In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic ingredients of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbertlattices, sothatthelatterareadirectconsequenceofanorthogonalitybased characterization of dimension. 1In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic ingredients of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbertlattices, sothatthelatterareadirectconsequenceofanorthogonalitybased characterization of dimension. 1

Jan 1, 20135 citations

Symmetric Quantum Sets and L-Algebras

Wolfgang Rump

Abstract Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$ , ${\mathbb{C}}$ , or ${\mathbb{H}}$ . Symmetric quantum sets are characterized as a class of $L$ -algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.Abstract Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

International Mathematics Research NoticesMay 17, 202022 citations

Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces

Thomas Vetterlein

Abstract The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.Abstract The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.

International Journal of Theoretical PhysicsMar 10, 202017 citations

Witnessing Irreducible Dimension

Wan Cong, Yu Cai, Jean-Daniel Bancal +1 more

The Hilbert space dimension of a quantum system is the most basic quantifier of its information content. Lower bounds on the dimension can be certified in a device-independent way, based only on observed statistics. We highlight that some such "dimension witnesses" capture only the presence of systems of some dimension, which in a sense is trivial, not the capacity of performing information processing on them, which is the point of experimental efforts to control high-dimensional systems. In order to capture this aspect, we introduce the notion of irreducible dimension of a quantum behavior. This dimension can be certified, and we provide a witness for irreducible dimension four.The Hilbert space dimension of a quantum system is the most basic quantifier of its information content. Lower bounds on the dimension can be certified in a device-independent way, based only on observed statistics. We highlight that some such "dimension witnesses" capture only the presence of systems of some dimension, which in a sense is trivial, not the capacity of performing information processing on them, which is the point of experimental efforts to control high-dimensional systems. In order to capture this aspect, we introduce the notion of irreducible dimension of a quantum behavior. This dimension can be certified, and we provide a witness for irreducible dimension four.

arXiv:1611.01258Physical Review LettersAug 25, 201754 citations

H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics

Samson Abramsky, Chris Heunen

A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within categorical axiomatizations of quantum mechanics. We develop a definition of H*-algebra that can be interpreted in any symmetric monoidal dagger category, reduces to the classical notion from functional analysis in the category of (possibly infinite-dimensional) Hilbert spaces, and hence provides a categorical way to speak about orthonormal bases and quantum observables in arbitrary dimension. Moreover, these algebras reduce to the usual notion of Frobenius algebra in compact categories. We then investigate the relations between nonunital Frobenius algebras and H*-algebras. We give a number of equivalent conditions to characterize when they coincide in the category of Hilbert spaces. We also show that they always coincide in categories of generalized relations and positive matrices.

arXiv:1011.6123Nov 29, 201025 citations

Biorthogonal quantum mechanics

Dorje C. Brody

The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

arXiv:1308.2609Journal of Physics A Mathematical and TheoreticalDec 24, 2013572 citations

>3

is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size

>3

are shown to be representable by Hilbert spaces over

{\mathbb{R}}

{\mathbb{C}}

, or

{\mathbb{H}}

. Symmetric quantum sets are characterized as a class of

L

-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.