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Disentanglement and the Bell Inequalities
In the year following the Einstein-Podsky-Rosen paper, Erwin Schrödinger looked more carefully at Einstein's "separability" assumption ( Years ago I pointed out that when two systems separate far enough to make it possible to experiment on one of them without interfering with the other, they are bound to pass, during the process of separation, through stages which were beyond the range of quantum mechanics as it stood then. For it seems hard to imagine a complete separation, whilst the systems are still so close to each other, that, from the classical point of view, their interaction could still be described as an unretarded Schrödinger says that the entangled system may become disentangled (Einstein's separation) and yet some perfect correlations between later measurements might remain. Note that the entangled system could simply decohere as a result of interactions with the environment, as proposed by decoherence theorists. The perfectly correlated results of Bell-inequality experiments might nevertheless be preserved, depending on the interaction.
Schrödinger tells us that the two-particle wave function Ψ
Bell's Theorem
Following David Bohm's version of EPR, John Bell considered two spin-1/2 particles in an entangled state with total spin zero.
| ψ > = (1/√2) | + - > - (1/√2) | - + > (1)
This is a superposition of two states, either of which conserves total spin zero. The minus sign ensures the state is anti-symmetric, changing sign under interchange of identical electrons. The coefficients 1/√2, when squared, give us the 1/2 probability of finding either state.
Schrödinger does not mention
Let's assume that Alice makes a measurement of a spin component of one particle, say the x-component. First, her measurement projects the entangled system randomly into either the If Alice measures the x-component and finds spin up, equation 1 becomes
| ψ > = | + - > (2)
Using Schrödinger's expansion of the two-particle wave function in products of single-particle wave functions, we can further write
| ψ > = | + >
_{x} | - >_{x} (3)
Schrödinger describes the entangled system as having separated,
The two particles continue to evolve apart. But now we can say that future measurements of Alice's particle are
But Bob will find his particle spin down with certainty
Bell's inequality was a study of how these perfect correlations decrease as a function of the angle between measurements by Alice and Bob. Bell predicted local hidden variables would produce a
Bell wrote that "Since we can predict in advance the result of measuring any chosen component of
David Bohm has shown that these values were not predetermined (they did not even exist according to the Copenhagen Interpretation)
According to Paul Dirac, Alice's Many commentators on Bell's theorem claim that Alice and Bob's spin component values can not pre-exist their measurements, certainly not (as Bohm showed) from their initial preparation in the total spin-zero entangled state (1). In that case, spin component values would have to pre-exist in all three dimensions, they say.
In his landmark 1985 article "Is the moon there
when nobody looks? Reality and the quantum theory," David Mermin wrote "it [pre-existing spins] amounts to the insistence that each particle has stamped on it in advance the outcome of the measurements of three different spin components corresponding to noncommuting observables
But Bohm showed that the single spin component in the x-direction is ψ and disentangles/decoheres it into Schrödinger's product of single-particle wave functions, _{12}ψ and _{1x}ψ ( | + >_{2x}_{x} and | - >_{x} ). The nonlocal collapse of the two-particle wave function conserves the total spin zero, so that Bob's now-independent x-component of spin is precisely opposite to that of Alice.
And it is that same spin x-component that Bob must measure to get the perfect anti-correlations needed for quantum cryptography keys. The other two (unmeasured) spin components have no definite values. They are still in random superpositions (their overall rotational symmetry still ensuring conservation of angular momentum).
