## Saturday, November 19, 2011

### New theorem: Quantum states are real

There's a new paper on arxiv called "The quantum state cannot be interpreted statistically".  It has a theorem which proves that, given a few basic assumptions, the quantum state (ie the wavefunction) must be real, rather than a merely statistical object.  Nature has an article which mostly just harps on how "seismic" the paper is.

Nature (correction: the article's author, not Nature itself) compares its importance to Bell's Theorem, which is a very important result indeed from 1964.  Bell's theorem proved that if there were "hidden variables" underneath the quantum state, then entangled particles must be communicating with each other faster than light.  I've explained Bell's theorem in the past.

I felt the news coverage left a lot of unanswered questions.  What do they even mean by the "statistical interpretation" of quantum mechanics?  Roughly how is it proven?  What is the difference between this and Bell's theorem?  I found the answers in the arxiv print, and will attempt to summarize them.

What does the "statistical interpretation" mean?

Let's say that we have two ways of flipping a coin.  The first method leads to a 50% chance of heads, and a 50% chance of tails.  The second method rigs it so the coin always comes up heads.  Let's say that I flipped a coin by one of these two methods, and showed you the result.  If the coin was heads, then you would not know which method I used.

Now say that I have two ways of preparing an electron.  And suppose that you measured the vertical spin component of the electron.  If I use the first method, there is a 50% chance the electron is spin up, and 50% chance spin down.  If I use the second method, the electron will always be spin up.  If I prepared the electron by one of these two methods, and you found that the electron is spin up, you would not know which method I used.

But electron spin is a little trickier than coin flips, because you can measure the spin component in any direction.  Suppose you had tried to measure the horizontal spin component, would you always be able to tell which method I used then?  The answer is no.  But perhaps there is yet another way to measure it?

The authors equate the "statistical interpretation" with the following: Given any two distinct ways to prepare a quantum state, there is a nonzero probability that the result is consistent with either method of preparation.  In other words, no matter what kind of measurement we make, there is a chance that we'll get an outcome that doesn't tell us anything.

What's the difference between this theorem and Bell's Theorem?

Bell's theorem requires that you take many measurements and compile statistics of these measurements.  Once you are confident enough in your statistics, you can show that the probabilities are incompatible with the "hidden variable" view of quantum mechanics.

This new theorem requires only one measurement.  One measurement, and you're done.  (Of course, if you have a noisy experiment, you may need to repeat it to build confidence in your result.)

Of course, the new theorem and Bell's theorem also have a slightly different set of assumptions, and slightly different conclusions.  But I think the primary difference is that the new theorem requires one measurement, while Bell's theorem requires compiling statistics.

Roughly how is it proven?

As an example, let's take the two methods of preparing an electron that I described above.  It turns out that no matter what measurement I make, there is a chance of an outcome that is consistent with either method A or method B.

But we can be tricky.  Let's duplicate the machine that prepares the electrons, and assume that these machines are independent of each other.  Now there are four methods of preparation:
1. A and A (ie both machines use method A)
2. A and B
3. B and A
4. B and B
Suppose that there is a chance that the first machine will produce an electron that is consistent with either method A or method B.  There is also a chance that the second machine will produce an electron that is consistent with either method A or method B.  Therefore, there is a chance that both machines produce electrons which are consistent with any of the four methods.

But it turns out that there is a measurement we can make with four possible outcomes. And each outcome is inconsistent with one of the methods.
• Outcome 1: inconsistent with method 1
• Outcome 2: inconsistent with method 2
• Outcome 3: inconsistent with method 3
• Outcome 4: inconsistent with method 4
What is this special measurement?  It's not straightforward.  In quantum mechanics, we can measure things like position, momentum, and spin.  But we can also measure things like helicity, which tells you whether the spin and momentum are in the same direction, without telling you what direction that is.  Similarly, we can measure whether the electrons have spin in the same direction or opposite directions.  The measurement described in the paper is sort of like that, but more complicated.

The same theorem can be generalized to any two methods of preparing a quantum state.  Suppose that one method always produces a spin up electron, and the other produces a spin up electron 99% of the time.  All you have to do is have N duplicates of the electron-producing machine (in this case, N=15 suffices), and take a special measurement.  No matter the outcome of this measurement is, it is inconsistent with one of the 2^N possible methods of preparation.

The conclusion is that any two distinct quantum states are not just "probably" different, but always different.  You just need a tricky measurement to show it.

Is this paper as groundbreaking as Nature claims?

I don't know.