Constraints on quantum mechanics()

Daniel Rohrlich

The axioms of quantum mechanics are well known, but most of them are abstract and mathematical. They are not clear physical statements like the two axioms of special relativity, defining a maximal signalling speed (the speed of light) and a fundamental symmetry (Lorentz invariance). Can we derive quantum mechanics from clear physical statements? Or, if we are not there yet, can we at least formulate clear physical statements that constrain quantum mechanics and any generalization of quantum mechanics?

Suprisingly - to say the least - the assumption that every physical value exists before we measure it (regardless of what goes on elsewhere) does not constrain quantum mechanics. This "local realism" assumption implies Bell's inequality, and quantum mechanics violates Bell's inequality. In particular, one form of Bell's inequality says that a certain combination \( C \) of measured correlations must not exceed \( 2 \) . But in quantum mechanics, \( C \) can reach \( 2\sqrt{2} \) . This fact is called "Tsirelson's bound", and it is a theorem of quantum mechanics.

Quantum mechanics does obey the no-signalling constraint (the first axiom of special relativity): quantum correlations are useless for sending faster-than-light signals. Does Tsirelson's bound follow from this constraint? No, it does not; as Sandu Popescu and I showed in Ref. \( [1] \) , hypothetical "superquantum" correlations (now also called Popescu-Rohrlich-box ("PR-box") correlations as in Ref. \( [2] \) ) can reach \( C = 4 \) without violating the no-signalling constraint. So where does Tsirelson's bound come from?

Quantum mechanics obeys another constraint, besides no-signalling: it has a classical limit, in which Planck's constant \( h \) vanishes and all physical values are measurable. In this limit, I have found 3 that PR-box correlations do not obey the no-signalling constraint. Generalized to all stronger-than-quantum correlations, this result is a derivation of Tsirelson's bound without assuming quantum mechanics. So we can derive at least a part of quantum mechanics from the two axioms of no-signalling and a classical limit, together with the negation of local realism.

For further details, see Refs. \( [3{-}5] \) . The work of \( [4] \) continues in collaboration with Avishy Carmi and Daniel Moskovich of the Mechanical Engineering Department and the Center for Quantum Information Science and Technology at BGU.

  1. S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Found. Phys. 24 (1994) 379.
  2. J. Barrett and S. Pironio, Popescu-Rohrlich Correlations as a Unit of Nonlocality, Phys. Rev. Lett. 95, 140401 (2005).
  3. D. Rohrlich, PR-box correlations have no classical limit, in Quantum Theory: A Two-Time Success Story (Yakir Aharonov Festschrift), eds. D. C. Struppa and J. M. Tollaksen (New York: Springer), 2013, pp. 205-211.
  4. D. Rohrlich, Stronger-than-quantum bipartite correlations violate relativistic causality in the classical limit.
  5. Video of my talk at the Aharonov-80 Conference in 2012 at Chapman University.