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CENTAUR: The system
 In Software Development Environments (SDE
, 1988
"... asymptotic normality for finite dimensional quantum ..."
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asymptotic normality for finite dimensional quantum
Local asymptotic normality in quantum statistics
, 2006
"... The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically pr ..."
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The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family ϕn θ0+u / √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all “local parameters ” u ∈ Rm such that θ = θ0 + u / √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ Rm. In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and show that weak and strong convergence are equivalent in the case of finite number of parameters for experiments based on type I algebras with discrete center. For reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory.
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 a ..."
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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
Markov triplets on CCRalgebras
 Acta Sci. Math. (Szeged
"... The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasifree states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasifree state is descr ..."
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The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasifree states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasifree state is described by the representing block matrix. The proof is based on results on the statistical sufficiency in the noncommutative case. The relation to classical Gaussian Markov triplets is also described.
Markov property and Strong additivity of von Neumann entropy for graded quantum systems
, 2006
"... It is easily verified that the quantum Markov property is equivalent to the strong additivity of von Neumann entropy for graded quantum systems. However, the structure of Markov states for graded systems is different from that for tensorproduct systems which have trivial grading. For threecomposed ..."
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It is easily verified that the quantum Markov property is equivalent to the strong additivity of von Neumann entropy for graded quantum systems. However, the structure of Markov states for graded systems is different from that for tensorproduct systems which have trivial grading. For threecomposed graded systems we have U(1)gauge invariant Markov states whose restriction to the marginal pair of subsystems is nonseparable. Key Words: Graded commutation relations. Strong subadditivity of von Neumann entropy. Sufficiency of conditional expectations. Quantum Markov property. Statistical independence for nonindependent systems. 1
EQUIVALENCE CLASSES AND LOCAL ASYMPTOTIC NORMALITY IN SYSTEM IDENTIFICATION FOR QUANTUM MARKOV CHAINS
"... Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up t ..."
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Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up to a ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. When the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard ’ rate n−1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e. it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check we prove that a parameter related to the ‘coordinate transformation ’ unitaries, has zero quantum Fisher information. 1.
Acta Sci. Math. (Szeged), 76(2010), 111–134. Markov triplets on CCRalgebras
"... The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasifree states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasifree state is descr ..."
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The paper contains a detailed computation about the algebra of canonical commutation relation, the representation of the Weyl unitaries, the quasifree states and their von Neumann entropy. The Markov triplet is defined by constant entropy increase. The Markov property of a quasifree state is described by the representing block matrix. The proof is based on results on the statistical sufficiency in the noncommutative case. The relation to classical Gaussian Markov triplets is also described.
Rev. Math. Phys. 23, 691–747 (2011). Quantum fdivergences and error correction
"... Quantum fdivergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz ’ quasientropies. Many wellknown distinguishability measures of quantum states are given by, or derived from, fdivergences; special examples include the quantum relative ent ..."
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Quantum fdivergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz ’ quasientropies. Many wellknown distinguishability measures of quantum states are given by, or derived from, fdivergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum fdivergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz ’ reversibility theorem for a large class of fdivergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable fdivergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex