interpreted statistically,” (2012), arXiv: quant-ph/1201.6554v1
2012
Later among the works it cites.

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