Indeterminism and Undecidability

Abstract

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born [8], can be proved from Chaitin’s follow-up to Gödel’s (first) incompleteness theorem. In comparison, Bell’s [2] theorem as well as the so-called free will theorem-originally due to Heywood and Redhead [44]-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in ’t Hooft’s cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e. almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories.

Dedicated to the memory of Michael Redhead (1929–2020).

Appendices

Appendix A: The Born Rule

The Born measure is a probability measure $\mu _a$ on the spectrum $\sigma (a)$ of a (bounded) self-adjoint operator a on some Hilbert space H, defined as follows by any state $\omega$ on B(H)³⁰:

Theorem A.1

Let H be a Hilbert space, let $a^*=a\in B(H)$, and let $\omega$ be a state on B(H). There exists a unique probability measure $\mu _a$ on the spectrum $\sigma (a)$ of a such that

$$\begin{aligned} \omega (f(a))=\int _{\sigma (a)} d\mu _a(\lambda )\, f(\lambda ), \,\, \mathrm { for\, all }\,\, f \in C(\sigma (a)). \end{aligned}$$

(3.5.1)

The Born measure is a mathematical construction; what is its relationship to experiment? This relationship must be the source of the (alleged) randomness of quantum mechanics, for the Schrödinger equation is deterministic. We start by postulating, as usual, that $\mu _a(\Delta )$ is the (single case) “probability” that measurement of the observable a in the state $\omega$ (which jointly give rise to the pertinent Born measure $\mu _a$) gives a result $\lambda \in \Delta \subset \sigma (a)$. Here I identify single-case “probabilities” with numbers (consistent with the probability calculus) provided by theory, upon which long-run frequencies provide empirical evidence for the theory in question, but do not define probabilities. The Born measure is a case in point: these probabilities are theoretically given, but have to be empirically verified by long runs of independent experiments. In other words, by the results reviewed below such experiments provide numbers whose role it is to test the Born rule as a hypothesis. This is justified by the following sampling theorem (strong law of large numbers): for any (measurable) subset $\Delta \subset \sigma (a)$ and any sequence $(x_n)\in \sigma (a)^{{\mathbb N}}$ we have $\mu _a^{\infty }$-almost surely:

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{1}{N}(1_{\Delta }(x_1)+\cdots + 1_{\Delta }(x_N))= \mu _a(\Delta ). \end{aligned}$$

(3.5.2)

Proof of Theorem 3.4.1. Let $a=a^*\in B(H)$, where H is a Hilbert space and B(H) is the algebra of all bounded operators on H, and let $\sigma (a)$ be the spectrum of a. For simplicity (and since this is enough for our applications, where $H={\mathbb C}^2$) I assume $\dim (H)<\infty$, so that $\sigma (a)$ simply consists of the eigenvalues $\lambda _i$ of a (which may be degenerate). Let us first consider a finite number N of identical measurements of a (a “run”). The first option in the theorem corresponds to a simultaneous measurement of the commuting operators

$$\begin{aligned} a_1&=a\otimes 1_H\otimes \cdots \otimes 1_H; \end{aligned}$$

(3.5.3)

$$\begin{aligned}&\cdots \nonumber \\ a_N&=1_H\otimes \cdots \otimes 1_H\otimes a, \end{aligned}$$

(3.5.4)

all defined on the N-fold tensor product $H^N\equiv H^{\otimes N}$ of H with itself.³¹ To put this in a broader perspective, consider any set $(a_1, \ldots , a_N)\equiv \underline{a}$ of commuting operators on any Hilbert space K (of which (3.5.3)–(3.5.4) is obviously a special case with $K=H^N$). These operators have a joint spectrum $\sigma (\underline{a})$, whose elements are the joint eigenvalues $\underline{\lambda }=(\lambda _1, \ldots , \lambda _N)$, defined by the property that there exists a nonzero joint eigenvector $\psi \in K$ such that $a_i\psi =\lambda _i\psi$ for all $i=1, \ldots , N$; clearly,

$$\begin{aligned} \sigma (\underline{a})=\{\underline{\lambda }\in \sigma (a_1)\times \cdots \times \sigma (a_n)\mid e_{\underline{\lambda }}\equiv e^{(1)}_{\lambda _1}\cdots e^{(n)}_{\lambda _n} \ne 0\} \subseteq \sigma (a_1)\times \cdots \times \sigma (a_N), \end{aligned}$$

