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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 21 (6 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 7 (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.
Structure of sufficient quantum coarsegrainings
"... B(K) be a coarsegraining and D1, D2 be density matrices on H. In this paper the consequences of the existence of a coarsegraining β: B(K) → B(H) satisfying βT(Ds) = Ds are given. (This means that T is sufficient for D1 and D2.) It is shown that Ds = ∑ r p=1 λs(p)S H s (p)R H (p) (s = 1, 2) shou ..."
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Cited by 6 (3 self)
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B(K) be a coarsegraining and D1, D2 be density matrices on H. In this paper the consequences of the existence of a coarsegraining β: B(K) → B(H) satisfying βT(Ds) = Ds are given. (This means that T is sufficient for D1 and D2.) It is shown that Ds = ∑ r p=1 λs(p)S H s (p)R H (p) (s = 1, 2) should hold with pairwise orthogonal summands and with commuting factors and with some probability distributions λs(p) for 1 ≤ p ≤ r (s = 1, 2). This decomposition allows to deduce the exact condition for equality in the strong subaddivity of the von Neumann entropy.
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 6 (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
A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality, Rev
 Math. Phys
"... 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 And ..."
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Cited by 6 (1 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. Supported by the grants VEGA 2/0032/09 and APVV 007106
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 4 (0 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
C.: On the power of twoparty quantum cryptography
, 2009
"... Abstract. We study quantum protocols among two distrustful parties. Under the sole assumption of correctness—guaranteeing that honest players obtain their correct outcomes—we show that every protocol implementing a nontrivial primitive necessarily leaks information to a dishonest player. This exten ..."
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Cited by 3 (1 self)
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Abstract. We study quantum protocols among two distrustful parties. Under the sole assumption of correctness—guaranteeing that honest players obtain their correct outcomes—we show that every protocol implementing a nontrivial primitive necessarily leaks information to a dishonest player. This extends known impossibility results to all nontrivial primitives. We provide a framework for quantifying this leakage and argue that leakage is a good measure for the privacy provided to the players by a given protocol. Our framework also covers the case where the two players are helped by a trusted third party. We show that despite the help of a trusted third party, the players cannot amplify the cryptographic power of any primitive. All our results hold even against quantum honestbutcurious adversaries who honestly follow the protocol but purify their actions and apply a different measurement at the end of the protocol. As concrete examples, we establish lower bounds on the leakage of standard universal twoparty primitives such as oblivious transfer.
Su ciency in quantum statistical inference
"... This paper attempts to develop a theory of su ciency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Su ciency of a coarsegraining means that all information is extracted about the mutual relation of a given family of states. In the paper su ci ..."
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Cited by 2 (2 self)
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This paper attempts to develop a theory of su ciency in the setting of noncommutative algebras parallel to the ideas in classical mathematical statistics. Su ciency 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 in nite 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. MSC: 46L53, 81R15, 62B05.