Access the full text.
Sign up today, get DeepDyve free for 14 days.
D. Petz (1986)
Quasi-entropies for finite quantum systemsReports on Mathematical Physics, 23
H. Umegaki (1959)
Conditional expectation in an operator algebra. III.Kodai Mathematical Seminar Reports, 11
Frank Hansen, G. Pedersen (2002)
Jensen's Operator InequalityBulletin of the London Mathematical Society, 35
V. Jaksic, Y. Ogata, Y. Pautrat, C. Pillet (2011)
Entropic Fluctuations in Quantum Statistical Mechanics. An IntroductionarXiv: Mathematical Physics
泉 英明 (2000)
Non-commutative Lp-spaces〔和文〕 (作用表環論の進展)
H. Kosaki (1992)
An inequality of Araki-Lieb-Thirring (von Neumann algebra case), 114
M. Takesaki (1979)
Theory of Operator Algebras II
Martin Muller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel (2013)
On quantum Rényi entropies: A new generalization and some propertiesJournal of Mathematical Physics, 54
Rahul Jain, J. Radhakrishnan, P. Sen (2002)
Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum statesThe 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
D. Petz (1985)
Quasi-entropies for States of a von Neumann AlgebraPublications of The Research Institute for Mathematical Sciences, 21
竹崎 正道 (2002)
Theory of operator algebras
H. Araki, T. Masuda (1982)
Positive Cones and L p -Spaces for von Neumann AlgebrasPublications of The Research Institute for Mathematical Sciences, 18
(2012)
Smooth entropies: A Tutorial
M. Berta, K. Seshadreesan, M. Wilde (2014)
Renyi generalizations of the conditional quantum mutual informationArXiv, abs/1403.6102
M. Mosonyi, T. Ogawa (2014)
Strong Converse Exponent for Classical-Quantum Channel CodingCommunications in Mathematical Physics, 355
H. Umegaki (1962)
Conditional expectation in an operator algebra. IV. Entropy and informationKodai Mathematical Seminar Reports, 14
N. Datta (2008)
Min- and Max-Relative Entropies and a New Entanglement MonotoneIEEE Transactions on Information Theory, 55
T. Ogawa, H. Nagaoka (1999)
Strong converse and Stein's lemma in quantum hypothesis testingIEEE Trans. Inf. Theory, 46
Donald Burkholder, Ronald Graham, William Johnson, Peter Jones (1996)
Transactions of the American Mathematical Society
KMR Audenaert, N Datta (2015)
$$\alpha $$ α -z-relative Renyi entropiesJ. Math. Phys., 56
M. Tomamichel (2015)
Quantum Information Processing with Finite Resources - Mathematical FoundationsArXiv, abs/1504.00233
M. Mosonyi, T. Ogawa (2013)
Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative EntropiesCommunications in Mathematical Physics, 334
M. Berta, F. Furrer, V. Scholz (2011)
The smooth entropy formalism for von Neumann algebrasJournal of Mathematical Physics, 57
M. Tomamichel, M. Wilde, A. Winter (2014)
Strong Converse Rates for Quantum CommunicationIEEE Transactions on Information Theory, 63
A. Lichnerowicz (2018)
Proof of the Strong Subadditivity of Quantum-Mechanical Entropy
Alexander Müller-Hermes, D. Reeb (2015)
Monotonicity of the Quantum Relative Entropy Under Positive MapsAnnales Henri Poincaré, 18
H Araki (1982)
339Publ. Res. Inst. Math. Sci. Kyoto Univ., 18
BH Araki (1976)
Relative entropy of states of von Neumann algebrasPubl. Res. Inst. Math. Sci. Kyoto Univ., 11
V. Jaksic, Y. Ogata, C. Pillet, R. Seiringer (2011)
Quantum Hypothesis Testing and Non-Equilibrium Statistical MechanicsArXiv, abs/1109.3804
M. Wilde, A. Winter, Dong Yang (2013)
Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative EntropyCommunications in Mathematical Physics, 331
H. Araki (1990)
On an inequality of Lieb and ThirringLetters in Mathematical Physics, 19
Salman Beigi (2013)
Sandwiched Rényi divergence satisfies data processing inequalityJournal of Mathematical Physics, 54
U. Haagerup, M. Junge, Quanhua Xu (2008)
A reduction method for noncommutative L_p-spaces and applicationsTransactions of the American Mathematical Society, 362
K. Audenaert, N. Datta (2013)
alpha-z-relative Renyi entropies
A. Uhlmann (1976)
The "transition probability" in the state space of a ∗-algebraReports on Mathematical Physics, 9
A. Uhlmann (1977)
Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theoryCommunications in Mathematical Physics, 54
H. Araki (1975)
Relative Entropy of States of von Neumann AlgebrasPublications of The Research Institute for Mathematical Sciences, 11
P. Alberti (1983)
A note on the transition probability over C*-algebrasLetters in Mathematical Physics, 7
K. Audenaert, N. Datta (2015)
α-z-Rényi relative entropiesJournal of Mathematical Physics, 56
J. Bisognano, E. Wichmann (1975)
On the duality condition for a Hermitian scalar fieldJournal of Mathematical Physics, 16
P. Bouwknegt, P. Jarvis, J. Links (2006)
Reports on Mathematical Physics
BH Araki (1976)
809Publ. Res. Inst. Math. Sci. Kyoto Univ., 11
M Tomamichel (2016)
Quantum Information Processing with Finite Resources—Mathematical Foundations. Volume 5 of Springer Briefs in Mathematical Physics
M Takesaki (2003)
Theory of Operator Algebras II. Volume 125 of Encyclopaedia of Mathematical Sciences
W. Stinespring (2010)
POSITIVE FUNCTIONS ON C*-ALGEBRAS
A. Jenčová (2016)
Rényi Relative Entropies and Noncommutative $$L_p$$Lp-SpacesAnnales Henri Poincaré, 19
E. Lieb, W. Thirring (2002)
Inequalities for the Moments of the Eigenvalues of the Schrodinger Hamiltonian and Their Relation to Sobolev Inequalities
H. Kosaki (1984)
APPLICATIONS OF THE COMPLEX INTERPOLATION METHOD TO A VON NEUMANN ALGEBRA: NON-COMMUTATIVE LP-SPACESJournal of Functional Analysis, 56
(1977)
Lp-Spaces Associated with an Arbitrary von Neumann Algebra
大矢 雅則, D. Petz (1993)
Quantum Entropy and Its Use
G. Pisier, Quanhua Xu (2003)
Chapter 34 - Non-Commutative Lp-Spaces, 2
京都大学数理解析研究所 (1969)
Publications of the Research Institute for Mathematical Sciences
W. Stinespring (1955)
Positive functions on *-algebras, 6
H Araki, T Masuda (1982)
Positive cones and Lp-spaces for von Neumann algebrasPubl. Res. Inst. Math. Sci. Kyoto Univ., 18
R. Frank, E. Lieb (2013)
Monotonicity of a relative Rényi entropyArXiv, abs/1306.5358
We show that Araki and Masuda’s weighted non-commutative vector-valued $$L_p$$ L p -spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter $$\alpha = \frac{p}{2}$$ α = p 2 . Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $$\alpha $$ α . We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $$\alpha \rightarrow \{\frac{1}{2},1,\infty \}$$ α → { 1 2 , 1 , ∞ } leading to minus the logarithm of Uhlmann’s fidelity, Umegaki’s relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz–Thorin theorem for Araki–Masuda $$L_p$$ L p -spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.
Annales Henri Poincaré – Springer Journals
Published: Mar 17, 2018
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.