(3.5.5)

where $e^{(i)}_{\lambda _i}$ is the spectral projection of $a_i$ on the eigenspace for the eigenvalue $\lambda _i\in \sigma (a_i)$. Von Neumann’s Born rule for the probability of finding $\underline{\lambda }\in \sigma (\underline{a})$ then simply reads

$$\begin{aligned} p_{\underline{a}}(\underline{\lambda })=\omega (e_{\underline{\lambda }}), \end{aligned}$$

(3.5.6)

where $\omega$ is the state on B(K) with respect to which the Born probability is defined.³² If $\dim (K)<\infty$, as I assume, we always have $\omega (a)=\text{ Tr }\,(\rho a)$ for some density operator $\rho$, and for a general Hilbert space K this is the case iff the state $\omega$ is normal on B(K). For (normal) pure states we have $\rho =|\psi \rangle \langle \psi |$ for some unit vector $\psi \in K$, in which case

$$\begin{aligned} p_{\underline{a}}(\underline{\lambda })=\langle \psi ,e_{\underline{\lambda }}\psi \rangle . \end{aligned}$$

(3.5.7)

The Born rule (3.5.6) is similar to the single-operator case ([52], §4.1)³³: the continuous functional calculus gives a Gelfand isomorphism of commutative C*-algebras

$$\begin{aligned} C^*(\underline{a},1_K)\cong C(\sigma (\underline{a})), \end{aligned}$$

(3.5.8)

under which the restriction of the state $\omega$, originally defined on B(K), to its commutative C*-subalgebra $C^*(\underline{a})$ defines a probability measure $\mu _{\underline{a}}$ on the joint spectrum $\sigma (\underline{a})$ via the Riesz isomorphism. This is the Born measure, whose probabilities are given by (3.5.6). For the case (3.5.3)–(3.5.4) we have equality in (3.5.5); since in that case $\sigma (a_i)=\sigma (a)$, we obtain

$$\begin{aligned} \sigma (\underline{a})=\sigma (a)^N, \end{aligned}$$

(3.5.9)

and therefore, for all $\lambda _i\in \sigma (a)$ and states $\omega$ on $B(H^N)$, the Born rule (3.5.6) becomes

$$\begin{aligned} p_{\underline{a}}(\lambda _1, \ldots , \lambda _N)=\omega (e_{\lambda _1}\otimes \cdots \otimes e_{\lambda _N}). \end{aligned}$$

(3.5.10)

Now take a state $\omega _1$ on B(H). Reflecting the idea that $\omega$ is the state on $B(H^N)$ in which N independent measurements of $a\in B(H)$ in the state $\omega _1$ are carried out, choose

$$\begin{aligned} \omega =\omega _1^N, \end{aligned}$$

(3.5.11)

the state on $B(H^N)$ defined by linear extension of its action on elementary tensors:

$$\begin{aligned} \omega _1^N(b_1\otimes \cdots \otimes b_n)=\omega _1(b_1) \cdots \omega _N(b_N). \end{aligned}$$

(3.5.12)

It follows that

$$\begin{aligned} \omega ^N(e_{\lambda _1}\otimes \cdots \otimes e_{\lambda _N})=\omega _1(e_{\lambda _1})\cdots \omega _1(e_{\lambda _N})=p_a(\lambda _1)\cdots p_a(\lambda _N), \end{aligned}$$

(3.5.13)

so that the joint probability of the outcome $(\lambda _1, \ldots , \lambda _N)\in \sigma (\underline{a})$ is simply

$$\begin{aligned} p_{\vec {a}}(\lambda _1, \ldots , \lambda _N)=p_a(\lambda _1)\cdots p_a(\lambda _N). \end{aligned}$$

(3.5.14)

Since these are precisely the probabilities for option 2 (i.e. the Bernoulli process), i.e.,

$$\begin{aligned} \mu _{\underline{a}}=\mu _a^N, \end{aligned}$$

(3.5.15)

this proves the claim for $N<\infty$. To describe the limit $N\rightarrow \infty$, let B be any C*-algebra with unit $1_B$; below I take $B=B(H)$, $B=C^*(a,1_H)$, or $B=C(\sigma (a))$. We now take

$$\begin{aligned} A_N=B^{\otimes N}, \end{aligned}$$

(3.5.16)

the N-fold tensor product of B with itself.³⁴ The special cases above may be rewritten as

$$\begin{aligned} B(H)^{\otimes N}&\cong B(H^N);\end{aligned}$$

(3.5.17)

$$\begin{aligned} C^*(a,1_H)^{\otimes N}&\cong C^*(a_1, \ldots , a_N,1_{H^N}); \end{aligned}$$

