# The PBR Theorem explained

The PBR theorem is another theorem of quantum mechanics, which could go alongside Bell’s Theorem and the Kochen-Specker Theorem.  I wrote this explanation in 2011, before the paper was officially published in Nature.  Since then, it’s been recognized as a moderately important theorem, and it has been named after its three authors (Pusey, Barrett, and Rudolph).  But at the time I didn’t really know whether it would become important.

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 were tails, you could figure out which of the methods I used, but if it were 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 different 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.

# My issues with queer-positive Christianity

In the recent discussion of antitheism, Alex Gabriel brought up his personal experience as a queer atheist:

I keep hearing from believers who take great pains to convince me they don’t hate gay people. Jesus never said anything about it, they tell me, and scripture has been misinterpreted, and the real sinners are homophobes, so for heaven’s sake let that be the end of it. I find that conversation hard, mainly because it never feels like it’s meant to be a conversation. I get the sense I’m expected to nod and sympathise, that my role in the discussion is to validate their feelings, not say what I actually think. It’s as if only part of me gets invited to speak: I’m allowed to oppose religious homophobia as a queer person, but not to critique religion in other forms as a queer atheist. I’m not being asked to participate in a dialogue—just to tell Christians what they want to hear.

As a queer atheist, this is an experience I share. And this is worth ranting about.

### A Catholic story

In high school, one of my best friends was gay. I didn’t have the slightest clue about it. I didn’t find out until several years later. He knew it himself, but he didn’t tell people, because my high school was Catholic. Instead, he only told his Catholic parents, and apparently they did not take it well.

# How to actually avoid generalizations

“Don’t make assumptions.” “Criticize the idea, not the person.” “Avoid generalizations.”

These are a few common rules about polite conversation. But they are broken so systematically that it raises the question of whether the rules are any good. One may vocally oppose generalizations, and in the next breath make a sweeping generalization of their own.

It seems that when someone else makes assumptions or generalizations, we hate it. But when we ourselves have the opportunity, we suddenly remember that assumptions and generalizations have some redeeming value after all. And when we next hear someone else make a generalization, we again forget what that value was.

I assert that the value of a generalization is quite simple. People like to state opinions, they like to hear opinions, and they like to use them to inform behavior. They also like to consider opinions and even disagree with them. And if the opinion is stronger by way of generalization, then all the better.

The question for me is not why we like generalizations, but why some generalizations turn out so wrong. What is the source of our aversion? And how can we avoid the kind of generalizations that produce such negative reactions?

# Stop telling me how horrible rape is

[cn: non-graphic discussion of rape, rape apology]

I and most people I know oppose rape and rape culture. One way for people to express this is by saying “Rape is a horrible crime.” While this is true enough, telling me how horrible rape is fails to actually reassure me. In fact, in some cases I find it to be a red flag, something that makes me less inclined to trust you. I do not know if other activists and survivors have similar reactions, but I will provide my own reasons.

Let us first consider a similar statement: “I am not a racist.” While this statement superficially expresses opposition to racism, it is not very convincing for the following reasons:

• Even people who are unambiguously racist can and will say the same thing.
• Rather than expressing dislike of racism, the statement instead expresses anxiety that someone (themselves) would be falsely accused of racism. Rather than doing something to address racism, they are instead creating barriers to other people who might try to address racism.
• The statement shows a misunderstanding of racism as something that is primarily located in a few bad individuals. It makes more sense to talk about racism on a societal level, rather than sorting individuals into the racist or non-racist box.

Each of these three points has an analogue when it comes to saying “Rape is a horrible crime.”

# The nice antitheist strategy

Alex Gabriel has an important essay, “My atheism will not be politically correct“, which discusses antitheism, and discusses the discussion surrounding antitheism. It’s common for many atheists to say that they are no longer antitheists, saying they now realize religion is not the most important problem in the world, and religion sometimes even helps people in times of tragedy. Furthermore, a lot of atheists are jerks and they find more allies among religious people.

Alex’s critique is that all these points, while they may have some merit, are unrelated to the issue of antitheism.  The only question is, would the world be a better place without religion in it?

At the surface, this might just seem to be a disagreement over how we define “antitheism”. But it’s more than that, it’s about how we choose that definition in the first place, and for what purpose. Many atheists choose to define “antitheism” as an extremist position, one that they contrast with their own position. This rhetorical strategy renders oneself more palatable to religious people, basically by throwing other atheists under the bus. Alex prefers a different strategy, where he doesn’t hold his tongue just to make religious people comfortable.

I also unhesitatingly identify as an antitheist, although for not quite the same reasons. I strive for a particular image: a radical queer atheist who is nonetheless very nice. In other words, I aim to break stereotypes. I do not think that this is something everyone needs to do; rather, I myself am well-positioned to do it, so why shouldn’t I do it? And an important part of breaking atheist stereotypes is making it clear that I am in fact an atheist, and why yes I even oppose the “nice” religions and do not think they are very nice at all.

# I liked Richard Carrier, past tense

If you hadn’t heard, Richard Carrier is suing FreeThought Blogs, Skepticon, The Orbit, and several individuals for two million dollars. To learn more, I recommend an episode on the Atheistically Speaking Podcast [eta: correction] about it. If you are interested in helping the defendants, you may contribute to the defense fund here. (Note that I am not personally liable since FreeThought Blogs is incorporated as an LLC.)

The primary subject of the lawsuit is defamation. Since I do not want to repeat any remarks that would risk me getting sued (and apparently merely referring to accusations against Carrier is sufficient), I will simply quote Richard Carrier himself.