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Fundamental properties of Tsallis relative entropy
 J. Math. Phys
"... Abstract. Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy g ..."
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Cited by 39 (10 self)
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Abstract. Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown. The generalized Tsallis relative entropy is defined and its subadditivity in the special case is shown by its joint convexity. As a byproduct, the superadditivity of the quantum Tsallis entropy for the independent systems in the case of 0 ≤ q < 1 is obtained. Moreover, the generalized PeierlsBogoliubov inequality is also proven.
Sufficiency in quantum statistical inference
"... This paper attempts to develop a theory of sufficiency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarsegraining means that all information is extracted about the mutual relation of a given family of states. In the paper s ..."
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Cited by 23 (4 self)
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This paper attempts to develop a theory of sufficiency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarsegraining means that all information is extracted about the mutual relation of a given family of states. In the paper su cient coarsegrainings are characterized in several equivalent ways and the noncommutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only infinite Hilbert space dimension, but proved here in generality: the KoashiImoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity.
Quantum graphical models and belief propagation
 Annals of Physics
, 2008
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A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality
 MATH. PHYS
, 2009
"... We introduce a generalization of relative entropy derived from the WignerYanaseDyson entropy and give a simple, selfcontained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K∗ApKB 1−p Lieb’s joint concavity in (A,B) for 0 < p < 1 a ..."
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Cited by 15 (3 self)
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We introduce a generalization of relative entropy derived from the WignerYanaseDyson entropy and give a simple, selfcontained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K∗ApKB 1−p Lieb’s joint concavity in (A,B) for 0 < p < 1 and Ando’s joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Lieb and Carlen for Tr1(Tr2 A p 12)1/p. In all cases the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.
Another short and elementary proof of strong subadditivity of quantum entropy
 Rep. Math. Phys
"... A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the CauchySchwarz inequality in elementary courses. Several consequences are p ..."
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Cited by 10 (2 self)
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A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the CauchySchwarz inequality in elementary courses. Several consequences are proved in a way which allow an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound for partial measurements is also included. 1
Capacities of quantum channels and how to find them
 Mathematical Programming
, 2003
"... Abstract: We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming. 1 ..."
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Cited by 9 (1 self)
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Abstract: We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming. 1
FROM JOINT CONVEXITY OF QUANTUM RELATIVE ENTROPY TO A CONCAVITY THEOREM OF LIEB
"... Abstract. This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen an ..."
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Cited by 9 (3 self)
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Abstract. This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb. 1.