Schrödinger’s Paradox and Proofs of Nonlocality Using Only Perfect Correlations

Abstract

Dedicated to our mentor, Joel L. Lebowitz, a master of statistical physics who, when it came to foundational issues of quantum mechanics, was always willing to listen. AbstractWe discuss proofs of nonlocality based on a generalization by Erwin Schrödinger of the argument of Einstein, Podolsky and Rosen. These proofs do not appeal in any way to Bell's inequalities. Indeed, one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations. First we explain that Schrödinger's argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems implies the existence of a non-contextual value-map for quantum observables; non-contextual means that the observable has a particular value before its measurement, for any given quantum system, and that any experiment "measuring this observable" will reveal that value. Then, we establish the impossibility of a non-contextual value-map for quantum observables without invoking any further quantum predictions. Combining this with Schrödinger's argument implies nonlocality. Finally, we illustrate how Bohmian mechanics is compatible with the impossibility of a non-contextual value-map.

EPR-Bell-Schrödinger proof of nonlocality using position and momentum

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Based on his extension of the classical argument of Einstein, Podolsky and Rosen, Schrödinger observed that, in certain quantum states associated with pairs of particles that can be far away from one another, the result of the measurement of an observable associated with one particle is perfectly correlated with the result of the measurement of another observable associated with the other particle. Combining this with the assumption of locality and some "no hidden variables" theorems, we showed in a previous paper [11] that this yields a contradiction. This means that the assumption of locality is false, and thus provides us with another demonstration of quantum nonlocality that does not involve Bell's (or any other) inequalities.

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“…In [11], following [24,25], we explained that, if one assumes locality, meaning that there is no effect whatsoever on the state of the second particle due to a measurement carried out on the first particle (when both particles are sufficiently spatially separated), there must exist what we call a "non-contextual value map" v which assigns to each observable A a value v(A) that pre-exists its measurement and is simply revealed by it.…”

“…This means that the assumption of locality is false, and thus provides us with another demonstration of quantum nonlocality that does not involve Bell's (or any other) inequalities. In [11] we introduced only "spin-like" observables acting on finite dimensional Hilbert spaces. Here we will give a similar argument using the variables originally used by Einstein, Podolsky and Rosen, namely position and momentum.…”

The outcomes of measurements in the de Broglie–Bohm theory

“…Depending on the spin state of the atom, one of the photons is absorbed and scattered in all directions, while the atom undergoes a recoil that pushes it either upwards of downwards. The other non absorbed photon is detected by either detector D 1 or D 2 , which provides a 1 The relation between this initial value and the result should then be contextual, as emphasized for instance in Refs [9,15,16].…”

“…where Z stands for Z p or Z n . The time evolution of the positions Z n is given by an expression similar to (16), where ∇ξ + (Z p , t ) is replaced by ∇ξ 0 (Z n , t ) and ∇ξ 0 (Z p , t ) by ∇ξ − (Z n , t ). Again, we see that the velocity of each pointer particle is a weighted average between the velocities associated with two wave packets, but for the pointers one of the wave packets is static.…”