Quantum Entanglement and Bell's Theorem

A simple description of the essential non-classical nature of quantum
entanglement is given in Entangled Choices.
For a more detailed and technical analysis, consider the quantum spin of
an electron (or any spin-1/2 particle), which is always found to have
either the value +h/2 or -h/2 in the direction of measurement, regardless of
the direction we choose to measure. (This was first shown for silver atoms
in the famous Stern-Gerlach experiment, and has subsequently been verified
for many other kinds of particles.) Thus an electron manifests one of only
two possible spin states, which we may call "spin up" and "spin down".
It's convenient to represent these states as orthogonal unit
vectors

In general an electron's spin state, Y, at any
instant can be represented by a linear combination of those two possible
observable states. The choice of a measurement direction is equivalent to
choosing a "basis" for expressing the spin components of the state Y. For any particular basis we can express the state in
the form

where c1 and c2 are complex constants. These
constants encode the probability that the chosen measurement will yield
either result. The probability is simply the norm of the respective
coefficient. For example, if c1 = a + ib and we let
c1* denote the complex conjugate, a -
ib, then the probability of a measurement on this basis yielding "spin up"
is equal to the norm, i.e., the product c1*c1 =
a2 + b2. Likewise the probability of finding "spin
down" on this basis is c2*c2. Since these are the
only two possilities, we have

c1*c1 + c2*c2
= 1

As always in quantum mechanics, each possible measurement basis is
associated with an operator whose eigenvalues are the possible results of
the respective measurement. For a given xyz basis of orthogonal space
coordinates we can represent the three principle measurements (i.e.,
measurements along the three axes) by the matrices

The eigenvalues of the measurement operator corresponding to whichever
measurement direction we choose determine the coefficients c1
and c2, which represent the probabilities of the possible
outcomes.

To see how this works, suppose Y1
is the initial state vector of the electron, and we decide to perform a
spin measurement corresponding to a particular operator . The
result is given by applying to
Y1, using ordinary matrix
multiplication, to give the new state vector Y2 as shown below:

Since Y2 is either pure "spin up"
or pure "spin down" in the direction of measurement represented by , it follows that a subsequent measurement in the same direction
must yield the same result, so Y2 must
be such that

for some constant l (since state vectors are
equivalent up to length). Thus the constant l is
an eigenvalue of the measurement operator , and
Y2 is the corresponding eigen-vector.

The outcome of (3) is unambiguous, because the eigenvector on the right
side is the same as the eigenvector on the left side. However, since the
arbitrary initial state Y1 in (2) is
not in general an eigenvector of , it
can yield either of the eigenvectors of . This
reveals the probabilistic aspect of quantum mechanics.

The eigenvectors of constitute a basis for the space of possible state vectors, so
Y1 can be expressed as a linear
combination of those eigenvectors. If we let Y2 and Y2' denote the eigenvectors, then we can
express Y1 as

Again, the norm of each complex coefficient gives the probability that
applied to Y1

will lead to the respective eigenstate.

For the three principle direction spin operators , , and presented above, the eigenvectors corresponding to +h/2 or -h/2 are as
shown below:

Each pair of eigenvectors constitutes a basis for the state space, so
we can express the electron's state vector as a linear combination of the
basis vectors for the desired measurement, and the coefficients give the
probability of that measurement yielding either "spin up" or "spin down".
We can think of these probabilities as the projections of the initial
state vector onto the orthogonal axes of the chosen measurement
basis.

Of course, we aren't restricted to measurements along one of the
principle axes. We can measure the spin of the electron along any spatial
axis, and each such measurement is represented by an operator. We also
note that these directions are purely relative to the state of the
particle in question. To illustrate, suppose a stream of electrons is
moving along the y axis and we perform spin measurements on these
particles in the z direction, which establishes the z component of the
particle's spin vector. If we filter out all those particles with "spin
down" in the z direction, we are left with a stream of particles all
having "spin up" in the z direction. Now suppose we perform a spin
measurement on the remaining particles along the direction in the xz plane
at an angle q with the positive z axis. In accord
with the interpretation of probabilities as the projections of the state
vector onto the basis axes, we infer that the measurement operator for
spin in this direction is given by the projections of the x and z
operators

Naturally the eigenvalues of this operator are +h/2 and -h/2, and it's
easy to show that the corresponding eigenvectors are

respectively. Each particle subjected to this measurement will have the
initial state vector indicating "spin up" in the z direction, and we can
express that initial vector as a linear combination of these basis
vectors

which implies that the coefficients are c1 = cos(q/2) and c2 = -sin(q/2). Consequently, the probabilities of "spin up" and
"spin down" for a measurement of such a particle along the q direction are cos(q/2)2 and sin(q/2)2 respectively.

