The armchair quantum physicist

If something happens in one location, it will influence the probability of something else happening elsewhere. This effect is instantaneous and works over any distance. We cannot explain this effect by some hidden information representing a shared past, but it is proof of the fundamental non-locality of Nature. Quantum mechanics predicts this non-locality, and experimenters have demonstrated it in practice.

At the same time, we live in a world where information cannot travel faster than the speed of light. If a ‘cause’ leads to an ‘effect,’ then we know there must be enough time to allow light to travel from the causes to the effect’s location. This limitation follows from Einstein’s theory of relativity.

In this post, we follow the work of Sandu Popescu and Daniel Rohrlich, who were the first to explore the relationship between non-locality and relativity. They asked what non-locality and relativistic causality together imply. Would relativity pose a limit on non-locality?

Popescu and Rohrlich identified a class of theories to which quantum mechanics belongs that yield non-local correlations while preserving relativistic causality [1]. So, non-locality and relativity do not uniquely describe quantum mechanics (and quantum mechanics is not unique in reconciling non-locality and causality). Is it possible to strengthen or extend the axioms of non-locality and causality to ultimately derive quantum mechanics? Since their original publication in 1994, this question has stirred a wide field of research, leading to new and fundamental insights until today.

Tsirelson bound

Non-locality means that there are correlations that can not be caused by the exchange of a ‘signal’ (a hypothetical signal should travel faster than light) and also not be caused by ‘local hidden information’ (because the correlations violate Bell’s inequalities) [2].

To exclude a ‘signal’ as a cause for correlation is relatively easy; we have to separate simultaneous events over a large distance. To prove that hidden information is not causing the correlation is more complicated. It took John Bell’s genius to prove that correlations caused by ‘local hidden variables’ could not exceed a specific value. Bell’s inequalities capture his conclusion. Based on his work, we now know that local hidden information cannot cause any correlation that exceeds the boundaries of these inequalities (or, in other words, violates Bell’s inequalities).

Suppose we observe correlations well separated in space and violating Bell’s inequalities. In that case, we have to conclude that these correlations are caused by non-locality and serve as evidence for the existence of non-locality. In his famous experiments, Alain Aspects (2022 Nobel Prize laureate) proved this for quantum mechanics [3].

In the 1980s, Boris Tsirelson discovered that the non-locality in quantum mechanics is also constrained below a maximum value [4]. For any classical theory, non-locality is bound by Bell’s inequality, and for quantum mechanics, non-locality cannot exceed the Tsirelson bound. For the famous CHSH correlation (a version of Bell’s inequalities), any value above 2 is incompatible with a local theory, and any value above 2√2 is incompatible with quantum mechanics.

The CHSH coin game

Relativistic causality is a principle in physics that states that any cause must precede its effect in time and that no information or influence can travel faster than the speed of light. We can apply this principle to the CHSH coin game (see our earlier post). In the game, two players each receive a cup covering a Two euro coin and a Single euro coin. They randomly chose to look at one of their coins and note whether it was heads up (result ‘+1’) or tails up (result ‘-1’). The other coin remains covered. If they find both coins in the same orientation, they win one cent, except when player one selects the S coin and player two the T coin; in that case, they only win if the two coins have opposite orientation. If they do not win one cent, they lose one cent (so for every turn, the payout is either +1 cent or -1 cent).

This coin game is closely related to the CHSH correlation, a non-locality metric. The average amount the players win or lose is directly related to the CHSH metric:

CHSH = <SS> + <ST> + <TS> -<TT>

In every turn, the players in our game have a 25% chance of selecting any of the combinations SS, ST, TS, or TT. The payout per round is then 25% of the CHSH correlation.

