C.S.UnnikrishnanIn 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 p, the state vector of the other particle suddenly becomes a (slightly damped) plane wave with the momentum -p. This statement is synonymous with the statement that a measurement of the momentum of the second particle would give the result -p, as follows from the conservation law for linear momentum. The same conclusion can be arrived at also by a formal calculation of the possible results of a joint measurement of the momenta of the two particles. One can go even further: instead of measuring the linear momentum of one particle, one can measure its angular momentum about a fixed axis. If this measurement yields the value mℏ, the state vector of the other particle suddenly becomes a cylindrical wave for which the same component of the angular momentum is -mℏ. This statement is again synonymous with the statement that a measurement of the said component of the angular momentum of the second particle certainly would give the value -mℏ. This can be inferred again from the conservation law of the angular momentum (which is zero for the two particles together) or by means of a formal analysis.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 Ψ12 of the two-particle system as in a 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
Ψ12 = 1/√2) | 1+2- > + 1/√2) | 1-2+ > (2)The probability amplitude wave function Ψ12 travels away from the source (at the speed of light or less). Let's assume that at t0 observer A finds an electron (e1) 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 Ψ12 has "collapsed" into the state | 1+2- >
(or into | 1-2+ >). 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. And conservation of linear momentum tells us that at t0 the second electron is equidistant from the source in the opposite direction. As with any wave-function "collapse", the probability amplitude information changes (it does not "travel" anywhere). Nothing really "collapses." Nothing is moving. Only information is changing. 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 | + - > and electron 2 "jumps" into a spin-down state with probability unity (certainty). The results of observer B's measurement at a later time t1 is therefore 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 | + - > or | - + > occurs is completely random. It is the result of "Nature's choice," as Paul Dirac described it. 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 + - > or | - + >. So Unnikrishnan need have no concern that measurement outcomes did not exist before the measurements. They do not. But the joint property of conserved total spin zero is true for either + - > or | - + >.