These quantum mechanical predictions (which have been well supported by
experiment) have some remarkable implications. In the preceding example we
established the initial state vector by performing a measurement in the z
direction and considering only particles that yielded "spin up" in that
direction. Then we subsequently performed a measurement at an angle q relative to the positive z direction, and the
probabilities of the outcomes were found to be a function of the angle
q between those two measurements. An essentially
equivalent experiment can be performed by examining the spins of two
spin-1/2 particles emitted in opposite directions from the decay of a
singlet state with zero total spin. In such a case conservation of angular
momentum requires that the spin state vectors of the individual particles
are precisely opposite, so if we measure the spin of one of these
particles along a certain direction and find "spin up" in that direction,
then the other particle must have pure "spin down" in that direction.
Thus, by measuring the spin of one particle and reducing its state vector
to one of the eigenvectors of the chosen measurement basis, we
automatically "collapse the wavefunction" of the other particle onto this
same basis.

At first this process may not seem very mysterious, since it's easy to
imagine that the two coupled particles are "programmed" with opposite
spins, such that they will always give opposite results when measured
along any given direction. However, a careful analysis of the quantum
mechanical predictions for all possible combinations of measurement angles
reveals the need for a profound shift in the classical view of the world.

If the measurement of one particle along a fixed direction (in the xz
plane) yields "spin down", then the other particle is purely "spin up" in
that direction. Consequently if we perform a measurement on the other
particle along a direction at an angle of q from
the first measurement, we've already seen that the probability of "spin
up" is cos(q/2)2 and the probability
of spin down in sin(q/2)2. In a
similar way we can show that if the measurement of the first particle
yields "spin up", then the other particle is purely "spin down" along that
direction, and a measurement of that other particle along a direction at
an angle q relative to the first will yield "spin
up" with probability sin(q/2)2 and
spin down with probability cos(q/2)2.

Hence the probability that the measurements of these two particles at
angles differing by q will both give the same
result (both up or both down) is sin(q/2)2, and the probability that they will
yield opposite results (one up and one down) is cos(q/2)2. The angle q
between the two measurements can be expressed as a - b
where a is the angle of the measurement performed
on one of the particles and b is the angle of the
measurement performed on the other. The two particles emitted from a
singlet state are said to be entangled, because regardless of how far
apart they travel before the spin measurements are made, the joint results
will exhibit these joint probabilities.

From a classical standpoint we would imagine that each particle emerges
from the singlet state with, in effect, a set of pre-programmed
instructions for what spin to exhibit at each possible angle of
measurement, or at least what the probability of each result should be.
The usual approach to analyzing these phenomena classically is to
stipulate that a particle's pre-programmed instructions for responding to
a measurement at a given angle must be definite and unambiguous (rather
than probabilistic) because we classically regard the two measurement
angles as independent, which implies that the measurement on the "other"
particle could be at the same angle as our measurement of this
particle, and the particles must give opposite results in that
case. Likewise the measurement on the "other" particle could be 180
degrees away from our measurement of this particle, and the particles
must give equal results in that case. Of course, the individual
measurements can each be either "spin up" or "spin down" in both of these
cases, so in principle they could still be probabilistic tendancies, but
classically we have no way of ensuring perfect correlation (or perfect
anti-correlation) of the joint results of spacelike separated events other
than by definitely pre-programming the spins of each particle for each
possible measurement angle.

