After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure induces a unique state of the von Neumann algebra. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure induces a unique state of the von Neumann algebra. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.

Advanced Science LettersJun 1, 200927 citations

A Topos Theoretic Notion of Entropy

Carmen-Maria Constantin, Andreas Doering

In the topos approach to quantum theory, the spectral presheaf plays the role of the state space of a quantum system. We show how a notion of entropy can be defined within the topos formalism using the equivalence between states and measures on the spectral presheaf. We show how this construction unifies Shannon and von Neumann entropy as well as classical and quantum Renyi entropies. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can (mathematically) reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension $3$ or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.In the topos approach to quantum theory, the spectral presheaf plays the role of the state space of a quantum system. We show how a notion of entropy can be defined within the topos formalism using the equivalence between states and measures on the spectral presheaf. We show how this construction unifies Shannon and von Neumann entropy as well as classical and quantum Renyi entropies. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can (mathematically) reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension $3$ or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.

arXiv:2006.03139Jun 4, 20201 citation

Self-adjoint Operators as Functions II: Quantum Probability

Andreas Doering, Barry Dewitt

In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.

arXiv:1210.5747Oct 21, 20127 citations

Topos quantum theory on quantization-induced sheaves

Kunji Nakayama

In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheaf-based one can be constructed on sheaves defined by the quantization-induced Lawvere-Tierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheaf-based expressions of physical propositions and truth objects, and thereby give a method of truth-value assignment to the propositions. Furthermore, we clarify the relationship to the presheaf-based quantum theory. We give translation rules between the sheaf-based ingredients and the corresponding presheaf-based ones. The translation rules have “coarse-graining” effects on the spaces of the presheaf-based ingredients; a lot of different proposition presheaves, truth presheaves, and presheaf-based truth-values are translated to a proposition sheaf, a truth sheaf, and a sheaf-based truth-value, respectively. We examine the extent of the coarse-graining made by translation.In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheaf-based one can be constructed on sheaves defined by the quantization-induced Lawvere-Tierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheaf-based expressions of physical propositions and truth objects, and thereby give a method of truth-value assignment to the propositions. Furthermore, we clarify the relationship to the presheaf-based quantum theory. We give translation rules between the sheaf-based ingredients and the corresponding presheaf-based ones. The translation rules have “coarse-graining” effects on the spaces of the presheaf-based ingredients; a lot of different proposition presheaves, truth presheaves, and presheaf-based truth-values are translated to a proposition sheaf, a truth sheaf, and a sheaf-based truth-value, respectively. We examine the extent of the coarse-graining made by translation.

arXiv:1404.2370Journal of Mathematical PhysicsOct 1, 20147 citations

Spectral presheaves as quantum state spaces

Andreas Döring

For each quantum system described by an operator algebra [Formula: see text] of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen-Specker theorem. This quantum state space is the spectral presheaf [Formula: see text]. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf [Formula: see text] geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.For each quantum system described by an operator algebra [Formula: see text] of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen-Specker theorem. This quantum state space is the spectral presheaf [Formula: see text]. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf [Formula: see text] geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering SciencesJun 29, 20152 citations

A Topos for Algebraic Quantum Theory

Chris Heunen, Nicolaas P. Landsman, Spitters, BAW Bas

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum Σ(A) in T (A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probabilityThe aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum Σ(A) in T (A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability

Jan 1, 2009118 citations

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or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.