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Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

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This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of \({\mathbb {C}} \) replaced by \({\mathbb {Z}}_{2}\). Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts all required the brackets to take values in \({\mathbb {Z}}_{2}\). But the usual QM brackets \(\left\langle \psi |\varphi \right\rangle \) give the “overlap” between states \(\psi \) and \(\varphi \), so for subsets \(S,T\subseteq U\), the natural definition is \(\left\langle S|T\right\rangle =\left| S\cap T\right| \) (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell’s Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over \({\mathbb {C}} \) and QM/Sets over \({\mathbb {Z}}_{2}\).

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  1. In full QM, the DeBroglie relations connect mathematical notions such as frequency and wave-length to physical notions such as energy and momentum. QM/sets is “non-physical” in the sense that it is a sets-version of the pure mathematical framework of (finite-dimensional) QM without those direct physical connections.

  2. The Spekkens toy model (Spekkens 2007) does not use vector spaces at all or utilize sets so it is not directly comparable.

  3. Instead of the fixed outcome set of classical probability theory, there is a vector space where each basis set plays the role of an outcome set or sample space. Since there are very different “incompatible” basis sets, “noncommutative” aspects of full QM appear in QM/sets.

  4. The mathematics can be generalized to the case where each point \(u_{i}\) in the sample space has a probability \(p_{i}\) but the simpler case of equiprobable points serves our conceptual purposes.

  5. Here \(\left\langle T|_{U}S\right\rangle =\left| T\cap S\right| \) takes values in the natural numbers \(\mathbb {N} \) outside the base field of \({\mathbb {Z}}_{2}\) just like, say, the Hamming distance function \(d_{H}\left( T,S\right) =\left| T+S\right| \) on vector spaces over \({\mathbb {Z}}_{2}\) in coding theory. McEliece (1977) Thus the “size of overlap” bra \(\left\langle T\right| _{U}:\wp \left( U\right) \rightarrow \mathbb {N} \) is not to be confused with the dual (“parity of overlap”) functional \(\varphi _{T}=\sum _{u_{j}\in T}\varphi _{u_{j}}:\wp \left( U\right) \rightarrow {\mathbb {Z}}_{2}\) where \(\varphi _{u_{j}}\left( \left\{ u_{k}\right\} \right) =\delta _{jk}\) for \(U=\left\{ u_{1},\ldots ,u_{n}\right\} \).

  6. One possible misinterpretation of QM/Sets is to misinterpret the transporting method as an embedding \({\mathbb {Z}}_{2}^{n}\rightarrow {\mathbb {C}} ^{n}\) defined by \(\left\{ u_{j}\right\} \longmapsto \left| u_{j}\right\rangle \) using a basis for each space. But such an embedding from a vector space over a field of finite characteristic to a vector space of characteristic zero cannot be linear. The repeated sum of a nonzero element in the domain space will eventually be 0 but its repeated nonzero image in the codomain space can never be 0. Indeed in QM/Sets, the brackets \(\left\langle T|_{U}S\right\rangle =\left| T\cap S\right| \) for \(T,T^{\prime },S\subseteq U\) should be thought of only as a measure of the overlap since they are not even linear, e.g., \(\left\langle T+T^{\prime }|_{U}S\right\rangle \ne \left\langle T|_{U}S\right\rangle +\left\langle T^{\prime }|_{U}S\right\rangle \) whenever \(T\cap T^{\prime }\ne \emptyset \).

  7. The term “\(\left\{ u_{j}\right\} \cap S^{\prime }\)” is not even defined in general since it is the intersection of subsets \(\left\{ u_{j}\right\} \subseteq U\) and \(S^{\prime }\subseteq U^{\prime }\) of two different universe sets U and \(U^{\prime }\).

  8. We use the double-line notation \(\left\| S\right\| _{U}\) for the U-norm of a set to distinguish it from the single-line notation \(\left| S\right| \) for the cardinality of a set. We also use the double-line notation \(\left\| \psi \right\| \) for the norm in QM although sometimes the single line notation \(\left| \psi \right| \) is used elsewhere.

