Locality in the EPR experiment

I. The von Neumann collapse postulate

In this section, we show that the postulate of von Neumann, that on measurement the wave function collapses to an eigenstate of the observable being measured, follows from Bayes's rule for conditioning probabilities in classical probability.

Suppose a system is described by the unit rays of the Hilbert space H and the self-adjoint operators on it. In some works, it is said that a measurement of an observable, say X, in quantum mechanics causes the state to jump to one of the eigenstates of X. To understand this, it is better to divide the measurement, and the state reduction, into two stages. Suppose for the time-being that the initial state of the system is given by the pure vector state (phi>, and that the eigenstates of X are (psi(n)>, with no multiplicity. Suppose that n runs over the set J of labels. The first stage is the interaction of the system with the measuring device, and the second stage is the seeing of the reading on the device by the observer. After just the first stage, the state of the system is the mixture of all the eigenstates of X, with weight given by the quantum mechanical transition probability

p(n):=|(psi(n)|phi)|^2.

Thus, the system is clearly in a mixed state. A good measuring device is a classical system in which the pointer of the device is 100% correlated with the eigenstate into which the system is projected. More, the details of the device do not affect the reading. Thus, a complete description of the device is given by the label n, an element of J. Although a classical concept, the observables of the measuring device (the readings) have a quantum description as multiplication operators chi(n) on the Hilbert space L2(J). Here, chi(n) is 1 at n, and zero elsewhere. The result of first stage of the measurement is thus described by the density operator

sumnp(n) chi(n) tensor |psi(n))(psi(n)|

on the tensor product of L2(J) and H. The classical nature of the measuring device means that the natural basis vectors labelled by j in J of L2(J) are separated by a superselection rule: no phase between these subspaces is observable.

The second stage of the measurement process is the reading of the instrument by the observer. We can imagine the case when the system itself has left the neighbourhood of the instrument. Since the instrument is classical, it can be observed without affecting the system further. Only classical probability is involved in this stage: the label n is observed with the probability p(n). After the observation of say the label m, the observer, say Alice, A, knows that the system must be in the state psi(m), since there was a 100% correlation between states and readings. By reading the instrument, A replaces (by Bayes's formula) the probability p(n) by the conditioned probability p(. |given m), and the state of the system is reduced to psi(m). Thus, von Neumann's projection postulate follows from the usual Bayes rule for classical probability. More generally, if A had made an incomplete observation of the reading, and could only be sure that n lies in some subset K of J, the classical probability of the reading would change from p to p(.|K). Then Bayes's theorem shows that the density matrix of the system, as observed by A, is reduced to

sum_{n\in K} p(n)|psi(n))(psi(n)|/[sum_{n\in K}p(n)]

as given by von Neumann. A similar argument can be given if some eigenvalues of X are not simple. If X has a continuous part to its spectrum, the classical theory (of conditional probability) is harder, but the argument can be given; it leads to von Neumann's projection postulate as well.

Thus, the first stage of the measurement process is caused by the physical interaction of the system with the device; the second stage, the reading, involves the conciousness of the observer, as remarked by Wigner. This does not mean that physics is subjective, any more than it does in classical probability. Indeed, once X has been singled out for measurement, and the measurement has been done, the theory reduces to classical probability in its techniques and interpretation.

II. Completely positive unital maps

Any self-adjoint operator X possesses a spectral measure P(lambda), which defines a projection-valued measure on the line R. This means that to any Borel subset E of the real line, is given a projection

P(E)=intE dP(lambda)

In the case of discrete spectrum, this becomes a family of projections, P(lambda), one for each point of the spectrum of X. Then we can summarise the von Neumann collapse postulate, derived above from Bayes's rule, by saying that the first stage of the measurement causes the state |phi), with density matrix |phi)(phi|, to change to

sum P(n)|phi)(phi|P(n)

More generally, by linearity, if the density matrix of a state before measurement is rho, then that after the first stage of measuring A is

sumn P(n) rho P(n)

By observing the reading, to be m \in J say, the observer picks out the projection onto the eigenstate |psi(m)).

