Quantum Measure Theory

Hamhalter, Jan

doi:10.1007/978-94-017-0119-8

Cited by 131 publications

(88 citation statements)

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“…Sometimes these allow simplifications of the preceding techniques, sometimes no progress has been achieved. (We refer to [3] and [1,6,7,11,24] for the basic definitions on quantum structures. ) Greechie diagrams have been first introduced in [4] as a tool for a construction of orthomodular lattices admitting no states.…”

Section: Motivation and Basic Notionsmentioning

confidence: 99%

Small Quantum Structures with Small State Spaces

Navara

2007

Int J Theor Phys

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Section: Motivation and Basic Notionsmentioning

confidence: 99%

Small Quantum Structures with Small State Spaces

Navara

2007

Int J Theor Phys

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“…The lattice of projections L(H) in a Hilbert space H generalizes to pre-Hilbert spaces in several ways (see [3], [6] and [10] etc.). Two ways seem most fruitful if H is replaced by a pre-Hilbert space S. This is the study of the class of orthogonally closed subspaces and the class of splitting subspaces of S respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Chapter 4 of A. Dvurečenskij [3] and Chapter 4 of J. Hamhalter [6] serve as a very good introduction on the subject. The following theorem gathers some important known results that will be used in the sequel.…”

Section: Introductionmentioning

confidence: 99%

“…The following theorem gathers some important known results that will be used in the sequel. For the proofs the reader can consult, for example, [3] and [6]. (iii) E(S) is a σ-complete orthomodular poset if and only if S is complete (see [2], [4] and [5]); (iv) (Amemiya-Araki Theorem) F (S) is orthomodular if and only if S is complete ( [1]).…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐splitting subspaces in a pre‐Hilbert space

Buhagiar

Chetcuti

2007

Mathematische Nachrichten

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Key words Pre-Hilbert space (=inner product space), quasi-splitting subspace, orthomodular poset, lattice MSC (2000) 03G12, 46C05, 81P10 Let S be a pre-Hilbert space. Two classes of closed subspaces of S that can naturally replace the lattice of projections in a Hilbert space are E(S) and F (S), the classes of splitting subspaces and orthogonally closed subspaces of S respectively. It is well-known that in general the algebraic structure of E(S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert space. In the present note we introduce the class Eq(S) of quasi-splitting subspaces of S. First it is shown that Eq(S) falls between E(S) and F (S). It is also shown that, in contrast to the other two classes, Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. At the end, the algebraic structure of Eq(S) is used to characterize Hilbert spaces.

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“…Gleason's Theorem was generalized and applied in many directions (see for example, the books [8,11] or the survey article [3]). An important application of Gleason's Theorem is a result by J. Hamhalter and P. Pták [12] proving that for an inner product space S, there exists a σ -additive state on the system of orthogonally closed subspaces F (S) if, and only if, S is complete (recall that a subspace A of S is orthogonally closed if A ⊥⊥ = A).…”

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confidence: 99%

On Gleason’s Theorem without Gleason

2008

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The original proof of Gleason's Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason's Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason's Theorem, e.g. that on the quantum logic L(H ) of all closed subspaces of a Hilbert space H, dim H ≥ 3, there is no finitely additive state whose range is countably infinite. In particular, if dim H = n, then on L(H ) there is a unique discrete state, namely m(A) = dim A/ dim H, A ∈ L(H ).

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Quantum Measure Theory

Cited by 131 publications

References 0 publications

Small Quantum Structures with Small State Spaces

Small Quantum Structures with Small State Spaces

Quasi‐splitting subspaces in a pre‐Hilbert space

On Gleason’s Theorem without Gleason

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