(3.5.18)

$$\begin{aligned} C(\sigma (a))^{\otimes N}&\cong C(\sigma (a) \times \cdots \times \sigma (a)), \end{aligned}$$

(3.5.19)

with N copies of H and $\sigma (a)$, respectively, and in (3.5.18) the $a_i$ are given by (3.5.3)–(3.5.4). We may then wonder if these algebras have a limit as $N\rightarrow \infty$. They do, but it is not unique and depends on the choice of observables, that is, of the infinite sequences $\mathbf {a}=(a_1,a_2, \ldots )$, with $a_N\in A_N$, that are supposed to have a limit. One possibility is to take sequences $\mathbf {a}$ for which there exists $M\in {\mathbb N}$ and $a_M\in A_M$ such that for each $N\ge M$,

$$\begin{aligned} a_N=a_M \otimes 1_B\cdots \otimes 1_B, \end{aligned}$$

(3.5.20)

with $N-M$ copies of $1_B$. On that choice, one obtains the infinite tensor product $B^{\otimes \infty }$, see Landsman [52], §C.14. The limit of (3.5.17) in this sense is $B(H^{\otimes \infty })$, where $H^{\otimes \infty }$ is von Neumann’s ‘complete’ infinite tensor product of Hilbert spaces,³⁵ in which $C^*(a,1_H)^{\otimes \infty }$ is the C*-algebra generated by $(a_1,a_2, \ldots )$ and the unit on $H^{\otimes \infty }$. The limit of (3.5.19) is

$$\begin{aligned} C(\sigma (a))^{\otimes \infty }\cong C(\sigma (a)^{{\mathbb N}}), \end{aligned}$$

(3.5.21)

where $\sigma (a)^{{\mathbb N}}$, which we previously saw as a measure space (as a special case of $X^{{\mathbb N}}$ for general compact Hausdorff spaces X), is now seen as a topological space with the product topology, in which it is compact.³⁶ As in the finite case, we have an isomorphism

$$\begin{aligned} C^*(a,1_H)^{\otimes \infty } \cong C(\sigma (a))^{\otimes \infty }, \end{aligned}$$

(3.5.22)

and hence, on the given identifications, we obtain an isomorphism of C*-algebras

$$\begin{aligned} C^*(a_1, a_2, \ldots , 1_{H^{\otimes \infty }})\cong C(\sigma (a)^{{\mathbb N}}). \end{aligned}$$

(3.5.23)

It follows from the definition of the infinite tensor products used here that each state $\omega _1$ on B defines a state $\omega _1^{\infty }$ on $B^{\otimes \infty }$. Take $B=B(H)$ and restrict $\omega _1^{\infty }$, which a priori is a state on $B(H^{\otimes \infty })$, to its commutative C*-subalgebra $C^*(a_1, a_2, \ldots , 1_{H^{\otimes \infty }})$. The isomorphism (3.5.23) then gives a probability measure $\mu _{\underline{a}}$ on the compact space $\sigma (a)^{{\mathbb N}}$, where the label $\underline{a}$ now refers to the infinite set of commuting operators $(a_1, a_2, \ldots )$ on $H^{\otimes \infty }$. To compute this measure, I use (3.5.1) and the fact that by construction functions of the type

$$\begin{aligned} f(\lambda _1, \lambda _2, \ldots )=f^{(N)}(\lambda _1, \ldots , \lambda _N), \end{aligned}$$

(3.5.24)

where $N<\infty$ and $f^{(N)}\in C(\sigma (a)^N)$, are dense in $C(\sigma (a)^{{\mathbb N}})$ (with respect to the appropriate supremum-norm), and that in turn finite linear combinations of factorized functions $f^{(N)}(\lambda _1, \ldots , \lambda _N)=f_1(\lambda _1)\cdots f_N(\lambda _N)$ are dense in $C(\sigma (a)^{N})$. It follows from this that

$$\begin{aligned} \mu _{\underline{a}}=\mu _a^{\infty }. \end{aligned}$$

(3.5.25)

Since this generalizes (3.5.15) to $N=\infty$, the proof of Theorem 3.4.1 is finished. $\Box$

Appendix B: 1-Randomness

In what follows, the notion of 1-randomness, originally defined by Martin-Löf in the setting of constructive measure theory, will be explained through an equivalent definition in terms of Kolmogorov complexity.³⁷ We assume basic familiarity with the notion of a computable function $f:{\mathbb N}\rightarrow {\mathbb N}$, which may be defined through recursion theory or Turing machines.

