A no-go theorem for models of quantum theory based on finite-speed causal influences

Stefano Pironio

Université Libre de Bruxelles, Belgium

Foundations of QM and relativistic space-time, Athens 2012

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Bell’s theorem quantum correlations cannot be explained without causal influence between particles

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entanglement

Bell’s 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 𝑣 > 𝑐 , but at a speed 𝑣 = ∞ !

Quantum non-locality is not a phenomenon that is continuous in (non-relativistic) space-time.

• Could the quantum correlations between entangled particles actually depend on their space-time location?

• Would it then be possible then to explain quantum non-locality as the result of causal influences propagating at a finite speed 𝑐 < 𝑣 < ∞?

“It 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.”

• Our result: Any model explaining quantum correlations through causal influences propagating at a finite speed v, for any 𝑣 < ∞, can be exploited for faster-than-light communication. even without knowledge of the underlying model / without access to any hidden physical quantities.

• If we do not believe in the possibility of supraluminal communication 𝑣 = ∞.

• Reference: Bancal, Pironio, Acin, Liang, Scarani, Gisin, arXiv:1110.3795

• 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 𝑃(𝑎𝑏|𝑥𝑦)

More parties 𝑃(𝑎𝑏𝑐𝑑|𝑥𝑦𝑧𝑤)

𝑣-causal models (generalize locally causal models)

Assume preferred reference frame. In this frame, causal influences propagate at most at speed 𝑣 > 𝑐.

𝑃 𝑎𝑏 𝑥𝑦 = ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥, 𝑏𝑦 𝑃𝜆(𝑏|𝑦) Reproduces the quantum correlations

𝑃 𝑎𝑏 𝑥𝑦 = ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥 𝑃𝜆(𝑏|𝑦) The correlations are “local”: ≠ quantum correlations

𝑣-causal models for more parties

• There is a complete chain of causal influences

𝑃 𝑎𝑏𝑐𝑑 𝑥𝑦𝑧𝑤= ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥 𝑃𝜆 𝑏 𝑦, 𝑎𝑥 𝑃𝜆 𝑐 𝑧, 𝑎𝑥𝑏𝑦 𝑃𝜆 𝑑 𝑤, 𝑎𝑥𝑏𝑦𝑐𝑧

The model reproduces the quantum correlations.

• There is no complete chain of causal influences

𝑃 𝑎𝑏𝑐𝑑 𝑥𝑦𝑧𝑤= ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥 𝑃𝜆 𝑏 𝑦, 𝑎𝑥 𝑃𝜆 𝑐 𝑧 𝑃𝜆 𝑑 𝑤, 𝑎𝑥𝑏𝑦𝑐𝑧

The models does not necessarily reproduces the quantum correlations.

Experimental tests of 𝑣-causal models

Looking for Bell violations in this configuration

Problems:

1) what is the preferred reference frame?

2) we can only lower-bound 𝑣

𝑃 𝑎𝑏 𝑥𝑦 = ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥 𝑃𝜆(𝑏|𝑦)

Salart, Baas, Branciard, Gisin, Zbinden, Nature 454, 861 (2008).

𝑣-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.:

𝑃 𝑎𝑏 𝑥𝑦 = ∫ 𝑑𝜆 𝜌 𝜆 𝑃𝜆 𝑎 𝑥, 𝑏𝑦 𝑃𝜆(𝑏|𝑦) Reproduces the quantum correlations no-signalling

𝑣-causal models and supraluminal no-signalling for more parties

Can we have a 𝑣-causal model such that:

• There is a complete chain of causal influences The model reproduces the quantum predictions.

• There is no complete chain of causal influences The predictions are arbitrary but satisfy supraluminal no-signalling.

Answer: NO, for any 𝑐 < 𝑣 < ∞

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From the definition of the model:

• ABD independent of C 𝑃 𝑎𝑏𝑑 𝑥𝑦𝑧𝑤 = 𝑃 𝑎𝑏𝑑 𝑥𝑦𝑤

• ACD independent of B 𝑃 𝑎𝑐𝑑 𝑥𝑦𝑧𝑤 = 𝑃 𝑎𝑐𝑑 𝑥𝑧𝑤

• BC|AD is « local » 𝑃 𝑏𝑐 𝑦𝑧, 𝑎𝑥 𝑑𝑤 = ∫ 𝑑𝜆𝜌 𝜆 𝑃𝜆 𝑏 𝑦, 𝑎𝑥𝑑𝑤 𝑃𝜆 𝑐 𝑧 𝑎𝑥𝑑𝑤

From no-signalling:

• ABC independent of D 𝑃 𝑎𝑏𝑐 𝑥𝑦𝑧𝑤 = 𝑃 𝑎𝑏𝑐 𝑥𝑦𝑧

• BCD independent of A 𝑃 𝑏𝑐𝑑 𝑥𝑦𝑧𝑤 = 𝑃(𝑏𝑐𝑑|𝑦𝑧𝑤)

𝑃 𝑎𝑏𝑐𝑑 𝑥𝑦𝑧𝑤

A’ •

These 5 mathematical conditions imply that a certain Bell expression 𝑆 satisfies the inequality 𝑆 ≤ 7. If 𝑆 > 7 for a 𝑣-causal model supraluminal signalling.

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• The inequality 𝑆 ≤ 7 is violated by QT.

• The Bell expression 𝑆 only involves de probabilities 𝑃(𝑎𝑏𝑑|𝑥𝑦𝑤) and 𝑃(𝑎𝑐𝑑|𝑥𝑧𝑤)

• By assumption, 𝑃′(𝑎𝑏𝑐𝑑|𝑥𝑦𝑧𝑤) is quantum 𝑃′ 𝑎𝑏𝑑 𝑥𝑦𝑤 is quantum

𝑃 𝑎𝑏𝑑 𝑥𝑦𝑤 = 𝑃′ 𝑎𝑏𝑑 𝑥𝑦𝑤 𝑃 𝑎𝑏𝑑 𝑥𝑦𝑤 is quantum

• In the same way 𝑃(𝑎𝑐𝑑|𝑥𝑧𝑤) is quantum

𝑃 𝑎𝑏𝑐𝑑 𝑥𝑦𝑧𝑤

C’ •

𝑣-causal models for quantum theory violate the inequality 𝑆 ≤ 7 in the above experiment

They are incompatible with no-signalling.

Conclusion

• Our result: 𝑣-causal models for quantum theory are incompatible with supraluminal signalling for any finite 𝑣 > 𝑐.

• It opens a new avenue of possibilities to test experimentally 𝑣-causal models (without weaknesses of standard Bell tests).

• This result illustrates the difficulty to modify quantum physics while maintining no-signalling.

• If we want to keep no-signalling 𝑣 = ∞ 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