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  • 1
    Electronic Resource
    Electronic Resource
    A uniqueness theorem for entanglement measures (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 2507-2512 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We obtain a mathematically simple characterization of all functionals coinciding with the von Neumann reduced entropy on pure states based on the Khinchin–Faddeev axiomatization of Shannon entropy and give a physical interpretation of the axioms in terms of entanglement. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1370954
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  • 2
    Electronic Resource
    Electronic Resource
    Symmetries of history quantum theories and decoherence functionals (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 5850-5855 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Recently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner's theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg's results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532598
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  • 3
    Electronic Resource
    Electronic Resource
    On tracial operator representations of quantum decoherence functionals (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 5643-5652 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532157
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  • 4
    Electronic Resource
    Electronic Resource
    A new class of entanglement measures (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 5306-5314 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We introduce new entanglement measures on the set of density operators on tensor product Hilbert spaces. These measures are based on the greatest cross norm on the tensor product of the sets of trace class operators on Hilbert space. We show that they satisfy the basic requirements on entanglement measures discussed in the literature, including convexity, invariance under local unitary operations and nonincrease under local quantum operations and classical communication. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1398062
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  • 5
    Electronic Resource
    Electronic Resource
    On the consistent effect histories approach to quantum mechanics (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 5368-5379 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A formulation of the consistent histories approach to quantum mechanics in terms of generalized observables (POV measures) and effect operators is provided. The usual notion of "history'' is generalized to the notion of "effect history.'' The space of effect histories carries the structure of a D-poset. Recent results of J. D. Maitland Wright imply that every decoherence functional defined for ordinary histories can be uniquely extended to a bi-additive decoherence functional on the space of effect histories. Omnès' logical interpretation is generalized to the present context. The result of this work considerably generalizes and simplifies the earlier formulation of the consistent effect histories approach to quantum mechanics communicated in a previous work of this author. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531709
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  • 6
    Electronic Resource
    Electronic Resource
    On Unentangled Gleason Theorems for Quantum Information Theory (2000)
    Springer
    Letters in mathematical physics 52 (2000), S. 239-245 
    Details
    ISSN: 1573-0530
    Keywords: unentangled Gleason theorem ; quantum multi-measures ; entanglement
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown here that a strengthening of Wallach's Unentangled Gleason Theorem can be obtained by applying results of the present authors on generalised Gleason theorems for quantum multi-measures arising from investigations of quantum decoherence functionals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007696209271
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  • 7
    Electronic Resource
    Electronic Resource
    Consistent histories and operational quantum theory (1996)
    Springer
    International journal of theoretical physics 35 (1996), S. 1581-1636 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02302260
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  • 8
    Electronic Resource
    Electronic Resource
    The Representation Theory of Decoherence Functionals in History Quantum Theories (2000)
    Springer
    International journal of theoretical physics 39 (2000), S. 871-884 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In the first part of this paper the general perspective of history quantum theoriesis reviewed. History quantum theories provide a conceptual and mathematicalframework for formulating quantum theories without a globally definedHamiltonian time evolution and for introducing the concept of space-time eventinto quantum theory. On a mathematical level a history quantum theory ischaracterized by the space of histories, which represent the space-time events, andby the space of decoherence functionals, which represent the quantum mechanicalstates in the history approach. The second part of this paper is devoted to thestudy of the structure of the space of decoherence functionals for some physicallyreasonable spaces of histories in some detail. The temporal reformulation ofstandard Hamiltonian quantum theories suggests to consider the case that thespace of histories is given by (i) the lattice of projection operators on someHilbert space or, slightly more generally, (ii) the set of projection operators insome von Neumann algebra. In the case (i) the conditions are identified underwhich decoherence functionals can be represented by, respectively, trace classoperators, bounded operators, or families of trace class operators on the tensorproduct of the underlying Hilbert space by itself. Moreover, we discuss thenaturally arising representations of decoherence functionals as sesquilinear forms.The paper ends with a discussion of the consequences of the results for thegeneral axiomatic framework of history theories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003635131465
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  • 9
    Electronic Resource
    Electronic Resource
    The Multi-form Generalised Gleason Theorem (1998)
    Springer
    Communications in mathematical physics 198 (1998), S. 705-709 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: Arising from investigations of decoherence functionals in the “histories” approach to quantum mechanics, the following result is given by the methods of [14]. Let A 1 and A 2 be von Neumann algebras without Type I 2 direct summands and let P(E j ), (j=1,2) be their lattices of projections. Let m\colon; P(A 1)×(P(A 2)→ℂ be a bounded quantum bi-measure. Then there is a unique bounded bilinear functional M on A 1×A 2 which extends m. In this note we use a different approach to establish a generalisation of this result to k-fold, vector valued, quantum multi-measures. This tool is needed for further investigations of decoherence functionals in quantum theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050491
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  • 10
    Electronic Resource
    Electronic Resource
    Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics (1999)
    Springer
    Communications in mathematical physics 204 (1999), S. 249-267 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050645
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