Non-commutative perspectives
We prove that the non-commutative perspective of an operator convex function is the unique extension of the corresponding commutative perspective that preserves homogeneity and convexity.
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arXiv subjects
Frank Hansen
Publications and source records attributed to Frank Hansen.
Trace functions with applications in quantum physics
We consider both known and not previously studied trace functions with applications in quantum physics. By using perspectives we obtain convexity statements for different notions of residual entropy, including the entropy gain of a quantum channel as studied by Holevo and others. We give new and simplified proofs of the Carlen-Lieb theorems concerning concavity or convexity of certain trace functions by making use of the theory of operator monotone functions. We then apply these methods in a study of new types of trace functions. Keywords: Trace function, convexity, entropy gain, residual entropy, operator monotone function.
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The fast track to Löwner's theorem
The operator monotone functions defined in the positive half-line are of particular importance. We give a version of the theory in which integral representations for these functions can be established directly without invoking Löwner's detailed analysis of matrix monotone functions of a fixed order or the theory of analytic functions. We found a canonical relationship between positive and arbitrary operator monotone functions defined in the positive half-line, and this result effectively reduces the theory to the case of positive functions. MSC2010 classification: 26A48; 26A51; 47A63. Key words and phrases: operator monotone function; integral representation; Löwner's theorem.
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WYD-like skew information measures
Following recent advances in the theory of operator monotone functions we introduce new classes of WYD-like skew information measures.
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Convexity of quantum $χ^2$-divergence
The quantum χ^2-divergence has recently been introduced and applied to quantum channels (quantum Markov processes). In contrast to the classical setting the quantum χ^2-divergence is not unique but depends on the choice of quantum statistics. In the reference [11] a special one-parameter family of quantum χ^2_α(ρ,σ)-divergences for density matrices were studied, and it was established that they are convex functions in (ρ,σ) for parameter values α\in [0,1], thus mirroring the classical theorem for the χ^2(p,q)-divergence for probability distributions (p,q). We prove that any quantum χ^2-divergence is a convex function in its two arguments.
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Metric adjusted skew information: Convexity and restricted forms of superadditivity
We give a truly elementary proof of the convexity of metric adjusted skew information following an idea of Effros. We extend earlier results of weak forms of superadditivity to general metric adjusted skew informations. Recently, Luo and Zhang introduced the notion of semi-quantum states on a bipartite system and proved superadditivity of the Wigner-Yanase-Dyson skew informations for such states. We extend this result to general metric adjusted skew informations. We finally show that a recently introduced extension to parameter values $ 1<p\le 2 $ of the WYD-information is a special case of (unbounded) metric adjusted skew information.
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Non-commutative Hardy inequalities
We extend Hardy's inequality from sequences of non-negative numbers to sequences of positive semi-definite operators if the parameter p satisfies 1 1. Applications to trace functions are given. We introduce the tracial geometric mean and generalize Carleman's inequality. Key words and phrases: Hardy's inequality, positive operator, trace function, geometric mean, Carleman's inequality.
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Some operator monotone functions
We prove that the functions t -> (t^q-1)(t^p-1)^{-1} are operator monotone in the positive half-axis for 0 < p < q < 1, and we calculate the two associated canonical representation formulae. The result is used to find new monotone metrics (quantum Fisher information) on the state space of quantum systems.
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Inequalities for quantum skew information
We study quantum information inequalities and show that the basic inequality between the quantum variance and the metric adjusted skew information generates all the multi-operator matrix inequalities or Robertson type determinant inequalities studied by a number of authors. We introduce an order relation on the set of functions representing quantum Fisher information that renders the set into a lattice with an involution. This order structure generates new inequalities for the metric adjusted skew informations. In particular, the Wigner-Yanase skew information is the maximal skew information with respect to this order structure in the set of Wigner-Yanase-Dyson skew informations. Key words and phrases: Quantum covariance, metric adjusted skew information, Robertson-type uncertainty principle, operator monotone function, Wigner-Yanase-Dyson skew information.
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Metric adjusted skew information
We extend the concept of Wigner-Yanase-Dyson skew information to something we call ``metric adjusted skew information'' (of a state with respect to a conserved observable). This ``skew information'' is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the ``lambda-skew information,'' parametrized by a lambda in (0,1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations. Key words: Skew information, convexity, monotone metric, Morozova-Chentsov function, lambda-skew information.
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Differential analysis of matrix convex functions II
We continue the analysis in [3] of matrix convex functions of a fixed order defined in a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus [5]. We amend and improve some points in the previously given presentation, and we give a number of simple but important consequences of matrix convexity of low orders.
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Jensen's operator inequality and its converses
We give a general formulation of Jensen's operator inequality for unital fields of positive linear mappings, and we consider different types of converse inequalities.
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The Wigner-Yanase entropy is not subadditive
Wigner and Yanase introduced in 1963 the Wigner-Yanase entropy defined as minus the skew information of a state with respect to a conserved observable. They proved that the Wigner-Yanase entropy is a concave function in the state and conjectured that it is subadditive with respect to the aggregation of possibly interacting subsystems. While this turned out to be true for the quantum-mechanical entropy, we negate the conjecture for the Wigner-Yanase entropy by providing a counter example.
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Jensen's inequality for conditional expectations
We study conditional expectations generated by an abelian $ C^* $-subalgebra in the centralizer of a positive functional. We formulate and prove Jensen's inequality for functions of several variables with respect to this type of conditional expectations, and we obtain as a corollary Jensen's inequality for expectation values.
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Extensions of Lieb's concavity theorem
The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As a special case we obtain a new proof of Lieb's concavity theorem for the function (A,B)\to\tr A^pK^*B^{q}K, where p and q are non-negative numbers with sum p+q\le 1. In addition, we prove concavity of the operator function (A,B)\to \tr(A(A+μ_1)^{-1}K^* B(B+μ_2)^{-1}K) on its natural domain D_2(μ_1,μ_2), cf. Definition 4.1
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Monotone trace functions of several variables
We investigate monotone operator functions of several variables under a trace or a trace-like functional. In particular, we prove the inequality τ(x_1... x_n)\leτ(y_1... y_n) for a trace τon a C^*-algebra and abelian n-tuples (x_1,...,x_n)\le (y_1,...,y_n) of positive elements. We formulate and prove Jensen's inequality for expectation values, and we study matrix functions of several variables which are convex or monotone with respect to the weak majorization for matrices.
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Differential analysis of matrix convex functions
We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of order n+1.
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Characterization of symmetric monotone metrics on the state space of quantum systems
The quantum Fisher information is a Riemannian metric, defined on the state space of a quantum system, which is symmetric and decreasing under stochastic mappings. Contrary to the classical case such a metric is not unique. We complete the characterization, initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed and tractable formula for the set of Morozova-Chentsov functions. In addition, we provide a continuously increasing bridge between the smallest and largest symmetric monotone metrics.
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