From this assumption it follows that the instructions to one particle
are just an inverted copy of the instructions to the coupled particle. In
other words, for each measurement angle from 0 to 2p the pre-programmed response to a spin measurement for
one particle is the opposite of the pre-programmed response of the other
particle at that angle. Furthermore, since we have perfect correlation if
our measurements are at angles that differ by 180 degrees, it follows that
the pre-programmed instructions for each particle are individually
anti-symmetric at a phase angle of 180 degrees. For example, if a
particle's programmed response for a measurement at angle a is "spin up", then the programmed response of that
same particle for a measurement at angle a +
p must be "spin down". Hence we can fully specify
the instructions to both particles by simply specifying the instructions
to one of the particles for measurement angles ranging from 0 to p .

The simplest function that satisfies the conditions stated so far is
constant "spin up" over the range from 0 to p for
one of the particles. This gives the total "response programs" of both
particles shown below:

Naturally we can take the absolute orientation of this profile as
arbitrary relative to our measurements. Unfortunately this simple profile
doesn't give the correct (quantum mechanical) joint correlations for
measurement angles that differ by amounts other than 0 or 180 degrees. If
we define the correlation exhibited by the measurements results of
a set of particle pairs as the number of agreements minus the number of
disagreements, all divided by the total number of pairs, then it's easy to
see that the correlation yielded by pairs with the above instruction
profiles (oriented randomly) varies linearly as a function of the
difference between the measurement angles. The correlation is -1 if the
angles differ absolutely by 0, the correlation is zero if the angles
differ by p /2 (90 degrees), and the correlation
is +1 if the angles differ by p (180 degrees). In
general, for measurement angles differing by q,
the correlation is C(q) = (2/p )|q | - 1. In contrast, since the quantum mechanical
predictions for agreement and disagreement are sin(q/2)2 and cos(q
/2)2 respectively, the quantum mechanical prediction for the
correlation is

A plot of the simple linear correlation profile and the quantum
mechanical profile is shown below.

These profiles agree only for measurements differing by 0, p /2, and p . For all other
cases the simple pre-programmed linear model fails to match the
predictions of quantum mechanics - which have been amply verified by
experiments. This raises the interesting question of whether any
pre-programmed response profile can reproduce the predictions of quantum
mechanics (and the experimental results). Suppose each particle is
programmed with a more complicated profile of responses as a function of
the measurement angle. In general we could partition the angular range
from 0 to p into n arbitrary segments,
alternating between "spin up" and "spin down". Together with the perfect
correlation and anti-correlation requirements at angluar differences of 0
and p , this suffices to determine the putative
response profiles of both particles over the entire range from 0 to 2p . Letting S(a) denote the
spin result (+1 for up and -1 for down) at the measurement angle a for one of the particles, and letting s(a) denote the spin result of the other particle, we
have S(a) = -s(a) and S(a) = -S(a +p). The two results agree for measurement angles a and b if S(a)s(b) = +1, and they disagree
if S(a)s(b) = -1. The absolute values of a
and b are taken to be arbitrary, so the
probability of agreement for all cases where b
- a = q for any fixed
q is given by integrating the quantity

from a = 0 to p .
(The result is symmetrical for a -b = q , and for the range
from p to 2p .)
Therefore, equating this with the quantum mechanical result, we must
have

which implies

This is just the correlation between the two measurements. If S(a) = +1 on the entire interval from a = 0 to p (and therefore
S(a) = -1 for all a from p to 2p ), the left hand side can be split into two
integrals, one extending from a = 0 to p -q with an argument of +1,
and the other from a = p
- q to p with an
argument of -1. Hence the left side reduces to (2/p)q -
1, as we saw earlier, and this does not match the quantum mechanical
prediction. In particular, we note that the rate of change of the quantum
correlation near q = 0 is zero, whereas the rate
of change of the correlation for based on our simple S function has a
positive slope at q = 0. This is because the
increase in correlation is proportional to the increase in q , arising from the transition at a = p - q
. Clearly this rules out every fixed S function, because if
any more transitions are added to the pre-programmed instructions, the
slope of the resulting correlation will increase proportionately. To
illustrate, consider the S function and the resulting correlation function
shown below:

The solid bars at the bottom signify the angular regions where "spin
up" has been pre-programmed for this particle. Since there are three times
as many transitions, the slope of correlation versus angle is three times
as great as in the basic case considered previously. This is true
regardless of how these transitions are distributed. For another example,
consider the S function and resulting correlation shown below:

This S function has nine transitions, so the initial slope is nine
times as great as the baseline case. We're forced to conclude that no
pre-programmed set of results for the range of possible measurement angles
can possibly reproduce the quantum mechanical predictions (and
experimental results). Equation (4) does have the formal
solution

but of course this S function isn't restricted to the discrete values
+1 and -1, so it can't be directly interpreted as a spin
indicator.