  9. It should be noted that the projection operator \(S\cap ():\wp \left( U\right) \rightarrow \wp \left( U\right) \) is not only idempotent but linear, i.e., \(\left( S\cap T_{1}\right) +(S\cap T_{2})=S\cap \left( T_{1}+T_{2}\right) \). Indeed, this is the distributive law when \(\wp \left( U\right) \) is interpreted as a Boolean ring with intersection as multiplication.

  10. In order for general real-valued attributes to be internalized as linear operators, in the way that characteristic functions \(\chi _{S}\) were internalized as projection operators \(S\cap ()\), the base field would have to be strengthened to \({\mathbb {C}} \) and that would take us, mutatis mutandis, from the probability calculus of QM/sets to that of full QM.

  11. Boole has been included along with Laplace in the name of classical finite probability theory since he developed it as the normalized counting measure on the elements of the subsets of his logic. Applying the same mathematical move to the dual logic of partitions results in developing the notion of logical entropy \(h\left( \pi \right) \) of a partition \(\pi \) as the normalized counting measure on the dit set \({\text {dit}}\left( \pi \right) \), i.e., \(h\left( \pi \right) =\frac{\left| {\text {dit}}\left( \pi \right) \right| }{\left| U\times U\right| }\). Ellerman (2009).

  12. Arthur Eddington made a very early use of the sieve idea:

    In Einstein’s theory of relativity the observer is a man who sets out in quest of truth armed with a measuring-rod. In quantum theory he sets out armed with a sieve (Eddington 1947, p. 267).

    This passage was quoted by Weyl (1949, p. 255) in his treatment of gratings.

  13. In Schumacher and Westmoreland’s modal quantum theory Schumacher and Westmoreland (2012), they also take the dynamics to be any non-singular linear transformation.


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Correspondence to David Ellerman.

Appendix: Transporting vector space structures

Appendix: Transporting vector space structures

It is important to rigorously understand the mathematics connecting finite-dimensional QM over \({\mathbb {C}} ^{n}\) to QM/Sets over \({\mathbb {Z}}_{2}^{n}\). There is a general method to transport some structures from a vector space V over a field \({\mathbb {K}}\) to a vector space \(V^{\prime }\) over a different field \({\mathbb {K}}^{\prime }\). Select a basis set U for the source space V and then consider a structure on V that can be characterized in terms of the basis set U. Then apply the free vector space over the field \({\mathbb {K}}^{\prime }\) construction to U to generate the target vector space \(V^{\prime }\). Since the source structure was defined in terms of the basis set U, it can be carried over or ”transported” to \(V^{\prime }\) via its basis set U.

This method can be stated in rigorous terms using category theory (Mac Lane 1998; Awodey 2006). The construction of the free vector space over a field \({\mathbb {K}}\) is a functor from the category Sets of sets and functions to the category \(Vect_{{\mathbb {K}}}\) of vector spaces over \({\mathbb {K}}\) and linear transformations. The functor will only be used here on finite sets where it takes a finite set U to the vector space \({\mathbb {K}}^{U}\). The primary structures being transported are direct-sum decompositions (DSD) of a finite-dimensional vector space V. A DSD a set \(\left\{ V_{i}\right\} \) of disjoint subspaces (i.e., only overlap is zero space) so that the whole space V is their direct sum, or, in terms of category theory, V is the coproduct \(V=\oplus V_{i}\) of the subspaces \(\left\{ V_{i}\right\} \). In the category Sets, a set \(\left\{ B_{i}\right\} \) of disjoint subsets of a set U is a set partition of U if \(\cup B_{i}=U\), or, in terms of category theory, U is the coproduct of the disjoint subsets \(\left\{ B_{i}\right\} \). The free vector space over \({\mathbb {K}}\) functor is a left adjoint, “left adjoints preserve colimits” (Awodey 2006, p. 197), and coproducts are a special type of colimit. Hence the free vector space functor carries a set partition \(\pi =\left\{ B_{i}\right\} _{i=1,\ldots ,m}\) to the DSD \(\left\{ V_{i}={\mathbb {K}}^{B_{i}}\right\} \) of \(V={\mathbb {K}}^{U}=\oplus {\mathbb {K}}^{B_{i}}\).