• For a local theory, the CHSH correlation should be two or less. The highest possible gain for the players in our coin game would be 0.50 cents per turn.
• We also know that the CHSH correlation cannot exceed 2√2 for quantum mechanics (the Tsirelson bound). The highest payout for our players would be around 0.70 cents per turn.
• The maximum conceivable payout would be 1 cent per round, equivalent to the CHSH value 4. This correlation is beyond quantum mechanics, and Popescu and Rohrlich’s article centers on whether relativistic causality would permit these correlations.

Relativistic causality

What is the limitation posed by relativistic causality on the outcome of this game? The probability that one player finds a coin heads up or tails up should be independent of whether the other player has chosen to look at the T or the S coin. Let us write as P(xy|ab) the probability for player 1 to find result ‘x’ and player 2 to find result ‘y’, given that player one selected coin ‘a’ and player two selected coin ‘b.’ So x and y represent the results for the players (either ‘+1’ or ‘-1’), and a and b represent their choices (to measure either the T coin or the S coin).

Relativistic causality requires that the probability for player one to find his S coin heads up is the same when player two selects the S or T. So, P(1?|SS) = P(1?|ST). Here, ‘?’ indicates that it does not matter whether player 2 finds his coin heads up or tails up for this statement. Analoguous requirements are P(-1?|SS) = P(-1?|ST), P(1?|TS) = P(1?|TT), and P(-1?|TS) = P(-1?|TT). So, the choice made by the second player does not influence the probabilities for the first player.

We can propose a scenario where:

P(-1,-1|SS) = P(1,1|SS) = 1/2

P(-1,-1|TS) = P(1,1|TS) = 1/2

P(-1,-1|ST) = P(1,1|ST) = 1/2

and where

P(-1,1|TT) = P(1,-1|TT) = 1/2

with all other probabilities equal to zero.

This scenario is fully compatible with relativistic causality. The payout for the players would be one cent per round. They would always win and never lose. If we calculate <SS> + <ST> + <TS> -<TT>, we find a value of 4, which is clearly above the Tsirelson bound. The correlations are allowed by the theory of relativity but outside the domain of quantum mechanics.

Thus, as Popescu and Rohrlich originally stated, “relativistic causality does not by itself constrain the maximum CHSH sum of quantum correlations” [2].

Playing dice

The combination of non-locality and causality means that our world, on the most fundamental level, has to be random. Imagine a player could control the orientation of his coins in the CHSH coin game. With the Popescu — Rohrlich correlations, the players could transmit information by selecting the orientation for their coins. As the correlation between the coins is instantaneous, the communication would not be limited by the speed of light and violate the constraints of relativity theory.

We accept that our world is non-local and that communication faster than the speed of light is impossible. We must also acknowledge that Nature is indeterministic at the most fundamental level. In Einstein’s words, we have to accept that God does play dice.

Conclusion

We see that the fundamental ‘axioms’ in physics are related and that one is needed to make the other possible. If we start with non-locality and relativistic causality, it automatically follows that Nature is indeterministic. But if we try to explain why Nature limits non-locality, we cannot find the answer in relativistic causality.

Popescu and Rohrlich have identified a class of non-local theories that comply with causality. Quantum mechanics is just one of these theories. Some of these theories go beyond quantum mechanics; they provide what we could call “superquantum” mechanics.

Is there anything that sets quantum mechanics apart from the other non-local theories? Is there any reason why “superquantumness” does not occur? These are the questions triggered by Popescu and Rohrlich in 1994. In upcoming posts, we will address how science has tried to answer these questions and to what extent scientists have been successful. In the meantime, please leave your comments.

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[1] S. Popescu, D. Rohrlich, “Quantum non-locality as an axiom,” Found Phys 24, 379 (1994).

[2] V. Scarani,” Feats, Features and Failures of the PR‐box,” AIP Conf. Proc. 844. 309 (2006).

[3] Aspect, J. Dalibard and G. Roger, “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Phys. Rev. Lett. 49, 1804 (1982).

[4] B. Tirelson, “Quantum generalizations of Bell’s inequality,” Letters in Mathematical Physics 4, 93, (1980).