Many treatments of quantum entanglement discuss "hidden variables", and
they derive inequalities involving the correlations as functions of these
putative variables. In a sense, these variables are redundant, because the
essence of the so-called "local realistic" premise is that each particle
emerges from the singlet state with definte instructions for the spin it
will exhibit for each possible measurement angle. This implies that the
only relevant free variable is the reference orientation, since the
instruction profile can be oriented in any direction (relative to the
measurements) with uniformly distributed probability. Integrating the
product of spins over the entire angular range with a fixed difference
angle must yield the correlation of the two spin measurements.

It might be argued that we need not assume any single particle exhibits
the quantum mechanical probabilities, because these probabilities can only
be evaluated by performing multiple measurements, so we might imagine that
it's necessary to evaluate the results over a sequence of particle pairs,
rather than just a single pair. Moreover, we might think that this
sequence could possess "memory", making the results of successive pairs
dependent. However, in principle we could prepare a large number of
particle pairs in an identical way, in spacelike-separated locations, and
perform measurements on the pairs independently (again based on the naive
conventional concept of local realism). According to quantum mechanics we
would still expect the combined results to satisfy the same correlations.
This implies that each particle pair must embody the overall propensities
that we expect to find manifested in multiple trials.

Nevertheless, it's worthwhile to review the traditional derivation of
Bell's inequality based on the premise that each particle emerging from
the singlet state possesses a definite spin as a function of the
measurement angle and some "hidden variable" m
associated with that pair. We assume an arbitrary probability density
function d (m), which
gives the probability density for any given pair of particles being
produced with that value of m. The observed spins
of the two particles are denoted by S(a,m) and s(b,m), where a and b are the angles of the respective measurements. As
before, the perfect anti-correlation of the two particles measured at the
same angle (according to quantum mechanics) implies that S(a,m) = -s(a,m). (In our previous discussion the implicit "hidden
variable" was simply the absolute reference angle for the measurements,
with a uniform probability density over the range from m = 0 to 2p , but we can argue
more generally for any hidden variable with any probability density over
any range.)

Now we consider the pairwise spin correlations for three different
angles, a, b, and f. If one member of each coupled pair of particles has
its spin measured at the angle a and the other is
measured at the angle f, the correlation between
the measurements is given by

where the integral is evaluated over the full range of m . Likewise if one particle is measured at b and the other at f, the
correlation is

Taking the difference between these two, we have

Since every spin value is either +1 or -1, it follows that S(a,m)2 = 1, so the
value of S(b,m) in this
expression can be written as S(a,m)2 S(b,m), and we can then factor out from the square brackets
to give

Now we take the absolute value of both sides, and make use of the
obvious fact that the absolute value of a definite integral of a function
is less than or equal to the integral of the absolute value of the
function, we have the inequality

Conventionally, a probability density function always has a
non-negative real value, so we stipulate that d(m) is non-negative. Also,
since the product S(a,m)
S(b,m) is always +1 or
-1, the quantity in the square brackets is always either +2 or 0. The
leading factors, s(f,m)
S(a,m), always equal +1
or -1, so they determine the sign of the integrand, which is discarded by
taking the absolute value. Hence the expression reduces to

Replacing S(b,m) by
-s(b,m), and splitting the integral into two parts,
give

The first integral is simply 1, since d(m) is a probability density function, and the second
integral is the correlation for two entangled particles measured at the
angles a and b . Hence
we can subtract this from both sides to give Bell's inequality

We can evaluate the left hand function assuming the quantum mechanical
correlation C(x,y) = -cos(x-y). This gives
|cos(a - f) - cos(b -
f)| + cos(a - b),
which is shown in the figure below. The colored regions represent that
cases where this quantity exceeds 1, and therefore violates the above
inequality. The maximum violations occur in the four white regions, where
the function attains the value of about 1.5.

