## Einstein's reality criterion

Sorry for the delay, this week I want to have a short post continuing to showcase the results of late Asher Peres which unfortunately are not well known and this is a shame.

In the famous EPR paper, Einstein introduced a reality criterion:

"If, without in any way disturbing a system, we can predict with certainty ... the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity"

Now it is generally accepted that the difference between classical and quantum mechanics is noncomutativity. While there are some subtle points to be made about this assertion (by the mathematical community), from the physics point of view the distinction is rock solid and we can build in confidence upon it.

Now consider again the  EPR-B experiment with its singlet state. Suppose the x and y components of the spin exists independent of measurement and let's call the measurement values: $$m_{1x}, m_{1y}, m_{2x}, m_{2y}$$. From experimental results we know:

$$m_{1x} = - m_{2x}$$
and
$$m_{1y} = - m_{2y}$$

And now for the singlet state $$\psi$$ let's compute:

$$(\sigma_{1x} \sigma_{2y} + \sigma_{1y} \sigma_{2x})\psi$$

which turns out to be zero. The beauty of this is that $$\sigma_{1x} \sigma_{2y}$$ commutes with $$\sigma_{1y} \sigma_{2x}$$ and by Einstein's reality criterion extended to commuting operators it implies that $$m_{1x} m_{2y} = - m_{1y} m_{2x}$$ which contradicts $$m_{1x} = - m_{2x}$$ and $$m_{1y} = - m_{2y}$$

This contradiction is in the same vein as the GHZ result, but it is not well known. The catch of this result is that measuring S1xS2y cannot be done at the same time with a measurement of  S1yS2x and so we are reasoning counterfactually. However, counterfactual reasoning is allowed in a noncontextual setting (in classical mechanics and in quantum mechanics for commuting operators) and the result is valid.

1. Florin,

You keep repeating the false assertion that classical mechanics is non-contextual. No chance you will ever abandon the billiard-ball caricature.

Classical electromagnetism (and all field theories for that matter) is a contextual theory. A different settings of the measurement instrument implies a different distribution of its internal particles which produce different electric and magnetic fields.

So, m1x equals -m2x only when you measure m1x and m2x.

If you measure m1x and m2y it is not necessary that m1x = -m2x.

I am still waiting for your local, non-realist explanation of this experiment.

Andrei

1. Andrei, you are confusing non-contextuality with initial conditions. Non-contextuality means the following: suppose you measure in experiment 1 the physical variables A and B, and suppose you measure in experiment 2 the physical variables B and C. Noncontextuality means that the the statistical model for B is the same in both experiment 1 and 2.

2. Florin, I don't make any confusion. In order to measure A and B you need a particular instrument setup, say AB. In order to measure B and C you need another setup, say BC. There needs to be a difference between those setups (different orientation of some magnets, or different instruments altogether), and that difference translates microscopically in a different distribution of the atoms that make up those instruments. But a different distribution of those atoms implies a different distribution of the electrons and quarks those atoms are made of and therefore a different electric and magnetic field acting on the measured particles.

So, you cannot assume that B measured in setup AB has to be the same as B measured in setup BC because the forces acting on B in setup AB are different that the forces acting on B in setup BC.

A very intuitive analogy can be found in Couder's experiments. Say you measure the momentum of an oil drop with a certain "instrument" that is inserted in the oil bath. The wave patterns on the oil surface that guide the drop are influenced by the geometry of the bath including the presence of the "instrument". Say that for the position 1 of the instrument you get a momentum P1. If you modify the position of the instrument (2) you will alter the wave patterns and therefore the momentum of the drop (P2). So, even if the oil drop has an objective, well-defined momentum for both measurements (1 and 2) the measured momentum, P1 and P2 are not identical, because of the influence of the instrument's position on the waves guiding the particle. This is the phenomenon behind the quantum contextuality that has been correctly pointed out by Bohr.

The above explanation is not specific to fluid mechanics (also a field theory) but can be applied to all field theories, including electromagnetism, GR, etc.

Andrei

3. Andrei,

OK, there is different confusion: boundary conditions vs. contextuality. Just because different cavities have different normal modes of vibration for example, this does nor mean that in classical physics the statistical model space for the vibrations is different.

4. I do not see your point. I have provided you with a pure classical example (fluid mechanics) where your counterfactual reasoning fails. I do not understand what you mean by "statistical model", but in the experiment I provided B measured with the setting AB is different than B measured in setting BC.

Still no non-realist local description of the experiment?

Andrei

2. Florin,
I thought, by reality, everyone means that "real objects should have unique properties before they are measured and (probably?) independent of the observer".Is Einstein's definition of reality different from other people's definition of reality? Also when you have time we might continue discussion of hidden variables in the previous thread. In your opinion,is the question settled once for all?

1. -Einstein's definition is sharper.
-The hidden variable issue is quite settled by now, but this is a fuzzy word meaning different things to different people. For a single particle there is a valid hidden variables model which completely reproduces the QM prediction and this model was found by Bell. The trouble starts when you consider 2 or more particles and this line of reasoning led Bell to his theorem.

3. I would view "(S1xS2y + S1yS2x)ψ = 0 implies that (m1xm2y + m1ym2x) = 0" and the counterfactual reasoning it supports as fallacious anyway but I'm not convinced that even a 'psiontologist' would allow it. Is it really okay (for them) to reason counterfactually about measurements that could not have been made (as well as ones which could've been but weren't)?

4. Florin,
I have also a doubt similar to Paul Hayes.You are probably using
"(S1xS2y + S1yS2x)ψ = 0 implies that (m1xm2y + m1ym2x) = 0". But ψ is not an eigenstate of S1xS2y and S1yS2x separately.
The last para
"The catch of this result is that measuring S1xS2y cannot be done at the same time with a measurement of S1yS2x and so we are reasoning counterfactually. However, counterfactual reasoning is allowed in a noncontextual setting (in classical mechanics and in quantum mechanics for commuting operators) and the result is valid" is the 64000 \$ question! Is there a general agreement on this in QM practitioners? I do believe in the end result though that QM is non realististic from the day wave-particle duality was discovered and there are no hidden variables. But I need some assurance!!

1. @Kashyap: What you insist on would require the angular momentum in the x and y directions be both in a basis that is simultaneously diagonalized. This does not happen.

LC

2. @Lawrence,
I was not insisting that S1x and S1y should be measureable simultaneously. I was trying to understand m1xm2y + m1ym2x = 0 without assuming this.
kashyap

5. Let me have a shared comment on the falaciusness criticism:

S1x and S2y commute and as such we can assign an element of reality m1xm2y. Similarly S1y and S2x commute and we assign an element of reality m1ym2x. Since S1xS2y commutes with S1yS2x, m1xm2y exists at the same time with m1ym2x *if we can extend the spirit of Einstein reality criterion into quantities which can be combined algebraically*. This is the key of the argument, and not the fact that S1xS2y and S1yS2x cannot be measured at the same time. So do we accept the extension of the spirit of Einstein reality criterion, or not? If yes, we have an algebraic contradiction.