# Understanding Bell’s theorem part 4: the counterfactual version

I was recently leafing through the great book Quantum Computation and Quantum Information, and noticed that the version of Bell’s theorem it presents is not any of those I wrote about in my three posts about Bell’s theorem, but rather the one I like the least, the counterfactual definiteness version. Annoyed, I checked another great book, Quantum Theory: Concepts and Methods, and saw that it also uses this god-forsaken version1. Nevertheless, I decided that the world has bigger problems to deal with and set it aside. Until I dropped by the Quantum Information lecture given by my boss, David Gross, and saw that he also prefers this undignified version. That’s it! The rise of fascism can wait. I need to set the record straight on counterfactual definiteness.

Before I start ranting about what I find so objectionable about it, I’ll present the proof of this version of Bell’s theorem the best I can. So, what is counterfactual definiteness? It is the assumption that not only the measurement you did in fact do has a definite answer, but also the measurement you did not do has a definite answer. If feels a lot like determinism, but it is not really the same thing, as the assumption is silent about how the result of the counterfactual measurement is determined, it just says that it is. To be more clear, let’s take a look at the data that comes from a real Bell test, the Delft experiment:2

N $x$ $y$ $a$ $b$
1 0 0 1 1
2 0 0 0 0
3 1 1 1 0
4 1 1 0 1
5 0 0 1 1
6 1 1 1 0
7 0 0 1 0
8 1 0 1 1
9 0 0 1 1
10 0 1 0 0

The first column indicates the rounds of the experiment, the $x$ and $y$ columns indicate the settings of Alice and Bob, and the $a$ and $b$ columns the results of their measurements. If one assumes counterfactual definiteness, then definite results must also exist for the measurements that were not made, for example in the first round there must exist results corresponding to the setting $x=1$ for Alice and $y=1$ for Bob. This data would then be just part of some more complete data table, for example this:

N $a_0$ $a_1$ $b_0$ $b_1$
1 1 0 1 1
2 0 1 0 1
3 1 1 0 0
4 1 0 1 1
5 1 1 1 1
6 1 1 0 0
7 1 1 0 0
8 1 1 1 0
9 1 0 1 0
10 0 0 1 0

In this table the column $a_0$ has the results of Alice’s measurements when her setting is $x=0$, and so on. The real data points, corresponding to the Delft experiment, are in black, and I filled in red the hypothetical results for the measurements that were not made.

What is the problem with assuming counterfactual definiteness, then? A complete table certainly exists. But it makes it possible to do something that wasn’t before: we can evaluate the entire CHSH game in every single round, instead of having to choose a single pair of settings. As a quick reminder, to win the CHSH game Alice and Bob must give the same answers when their settings are $(0,0)$, $(0,1)$, or $(1,0)$, and give different answers when their setting is $(1,1)$. In other words, they must have $a_0=b_0$, $a_0=b_1$, $a_1=b_0$, and $a_1 \neq b_1$. But if you try to satisfy all these equations simultaneously, you get that $a_0=b_0=a_1 \neq b_1 = a_0$, a contradiction. At most, you can satisfy 3 out of the 4 equations3. Then since in every row the score in the CHSH game is at most $3/4$, if we sample randomly from each row a pair of $a_x,b_y$ we have that
$\frac14(p(a_0=b_0) + p(a_0=b_1) + p(a_1=b_0) + p(a_1\neq b_1)) \le \frac34,$
which is the CHSH inequality.

But if you select the actual Delft data from each row, the score will be $0.9$. Contradiction? Well, no, because you didn’t sample randomly, but just chose $1$ out of $4^{10}$ possibilities, which would happen with probability $1/4^{10} \approx 10^{-6}$ if you actually did it randomly. One can indeed violate the CHSH inequality by luck, it is just astronomically unlikely.

Proof presented, so now ranting: what is wrong with this version of the theorem? It is just so lame! It doesn’t even explicitly deal with the issue of locality, which is fundamental in all other versions of the theorem4! The conclusion that one takes from it, according to Asher Peres himself, is that “Unperformed experiments have no results”. To which the man in the street could reply “Well, duh, of course unperformed experiments have no results, why are you wasting my time with this triviality?”. It leaves the reader with the impression that they only need to give up the notion that unperformed experiments have results, and they are from then on safe from Bell’s theorem. But this is not true at all! The other proofs of Bell’s theorem still hold, so you still need to give up either determinism or no action at a distance, if you consider the simple version, or unconditionally give up local causality, if you consider the nonlocal version, or choose between generalised local causality and living in a single world, if you consider the Many-Worlds version.

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### 6 Responses to Understanding Bell’s theorem part 4: the counterfactual version

1. Danylo says:

From time to time I think about crazy idea that actually we don’t need to drop local causality. See, when we compare measurements of Alice and Bob – we do it in our own mind, that is locally :) So, there’s still local causality but it goes backwards in time.

Unfortunately, I didn’t find any usage of this idea :)

2. Mateus Araújo says:

Well, but that is true of anything, even actual faster-than-light signalling. The question is not what happens in our minds, but out there in the world.

