Quasi-entropies for States of a von Neumann Algebra

@article{Petz1985QuasientropiesFS,
  title={Quasi-entropies for States of a von Neumann Algebra},
  author={D{\'e}nes Petz},
  journal={Publications of The Research Institute for Mathematical Sciences},
  year={1985},
  volume={21},
  pages={787-800}
}
  • D. Petz
  • Published 31 August 1985
  • Mathematics
  • Publications of The Research Institute for Mathematical Sciences
In a general von Neumann algebra context the relative entropy of two states was defined and investigated by Araki ([3], see also [5]). When <p and w are normal states on a von Neuman algebra M the relative entropy S(<p,<u) is defined by means of the relative modular operator A (cp, co). r-<logJ(p,a00,0> if J(0>)>j(o0. S(<p 9 co) = \. v + oo otherwise. where Q is the representing vector for co in the natural positive cone of the standard form of M and $(•) denotes the support of a functional. We… 
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  • G. Gour, M. Wilde
  • Computer Science
    2020 IEEE International Symposium on Information Theory (ISIT)
  • 2020
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References

SHOWING 1-10 OF 30 REFERENCES
Positive Cones and L p -Spaces for von Neumann Algebras
The Lp-space LP(M, if) for a von Neumann algebra M with reference to its cyclic and separating vector T? in the standard representation Hilbert space H of M is constructed either as a subset of H
Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of von Neumann algebras in standard form
AbstractIt is shown that for normal states ρ and φ of a W*-algebra $$A, P(\rho ,\phi ) \leqslant (\xi (\rho ),\xi (\phi )) \leqslant P(\rho ,\phi )^{1/2} $$ , where P(.,.) is the transition
Quasi-entropies for finite quantum systems
Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory
We show that the Wigner-Yanase-Dyson-Lieb concavity is a general property of an interpolation theory which works between pairs of (hilbertian) seminorms. As an application, the theory extends the
General properties of entropy
TLDR
This paper discusses properties of entropy, as well as related concepts such as relative entropy, skew entropy, dynamical entropy, etc, in detail with reference to their implications in statistical mechanics, to get a glimpse of systems with infinitely many degrees of freedom.
Interpolation theory and the Wigner-Yanase-Dyson-Lieb concavity
The Wigner-Yanase-Dyson-Lieb concavity is naturally captured in the frame of interpolation theory. Among other results, a certain generalization (involving operator monotone functions) of this
Form convex functions and the wydl and other inequalities
We show that the functional calculus of sesquilinear forms derived in [3] leads to a general theory of the Wigner, Yanase, Dyson-Lieb-type inequalities. In particular, we obtain the joint convexity
Modular Theory in Operator Algebras
TLDR
A bale and grain feeder device includes an elongate conveyor and table structure which is mounted on the mixer-grinder vehicle chassis and cooperates with the conveyor paddles to convey the slabs of hay along the table where the hay or grain is directed to the hammer mill by paddles.
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