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  • 1
    Digitale Medien
    Digitale Medien
    Geometries of quantum states (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 2662-2673 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension. © 1996 American Institute of Physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.531535
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  • 2
    Digitale Medien
    Digitale Medien
    Geometry of canonical correlation on the state space of a quantum system (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 780-795 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the "more mixed'' (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér–Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.530611
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  • 3
    Digitale Medien
    Digitale Medien
    Stationary quantum source coding (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 4857-4864 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: In this article the quantum version of the source coding theorem is obtained for a completely ergodic source. This result extends Schumacher's quantum noiseless coding theorem for memoryless sources. The control of the memory effects requires some earlier results of Hiai and Petz on high probability subspaces. Our result is equivalently considered as a compression theorem for noiseless stationary channels. © 2001 American Institute of Physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.1398335
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  • 4
    Digitale Medien
    Digitale Medien
    McMillan type convergence for quantum Gibbs states (1995)
    Springer
    Archiv der Mathematik 64 (1995), S. 154-158 
    Details
    ISSN: 1420-8938
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01196636
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  • 5
    Digitale Medien
    Digitale Medien
    On the equality in Jensen's inequality for operator convex functions (1986)
    Springer
    Integral equations and operator theory 9 (1986), S. 744-747 
    Details
    ISSN: 1420-8989
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Let A and B be C*-algebras with unit and assume that ϕ∶A→B is a positive unit preserving linear mapping. Choi proved that $$f(\Phi (a)) \leqq \Phi (f(a))$$ if a=a*∈A and Sp(a)⊂(α, β) for every operator convex function f: (α, β) → ℝ. We prove that the equality holds if and only if ϕ restricted to the subalgebra generated by {a} is multiplicative. An example is shown as an application.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01195811
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  • 6
    Digitale Medien
    Digitale Medien
    The proper formula for relative entropy and its asymptotics in quantum probability (1991)
    Springer
    Communications in mathematical physics 143 (1991), S. 99-114 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract Umegaki's relative entropyS(ω,ϕ)=TrD ω(logD ω−logD ϕ) (of states ω and ϕ with density operatorsD ω andD ϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality TrA logAB ≧TrA(logA+logB) is obtained for positive definite matricesA andB.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02100287
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  • 7
    Digitale Medien
    Digitale Medien
    Sufficient subalgebras and the relative entropy of states of a von Neumann algebra (1986)
    Springer
    Communications in mathematical physics 105 (1986), S. 123-131 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract A subalgebraM 0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM 0. The main result says thatM 0 is weakly sufficient for (φ,ω) if and only ifM 0 contains the Radon-Nikodym cocycle [Dφ,Dω] t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01212345
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  • 8
    Digitale Medien
    Digitale Medien
    A variational expression for the relative entropy (1988)
    Springer
    Communications in mathematical physics 114 (1988), S. 345-349 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract We prove that for the relative entropy of faithful normal states ϕ and ω on the von Neumann algebraM the formula $$S(\varphi ,\omega ) = \sup \{ \omega (h) - \log \varphi ^h (I):h = h^* \in M\}$$ holds.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01225040
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  • 9
    Digitale Medien
    Digitale Medien
    Properties of Free Entropy Related to Polar Decomposition (1999)
    Springer
    Communications in mathematical physics 202 (1999), S. 421-444 
    Details
    ISSN: 1432-0916
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik , Physik
    Notizen: Abstract: The free entropies of non-selfadjoint random variables and of unitary random variables are introduced and discussed by the methods of Voiculescu's free analysis. The additivity is shown to be equivalent to freeness. The relation among , and χ is investigated in the case when is the polar decomposition. The subadditivity is proven and applications to some maximization problems for are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002200050589
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  • 10
    Digitale Medien
    Digitale Medien
    A de Finetti-type theorem withm-dependent states (1990)
    Springer
    Probability theory and related fields 85 (1990), S. 65-72 
    Details
    ISSN: 1432-2064
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper certain translation invariant states on the infinite tensor productC *-algebra are considered. Form ∈ ℤ+ a state ϕ onA ism-dependent if $$\varphi (a_1 \otimes a_2 \otimes ... \otimes a_l ) = \varphi (a_1 \otimes ... \otimes a_k )\varphi (a_{k + m + 1} \otimes ... \otimes a_l )$$ wheneverl〉k+m anda k + 1 =a k + 2 = ... =a k +m = 1. The closed convex hull of the stationarym-dependent states is characterized by a symmetry condition. The case ofm=0 corresponds to independence and the result reduces to aC *-algebraic version, due to E. Størmer, of the classical de Finetti's theorem on exchangeable sequences.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01377629
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