A string is a finite succession of bits (i.e. zeros and ones). The length of a string $\sigma$ is denoted by $|\sigma |$. The set of all strings of length N is denoted by $\underline{2}^N$, where $\underline{2}=\{0,1\}$, and

$$\begin{aligned} \underline{2}^*=\bigcup _{N\in {\mathbb N}}\underline{2}^N \end{aligned}$$

(3.5.26)

denotes the set of all strings. The Kolmogorov complexity $K(\sigma )$ of $\sigma \in \underline{2}^*$ is defined, roughly speaking, as the length of the shortest computer program that prints $\sigma$ and then halts. We then say, again roughly, that $\sigma$ is Kolmogorov random if this shortest program contains all of $\sigma$ in its code, i.e. if the shortest computable description of $\sigma$ is $\sigma$ itself.

To make this precise,³⁸ fix some universal prefix-free Turing machine U, seen as performing a computation on input $\tau$ (in its prefix-free domain) with output $U(\tau )$, and define

$$\begin{aligned} K(\sigma )=\min _{\tau \in \underline{2}^*} \{|\tau |: U(\tau )=\sigma \}. \end{aligned}$$

(3.5.27)

The function $K:\underline{2}^*\rightarrow {\mathbb N}$ is uncomputable, but that doesn’t mean it is ill-defined. The choice of U affects $K(\sigma )$ up to a $\sigma$-independent constant, and to take this dependency into account we state certain results in terms of the “big-O” notation familiar from Analysis.³⁹ For example, if $\sigma$ is easily computable, like the first $|\sigma |$ binary digits of $\pi$, then

$$\begin{aligned} K(\sigma )=O(\log |\sigma |), \end{aligned}$$

(3.5.28)

with the logarithm in base 2 (as only the length of $\sigma$ counts). However, a random $\sigma$ has

$$\begin{aligned} K(\sigma )=|\sigma |+ O(\log |\sigma |). \end{aligned}$$

(3.5.29)

We say that $\sigma$ is c-Kolmogorov random, for some $\sigma$-independent constant $c\in {\mathbb N}$, if

$$\begin{aligned} K(\sigma )\ge |\sigma |-c. \end{aligned}$$

(3.5.30)

Simple counting arguments show that as $|\sigma |=N$ gets large, the overwhelming majority of strings in $\underline{2}^N$ (and hence in $\underline{2}^*$) is c-random.⁴⁰ The following theorem, which might be called Chaitin’s first incompleteness theorem, therefore shows that randomness is elusive⁴¹:

Theorem B.1

For any sound mathematical theory T containing enough arithmetic there is a constant $C\in {\mathbb N}$ such that T cannot prove any sentence of the form $K(\sigma )>C$ (although infinitely many such sentences are true), and as such T can only prove (Kolmogorov) randomness of finitely many strings (although infinitely many strings are in fact random).

The proof is quite complicated in its details but it is based on the existence of a computably enumerable (c.e.) list $\mathsf {T}=(\tau _1,\tau _2,\ldots )$ of the theorems of T, and on the fact that after Gödelian encoding by numbers, theorems of any given grammatical form can be computably searched for in this list and will eventually be found. In particular, there exists a program P (running on the universal prefix-free Turing machine U used to define $K(\cdot )$) such that P(n) halts iff there exists a string $\sigma$ for which $K(\sigma )>n$ is a theorem of T. If there is such a theorem the output is $P(n)=\sigma$, where $\sigma$ appears in the first such theorem of the kind (according to the list $\mathsf {T}$). By definition of $K(\cdot )$, this means that

$$\begin{aligned} K(\sigma )\le |P|+|n|. \end{aligned}$$

(3.5.31)

Now suppose that no C as in the above statement of the theorem exists. Then there is $n\in {\mathbb N}$ large enough that $n>|P|+|n|$ and there is a string $\sigma \in \underline{2}^*$ such that T proves $K(\sigma )>n$. Since T is sound this is actually true,⁴² which gives a contradiction between

$$\begin{aligned} K(\sigma )>n>|P|+|n|;&K(\sigma )\le |P|+|n|. \end{aligned}$$

(3.5.32)

Note that this proof shows that a proof in T of $K(\sigma )>n$ (if true) would also identify $\sigma$.