Indeed, should Bob measure a different angle from Alice, we get the angle-dependent results of the Bell inequalities. Bell predicted a The cosine squared dependence of the intensity of light passing through crossed polarizers was discovered in 1809 by Étienne Louis Malus. It is known as the "law of Malus." In the important case where Alice and Bob measure at the same angle, the cosine of zero is 1 and the correlation is perfect. This corresponds to Paul Dirac's assertion that there are some cases where quantum mechanical experiments involve no indeterminism. Dirac mentions the case of measuring a photon passing through a vertical polarizer that has been prepared in a vertical state of polarization. The probability is unity or certainty, Dirac says. Only when the initial state and the observation are such that there is a probability unity, i.e. a certainty, for one particular result is it possible that the observation may produce no change of state... This is the origin of the quantum Zeno effect. We can visualize the angles in the Bell experiments that correspond to measuring the same angles, or directly opposite angles, which according to Dirac give equally certain outcomes. We can also show Bell's straight-line predictions for local hidden variables, the sides of the square of the local hidden variables "polytope." The certain outcomes are at the corners of the square, 0°, 90°, 180°, 270°, where Bell somewhat unrealistically predicted there would be "kinks" instead of the smooth curvature of the cosines at the corners that are predicted by quantum mechanics.
This figure shows the straight-line predictions of Bell's inequalities for local hidden variables, the cosine curves predicted by quantum mechanics and conservation of angular momentum, and the odd "kinks" at angles 0°, 90°, 180°, and 270°, in what may be called a "Popescu-Rorhlich box." The "PR Box" shows Bell’s local hidden variables prediction as four straight lines of the inner square. The circular region of quantum mechanics correlations are found outside Bell's straight lines, "violating" his inequalities. Quantum mechanics and Bell's inequalities meet at the corners, where Bell's predictions show a distinctly non-physical right-angle that Bell called a "kink." All experimental results have been found to lie along the curved quantum predictions called the "Tsirelson bound." (Tsirelson, 1980)
In 1976, Bell gave us this diagram of the "kinks" in his local hidden variables inequality. He says, Unlike the quantum correlation, which is stationary in
Violating Tsirelson's Bound?
In 1994, Sandu Popescu and Daniel Rohrlich proposed two new axioms of quantum mechanics, Popescu and Rohrlich proposed that entangled Bell states might exhibit nonlocal correlations even beyond Tsirelson's bound, a theory they called "superquantum mechanics," one more nonlocal than quantum theory. They would not only violate Bell's inequalities but also violate quantum mechanics, by going beyond Tsirelson's bound. These hypothetical nonlocal correlations, found in the outermost square, would violate quantum mechanics but not the principle of relativity, because there is no signaling. Whereas Bell's inequalities predict the sum of correlations is ≤2 for local hidden variables, and quantum mechanics predicts ≤2√2, "superquantum" correlations are ≤4. Popescu and Rohrlich asked "Where does this bound come from? It derives from the Hilbert space structure of quantum mechanics, but what does it mean?" (1994, p.382) One answer comes from C.S. Unnikrishnan (2005), "The correlation function allowed by the basic assumption of validity of conservation law is unique, and surprisingly it is identical to the quantum mechanical correlation function. Therefore, a physical system with discrete observable values can show correlations different from what is predicted by quantum mechanics only by violating a fundamental conservation law" The perfect correlations of entangled spin states are the result of the conservation of total electron spin zero. The fall-off in correlations when Alice and Bob measure at different angles is a consequence of the "law of Malus."
Popescu-Rohrlich Boxes
In classic Bell experiments, Alice and Bob make a series of measurements with outcomes that can be described with binary numbers, (1,0) = | + >|- >, (0,1) = | - >| + >, (1,1) = | +>| + >, (0,0) = | - >| - >. They count the numbers of different outcomes and relate them to the probabilities of those outcomes, P(a,b|x,y), where x,y = ±1.
In the jargon of quantum nonlocality,
Nicolas Gisin (2014, p.13) and Jeffrey Bub (2016, p.89) both describe their boxes as describable by a simple equation
a
+ b = x·y
where the addition operation
Decoherence and Disentanglement?
Might decoherence by environmental interactions cause the superposition of states | + - > - | - + > (equation 3) to collapse into one of these states and then into a product of single particle states, as Schrödinger told Einstein would happen for disentanglement?