If instead of measuring one observable, the device is designed to simultaneously measure commuting observables X(1), X(2), ..., X(r). then we make use of the joint spectral measure P(lambda(1),..., lambda(r)) of this abelian set, and the first stage of the process is given by the map

rho mapsto sum_{lambda(1),...,lambda(r)}P{lambda(1),...) rho P(lambda(1),...}

This defines a completely positive unital map on the space of operators. A unital map is simply one that maps the unit operator to itself; that this is so here follows at once by putting rho = I, and using the fact that P2=P for any projection. Then we complete the proof by noting that for any spectral family, the sum adds up to one:

sum_{lambda(1),..., lambda(r)}P{lambda(1),...,lambda(r)}=1.

A positive map is a linear map that takes a positive operator to a positive operator (positive means positive semidefinite). It is said to be completely positive if its tensor with the identity In is positive for all natural numbers n (here, In denotes the unit operator on the n-dimensional complex Hilbert space Cn).

Thus, experts in measurement theory postulate that the first stage of any measurement of a compatible set of observables is, generally, a completely positive unital map on the set of density operators. To illustrate the EPR experiment (I do not call it a paradox), we do not need the generalisation proposed by Davies and Lewis, which involves the introduction of positive-operator-valued measures.

III. The observation of entangled states.

States consisting of two parts, that are spatially distant but nevertheless are correlated, can occur in classical probability. Bell calls this the paradox of Dr. Bertelmann's socks. If Dr. Bertelmann shows us the colour of one of his socks, say it is red, by raising his trouserleg, we can with some certainty guess that the colour of his other sock is red. Our guess will be more reliable after he has shown us the info than before. We would not try to argue that our seeing the red of one sock had any physical influence on the other. No; our seeing it simply reveals what was there; it is our assessment of the probability that is changed. See my experience of a lecture on this by J. S. Bell. Similarly, in a game of cards, when I see that my hand contains an ace of spades, I change the probabilities, from what they were before I saw the hand. This information does not change the objective fact of my opponent's hand as viewed by him. My seeing the ace of spades does NOT change the assessment that my opponent makes of my chances of holding this card (unless I send him some signal concerning my hand). In the same way, a measurement of the spin of one of a pair of EPR-electrons does not alter the state of the other, as seen by the other observer. We can use this classical argument because the measurement of a complete commuting set of observables (A's spin S(1) and B's spin S(2) ) has set up a classical probability model in the sense of Kolmogorov. In his book, Mind, Matter and Quantum Mechanics [Springer-Verlag, 1993, 2004], p. 29, Stapp says that this result in classical probability is entirely non-problematic, and does not require that there be any communication between the players to ensure that the hands are correlated. He says that the situation is quite different in quantum mechanics, but does not explain why. Penrose also gives a similar example, and also says that the classical situation does not require any instantaneous signal to travel between the observers. Again, he claims that the situation in quantum measurement is different. Our view is that the interpretation of quantum mechanics is precisely that of the classical probability set up by ther measurement using the chosen complete set of commuting observables. As remarked by Peierls, quantum mechanics is the Copenhagen interpretation.

We now prove in detail that B's assessment of his state is unchanged by A's measurement. The reader will see that an attitude similar to that necessary in the theory of games has been adopted. Indeed, the analysis below is a simple example of a quantum game. This point of view is outlined in my review of Mielnik's article `The paradox of two bottles in quantum mechanics', Found. Phys. 20, 745-755, 1990, and has been advocated in my book, Statistical Dynamics.