As an idealization of a long (binary) string, a (binary) sequence $x=x_1x_2\cdots$ is an infinite succession of bits, i.e. $x\in \underline{2}^{{\mathbb N}}$, with finite truncations $x_{|N}=x_1\cdots x_N\in \underline{2}^N$ for each $N\in {\mathbb N}$. We then call x Levin–Chaitin random if each truncation of x is c-Kolmogorov random for some c, that is, if there exists $c\in {\mathbb N}$ such that

$$\begin{aligned} K(x_{|N})\ge N - c \end{aligned}$$

for each $N\in {\mathbb N}$. Equivalently,⁴³ a sequence x is Levin–Chaitin random if eventually $K(x_{|N})>>N$, in that

$$\begin{aligned} \lim _{N\rightarrow \infty } (K(x_{|N})-N)=\infty . \end{aligned}$$

(3.5.33)

Apart from having the same intuitive pull as Kolmogorov randomness (of strings), this definition gains from the fact that it is equivalent to two other appealing notions of randomness, namely patternlessness and unpredictability, both also defined computationally. In view of these equivalences we simply call a Levin–Chaitin random sequence 1-random.⁴⁴

A sequence $x\in \underline{k}^{{\mathbb N}}$ is Borel normal in base k if each string $\sigma$ has frequency $k^{-|\sigma |}$ in x. Any hope of defining randomness as Borel normality in base 10 is blocked by Champernowne’s number $0123456789101112131 \ldots$, which is Borel normal but clearly not random in any reasonable sense (this is also true in base 2). The decimal expansion of $\pi$ is also conjectured to be Borel normal in base 10 (with huge numerical support), although $\pi$ clearly is not random either. However, Borel normality seems a desirable property of truly random numbers on any good definition, and so we are fortunate to have:

Proposition B.2

A 1-random sequence is Borel normal (in base 2, but in fact in any base) and hence (“monkey typewriter theorem”) contains any finite string infinitely often.⁴⁵

Another desirable property comes from the following theorem due to Martin-Löf, in which P is the 50-50 probability on $\{0,1\}$ and $P^{\infty }$ is the induced probability measure on $\underline{2}^{{\mathbb N}}$:

Theorem B.3

With respect to $P^{\infty }$ almost every outcome sequence $x\in \underline{2}^{{\mathbb N}}$ is 1-random.

This implies that the 1-random sequences form an uncountable subset of $\underline{2}^{{\mathbb N}}$,⁴⁶ although topologically this subset is meagre (i.e. Baire first category).⁴⁷ Chaitin’s incompleteness theorem for (finite) strings has the following counterpart for (infinite) sequences:

Theorem B.4

If $x\in \underline{2}^{{\mathbb N}}$ is 1-random, then ZFC (or any sufficiently comprehensive mathematical theory T meant in Theorem B.1) can compute only finite many digits of x.⁴⁸

This clearly excludes defining a 1-random number by somehow listing its digits, but some can be described by a formula. One example is Chaitin’s $\Omega$, or more precisely $\Omega _U$,⁴⁹ which is the halting probability of some fixed universal prefix-free Turing machine U, given by

$$\begin{aligned} \Omega _U:=\sum _{\tau \in \underline{2}^*\mid U(\tau )\downarrow }2^{-|\tau |}. \end{aligned}$$

(3.5.34)

Appendix C: Bell’s Theorem and Free Will Theorem

In support of the analysis of hidden variable theories in the main text, this appendix reviews Bell’s [2] theorem and the free will theorem, streamlining earlier expositions ([17, 52], Chap. 6) and leaving out proofs and other adornments.⁵⁰ In the specific context of ’t Hooft’s theory (where the measurement settings are determined by the hidden state) and Bohmian mechanics (where they are not, as in the original formulation of Bell’s theorem and in most hidden variable theories) an advantage of my approach is that both free (uncorrelated) und correlated settings fall within its scope; the former are distinguished from the latter by an independence assumption.⁵¹

As a warm-up I start with a version of the Kochen–Specker theorem, whose logical form is very similar to Bell’s [2] theorem and the free will theorem, as follows:

Theorem C.1

Determinism, qm, non-contextuality, and free choice are contradictory.

