More on the Statistical Interpretation

After sleeping on it, the suggestion of Pusey et al that two completely different preparation methods can leave a photon in physically identical states no longer seems quite so improbable, particularly when looked at through the somewhat astigmatic lens of the statistical interpretation, which after all says that two photons that have been prepared in identical ways could be in completely different physical states.

Even granted this, there are still things about the argument of “The quantum state cannot be interpreted statistically” by Matthew F. Pusey, Jonathan Barrett and Terry Rudolph that I don’t understand, although my well-known lack of intellectual capability may be holding me back here.

So given two photons, both of which are in physical state λ, one of which has been prepared by… well, I don’t want to use the word “state” here, because that’s one of the things that makes the paper confusing to my small mind. So let’s say the photon has been “ketted”, which is to the best of my knowledge the first use of “ket” as a verb.

Using this terminology we can say that although the photons are both in physical state λ they have been ketted into it by different means, one via |0> and one via |+>. Now according to the statistical interpretation of quantum mechanics, being ketted into a state via the |0> or |+> operation (which are really the <0| and <+| operators, but “braed” is unpronounceable and ugly) results in different statistics for outcomes on future measurements, which is OK because the photons are only in identical states some fraction q of the time, by hypothesis.

Now here’s a thing: so far I don’t see anything that isn’t consistent with both the statistical interpretation or the physical reality interpretation. The statistical interpretation does not entail that two photons prepared differently may be in the same state and the physical reality interpretation does not exclude it: the author’s own coin-flipping example demonstrates the latter, and the existence of real numbers demonstrates the former. It may very well be, under the statistical interpretation, that no two photons are ever in the identical physical state.

So nothing thus far is in any way dependent on either interpretation of the wavefunction. Ergo, it is not clear–at least not to me–what is being contradicted when the authors derive their contradiction.

However, since the photons are in identical physical states we should be able to predict with certainty what the outcome of any measurement on them will be, and this the authors proceed to do. They imagine an apparatus that performs some kind of joint measurement on the two photons, projecting the pair according to the operators:

<E1| = <0,1| + <1, 0|
<E2| = <0,-| + <1, +|
<E3| = <+,1| + <-, 0|
<E4| = <+,-| + <-, +|

where I have suppressed the normalization factors of 1/√2 and fiddled with the notation a bit. Since the interesting cases are all zeros the normalization hardly matters.

Now suppose the photons have been ketted into |0,0>. This will produce <E1|0,0> = 0. But at least q² of the time the photons will both be in λ, and since the authors assume that λ alone (and the state of the apparatus) determines the outcome of the experiment, and λ contains no information about how it got there (by hypothesis) it is possible that the result of <E1| will be non-zero. And the same argument goes for <E2| and so on.

Without going into the detailed statistics of this simple case the authors conclude, “This leads immediately to the desired contradiction. At least q² of the time, the measuring device is uncertain which of the four possible preparation methods was used, and on these occasions it runs the risk of giving an outcome that quantum theory predicts should occur with probability 0.”

They then say: “This argument shows that no physical state λ of the system can be compatible with both of the quantum states |0> and |+>. If the same can be shown for any pair of quantum states |φ₀> and |φ₁>, then the quantum state can be inferred uniquely from λ. In this case, the quantum state is a physical property of the system, and the statistical view is false.”

So the general program of the argument is: if we can show that no physical state λ is compatible with being ketted into different quantum states, then there must be a one-to-one relationship between the ketting process and the physical state. The statistical interpretation says that ketting creates a distribution of physical states that result in the diverse experimental outcomes we see, so it must be false because we have proven that ketting by different operators must create different physical states.

OK, at this point the argument no longer seems completely crazy.

But…

It seems to me there is a fundamental flaw, or loophole: the statistical interpretation does not entail that differently ketted photons will sometimes be in the same physical state λ.

That is, “Preparing a photon in the same quantum state will sometimes result in photons in different physical states” does not imply “Preparing a photon in different quantum states will sometimes result in photons that are in the same physical state”. The former proposition is the statistical interpretation. The latter is the assumption that the author’s argument depends on.

On this basis I would say the argument is certainly intriguing, but equally certainly incomplete. If the state λ is rich in real numbers–and why wouldn’t it be?–it is entirely plausible that differently prepared photons never have the same physical state, and at that point the argument simply does not work, which is really too bad.