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Journal of Physics A Mathematical and Theoretical

General quantum theory—unification of classical and modal quantum theories

Koen Thas

Published Jul 6, 2020• 1 citation
Publisher

Abstract

Abstract Inspired by classical (‘actual’) quantum theory over C and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a ( σ , 1)-Hermitian form φ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over C and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [15], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U ( V , φ ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over C , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over C at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over Q ¯ . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.

DOI
10.1088/1751-8121/aba307

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Abstract Inspired by classical (‘actual’) quantum theory over C and modal quantum theory (MQT), which is a model of quantum theory over certain finite fields, we introduce general quantum theory as a quantum theory—in the København interpretation—over general division rings with involution, in which the inner product ‘is’ a ( σ , 1)-Hermitian form φ . This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical quantum theory over C and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [15], and generalized probability theories. On the other hand, in many general quantum theories, a geometrical object which we call ‘quantum kernel’ arises, which is invariant under the unitary group U ( V , φ ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over C , but it is present, for instance, in all known MQTs. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 0, every general quantum theory over an algebraically closed field behaves like classical quantum theory over C at many levels, and that all such theories share one model, which we pin down as the ‘minimal model,’ which is countable and defined over Q ¯ . Moreover, to make the analogy with classical quantum theory even more striking, we show that Born’s rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for general quantum theories in characteristic 0 which allows one to extend any such theory over algebraically closed fields (such as classical complex quantum theory) to larger theories in which a quantum kernel is present. In this sense, these singular objects are always virtually around in abundance.
Inspired by classical ("actual") Quantum Theory over C\mathbb{C}C and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the København interpretation -- over general division rings with involution, in which the inner product "is" a (σ,1)(σ,1)(σ,1)-Hermitian form φφφ. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over C\mathbb{C}C and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group U(V,φ)\mathbf{U}(V,φ)U(V,φ), and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic 000, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over C\mathbb{C}C at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over Q‾\overline{\mathbb{Q}}Q​. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic 000 which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.
In a recent paper, Chang et al. have proposed studying “quantum F u n ”: the q ↦ 1 limit of modal quantum theories over finite fields F q , motivated by the fact that such limit theories can be naturally interpreted in classical quantum theory. In this letter, we first make a number of rectifications of statements made in that paper. For instance, we show that quantum theory over F 1 does have a natural analogon of an inner product, and so orthogonality is a well-defined notion, contrary to what was claimed in Chang et al. Starting from that formalism, we introduce time evolution operators and observables in quantum F u n , and we determine the corresponding unitary group. Next, we obtain a typical no-cloning result in the general realm of quantum F u n . Finally, we obtain a no-deletion result as well. Remarkably, we show that we can perform quantum deletion by almost unitary operators, with a probability tending to 1. Although we develop the construction in quantum F u n , it is also valid in any other quantum theory (and thus also in classical quantum theory in complex Hilbert spaces).
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.
We give a brief introduction to the so-called general boundary formulation (GBF) of quantum theory. This new axiomatic formulation provides a description of the quantum dynamics which is manifestly local and does not rely on a metric background structure for its definition. We present the basic ingredients of the GBF, in particular we review the core axioms that assign algebraic structures to geometric ones, the two quantisation schemes so far developed for the GBF and the probability interpretation which generalizes the standard Born rule. Finally we briefly discuss some of the results obtained studying specific quantum field theories within the GBF.
We start by rewriting classical mechanics in a quantum mechanical fashion and point out that the only difference with quantum theory re- sides at one point. There is a classical analogon of the collapse of the wavefunction and an extension of the usual Born rule is proposed which might solve this problem. We work only with algebra’s over the real (com- plex) numbers, general number fields of finite characteristic allowing for finite dimensional representations of the commutation relations are not considered given that such fields are not well ordered and do not give rise to a well defined probability interpretation. Our theory generalizes however to discrete spacetimes and finite dimensional algebra’s. Looking at physics this way, spacetime itself distinguishes itself algebraically by means of well chosen commutation relations and there is further nothing special about it meaning it has also a particle interpretation just like any other dynamical variable. Likewise, there is no reason for the dynamics to be Hamiltonian and therefore we have a nonconservative formulation of quantum physics at hand. The harmonic oscillator (amongst few oth- ers) distinguishes itself because the algebra forms a finite dimensional Lie algebra; the classical and quantum (discrete) harmonic oscillator are stud- ied in a some more generality and examples are given which are neither classical, nor quantum.
We introduce a realist, unextravagant interpretation of quantum theory that builds on the existing physical structure of the theory and allows experiments to have definite outcomes, but leaves the theory's basic dynamical content essentially intact. Much as classical systems have specific states that evolve along definite trajectories through configuration spaces, the traditional formulation of quantum theory asserts that closed quantum systems have specific states that evolve unitarily along definite trajectories through Hilbert spaces, and our interpretation extends this intuitive picture of states and Hilbert-space trajectories to the case of open quantum systems as well. We provide independent justification for the partial-trace operation for density matrices, reformulate wave-function collapse in terms of an underlying interpolating dynamics, derive the Born rule from deeper principles, resolve several open questions regarding ontological stability and dynamics, address a number of familiar no-go theorems, and argue that our interpretation is ultimately compatible with Lorentz invariance. Along the way, we also investigate a number of unexplored features of quantum theory, including an interesting geometrical structure---which we call subsystem space---that we believe merits further study. We include an appendix that briefly reviews the traditional Copenhagen interpretation and the measurement problem of quantum theory, as well as the instrumentalist approach and a collection of foundational theorems not otherwise discussed in the main text.