Now start with the structure of a DSD \(\left\{ V_{i}\right\} \) on \(V\in Vect_{{\mathbb {K}}}\). What we previously called “characterizing the structure in terms of a basis set U” is rigorously interpreted to mean, in this case, finding a basis U and a partition \(\left\{ B_{i}\right\} \) on U so that the given DSD \(\left\{ V_{i}\right\} \) is the image of the free vector space functor, i.e., \(V={\mathbb {K}}^{U}=\oplus {\mathbb {K}}^{B_{i}}=\oplus V_{i}\). But then the free vector space functor over a different field \({\mathbb {K}}^{\prime }\) can be applied to the same set partition \(\left\{ B_{i}\right\} \) of the set U to generate a DSD \(\left\{ V_{i}^{\prime }={\mathbb {K}}^{\prime B_{i}}\right\} \) of \(V^{\prime }={\mathbb {K}}^{\prime U}\). That is how to rigorously describe “transporting” a set-based structure on a vector V over \({\mathbb {K}}\) to a vector space \(V^{\prime }\) over a different field \({\mathbb {K}}^{\prime }\).

To show that any given DSD \(\left\{ V_{i}\right\} \) of V is in the image of the free vector space over \({\mathbb {K}}\) functor, pick basis set \(B_{i}\) of \(V_{i}\). The sets \(B_{i}\) are disjoint and since \(\left\{ V_{i}\right\} \) is a DSD, the union \(U=\cup B_{i}\) is a basis for V so \(V_{i}={\mathbb {K}}^{B_{i}}\) and \(V={\mathbb {K}}^{U}=\oplus {\mathbb {K}}^{B_{i}}\).

This method is applied to the transporting of self-adjoint operators from \(V={\mathbb {C}} ^{n}\) to \(V^{\prime }={\mathbb {Z}}_{2}^{n}\) that motivates QM/Sets. A self-adjoint operator \(F:{\mathbb {C}} ^{n}\rightarrow {\mathbb {C}} ^{n}\) has a basis \(U=\left\{ u_{1},\ldots ,u_{n}\right\} \) of orthonormal eigenvectors and it has real distinct eigenvalues \(\left\{ \phi _{i}\right\} _{j=1,\ldots ,m}\), so it defines the real eigenvalue function \(f:U\rightarrow {\mathbb {R}} \) where for \(u_{j}\in U\), \(f\left( u_{j}\right) \) is one of the distinct eigenvalues \(\left\{ \phi _{i}\right\} _{i=1,\ldots , m}\). For each distinct eigenvalue \(\phi _{i}\), there is the eigenspace \(V_{i}\) of its eigenvectors and \(\left\{ V_{i}\right\} _{i=1,\ldots ,m}\) is a DSD on \(V={\mathbb {C}} ^{n}\). The inverse-image \(\pi =\left\{ B_{i}=f^{-1}\left( \phi _{i}\right) \right\} _{i=1,\ldots ,m}\) of the eigenvalue function \(f:U\rightarrow {\mathbb {R}} \) is a set partition on U.

Thus the set-based structure we have is the set U with a partition \(\left\{ B_{i}=f^{-1}\left( \phi _{i}\right) \right\} _{i}\) on U induced by a real-value function \(f:U\rightarrow {\mathbb {R}} \) on U. That set-based structure is sufficient to reconstruct the DSD \(\left\{ V_{i}={\mathbb {C}} ^{B_{i}}\right\} _{i}\) on \(V={\mathbb {C}} ^{n}\cong {\mathbb {C}} ^{U}=\oplus {\mathbb {C}} ^{B_{i}}\) as well as the original operator F. The operator F is defined on the basis U by \(Fu_{j}=f\left( u_{j}\right) u_{j}\) for \(j=1,\ldots ,n\). That process of going from the function \(f:U\rightarrow {\mathbb {R}} \) on a basis set U of \({\mathbb {C}} ^{U}\) to an operator on \({\mathbb {C}} ^{U}\) might be called internalizing the function \(f:U\rightarrow {\mathbb {R}} \) in \({\mathbb {C}} ^{U}\).

