In the three editions of his first book Quantum Non-Locality and Relativity (1994-2011), Maudlin describes the EPR Paradox and Bell's Theorem in great detail, but with few of the equations and diagrams that explain Bell's inequality predictions and why they are violated by experiments, agreeing with quantum mechanics. See our John Bell page.

In his most recent book, Philosophy of Physics: Quantum Theory (2019), Maudlin returns to summarize the situation with entangled particles...

Einstein, Podolsky, and Rosen never took the possibility of such a nonlocal physical interaction between the socks (or the electrons) seriously. In fact, they thought the idea so absurd that they never imagined anyone would entertain it. What the EPR article pointed out was that to avoid such a strange “spooky action at- a-distance” (in Einstein's famous phrase), one has to postulate that the two electrons described above have definite dispositions concerning how they would react to the magnets from the moment they are produced and separate from each other. One of the electrons has to be z-spin up and the other z-spin down from the outset. Otherwise, how could either be sensitive to the behavior of the other in the right way to preserve the perfect anticorrelation?

Philosophy of Physics: Quantum Theory, p. 27.

Maudlin's statement "One of the electrons has to be z-spin up and the other z-spin down from the outset" is close, but not quite correct. The same could be said of x-spin and y-spin, but spin can not be in all three directions at the same time. The correct statement is "If z-spins are measured for the two particles, they will be found opposite." The same is true of x-spin and y-spin. But not because they had those spins "from the outset." Only that the total spin must always be zero, by conservation of total spin zero in all directions, when they are measured in any direction.

The conservation of total spin zero is a condition of the initial entanglement that we regard as a "hidden constant of the motion" or a "common cause."

In the final pages of the 2011 edition of Quantum Non-Locality and Relativity, Maudlin explained that the quantum state of a "composite system" of entangled electrons (or photons) can not be described as the product of the individual states of the two particles. In 1935 Erwin Schrödinger reacted to the EPR paper by telling Albert Einstein there was a flaw in Einstein's separability argument about the EPR Paradox. Einstein assumed that the particles could "separate" as they travel away from each other and become independent, single-particle quantum states. Schrödinger said that Einstein's "separability principle" (Trennungsprinzip) was simply wrong. The correct "wave function" that "entangles" two particles with one another (as describable by Schrödinger's wave equation) is a two-particle, interdependent wave function. Only after a measurement "collapses" the two-particle wave function can the particles be described as the product of single-particle wave functions.

Maudlin writes the spin state of the pair of entangled particles as

1/√2 | ↑_z > _L | ↓ _z > _R - 1/√2 | ↓ _z > _L | ↑_z > _R

This is not the Cartesian product of vectors in H _L and H_R, so neither electron has any definite spin.

But what if we ask about the operator which represents the product of the z-spin of particle L and the z-spin of particle R? Note that | ↑_z > _L | ↓ _z > _R is an eigenstate of the product operator with eigenvalue -1 since if the pair is in this state the left particle has z-spin +1 and the right z-spin -1. Similarly, | ↓ _z > _L | ↑_z > _R is an eigenstate of the same operator, also with eigenvalue -1 (the operator is therefore degenerate). The linearity of the operator guarantees that the linear sum 1/√2 | ↑_z > _L | ↓ _z > _R - 1/√2 | ↓ _z > _L | ↑_z > _R is also an eigenstate, with eigenvalue -1. Thus the z-spins of the particles are certain to have opposite values even though neither particle has a determinate z-spin.

Quantum Non-Locality and Relativity (2011), p. 283.

Few philosophers of science include these mathematical details of quantum mechanics as does Maudlin. We can compare his quantum mechanics math to that of David Bohm, who in 1957 first formulated the quantum mechanics of entangled particles (atoms),

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

ψ = (1/√2) [ ψ₊ (1) ψ_- (2) - ψ_- (1) ψ₊ (2) ]

where ψ₊ (1) refers to the wave function of the atomic state in which one particle (A) has spin +ℏ/2, etc. The two atoms are then separated by a method that does not influence the total spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle (A) is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A.

"Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Physical Review, vol.108, no.4. p.1070, 1957

Note that when Bohm says "because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A," he is implicitly using the conservation of total spin.

In 1964, John Bell followed David Bohm. As with Bohm, Bell does not explicitly mention conservation of angular momentum, but he describes spin components measured in the same direction...

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 σ₁ and σ₂. If measurement of the component σ₁ • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ₂ • a must yield the value — 1 and vice versa.

("On the Einstein-Podolsky-Rosen Paradox," Physics, 1.3, p.195.)

In his most recent description of two-particles that may be entangled, Maudlin first describes the two particles in separate single-particle quantum states...

We now have all the pieces in place to apply the recipe to Bohm’s version of the EPR experiment and to derive predictions of violations of Bell’s inequality...As usual, we construct the entangled state by starting with unentangled product states. Suppose we have a pair of electrons that begin in the same location, with one traveling off to the right the other to the left. The one going to the right can have the spinor |z↑> and the one traveling to the left |z↓>. The resulting product state could be symbolized as | z↑, right> and |z ↓, left>...There is no entanglement in either of these states, and making predictions from them is easy. For example, in the first state, if both particles are passed through z-oriented magnets, the right-moving particle will be deflected up and the left-moving one down. If they are both passed through x-oriented magnets, then each has a 50-50 chance of being deflected either way, with no correlations predicted between them. That is, finding out which direction one goes will not change the prediction about the other. It will still be 50-50.

Philosophy of Physics: Quantum Theory, pp. 69-70.

Now Maudlin goes on to describe the entangled two-particle quantum system, which Erwin Schrödinger describes as a superposition of the above two states..

By the superposition principle, we can form from this pair of states the entangled state | z↑, right> |z ↓, left> - | z ↓, right> | z ↑, left>... This is called the singlet state of spin...

What should we predict if we pass both electrons through z-oriented magnets followed by a phosphorescent screen?

...[We] predict a 50% chance of the right-hand flash occurring up and the left down, and a 50% chance of the right-hand flash being down and the left up. There is no chance that both will be up or both down. In short it is certain that the location of one flash will be up and the other down, but completely uncertain which will be up and which down. Observing either flash renders one completely sure of where the other will be.

Einstein argued that in this case, where the two electrons can be arbitrarily far apart from each other, we cannot accept that what happens to one electron can have any physical influence or effect on the other. But absent such “spooky action-at-a-distance,” it follows that each electron must be predisposed all along to be deflected the way it is: otherwise, how could the second electron, uninfluenced by the first, always behave the opposite way?

Philosophy of Physics: Quantum Theory, pp.70-71

Like Bohm and Bell, Maudlin also does not mention the law of conservation of angular momentum (spin), but what if the two entangled particles remain at all times before a measurement in that singlet state with total spin zero? Can we still say, as Maudlin does, that there is still a 50-50 chance that either particle will be measured up or down individually as required by quantum mechanical indeterminism, but that the two-particle wave function has had a joint property that the total spin must remain zero at all times since the particles were originally entangled?

So between the initial entanglement and the measurement, we cannot say one of the particles always has the same spin that will be found at measurement. It does not, as Einstein feared, follow "that each electron must be predisposed all along to be deflected the way it is."

But can we say that the two spins, however indeterministic (50-50) they may be individually, at all times between the initial entanglement and measurement, must be exactly opposite to one another when they are measured at the same angle. Otherwise a law of conservation, a deeper principle than either quantum or classical mechanics themselves, would be violated.

As Maudlin correctly described this situation above, "the z-spins of the particles are certain to have opposite values even though neither particle has a determinate z-spin."

Apart from the theory that properties like angular and linear momentum and energy are perfectly conserved quantities, the experiments of entangled particles spin always find spins exactly opposite when measured at the same angle. Theory and experiment together agree that conservation of the sum of two spins keeps the total spin zero, one up and one down, whatever the indeterministic nature might be of individual spin measurements.

As Maudlin also says above, "there is no chance that both will be up or both down."

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