Of course, this unusual formulation hinges on the precise meaning of these terms.

• determinism is the conjunction of the following two assumptions.

  1. There is a state space X with associated functions $A: X\rightarrow S$ and $L:X\rightarrow O$, where S is the set of all possible measurement settings Alice can choose from, namely a suitable finite set of orthonormal bases of ${\mathbb R}^3$ (11 well-chosen bases will do to arrive at a contradiction),⁵² and O is some set of possible measurement outcomes. Thus some $x\in X$ determines both Alice’s setting $a=A(x)$ and her outcome $\alpha =L(x)$.

  2. There exists some set $\Lambda$ and an additional function $H:X\rightarrow \Lambda$ such that

  $$\begin{aligned} L=L(A,H), \end{aligned}$$

  (3.5.35)

  in the sense that for each $x\in X$ one has $L(x)=\hat{L}(A(x),H(x))$ for a certain function $\hat{L}:S \times \Lambda \rightarrow O$. This self-explanatory assumption just states that each measurement outcome $L(x)=\hat{L}(a,\lambda )$ is determined by the measurement setting $a=A(x)$ and the “hidden” variable or state $\lambda =H(x)$ of the particle undergoing measurement.

• qm fixes $O=\{(0,1,1), (1,0,1), (1,1,0)\}$, which is a non-probabilistic fact of quantum mechanics with overwhelming (though indirect) experimental support.

• non-contextuality stipulates that the function $\hat{L}$ just introduced take the form

  $$\begin{aligned} \hat{L}((\vec {e}_1,\vec {e}_2,\vec {e}_3),\lambda )=(\tilde{L}(\vec {e}_1,\lambda ), \tilde{L}(\vec {e}_2,\lambda ), \tilde{L}(\vec {e}_3,\lambda )), \end{aligned}$$

  (3.5.36)

  for a single function $\tilde{L}:S^2\times \Lambda \rightarrow \{0,1\}$ that also satisfies $\tilde{L}(-\vec {e},\lambda )=\tilde{L}(\vec {e},\lambda )$.⁵³

• free choice finally states that the following function is surjective:

  $$\begin{aligned} A\times H:X\rightarrow S\times \Lambda ;&x\mapsto (A(x),H(x)). \end{aligned}$$

  (3.5.37)

  In other words, for each $(a,\lambda )\in S\times \Lambda$ there is an $x\in X$ for which $A(x)=a$ and $H(x)=\lambda$. This makes A and H “independent” (or: makes a and $\lambda$ free variables).

See Landsman [52], §6.2 for a proof of the Kochen–Specker theorem in this language.⁵⁴

Bell’s [2] theorem and the free will theorem both take a similar generic form, namely:

Theorem C.2

Determinism, qm, local contextuality, and free choice, are contradictory.

Once again, I have to explain what these terms exactly mean in the given context.

• determinism is a straightforward adaptation of the above meaning to the bipartite “Alice and Bob” setting. Thus we have a state space X with associated functions

  $$\begin{aligned} A: X\rightarrow S;&B: X\rightarrow S;&L:X\rightarrow O&R: X\rightarrow O, \end{aligned}$$

  (3.5.38)

  where S, the set of all possible measurement settings Alice and Bob can each choose from, differs a bit between the two theorems: for the free will theorem it is the same as for the Kochen–Specker theorem above, as is the set O of possible measurement outcomes, whereas for Bell’s theorem (in which Alice and Bob each measure a 2-level system), S is some finite set of angles (three is enough), and $O=\{0,1\}$.

  • In the free will case, these functions and the state $x\in X$ determine both the settings $a=A(x)$ and $b=B(x)$ of a measurement and its outcomes $\alpha =L(x)$ and $\beta =R(x)$ for Alice on the Left and for Bob on the Right, respectively.