Suppose that an atom emits two electrons in a singlet spin state in opposite directions, which are observed by Alice (A) and Bob (B) at two far-separated sites. The state of the spin is pure, and is given by the vector

psi = 2(-1/2){ |+)|-) - |-)|+) }

The spin measurement, in the third direction, by A is achieved by the completely positive unital map which, on a state rho, is given by

M(3) rho = P(+) rho P(+) + P(-) rho P(-)

where P(+) and P(-) are the projections onto the two eigenstates of J(3), the operator representing the spin of A's particle in the third direction. From the point of view of the total system, A's measurement uses the completely positive stochastic map M(3) tensor I. Because this operator consists of the identity on B's Hilbert space, the measurement by A does not alter the partial state seen by B, namely, the restriction, rho| alg(B), of rho to alg(B), the observable algebra of B:

[(M(3) tensor I) rho]| alg(B) = rho| alg(B)

As for the total state, the density operator rho(psi) of the pure state psi, on measurement it changes as follows:

(M(3)tensor I)rho(psi) = 1/2[ P(+) tensor P(-) + P(-) tensor P(+) ]........................(1)

Thus the measurement by Alice, of J(3) of her particle, without observing the result, leads to the classical mixture with equal weights, of the two possible results, spin up and spin down, 100% anti-correlated with the spin at B. The pointer of her instrument is a classical random variable, not in the microsystem; it is 100% correlated with Alice's spin. By looking at the reading of her pointer, Alice will condition the microstate by knowledge, and will produce a pure state, and will also find out what the the spin of B's particle would be, if measured in the 3-direction.

If A has measured S(3) and has seen her result, then the state she assigns to the whole algebra, the conditioned state, is the pure state, and B's result, if he measures it, is sure for Alice (though not for Bob). Some people regard this as a state-preparation by Alice for Bob. This makes sense only if he is informed of her result by Alice. B can then measure his spin, confirming the conservation law. His measurement brings no new info from the point of view of the total system; this is why his measurement alters his state but not that assigned by both of them if they share info. But as long as there is no signal from A, B's initial state is the completely mixed state of the unpolarised electron, and his spin measurement (with no looking at the result) does not alter his partial state.

If A makes two measurements, of J(3) and then J(2) say, in that order, then the first, J(3), will tell her the result that B would get if he measured his J(3), but the second would not tell her the result that B would get if he measured his J(2): he would not necessarily get the opposite result from hers. Roughly, this is because the first measurement interfered with the system, spoiling the conservation of total spin in the 2-direction. Don't take my word for this; just look at the calculation given now. After A's measurement of J(3), the state is as above:

[M(3) tensor I] rho = 1/2[P(-) tensor P(+) + P(+) tensor P(-)]........................(1)

When we apply [M(2) tensor I] to this, we do not alter the partial state of B, because of the unit factor in the operation. Thus we do not alter the fact that the value of J(3) that B would find if he measured it is anti-correlated 100% with the quantum record held by A; but if Bob as well as Alice measure J(2), there will be no correlation between the results for J(2). If A's and B's second measurement is not J(1) or J(2), but contains a small component along J(3), then there will be a small but not 100% anticorrelation between the second measurements. The theory tells us exactly what to expect. Let us do the case when both measure J(2).

Because the state is now given by eq (1) above, we need to find [M(2) tensor I] [P(+) tensor P(-)] and [M(2) tensor I] [P(-) tensor P(+)] to find the effect of A's second measurement. On A's Hilbert space, this reduces to finding M(2)P(+), since M(2)P(-) = 1 - M(2)P(+). This is easy:

M(2)P(+) = P{J(2) = +}P{J(3) = +}P{J(2) = +} + P{J(2) = -}P{J(3) = +}P{J(2) = -} = 1/2[P{J(2) = +} + P{J(2) = -}].

Here, we have denoted by P{J(2) = +} the spectral projection onto its eigenvalue +1/2, and so on, and for clarity have used the same notation for what we called P(+) and P(-), namely P{J(3) = +} for P(+) etc.

Put this in the formula, and we see that the state after both A and B have made their second measurement is the equal mixture of the four possible pure states, the projections onto the four eigenstates of J(2) tensor J(2): there is now no correlation. With probability 1/4, both A and B could find that J(2) for their sample is +1/2, and the total is not zero. This violation of the law of conservation of angular momentum is caused by the macroscopic intervention of the measuring device used by A in her first measurement, of J(3). This device did not interfere with the conservation of J(3), but did interfere with the law for J(2).