And does this also lead to the If so, conservation of angular momentum is all that is necessary for perfectly correlated quantum key distribution via entangled Bell states, without any "spooky action-at-a-distance," without any "influence" of one particle on the other at faster-than light speeds, just as Schrödinger thought in 1936. Alice's "first" measurement of a spin x-component would still be the "cause" of Bob's perfect correlation (assuming Bob measures in the same x-direction, of course). Correlations would be because the two spins were in a superposition of perfectly opposed directions before Alice's measurement, both conserving angular momentum. (Unnikrishnan, 2005) According to Asher Peres (1998), Nathan Rosen in 1931 described the normal hydrogen molecule with a two-particle wave function that he years later recognized was a Schrödinger "entangled state." Rosen wrote ψ = ψ(al)ψ(b2) + ψ(b1)ψ(a2).
Can Perfect Correlations Be Explained by Conservation Laws?
David Bohm, Eugene Wigner, and even John Bell suggested that conservation of angular momentum (or particle spin) tells us that if one spin-1/2 electron is measured up, the other must be down. Albert Einstein used conservation of linear momentum in his development of the EPR Paradox. Bohm changed from the continuous variables position and momentum to the discrete quantum variable of electron spin.
Bohm wrote in his 1952 book
The system containing the spin of two atoms has four basic wave functions, from which an arbitrary wave function can be constructed.
[Because the wave function] has definite phase relations between ψ
Bohm explained that in classical theory, the spin correlations are produced because when the atoms of the original molecule separated, each atom would continue to have every component of its spin angular momentum opposite to that of the other. He says this is Now, if the spin were a classical angular momentum variable, the interpretation of this process would be as follows: While the two atoms were together in the form of a molecule, each component of the angular momentum of each atom would have a definite value that was always opposite to that of the other, thus making the total angular momentum equal to zero. When the atoms separated, each atom would continue to have every component of its spin angular momentum opposite to that of the other. The two spin-angular-momentum vectors would therefore be correlated. These correlations were originally produced when the atoms interacted in such a way as to form a molecule of zero total spin, but after the atoms separate, the correlations are maintained by the deterministic equations of motion of each spin vector separately, which bring about conservation of each component of the separate spin-angular- momentum vectors. With his colleague, Yakir Aharonov, in 1957 Bohm reiterated his model We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore Eugene Wigner wrote in 1962
Writing a few years after Bohm, and one year before Bell, Wigner explicitly describes Einstein's work with the conservation of linear momentum as well as Bohm's conservation of angular momentum (spin) that explains perfect correlations between angular momentum (spin) components measured in the same direction
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 John Bell wrote in 1964, With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins
Where Bohm and Wigner are explicit, Bell is
Albert Einstein made the same implicit argument in 1933, shortly before EPR, though again with conservation of Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is.
Supporters of the Copenhagen Interpretation claim that the properties of the particles (like angular or linear momentum) In our case, the entangled particles have been prepared in a superposition of [Bell] states, both of which have total spin zero.
ψ = (1/√2) [ _{12}ψ]
_{+} (1) ψ_{-} (2) - ψ_{-} (1) ψ_{+} (2)
So whichever of these two states is Einstein famously maintained that the strongest theories are those built on universal principles. Surely conservation principles, which Emmy Noether showed are built on the still deeper and simpler concept of symmetry, should be a part of any basis for physics, classical or quantum.
Conclusion
As Emmy Noether showed, conservation laws arise from deeper principles of symmetry.
The perfect symmetry between the indistinguishable electrons in Bohm's entanglement experiment, the perfect symmetry of their two-particle wave function, and the change of sign of that wave function under exchange of the two identical particles, all are evidence of
They are
They are now described by the product of two single-particle wave functions. The original pure state superposition has decohered. They are now The symmetry and angular momentum conservation of entangled particles is theoretically deeper than classical and quantum physics. The theory of entanglement is confirmed by all experiments that measure spins in exactly the same direction, as Bohm, Bell, and Wigner all agree. Unlike most quantum experiments, the results are not a statistical distribution around some expectation value. Conservation is deeper than quantum mechanics.
This entanglement is the basis for generation of It is also the basis for the instantaneous "teleportation" of quantum information.
Conservation is not a
References
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