  • All of this is also true in the Bell case, but since his theorem relies on impossible measurement statistics (as opposed to impossible individual outcomes), one in addition assumes a probability measure $\mu$ on X.⁵⁵

  Furthermore, there exists some set $\Lambda$ and some function $H:X\rightarrow \Lambda$ such that

  $$\begin{aligned} L=L(A,B,H);&R=R(A,B,H), \end{aligned}$$

  (3.5.39)

  in the sense that for each $x\in X$ one has functional relationships

  $$\begin{aligned} L(x)=\hat{L}(A(x),B(x),H(x));&R(x)=\hat{R}(A(x),B(x),H(x)), \end{aligned}$$

  (3.5.40)

  for certain functions $\hat{L}:S \times S\times \Lambda \rightarrow O$ and $\hat{R}:S \times S\times \Lambda \rightarrow O$.

• qm reflects elementary quantum mechanics of correlated 2-level and 3-level quantum systems for the Bell and the free will cases, respectively, as follows⁵⁶:

  • In the free will theorem, $O=\{(0,1,1), (1,0,1), (1,1,0)\}$ is the same as for the Kochen–Specker theorem. In addition perfect correlation obtains: if $a=(\vec {e}_1,\vec {e}_2,\vec {e}_3)$ is Alice’s orthonormal basis and $b=(\vec {f}_1,\vec {f}_2,\vec {f}_3)$ is Bob’s, one has

    $$\begin{aligned} \vec {e}_i=\vec {f}_j\, \Rightarrow \, \hat{L}_i(a,b,z)=\hat{R}_j(a,b,z), \end{aligned}$$

    (3.5.41)

    where $\hat{L}_i, \hat{R}_j: S \times S\times \Lambda \rightarrow \{0,1\}$ are the components of $\hat{L}$ and $\hat{R}$, respectively. Finally,⁵⁷ if $(a',b'$) differs from (a, b) by changing the sign of any basis vector,

    $$\begin{aligned} \hat{L}(a',b',\lambda )=\hat{L}(a,b,\lambda );&\hat{R}(a',b',\lambda )=\hat{R}(a,b,\lambda ). \end{aligned}$$

    (3.5.42)

  • In Bell’s theorem, $O=\{0,1\}$, and the statistics for the experiment is reproduced as conditional joint probabilities given by the measure $\mu$ through

    $$\begin{aligned} P(L\ne R|A=a,B=b)=\sin ^2(a-b). \end{aligned}$$

    (3.5.43)

• local contextuality, which replaces and weakens non-contextuality, means that

  $$\begin{aligned} L(A,B,H)=L(A,H);&R(A,B,H)=G(B,H). \end{aligned}$$

  (3.5.44)

  In words: Alice’s outcome given $\lambda$ does not depend on Bob’s setting, and vice versa.

• free choice is an independence assumption that looks differently for both theorems:

  • In the free will theorem it means that each $(a,b,\lambda )\in S\times S\times \Lambda$ is possible in that there is an $x\in X$ for which $A(x)=a$, $B(x)=b$, and $H(x)=\lambda$.

  • In Bell’s theorem, (A, B, H) are probabilistically independent relative to $\mu$.⁵⁸

This concludes the joint statement of the free will theorem and Bell’s [2] theorem in the form we need for the main text. The former is proved by reduction to the Kochen–Specker theorem, whilst the latter follows by reduction to the usual version of Bell’s theorem via the free choice assumption; see Landsman [52], Chap. 6 for details.

For our purposes these theorems are equivalent, despite subtle differences in their assumptions. Bell’s theorem is much more robust in that it does not rely on perfect correlations (which are hard to realize experimentally), and in addition it requires almost no input from quantum theory. On the other hand, Bell’s theorem uses probability theory in a highly nontrivial way: like the hidden variable theories it is supposed to exclude it relies on the possibility of fair sampling of the probability measure $\mu$. The factorization condition defining probabilistic independence passes this requirement of fair sampling on to both the hidden variable and the settings, which brings us back to the main text.

Different parties may now be identified by the assumption they drop: Copenhagen quantum mechanics rejects determinism, Valentini [86] rejects the Born rule and hence qm, Bohmians rejects local contextuality, and finally ’t Hooft rejects free choice. However, as we argue in the main text, even the latter two camps do not really have a deterministic theory underneath quantum mechanics because of their need to randomly sample the probability measure they must use to recover the predictions of quantum mechanics.