Two Subspaces

Halmos, Paul R.

doi:10.1007/978-1-4613-8208-9_26

Cited by 91 publications

(169 citation statements)

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“…This result is due to Dixmier [24]. An elegant proof is given in Halmos [31]. See also the survey Böttcher-Spitkovsky [18].…”

Section: Basic Facts About Quasifree Statesmentioning

confidence: 91%

Determinantal Point Processes and Fermion Quasifree States

Olshanski

2020

Commun. Math. Phys.

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Determinantal point processes are characterized by a special structural property of the correlation functions: they are given by minors of a correlation kernel. However, unlike the correlation functions themselves, this kernel is not defined intrinsically, and the same determinantal process can be generated by many different kernels. The non-uniqueness of a correlation kernel causes difficulties in studying determinantal processes.We propose a formalism which allows to find a distinguished correlation kernel under certain additional assumptions. The idea is to exploit a connection between determinantal processes and quasifree states on CAR, the algebra of canonical anticommutation relations.We prove that the formalism applies to discrete N -point orthogonal polynomial ensembles and to some of their large-N limits including the discrete sine process and the determinantal processes with the discrete Hermite, Laguerre, and Jacobi kernels investigated by Borodin and the author in [16]. As an application we resolve the equivalence/disjointness dichotomy for some of those processes.

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“…This result is due to Dixmier [24]. An elegant proof is given in Halmos [31]. See also the survey Böttcher-Spitkovsky [18].…”

Section: Basic Facts About Quasifree Statesmentioning

confidence: 91%

Determinantal Point Processes and Fermion Quasifree States

Olshanski

2020

Commun. Math. Phys.

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“…Proof. Formula (15) follows by (14) and the above discussion. If A H is L p as a complex linear operator, then A H is L p as a real linear operator, hence…”

Section: Angle Between Subspaces and Type I Propertymentioning

confidence: 97%

Von Neumann Entropy in QFT

Longo

2020

Commun. Math. Phys.

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In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones O ⊂ O of the spacetime, where the closure of O is contained in O. Given a QFT net A of local von Neumann algebras A(O), we consider the von Neumann entropy S A (O, O) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras A(O) ⊂ A( O) (split property). We show that this canonical entanglement entropy S A (O, O ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy S A (O, O), the infimum of the vacuum von Neumann entropy of F , where F here runs over all the intermediate, discrete type I von Neumann algebras. We prove that S A (O, O) is finite for the local chiral conformal net generated by finitely many commuting U (1)-currents.Von Neumann entropy is the basic concept in quantum information and extends the classical Shannon's information entropy notion to the non commutative setting.As is well known, a state ω on a matrix algebra M is given by a density matrix ρ, namely ω(T ) = Tr(ρT ), T ∈ M . The von Neumann entropy of ω is given byEntanglement entropy is a measure of quantum information by the degree of quantum entanglement of a quantum state.Let's consider a bipartite, finite-dimensional quantum system M = A ⊗ B, where A and B are matrix algebras. Given a pure state ω on the matrix algebra M , let ρ A and ρ B the density matrices associated with the restrictions of ω to A and B respectively on A and B. The entanglement entropy of ω is defined as(1)The above definition directly extends to the case M = k A k ⊗ B k is a direct sum, where A k , B k are matrix algebras and the restrictions ω| A k ⊗B k are pure (not normalised). It also extends to the infinite-dimensional case where A k , B k are type I factors; in this last case, however, the entanglement entropy may be infinite. Entanglement is certainly one of the main feature of quantum physics and there is a famous, long standing debate on its interpretation (EPR paradox, Bell's inequalities, etc.). An overview of the matter lies beyond the purpose of this introduction.The role of entanglement in Quantum Field Theory is more recent and increasingly important; it represents a piece of the quantum information framework in this subject. It appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc.

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“…Proof. These properties of close projections are easily obtained using Halmos's representation of two subspaces [9]. This representation provides 2 x 2 matrix forms for P and Q, with commuting positive operator entries.…”

Section: Close Projectionsmentioning

confidence: 99%

“…Furthermore, it can be readily verified that ran P 1 and ran Q Y are in generic position as subspaces of M; that is, in the lattice 0>{Ji) any two of the projections P V P^, Q x and Q± are complementary. So, according to Halmos [9]…”

Section: Close Projectionsmentioning

confidence: 99%

“…It will be convenient to assume also that M x n M 2 and M x 0 M 2 have the same dimension. Then, by an easy generalisation of a result of due to Halmos [9], we may assume that, up to unitary equivalence, the underlying Hilbert space is X © X, the orthogonal direct sum of two copies of a Hilbert space X, and that the matrices of the projections Q x and Q 2 are given by Let P = E(u) © 0, and let x be a non-zero vector in the range of E{u) L . Then P(Cx © Sx) = P(Cx ®-Sx) = 0.…”

Section: Complementary Subspacesmentioning

confidence: 99%

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Distance Formulae for Subspace Lattices

Davidson

Harrison

1989

Journal of the London Mathematical Society

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Since 1975, when Arveson [2] established a distance formula for nest algebras, distance formulae have become increasingly important to our understanding of reflexive operator algebras. Operator algebras which have distance formulae are said to be hyperreflexive [3]. In this paper we study a dual notion of hyperreflexivity for subspace lattices. We show that every commutative subspace lattice is hyperreflexive, and obtain an estimate of the associated distance constant, called the constant of hyperreflexivity. Sharper estimates are obtained in special cases such as nests and the projection lattices of certain von-Neumann algebras. We study an asymptotic variant of hyperreflexivity, which illustrates one of the differences between hyperreflexivity for subspace lattices and for operator algebras. Then we construct some reflexive, but not hyperreflexive subspace lattices. These are atomic Boolean algebras of subspaces, obtained by piecing together hyperreflexive subspace lattices with arbitrarily large constants of hyperreflexivity. Finally we examine 4-element Boolean algebras generated by pairs of complementary subspaces. According to Halmos [11], such subspace lattices are reflexive, but we show that hyperreflexivity depends on the spatial alignment of the subspaces. Hyperreflexive subspace latticesLet ^{JF) and ^{J^) denote, respectively, the algebra of all (bounded linear) operators, and the lattice of all (self-adjoint) projections, on a Hilbert space 3tf. A weakly closed subalgebra of ^(Jf), containing the identity operator 1, will be called an operator algebra, and a strongly closed sublattice of ^(«#), containing the trivial projections 0 and 1, will be called a subspace lattice. For any subset si of $?(Jf), lat si denotes the subspace lattice consisting of all projections P for which P l AP = 0 for all A in si (where P 1 = 1 -P); dually, for any subset <£ of ^f ) , we denote by alg J£? the operator algebra consisting of all operators A for which P L AP = 0 for all P in if. We say that si is reflexive if si = alg lat si, and J? is reflexive if if = lat alg if [10].Suppose that P is a projection and if is a subspace lattice. The Arveson distance of P from if, namely p(P, if), is defined by 0(P,J?) = supQP'-APW :/fe(alg S£ )J, where (alg if ) x denotes the set of all contractions in alg if.We are concerned with the relationship between /?(/*, if) and d(P,J£), the ordinary metric distance of P from SC, defined by

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Two Subspaces

Cited by 91 publications

References 1 publication

Determinantal Point Processes and Fermion Quasifree States

Determinantal Point Processes and Fermion Quasifree States

Von Neumann Entropy in QFT

Distance Formulae for Subspace Lattices

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