Uncertainty Principle Determines Nonlocality

This article was written by Florin Moldoveanu wrote on Nov. 26, 2010 in (http://www.fqxi.org/community/forum/topic/786)

In 1935, Einstein, Podolski, and Rosen published their seminal paper aimed at proving that quantum mechanics is incomplete. Quantum mechanics prohibits the simultaneous measurement of certain properties like, for example, position and momentum. But is this due to the inherently clumsiness of us, macroscopic objects? Or is this uncertainty an inherent property of nature?

Einstein, Podolski, and Rosen thought that at its core nature is deterministic and to prove it they devised a clever thought experiment: have a quantum system that splits into two parts. For example have an unstable particle initially at rest split into two identical particles flying apart in opposite directions. Then measure the position on the left particle, and the momentum on the right particle. Because of momentum conservation, the left particle should have the opposite value of momentum, and since we can measure both position and momentum (on different systems) with arbitrary precision this would violate the uncertainty principle, making quantum mechanics an incomplete theory. On the other hand, taking the point of view that quantum mechanics is indeed a complete theory would imply an unpalatable “spooky action at a distance” where correlations between two spatially separated systems would occur in a way incompatible with any local classical description as shown by John Bell.

It turns out however, that this “spooky action at a distance”, or nonlocality is really how nature behaves, and this was experimentally settled by the Aspect experiment. But there is more: Einstein, Podolski, and Rosen had wanted to prove the uncertainty principle wrong because of “unphysical” nonlocality, but in a recent Science paper, Jonathan Oppenheim and Stephanie Wehner showed––in a bit of an ironic twist––that the “spooky action at a distance” is determined in part by the uncertainty principle.

Now, why is this important? Because as much as quantum mechanics is strange and predicts many counter-intuitive phenomena, it is not as strange as it could be allowed by the no-signaling condition of relativity. In other words, there could be stronger correlations then those predicted by quantum mechanics between the two measurements, while still obeying the condition that whatever I do “over here” does not send any signal “over there.” One example of this is the so-called Popescu-Rohrlich (PR) box, a hypothetical unphysical device able to achieve the maximum correlations between two spatially separated systems. So why is a PR box not allowed by nature? A physical implementation of a PR box would be a hacker’s dream come true because it would allow unrestricted eavesdropping on over-correlated data.

But how can one reason meaningfully over unphysical situations? To what degree do we have to create “simulated realities” and how confident can we be that the conclusions one reaches are not just the author’s fantasies? Fortunately there is a clear answer: Discuss quantum mechanics and hypothetical theories in the language of information and game theory. This approach provides a platform which is guaranteed to reduce itself to quantum and classical mechanics in the appropriate limits, and also generate meaningful conclusions to all other possible physical theories. Of course, when additional requirements of standard axiomatizations of quantum mechanics are imposed (like projective geometries of Jordan algebras, for example) this continuum of potential theories reduces itself to a handful of discrete possible cases. But casting quantum mechanics in the new framework can add new insights and clarifications for old puzzles.

So how do Oppenheim and Wehner go about proving that the uncertainty principle determines nonlocality? In a nutshell, it goes like this. First, the uncertainty principle is expressed as a Shannon entropy inequality and then as a Deutsch min-entropy inequality. Then the typical Alice-Bob pair is set to play an “XOR retrieval game” proving that any violation of the Tsirelson’s bound implies a violation of the min-entropic uncertainty relations (for details, please see Oppenheim and Wehner’s paper arXiv:1004.2507v1). Now let’s dig in and explain all this.

John Bell proved that any local deterministic theory should obey what are now called Bell inequalities. Quantum mechanics violates those inequalities and Tsirelson asked something more: What is the maximal possible amount of those violations? Any larger violation of Bell’s inequality over the Tsirelson bound up to the maximal PR box correlation would represent an even spookier theory of nature.

In an XOR game, Alice and Bob are in states “s” and “t” respectively, and they generate the answers “a” and “b” (a and b are zero or one). Winning the game is determined by the XOR of a and b = a+b mod 2. The players are allowed to choose any strategy they want to maximize the chance of winning, but they cannot communicate with one another during the actual game. Classically, the best chance of winning is 3/4, but using quantum mechanics the odds can be increased up to 1/2 + 1/(2sqrt(2)). The way of doing it is by Alice performing a measurement in a preferred “eigenstate basis” and this “steers” the state of Bob to a maximally certain state. Discovered by Schrödinger, “steerability” allows Alice to influence the outcome of Bob’s experiments in a non-trivial way and, still, it does not transmit any information.

Ultimately, the best chance of winning an XOR game is given by the interplay of steerability and uncertainty relations. Intuitively, the larger the uncertainty, the worst are the odds, but this flies in the face of the common quantum mechanics intuition because the best outcome in quantum mechanics achieving the Tsirelson bound occurs precisely when incompatible measurements are chosen. But this is only an artifact of the power of steerability. Quantum mechanics already achieves maximum steerability and going above the Tsirelson bound requires LESS uncertainty (a PR box would have perfect steerability and no uncertainty). Classically, any hidden variable theory would have no uncertainty, but its steerability would be limited to the trivial case.

At this point the following problem presents itself. Quaternionic quantum mechanics can go over the Tsirelson bound and this representation does not have less uncertainty than standard quantum mechanics over the complex numbers because its physical predictions are the same. Since steerability is already at maximum in complex quantum mechanics, this means that either the Tsirelson bound is dependent on the particular number system representation of quantum mechanics, or there is another parameter at play determining nonlocality. The question is open at this time.

Oppenheim and Wehner suggest another open problem: Clarify the relationship between uncertainty and complementarity because one can imagine theories with less degrees of complementarity than quantum mechanics but with the same degree of non-locality and uncertainty. Here complementarity measures the degree to which one measurement disturbs the next measurement.

Reasoning about hypothetical theories in an information theory and game approach opens the door to counter examples and new fruitful insights which could demystify the spooky part of quantum mechanics. Two thousand years ago, people held contests of large number multiplication using abacuses. To them, the fact that someday elementary school children would perform those multiplications with ease and faster than they could ever do it, looked spooky. All this was possible because of the transition from Roman to Arabic numerals. Maybe in the distant future the only thing spooky thing about quantum mechanics would be how spooky it looked to us with our “primitive” tools like Hilbert spaces and non-commutative observables. So let’s loosen up, play some games, and achieve a better intuition about nature:

“- Billy, your computer game time is up. Go upstairs and do your homework!”

“- But Mom, I AM doing my homework. I am developing a quantum mechanics intuition right now!”

Gathered by: Sh.Barzanjeh(shabirbarzanjeh@gmail.com)