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

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
L^{2}(*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

on the tensor product of L^{2}(*J*) and *H*.
The classical nature of the measuring device means that the natural basis
vectors labelled by *j* in *J* of *L*^{2}(*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

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.

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

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

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

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

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 P^{2}=P for any projection. Then we complete the proof by
noting that for any spectral family, the sum adds up to one:

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 I_{n}
is positive for
all natural numbers *n* (here, I_{n} denotes the unit operator on the
*n*-dimensional complex Hilbert space **C _{n}**).

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.

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

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

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:

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

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:

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:

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.