We see that the idea of Einstein, Podolski and Rosen, that one can measure a property of B's particle in this set-up by measuring the same property of A's particle, and then using a conservation law, works only for the first measurement.

In the relativistic case, in place of the tensor product we have a local C*-algebra with local structure, along the lines of a Haag field. A and B will now be space-like separated, and the measurement of a local property by A or B will be done using a CP unital map M(A) or M(B) which acts on the whole algebra, but which is the identity map when restricted to algebras space-like to the region of space-time in which the measurement takes place. The two maps must commute if Alice and Bob are space-like separated. The calculation can then go exactly as for the non-relativistic case above. See my review article for more details.

The recent article in the NEW SCIENTIST entitled "Quantum Entanglement: How the future can influence the past", by Michael Brooks, 27 March 2004, is completely wrong. The future cannot influence the past. Nor is there such a thing as "remote control", as claimed in the article on page 32. It is not true that "if something affects the quantum state of one particle, it will inevitably affect the quantum state of the other [entangled particle], no matter how far apart they are" [page 32]. As we proved above, the state of the second particle (as viewed by Bob) is unchanged by Alice's measurement.

The claim that the future can change the past is based on a slip, common in the interpretation of statistical correlations. Brooks talks about measuring the spin of a photon, and then measuring it again later, and getting a different result [page 35]: "...the very act of measuring the photon a second time can affect how it was polarised earlier on". This is of course impossible if the first measurement was recorded on a classical instrument. There will be correlations between the two results, but this merely allows us to conclude that there is an association between the spins, not that the later spin-value was the cause of the earlier one. Indeed, the criterion of priority, one of three needed to come to this conclusion, is not satisfied. The other criterion, that of direction, also fails. See my article EPR, cot deaths and the dangers of cannabis.

Brooks's article speculates that this property, entanglement, might be behind life. This idea is not new, as it can be found in "The Emperor's New Mind", a book by Roger Penrose, and is also in the article on the mind by Stapp, reviewed by me here. This article is a summary of Stapp's book, "Mind, Matter and Quantum Mechanics", Springer-Verlag, 1993, 2004. The arguments for the idea were wrong when Penrose wrote, and wrong when Stapp wrote; they remain wrong when the New Scientist writes them.

D'Ariano argues that the collapse of the wave-function takes place simultaneously over all space. Taken literally, this would imply the instantaneous transfer of information. We see that the solution to this problem is to assign information algebras to each observer, as in the theory of games, so that different observers assign different states to the same physical system. This is done only after a measurement. The argument that physics becomes subjective instead of objective has no more force here than in classical probability. Indeed, after a measurement, the quantum record (the classical pointers of the measuring instruments) is an objective fact; the state at this stage merely describes Alice's or Bob's knowledge of the event, and this depends on which of them can see the pointers. The description of the quantum results by classical events depends of which complete set of commuting observables was chosen by Alice and Bob. Thus, a different classical model is needed for each context. Note that it is the classical description that is contextual: the assignment of random variables to observables depends on the context; the mapping between observables and self-adjoint operators is the same whatever complete commuting set is contemplated. So I claim that quantum theory is non-contextual.

If an incomplete set of commuting observables is measured, then the description of the resulting state is only partially classical. For example, if Alice measures J(3), and sees the result, it would be wrong for her to claim that Bob's particle has the opposite value of J(3) from hers. She does NOT know that there is a classical pointer showing the opposite result from her pointer; the experiment might not have been done. Indeed, for all she knows, Bob has already measured his J(2), and has a pointer to prove it, before she did her measurement. Thus, the only safe claim for Alice is that Bob would find, or would have found, the opposite value of J(3) if he were to make, or has made, the measurement of his J(3).

In most first courses in quantum mechanics, and of probability, only one observer is discussed, and so D'Ariano's point of view is correct in that case. The EPR experiment pin-points the need for subjectivity in quantum probability; the same need in classical probability has been known and used since Bayes. A similar stand is taken by E. B. Davies in his book "Science in the Looking Glass", OUP, 2004.




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© by Ray Streater, 2 Oct 2003.