Building on established literature and recent developments in the graph-theoretic characterisation of its CPM category, we provide a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations. On the way, we highlight the wealth of exotic beasts that hide amongst the extensive operational and structural similarities that the theory shares with more traditional arenas of categorical quantum mechanics, such as the category fdHilb. We conclude our journey by proving that fRel is local, but not without some unexpected twists.
Building on established literature and recent developments in the graph-theoretic characterisation of its CPM category, we provide a treatment of pure state and mixed state quantum mechanics in the category fRel of finite sets and relations. On the way, we highlight the wealth of exotic beasts that hide amongst the extensive operational and structural similarities that the theory shares with more traditional arenas of categorical quantum mechanics, such as the category fdHilb. We conclude our journey by proving that fRel is local, but not without some unexpected twists.
Mixed states are of interest in quantum mechanics for modelling partial information. More recently categorical approaches to linguistics have also exploited the idea of mixed states to describe ambiguity and hyponym / hypernym relationships. In both these application areas the category Rel of sets and binary relations is often used as an alternative model. Selinger's CPM construction provides the setting for mixed states in Hilbert space based categorical quantum mechanics. By analogy, applying the CPM construction to Rel is seen as introducing mixing into a relational setting. We investigate the category CPM(Rel) of completely positive maps in Rel. We show that the states of an object in CPM(Rel) are in bijective correspondence with certain families of graphs. Via map-state duality this then allows us provide a graph theoretic characterization of the morphisms in CPM(Rel). By identifying an appropriate composition operation on graphs, we then show that CPM(Rel) is isomorphic to a category of sets and graphs between them. This isomorphism then leads to a graph based description of the complete join semilattice enriched dagger compact structure of CPM(Rel). These results allow us to reason about CPM(Rel) entirely in terms of graphs. We exploit these techniques in several examples. We give a closed form expression for the number of states of a finite set in CPM(Rel). The pure states are characterized in graph theoretic terms. We also demonstrate the possibly surprising phenomenon of a pure state that can be given as a mixture of two mixed states.
We present a uniform framework for the treatment of a large class of toy models of quantum theory. Specifically, we will be interested in theories of wavefunctions valued in commutative involutive semirings, and which give rise to some semiring-based notion of classical non-determinism via the Born rule. The models obtained with our construction possess many of the familiar structures used in Categorical Quantum Mechanics. We also provide a bestiary of increasingly exotic examples: some well known, such as real quantum theory and relational quantum theory; some less known, such as hyperbolic quantum theory, p-adic quantum theory and "parity quantum theory"; and some entirely new, such as "finite-field quantum theory" and "tropical quantum theory". As a further bonus, the measurement scenarios arising within these theories can be studied using the sheaf-theoretic framework for non-locality and contextuality. Their computational complexity can similarly be studied within existing frameworks for affine and unitary circuits over commutative semirings.
In a recent paper [12], we discussed the serious inconsistency present within the operational and mathematical definition(s) of the notion of pure state. Continuing this analysis, in this work we attempt to address the role of 'purity' and 'mixtures' within two different categorical approaches to QM, namely, the topos approach originally presented by Chris Isham and Jeremy Butterfield [27, 28, 29] and the more recent logos categorical approach presented by the authors of this article [10, 11, 13]. While the first approach exposes the difficulties to produce a consistent understanding of pure states and mixtures, the latter approach presents a new scheme in which their reference is erased right from the start in favor of an intensive understanding of projection operators and quantum superpositions. This new account of the theory, grounded on an intensive interpretation of the Born rule, allows us not only to avoid the orthodox interpretation of projection operators --either as referring to definite valued properties or measurement outcomes-- but also to consider all matrices (of any rank) on equal footing. It is from this latter standpoint that we conclude that instead of distinguishing between pure and mixed states it would be recommendable --for a proper understanding of the theory of quanta-- to return to the original matrix formulation of quantum mechanics presented by Werner Heisenberg in 1925.
We review current research in the field of categorical quantum mechanics. We place the category of relations within the same categorical framework and point out the similarities with the category of Hilbert spaces. Subsequently, we provide a categorical model for investigating the spectral theorem (as provided in [22]), within the context of an internally diagonalisable element and apply these ideas to both categories mentioned. We find out that while all normal operators can be internally diagonalisable in the category of Hilbert spaces – this is not the case for the category of Relations. We conclude by providing insights on how we can separate the internally diagonalisable elements within the set of normal operators .
In the tradition of toy models of quantum mechanics in vector spaces over finite fields (e.g., Schumacher and Westmoreland's "modal quantum theory"), one finite field stands out, 2, since vectors over 2 have an interpretation as natural mathematical objects, i.e., sets. This engages a sets-to-vector-spaces bridge that is part of the mathematical folklore to translate both ways between set concepts and vector space concepts. Using that bridge, the mathematical framework of (finite-dimensional) quantum mechanics can be transported down to sets resulting in quantum mechanics over sets or QM/sets. This approach leads to a different treatment of Dirac's brackets than in "modal quantum theory" (MQT), and that gives a full probability calculus (unlike MQT that only has zero-one modalities of impossible and possible). That, in turn, leads to a rather fulsome theory of QM over sets that includes "logical" models of the double-slit experiment, Bell's Theorem, quantum information theory, quantum computing, and much else. Indeed, QM/sets is proposed as the "logic" of QM in the old-fashioned sense of "logic" as giving the simplified essentials of a theory. QM/sets is also a key part of a broader research program to provide an interpretation of QM based on the notion of "objective indefiniteness," a program that grew out the recent development of the logic of partitions mathematically dual to the usual Boolean logic of subsets.
Ongoing work in quantum information emphasises the need for a structural understanding of quantum speedups: in this work, we focus on the quantum Fourier transform and the structures in quantum theory that enable it. We elucidate a general connection in any process theory between the Fourier transform and strongly complementary observables, i.e. Hopf algebras in dagger symmetric monoidal categories. We generalise the necessary tools of representation theory from fdHilb to arbitrary dagger symmetric monoidal categories. We define groups, characters and representations, and we prove their relation to strong complementarity. The Fourier transform is then defined in terms of pairs of strongly complementary observables, in both the abelian and non-abelian case. In the abelian case, we draw the connection with Pontryagin duality and provide categorical proofs of the Fourier Inversion Theorem, the Convolution Theory, and Pontryagin duality. Our work finds application in the novel characterisation of the Fourier transform for the category fRel of finite sets and relations. This is a result of interest for the study of categorical quantum algorithms, as the usual construction of the quantum Fourier transform in terms of Fourier matrices is shown to fail in fRel. Despite this, the process theoretic perspective on the Fourier transform is sensible in this setting. Furthermore, our categorical setting provides a generalisation of the abelian Fourier transform from finite-dimensional Hilbert spaces to finite-dimensional modules over arbitrary semirings, as well as a further generalisation to finite non-abelian groups, including a fully categorical generalisation of the Gelfand-Naimark theorem.