Given the set-based structure of a real-valued function \(f:U\rightarrow {\mathbb {R}} \), which determines the set partition \(\left\{ f^{-1}\left( \phi _{i}\right) \right\} _{i=1,\ldots ,m}\) on U, we then apply the free vector space over \({\mathbb {Z}}_{2}\) functor to construct the vector space \({\mathbb {Z}}_{2}^{U}\). That vector space is more familiar in the form of the powerset \(\wp \left( U\right) \cong {\mathbb {Z}}_{2}^{U}\) since each function \(U\rightarrow {\mathbb {Z}}_{2}=\left\{ 0,1\right\} \) in \({\mathbb {Z}}_{2}^{U}\) is the characteristic function \(\chi _{S}\) of a subset \(S\in \wp \left( U\right) \). The free vector space functor \({\mathbb {Z}}_{2}^{()}\) takes the coproduct \(U=\cup _{i=1}^{m}f^{-1}\left( \phi _{i}\right) \) to the DSD \(\left\{ \wp \left( f^{-1}\left( \phi _{i}\right) \right) \right\} \) of \(\wp \left( U\right) \). The attempt to internalize the real function \(f:U\rightarrow {\mathbb {R}} \) would only work if f took values in \({\mathbb {Z}}_{2}=\left\{ 0,1\right\} \subseteq {\mathbb {R}} \) in which case f would be a characteristic function \(\chi _{S}\) for some subset \(S\in \wp \left( U\right) \). In that special case, the internalized operator would be the projection operator \(P_{S}:{\mathbb {Z}}_{2}^{U}\rightarrow {\mathbb {Z}}_{2}^{U}\) which in terms of the basis U has the action \(P_{S}\left( T\right) =S\cap T\) taking any subset \(T\in \wp \left( U\right) \) to \(S\cap T\in \wp \left( S\right) \).

Hence outside of characteristic functions, the real-valued functions \(f:U\rightarrow {\mathbb {R}} \) cannot be internalized as operators on \({\mathbb {Z}}_{2}^{U}\). But that is fine since the idea of the model QM/Sets is that given a basis U of \({\mathbb {Z}}_{2}^{n}\), the quantum probability calculus will just be the classical finite probability calculus with the outcome set or sample space U where \(f:U\rightarrow {\mathbb {R}} \) is a real-valued random variable. We have illustrated the transporting of set-based structures on \({\mathbb {C}} ^{n}\) to \({\mathbb {Z}}_{2}^{n}\) using a basis set U, but in the stand-alone model QM/Sets, we cut the umbilical cord to \({\mathbb {C}} ^{n}\) and work with any other basis \(U^{\prime }\) of \({\mathbb {Z}}_{2}^{n}\) and real-valued random variables \(g:U^{\prime }\rightarrow {\mathbb {R}} \) on that sample space.

Other structures can be transported across the bridge from \({\mathbb {C}} ^{n}\) to \({\mathbb {Z}}_{2}^{n}\). QM/Sets differs from the other four attempts to define some toy version of QM on sets by the treatment of the Dirac brackets. Starting with our orthonormal basis U on a finite-dimensional Hilbert space \({\mathbb {C}} ^{n}\) (where the bracket is the inner product), we need to define the transported brackets applied to two subsets \(S,T\subseteq U\) in \(\wp \left( U\right) \). The two subsets define the vectors \(\psi _{S}=\sum _{u\in S}\left| u\right\rangle \) and \(\psi _{T}=\sum _{u\in T}\left| u\right\rangle \) in \({\mathbb {C}} ^{n}\) which have the bracket value \(\left\langle \psi _{S}|\psi _{T}\right\rangle =\left| S\cap T\right| \). Since that value is defined just in terms of the subsets \(S,T\subseteq U\) as the cardinality of their overlap, that value can be transported to \(\wp \left( U\right) \) as the real-valued basis-dependent brackets \(\left\langle S|_{U}T\right\rangle =\left| S\cap T\right| \).

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Ellerman, D. Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus. Synthese 194, 4863–4896 (2017).

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