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C.S.Unnikrishnan
In 2004, C.S.Unnikrishnan of the Tata Institute of Fundamental Research in Mumbai, India proposed that the conservation law of angular momentum can correlate measurements of entangled electrons, explaining the perfect correlations of entangled particles, without the faster-than-light interactions-at-a-distance or "hidden viable" often invoked to explain nonlocaiity. Unnikrishnan wrote Bell’s inequalities can be obeyed only by violating a conservation law. Unnikrishnan argues that conservation of angular momentum (electron spin) produces the same perfect correlations (or anti-correlations) found in all Bell test experiments when both experimenters measure at the same (pre-agreed upon) measurement angle. Unnikrishnan is concerned that "For individual measurements of the two-point correlation, the conservation law cannot be invoked, since only the conditional probabilities are predicted by quantum mechanics." He uses instead averages of measurements. Apparently Unnikrishnan's concern is that individual measurements will have random outcomes of up-down, down-up, and even some up-up and down-down, since quantum mechanics predicts only probabilities for each electron. In an important article written before Bell's Theorem paper, Eugene Wigner in 1963 cited the conservation of linear momentum (for the EPR paper) and conservation of angular momentum (for David Bohm's 1952 version of nonlocality with electron spins). Wigner wrote If a measurement of the momentum of one of the particles is carried out — the possibility of this is never questioned — and gives the result Conservation laws are the consequence of extremely deep properties of nature that arise from simple considerations of symmetry. We regard these laws as "cosmological principles." Physical laws do not depend on the absolute place and time of experiments, nor their particular direction in space. Conservation of linear momentum depends on the translation invariance of physical systems, conservation of energy the independence of time, and conservation of angular momentum the invariance under rotations. Conservation laws are the consequence of symmetries, as explained by Emmy Noether. The Bohm version of the EPR experiment starts with two electrons (or photons) prepared in an entangled state that is a mixture of two-particle states, each of which conserves the total angular momentum and, of course, conserves the linear momentum as in Einstein's original EPR example. This information about the linear and angular momenta is established by the initial state preparation.
Quantum mechanics describes the probability amplitude wave function superposition of two-particle states. It is not a product of single-particle states, and there is no information about the identical indistinguishable electrons traveling along distinguishable paths. With slightly different notation, we can write equation (1) as
Ψ = 1/√2) | _{12}1 > + 1/√2) | _{+}2_{-}1 > (2)
_{-}2_{+}
The probability amplitude wave function t observer A finds an electron (e_{0}_{1}) with spin up.
At the time of this "first" measurement, by observer A or B, new information comes into existence telling us that the wave function 1 > _{+}2_{-}(or into | 1 >). Just as in the two-slit experiment, probabilities have now become certainties, one possibility is now an actuality. If the first measurement finds a particular component of electron 1 spin is up, so the same spin component of entangled electron 2 must be down to conserve angular momentum.
_{-}2_{+}
And conservation of linear momentum tells us that at
If the measurement finds an electron (call it electron 1) as spin-up, then at that moment of new information creation, the two-particle wave function collapses to the state | determined to be spin down.
Notice that Einstein's intuition that the result seems already "determined" or "fixed" before the second measurement is in fact correct. The result is determined by the law of conservation of momentum.
Note the quantum mechanics claim that the particular spin values did not exist is correct. Which of the two-particle quantum states | Note also that before the measurement the two-particle wave function was rotationally symmetric, with no preferred angular direction. The preferred angle comes into existence as a result of what Werner Heisenberg called the "free choice" of the experimenter.
This choice of measurement angle breaks the rotational symmetry of the two-particle wave function. As Erwin Schrödinger described it to Einstein in his 1935 response to the EPR paper, the measurement disentangles the particles and projects the pure-state superposition into a mixed-state product of single-particle wave functions, either
So Unnikrishnan need have no concern that measurement outcomes did not exist before the measurements. They do not. But the |