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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 24 (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 7 (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.
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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Cited by 4 (2 self)
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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.
Properties of nonfreeness: an entropy measure of electron correlation
 Int. J. of Quantum Information
, 2007
"... “Nonfreeness ” is the (negative of the) difference between the von Neumann entropies of a given manyfermion state and the free state that has the same 1particle statistics. It also equals the relative entropy of the two states in question, i.e., it is the entropy of the given state relative to the ..."
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Cited by 3 (3 self)
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“Nonfreeness ” is the (negative of the) difference between the von Neumann entropies of a given manyfermion state and the free state that has the same 1particle statistics. It also equals the relative entropy of the two states in question, i.e., it is the entropy of the given state relative to the corresponding free state. The nonfreeness of a pure state is the same as its “particlehole symmetric correlation entropy”, a variant of an established measure of electron correlation. But nonfreeness is also defined for mixed states, and this allows one to compare the nonfreeness of subsystems to the nonfreeness of the whole. Nonfreeness of a part does not exceed that in the whole; nonfreeness is additive over independent subsystems; and nonfreeness is superadditive over subsystems that are independent on the 1particle level. The word “correlation ” in the context of manyelectron systems is somewhat overcharged and ambiguous, except when used in the expression “correlation energy, ” where it refers to the difference between the energy of the true ground state of a manyelectron system and the
Contents
, 2009
"... This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some selfadjoint operators with a gap in their essential spectrum, occuring in Quantum Mechanics. First we consider Galerkin bases which preserve the decomposition of the ambient Hilbert space ..."
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Cited by 2 (0 self)
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This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some selfadjoint operators with a gap in their essential spectrum, occuring in Quantum Mechanics. First we consider Galerkin bases which preserve the decomposition of the ambient Hilbert space into a direct sum H = PH ⊕ (1 − P)H given by a fixed orthogonal projector P, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (we show that pollution is absent in a Wanniertype basis), and to Dirac operators (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in PH and vectors in (1 − P)H. Abstract results are proved and applied to several practical methods like the famous kinetic balance condition of relativistic Quantum Mechanics.
Accessible versus Holevo information for a binary random variable
, 2006
"... The accessible information Iacc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds ..."
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The accessible information Iacc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed χ(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the Iacc(E) and χ(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information Iacc(E) of an ensemble E ∆ = {(p, ρ0), (1−p, ρ1)}, with 0 ≤ p ≤ 1, as functions of p and χ(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states ρ0, ρ1. As an intermediate step, therefore, we give new relations between the two quantities Iacc(E), χ(E) and the fidelity B(ρ0, ρ1). 1