A different approach to analyzing quantum entanglement is to assume
that each particle is pre-programmed not with a definite result for each
measurement angle, but with a probability of yielding specific results.
Needless to say, this model will be incompatible with the premise of
independence of measurement angles, because a probabilistic model can't
automatically enforce the perfect correlations and anti-correlations
predicted by quantum mechanics at q = 0 and p . Still, the previous discussion has shown that the
naive premises of "local realism and free choice" are untenable anyway, so
it's worthwhile to examine a probabilistic scheme.

Thus we imagine that the first particle contains a function f(a) that represents the probability of' yielding spin up
for the measurement angle a . Likewise the second
particle is programmed with the function g(b)
that represents the probability of yielding "spin up" for the measurement
angle b . In these terms we can express the
probabilities of agreement and disagreement as

If we double the "disagree" equation and subtract 1 from both sides we
get

Incidentally, if we expand the right-hand side using the trigonometric
angle addition formula, we get the interesting expression

which could be regarded as an "inner product" of vectors with the
components of the form sin(a), cos(a), and 1 - 2f(a).

Returning to the original "disagree" equation, if we assume the
functions f and g individually possess derivatives, and if a and b are independent
variables (as is classically the case for two freely chosen measurement
angles), we can take the partial derivative with respect to b at any constant a to
give

Solving for 1 - 2f(a), substituting into equation (5), and re-arranging
terms gives

Similarly we can derive the relation

It's interesting that the expressions on the left side of these
equations are formally the same as the queing theory "transition rates" of
the functions 2g and 2f if we identify these functions with the state
probabilities and the arguments a and b with the model time.

In any case, notice that these equations theoretically imply that each
of the two entangled particles contains enough information to enable
someone to compute the measurement angle at the other particle. For
example, the second equation can be solved for b
to give

Hence, assuming the particle contains the complete function f, and
assuming this function possesses derivatives, it follows that the
measurement angle b at the other particle
is a deterministic function of the measurement angle a at this particle. This contradicts the premise
that a and b are
independent variables, so the only possible conclusion is that one or more
of our basic assumptions were wrong. One questionable assumption was that
not only does each particle "contain" its respective function (f or g),
but that these functions are differentiable. This need not be the case.
Continuous but nowhere-differentiable functions are well-known in
mathematics, and we could postulate that f and g are such
functions.

On the other hand, if we rule out such exotic functions, it seems that
the only remaining possibility is that, even though we may believe we are
free to choose the two separate measurement angles "freely" and
independently, they are actually deterministically linked. In other words,
not only are the particles entangled, but so are our "free" choices of
measurement bases for those particles. This analysis suggests a much more
deterministic universe than is commonly conceived, regardless of whether
it's possible to extract the information about f and its derivative from
the particle. (These "free" choices may even be spacelike separated, but
of course this does not imply any violation of special relativity, because
it is only necessary to assume that the choice events share a common event
in their causal pasts, which is always the case provided we are willing to
go back far enough.)

Suppose we differentiate equation (6) again with respect to b at constant a to
give

The left hand side is equal to (1-2f)(1-2g), so we can make this substitution, cancel the
factor of (1-2f), and re-arrange terms to arrive
at

This has the particular solution g(b) = 1/2,
and we can add this to the homogeneous solution to give the total general
solution

Since this function signifies the probability of "spin up", we expect
it to be real-valued and in the range from 0 to 1. Hence we can set
C1 = C2 = 1/4 to give

Applying the same analysis to f(a) and setting
C1 = C2 = -1/4
gives

If we then substitute these expressions back into the original "agree"
and "disagree" formulas, we find that in both cases they reduce
to

Comparing this with the trigonometric identity

we see that it is not generally satisfied unless a or b is a multiple of p , in which case it is trivially satisfied. One way
around this impasse is to postulate slightly more general functions for f
and g, namely

where a0 and b0 are constant phase angles for a given
pair of particles. Substituting these functions into the basic "agree" and
"disagree" equations gives the single condition

The can also be expressed as a null line element

where the determinant of the coefficient matrix is 1 + cos(a0 -
b0). It can also be expressed in the form of a Mobius
transformation between the tangents of a and b as follows

The similarity parameterof the transformation is 1 - cos(a0 - b0), whereas the normalized trace is
tan((a0 -
b0)/2).

Return to MathPages
Main Menu