3. Danylo says:

But if there is no observer-independent facts, then what is objective? What is “out there in the world”? I am completely lost. No joke.

4. Mateus Araújo says:

There are observer-independent facts. What is observer-dependent are the measurement results. And observer-dependent does not mean subjective: even in classical mechanics the speed of a car depends on the reference frame of the observer. But it is a perfectly objective fact that in a given reference frame the speed is 100 km/h.

5. David Gross says:

Hi Mateus,

Since you implied that my teaching choices are worse than the rise of fascism, I feel like I should defend CFD as a way for presenting Bell. :)

The CFD approach obviously does not give you the strongest version of Bell. But it uses only elementary arguments and already tells you something highly non-trivial about Nature.

By “elementary” I mean: Arguments that a mathematically inclined high school student can understand after less than one hour (I’ve tried). Keep in mind: Even the second year bachelor students that attend the Introductory Quantum Mechanics lecture here have at this point not been formally introduced to probability theory. They might have an intuitive understanding of probabilities, but will not be comfortable with manipulations such as the law of total probability (required in the “simple version” you link to in your post).

I mean, just squint and compare your own write-up of the “simple version” with your own write-up of the CFD argument. The former is “math” with capital sigmas, Greek letters, and products of conditional probabilities. The other is a table with black and red 0s and 1s. It is accessible – I think – to a much larger audience.

Your concern was that too much is lost this way. But let’s tell the CFD story fully, and see what we can get out of it using just high-school level arguments.

You quote Peres as saying that “Unperformed experiments have no results”, to which a normal person would reply “Duh”. For a more dramatic effect, proceed this way:

First, remind people of the kitchen-philosophy question of whether or not the moon is there when nobody looks. Say that this seems like one of these problems which you can endlessly discuss over copious quantities of red wine, without getting anywhere. No comes the mind-blowing realization: One can today perform experiments that have a bearing on this question. And their results have implications even more radical than “the moon isn’t there without an observer” would have been (lame…). Look, there’s a conceptually simple experiment that shows that even assuming that “the moon is either there or not, independently of any observations” gets us into trouble. That’s every bit as dramatic as it sounds. Again: to assume that physical quantities (like “is it there?”) have a value that is independent of observations is incompatible with experimental results – regardless of which value one assumes to be realized.

Now, Mateus, I don’t know what kind of people you interact with in the streets – but I have yet to meet someone who said “Duh”. :-)

Of course, then comes the usual disclaimer that you can’t actually do that with the moon, as it is constantly being “observed”, e.g. by interaction with background radiation or sun light. Because of these observations, it’s OK to ascribe a state to it. But in the microscopic world, … you get the drift.

To proof that “one cannot assign values to unmeasured physical quantities”, one produces the argument like you did.

One can than spend some time reiterating that $p\le 3/4$ (or $\le 2$) is a simple identity that holds for any possible assignment. This is the strength of the result! It falsifies an entire huge class of theories in one go (cue talking about Einstein's wasted years…).

After the students have digested this, the subset of the audience that has attained some mathematical maturity can then be presented with the sharpened version that you wrote up in your 2016 post. (Today, I would explain it using the language of causal graphs that Rafael taught me to appreciate). But I think by forgoing the easier CFD argument, one looses a significant part of the audience.

The "worse than fascism" lecture was admittedly a master's level course. So I guess I was just being lazy recycling material I prepared for undergraduate teaching. So, well, in that sense, your point stands.

I've once presented the CFD version at an event for physics teachers. In the audience was a highly experienced colleague who has done some state-of-the-art Bell experiments. After the lecture he complained to me that I forgot to mention the locality assumption. That mildly irked me – because obviously he didn't follow my argument, but instead just compared the main message to the common narrative. Anyway, it shows that if one does present the CFD argument, it's important to say that there are (slightly) more complicated and stronger versions out there, in order to avoid later confusion.

6. Mateus Araújo says:

Hi David,

Thanks for the comment. Ok, to be fair, the “Duh” reaction comes from people that simply read the slogan “Unperformed experiments have no results” without actually understanding what it means; it is indeed nontrivial, shocking even, to realize that one cannot assign any value at all to a measurement result if you haven’t actually done the measurement.

Nevertheless I maintain that, if not worse than fascism, the CFD version is the lamest version of Bell’s theorem. Come on, what feels more dramatic to you, concluding that the world is not deterministic, or that you actually need to measure the position of the Moon for it to exist? Or concluding that the world is not local? Or that there are several worlds?

But besides this lack of drama, it really bothers me the deeply misleading conclusion that many people take from this CFD version: that any physical quantity at all cannot have a value that is independent of observations. No, not any physical quantity, only measurement results. This makes all the difference in the world, because it still allows us to have an observer-independent reality. It is just that this reality is made of wavefunctions, not point particles with well-defined position and momentum.

The actual Moon, the one made of atoms with wavefunctions, would still be definitely there, even if it weren’t being continuously measured. What cannot be either there or not there is this classical model of the Moon as a set of points in phase space.