A no-go theorem for models of quantum theory based on finite-speed causal influences
Stefano Pironio
UniversitΓ© Libre de Bruxelles, Belgium
Foundations of QM and relativistic space-time, Athens 2012
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measurement
outcome entanglement
Bellβs theorem quantum correlations cannot be explained without causal influence between particles
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entanglement
Bellβs theorem quantum correlations cannot be naively explained without causal influence between particles superluminal
The quantum predictions for measurements on entangled particles are independent of their space-time location.
Causal influences between particles must propagate not only at a speed π£ > π , but at a speed π£ = β !
Quantum non-locality is not a phenomenon that is continuous in (non-relativistic) space-time.
β’ Could the quantum correlations between entangled particles actually depend on their space-time location?
β’ Would it then be possible then to explain quantum non-locality as the result of causal influences propagating at a finite speed π < π£ < β?
βIt is quite possible that quantum nonlocal connections might be propagated, not at infinite speeds, but at speeds very much greater than that of light. In this case, we could expect observable deviations from the predictions of current quantum theory.β
β’ Our result: Any model explaining quantum correlations through causal influences propagating at a finite speed v, for any π£ < β, can be exploited for faster-than-light communication. even without knowledge of the underlying model / without access to any hidden physical quantities.
β’ If we do not believe in the possibility of supraluminal communication π£ = β.
β’ Reference: Bancal, Pironio, Acin, Liang, Scarani, Gisin, arXiv:1110.3795
β’ Answers a question first raised in Scarani, Gisin, Phys. Lett. A 295, 167 (2002).
Bell-type experiments
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Experiment completely characterized by π(ππ|π₯π¦)
More parties π(ππππ|π₯π¦π§π€)
π£-causal models (generalize locally causal models)
Assume preferred reference frame. In this frame, causal influences propagate at most at speed π£ > π.
π ππ π₯π¦ = β« ππ π π ππ π π₯, ππ¦ ππ(π|π¦) Reproduces the quantum correlations
π ππ π₯π¦ = β« ππ π π ππ π π₯ ππ(π|π¦) The correlations are βlocalβ: β quantum correlations
π£-causal models for more parties
β’ There is a complete chain of causal influences
π ππππ π₯π¦π§π€= β« ππ π π ππ π π₯ ππ π π¦, ππ₯ ππ π π§, ππ₯ππ¦ ππ π π€, ππ₯ππ¦ππ§
The model reproduces the quantum correlations.
β’ There is no complete chain of causal influences
π ππππ π₯π¦π§π€= β« ππ π π ππ π π₯ ππ π π¦, ππ₯ ππ π π§ ππ π π€, ππ₯ππ¦ππ§
The models does not necessarily reproduces the quantum correlations.
Experimental tests of π£-causal models
Looking for Bell violations in this configuration
Problems:
1) what is the preferred reference frame?
2) we can only lower-bound π£
π ππ π₯π¦ = β« ππ π π ππ π π₯ ππ(π|π¦)
Salart, Baas, Branciard, Gisin, Zbinden, Nature 454, 861 (2008).
π£-causal models and supraluminal no-signalling
Even if they are supraluminal causal influences, they do not necessarily need to be manifest at the observational level. E.g.:
π ππ π₯π¦ = β« ππ π π ππ π π₯, ππ¦ ππ(π|π¦) Reproduces the quantum correlations no-signalling
π£-causal models and supraluminal no-signalling for more parties
Can we have a π£-causal model such that:
β’ There is a complete chain of causal influences The model reproduces the quantum predictions.
β’ There is no complete chain of causal influences The predictions are arbitrary but satisfy supraluminal no-signalling.
Answer: NO, for any π < π£ < β
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From the definition of the model:
β’ ABD independent of C π πππ π₯π¦π§π€ = π πππ π₯π¦π€
β’ ACD independent of B π πππ π₯π¦π§π€ = π πππ π₯π§π€
β’ BC|AD is Β« local Β» π ππ π¦π§, ππ₯ ππ€ = β« πππ π ππ π π¦, ππ₯ππ€ ππ π π§ ππ₯ππ€
From no-signalling:
β’ ABC independent of D π πππ π₯π¦π§π€ = π πππ π₯π¦π§
β’ BCD independent of A π πππ π₯π¦π§π€ = π(πππ|π¦π§π€)
π ππππ π₯π¦π§π€
Aβ β’
These 5 mathematical conditions imply that a certain Bell expression π satisfies the inequality π β€ 7. If π > 7 for a π£-causal model supraluminal signalling.
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β’ The inequality π β€ 7 is violated by QT.
β’ The Bell expression π only involves de probabilities π(πππ|π₯π¦π€) and π(πππ|π₯π§π€)
β’ By assumption, πβ²(ππππ|π₯π¦π§π€) is quantum πβ² πππ π₯π¦π€ is quantum
π πππ π₯π¦π€ = πβ² πππ π₯π¦π€ π πππ π₯π¦π€ is quantum
β’ In the same way π(πππ|π₯π§π€) is quantum
π ππππ π₯π¦π§π€
Cβ β’
π£-causal models for quantum theory violate the inequality π β€ 7 in the above experiment
They are incompatible with no-signalling.
Conclusion
β’ Our result: π£-causal models for quantum theory are incompatible with supraluminal signalling for any finite π£ > π.
β’ It opens a new avenue of possibilities to test experimentally π£-causal models (without weaknesses of standard Bell tests).
β’ This result illustrates the difficulty to modify quantum physics while maintining no-signalling.
β’ If we want to keep no-signalling π£ = β that is, we must accept that quantum non-locality arises discontinuously between systems that arbitrarily distant.
Bancal, Pironio, Acin, Liang, Scarani, Gisin, arXiv:1110.3795