When quantum mechanics fails in EPR experiments
The Fair Sampling Assumption is invoked when experiments are inefficient and a sizable portion of the events that should be detected are not. Intuition tells us that if we improve detection efficiency and build better experiments the number of detected events will increase until, at 100% efficiency, Fair Sampling would be verified because all events would be recorded. This fails, however, to take into account the Heisenberg Uncertainty Principle. Fair Sampling is always valid but it changes between quantum and classical systems.
The Fair Sampling assumption is used in EPR experiments to argue that the actual events counted are representative of the total number emitted. Most quantum mechanical experiments, especially with photons, are inefficient and so this remains a loophole to ruling out local realism. The general consensus about the validity of the Fair Sampling assumption is Bell’s frequently quoted comment:
…it is hard for me to believe that quantum mechanics works so nicely for inefficient practical set-ups and is yet going to fail badly when sufficient refinements are made.
Even though Bell refers to quantum mechanics, he was thinking classically like in the first paragraph above. Here I want to point out that the Fair Sampling assumption must distinguish between “EPR pairs detected” and “EPR coincidences detected” when an observable depends upon operators that do not commute.
Consider a two level quantum system, (e.g. a spin ½), and some property of the system that depends upon the Pauli spin operators, A=A(σx,σy,σz,). Now assume that the operator is represented in the usual z basis,|±,z>,
The quantities (ax,ay,az) are real numbers and are the various matrix elements of the three Pauli spin operators with A.
Clearly only the diagonal terms of the z component can be measured because in this representation σz is an eigenoperator on |±,z>. Since the Pauli spin operators do not commute, [σx,σz] = iσy, the Heisenberg Uncertainty Principle states that if the z component is measured, it is impossible to simultaneously measure the other two components, σx, σy , and cyclically.
Not being able to detect something does not mean it is not there. Rather it means we cannot know if it there from that particular experiment. Measuring in the z direction cannot give us information about properties that are off-diagonal. We cannot know if the other components are present or not.
Additional experiments are needed. The off-diagonal terms can be measured by turning the filter to a different direction, from in the z direction to the x to the y. Quantum mechanics requires transforming to that direction. In the x and y representations diagonal and off-diagonal elements are swapped. Using the unitary transformations,
these alternate representations of the operator, A, are
where as usual,
In the x representation the diagonal components correspond to the x axis whereas the z and y components become off-diagonal and cannot be measured. Likewise the y axis can be diagonal and measurable but then z and x cannot be measured.
This nicely illustrates the well-known fact: if non-commuting operators define a property of a system, then it is not possible to completely characterize that property with one experiment. It is not generally thought, however, that another consequence of Heisenberg Uncertainty is that properties can be missed.
Let us assume that the three spin axes are equivalent (ax = ay = az) . There is no experimental way to know if a spin ½ has more than one axis of quantization. Performing complementary experiments by rotating to other directions would be inconclusive. The obvious interpretation of that is a single axis reorients and lines up along the new direction. On the other hand, an equally valid interpretation is the existence of two or three equivalent axes. Those properties lie beyond experimental verification, but if they exist then only a fraction of the actual number of events are detectable.
Consider the following gedanken experiment: let us suppose that experiment can simultaneously detect both the x and z spin components. If there are two axes of quantization then EPR coincidences will be detected in both the z channels and simultaneously in the x channels. That is each EPR pair is capable of two simultaneous coincidences.
If the Fair Sampling assumption means that EPR pairs detected are representative of those emitted, then no problem. But if it means that the number of EPR coincidences observed is representative of the total number emitted, the it fails for non-commuting operators.
It is impossible to count events that are not detected so it is impossible to determine experimentally if missed coincidences are actually present. The undetectable coincidences associated with the components that are off-diagonal are, however, as real as those measured, only experiment is blind to them.
In a bit more detail let’s see the consequences of making the distinction between EPR pairs and EPR coincidences.
EPR experiments are based upon measuring coincidence events from photon pairs that were entangled (e.g. in a singlet state) in the past and at some instant separated, one moving toward Alice and the other toward Bob. An EPR pair gives two clicks at the same instant, one at Alice and the other at Bob. These simultaneous clicks are assumed to be one coincidence. Depending on whether the photons are transmitted, +, or absorbed, -, four types can occur. When passed through polarizers oriented at “a” on Alice’s side and “b” on Bob’s, coincidences are counted over a statistically large sample giving the numbers of EPR pairs in those coincident states, N++, N—, N+-, N-+. From these the correlation between the EPR pairs, E(a,b), can be measured using
But from the above, this expression tacitly assumes that spin has only one axis of quantization and therefore the number of EPR pairs is equal the number of EPR coincidences. If however there were two axes of quantization, σx, σz, then for the total number of coincidences is double the number of EPR pairs. Therefore the correlation becomes,
By the same argument if all three Pauli spin components are present in an observable, then there are three times more coincidences possible than EPR pairs counted, and the correlation becomes,
These equations mean that each axis contributes a fraction of the correlation. The total correlation is the sum of the correlation from each axis obtained from complementary experiments.
Of course the usual conclusion is that spin only has a single axis of quantization. This is one of Bell’s assumptions in deriving his inequalities. However there is no experimental way to confirm that this is correct.