Yesterday I saw with disappointment a new paper on the arXiv by Hossenfelder and Palmer, Rethinking Superdeterminism. There they argue that physics took a wrong turn when we immediately dismissed superdeterminism; instead it is a solution to the conundrum of nonlocality and the measurement problem.

No. It’s not. It’s a completely sterile idea. I’ll show why, by fleshing out the calculations of the smoking and cancer example they quote in the paper, and then examining the case of the Bell test.

Let’s suppose you do the appropriate randomized trial, and measure the conditional probabilities1

\[ p(\text{cancer}|\text{smoke}) = 0.15\quad\text{and}\quad p(\text{cancer}|\neg\text{smoke}) = 0.01,\]a pretty damning result. A tobacco company objects to the conclusion, saying that the genome of the subjects was correlated with whether you forced them to smoke2, such that you put more people predisposed to have cancer in the smoking group.

It works like this: the law of total probability says that

\[ p(a|x) = \sum_\lambda p(\lambda|x)p(a|x,\lambda),\] where in our case $a \in \{\text{cancer},\neg\text{cancer}\}$, $x \in \{\text{smoke},\neg\text{smoke}\}$, and $\lambda \in \{\text{predisposed},\neg\text{predisposed}\}$ is the hidden variable, in this case the genome determining whether the person will have cancer anyway. The tobacco company says that your results are explained by the conspiracy $p(\text{predisposed}|\text{smoke}) = 0.15$ and $p(\text{predisposed}|\neg\text{smoke}) = 0$, from which we can calculate the actual cancer rates to be

\begin{gather*}

p(\text{cancer}|\text{smoke},\neg\text{predisposed}) = 0 \\

p(\text{cancer}|\neg\text{smoke},\neg\text{predisposed}) = 0.01,

\end{gather*}so the same data indicates that smoking *prevents* cancer! If you assume, though, that $p(\text{predisposed}|\text{smoke}) = p(\text{predisposed}|\neg\text{smoke})$, then the absurd conclusion is impossible.

With this example I want to illustrate two points: first, that assuming $p(\lambda|x) \neq p(\lambda)$ is just a generic excuse to dismiss any experimental result that you find inconvenient, be it that smoking causes cancer or that Bell inequalities are violated. Second, that without assuming $p(\lambda|x) = p(\lambda)$ 3 you can’t conclude anything from your data.

In their paper, Hossenfelder and Palmer dismiss this example as merely classical reasoning that is not applicable to quantum mechanics. It’s not. One can always use the law of total probability to introduce a hidden variable to explain away *any* correlation, whether it was observed in classical or quantum contexts. Moreover, they claim that while $p(\lambda|x) = p(\lambda)$ is plausible in classical contexts, it shouldn’t be assumed in quantum contexts. This is laughable. I find it perfectly conceivable that tobacco companies would engage in conspiracies to fake results related to smoking and cancer, but to think that Nature would engage in a conspiracy to fake the results of Bell tests? Come on.

They also propose an experiment to test their superdeterministic idea. It is nonsense, as any experiment about correlations is without the assumption that $p(\lambda|x) = p(\lambda)$. Of course, they are aware of this, and they assume that $p(\lambda|x) = p(\lambda)$ would hold for their experiment, just not for Bell tests. Superdeterminism for thee, not for me. They say that when $x$ is a measurement setting, changing it will necessarily cause a large change in the state $\lambda$, but if you don’t change the setting, the state $\lambda$ will not change much. Well, but what is a measurement setting? That’s human category, not a fundamental one. I can just as well say that the time the experiment is made is the setting, and therefore repetitions of the experiment done at different times will probe different states $\lambda$, and again you can’t conclude anything about it.

Funnily, they say that “…one should make measurements on states prepared as identically as possible with devices as small and cool as possible in time-increments as small as possible.” Well, doesn’t this sound like a very common sort of experiment? Shouldn’t we have observed deviations from the Born rule a long time ago then?

Let’s turn to how superdeterministic models dismiss violations of Bell inequalities. They respect *determinism* and *no action at a distance*, but violate *no conspiracy*, as I define here. The probabilities can then be decomposed as

\[ p(ab|xy) = \sum_\lambda p(\lambda|xy)p(a|x,\lambda)p(b|y,\lambda),\]and the dependence of the distribution of $\lambda$ on the settings $x,y$ is used to violate the Bell bound. Unfortunately Hossenfelder and Palmer4 do not specify $p(\lambda|xy)$, so I have to make something up. It is trivial to reproduce the quantum correlations if we let $\lambda$ be a two-bit vector, $\lambda \in \{(0,0),(0,1),(1,0),(1,1)\}$, and postulate that it is distributed as

\[p((a,b)|xy) = p^Q(ab|xy),\] where $p^Q(ab|xy)$ is the correlation predicted by quantum mechanics for the specific experiment, and the functions $p(a|x,\lambda)$ and $p(b|y,\lambda)$ are given by

\[p(a|x,(a’,b’)) = \delta_{a,a’}\quad\text{and}\quad p(b|y,(a’,b’)) = \delta_{b,b’}.\] For example, if $p^Q(ab|xy)$ is the correlation maximally violating the CHSH inequality, we would need $\lambda$ to be distributed as

\[ p((a,b)|xy) = \frac14\left(1+\frac1{\sqrt2}\right)\delta_{a\oplus b,xy}+\frac14\left(1-\frac1{\sqrt2}\right)\delta_{a\oplus b,\neg(xy)}.\]The question is, why? In the quantum mechanical case, this is explained by the quantum state being used, the dynamical laws, the observable being measured, and the Born rule. In the superdeterministic theory, what? I have never seen this distribution be even mentioned, let alone justified.

More importantly, why should this distribution be such that the superdeterministic correlations reproduce the quantum ones? For example, why couldn’t $\lambda$ be distributed like

\[ p((a,b)|xy) = \frac12\delta_{a\oplus b,xy},\] violating the Tsirelson bound?5 Even worse, why should the superdeterministic distributions respect even no-signalling? What stops $\lambda$ being distributed like

\[ p((a,b)|xy) = \delta_{a,y}\delta_{b,x}?\]

In their paper, Hossenfelder and Palmer define a superdeterministic theory as a local, deterministic, reductionist theory that reproduces quantum mechanics approximately. I’m certain that such a theory will never exist. Its dynamical equations would need to correlate 97,347,490 human choices with the states of atoms and photons in 12 laboratories around the planet to reproduce the results of the BIG Bell test. Its dynamical equations would need to correlate the frequency of photons emitted by other stars in the Milky Way with the states of photons emitted by a laser in Vienna to reproduce the results of the Cosmic Bell test. Its dynamical equations would need to correlate the bits of a file of the movie “Monty Python and the Holy Grail” with the state of photons emitted by a laser in Boulder to reproduce the results of the NIST loophole-free Bell test. It cannot be done.