arXiv subjects

Anna Jencova

Publications and source records attributed to Anna Jencova.

At least 19 recordsLinked to original sources

arXiv2020

Rényi relative entropies and noncommutative $L_p$-spaces

We propose an extension of the sandwiched Rényi relative $α$-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values $α>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces with respect to a state. We show that these extensions coincide with the previously defined Araki-Masuda divergences [M. Berta et al., Annales Henri Poincaré, 19:1843--1867, 2018] and prove some of their properties, in particular the data processing inequality with respect to positive normal unital maps. As a consequence, we obtain monotonicity of the Araki relative entropy with respect to such maps, extending the results of [A. Müller-Hermes and D. Reeb. Annales Henri Poincaré,18:1777--1788, 2017] to arbitrary von Neumann algebras. It is also shown that equality in data processing inequality characterizes sufficiency (reversibility) of quantum channels.

quant-ph↗

arXiv2019

Tensor product of dimension effect algebras

Dimension effect algebras were introduced in (A. Jencova, S. Pulmannova, Rep. Math. Phys. 62 (2008), 205-218), and it was proved that they are unit intervals in dimension groups. We prove that the effect algebra tensor product of dimension effect algebras is a dimension effect algebra, which is the unit interval in the unital abelian po-groups tensor product of the corresponding dimension groups.

math.RA↗

arXiv2017

Preservation of a quantum Renyi relative entropy implies existence of a recovery map

It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched Rényi relative $α$-entropy with $α>1$ is also equivalent to this property. For the proof, we use an interpolating family of $L_p$-norms with respect to a state.

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arXiv2017

A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $ξ_a$ on the $σ$-orthomodular lattice of projections in $A$ and that $ξ_a$ determines the spectral resolution of $a$. Also, if $f$ is a continuous function defined on the spectrum of $a$, we formulate a definition of $f(a)$, thus obtaining a continuous functional calculus for $A$.

math.RA↗

arXiv2016

States and synaptic algebras

Different versions of the notion of a state have been formulated for various so-called quantum structures. In this paper, we investigate the interplay among states on synaptic algebras and on its sub-structures. A synaptic algebra is a generalization of the partially ordered Jordan algebra of all bounded self-adjoint operators on a Hilbert space. The paper culminates with a characterization of extremal states on a commutative generalized Hermitian algebra, a special kind of synaptic algebra.

math-ph↗

arXiv2016

A projection and an effect in a synaptic algebra

We study a pair p,e consisting of a projection p (an idempotent) and an effect e (an element between 0 and 1) in a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra). We show that some of Halmos's theory of two projections (or two subspaces), including a version of his CS-decomposition theorem, applieas on this settinh, and we introduce and study two candidates for a commutator for p and e.

math.FA↗

arXiv2016

Vector lattices in synaptic algebras

A synaptic algebra $A$ is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace $V$ of $A$ in regard to the question of when $V$ is a vector lattice. Our main theorem states that if $V$ contains the identity element of $A$ and is closed under the formation of both the absolute value and the carrier of its elements, then $V$ is a vector lattice if and only if the elements of $V$ commute pairwise.

math.RA↗

arXiv2016

Every synaptic algebra has the monotone square root property

A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW*-algebra. In this paper we prove that a synaptic algebra A has the monotone square property, i.e., if a and b are positive elements, then if a is less or equal than b, then the square root of a is less or equal than the square root of b.

math.OA↗

arXiv2016

Comparison of quantum channels and statistical experiments

For a pair of quantum channels with the same input space, we show that the possibility of approximation of one channel by post-processings of the other channel can be characterized by comparing the success probabilities for the two ensembles obtained as outputs for any ensemble on the input space coupled with an ancilla. This provides an operational interpretation to a natural extension of Le Cam's deficiency to quantum channels. In particular, we obtain a version of the randomization criterion for quantum statistical experiments. The proofs are based on some properties of the diamond norm and its dual, which are of independent interest.

quant-ph↗

arXiv2016

Comparison of quantum channels and statistical experiments

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arXiv2015

On the convex structure of process POVMs

Measurements on quantum channels are described by so-called process operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger that the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structure. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We give a characterization of A-extremal and A-pure POVMs in the Appendix.

quant-ph↗

arXiv2015

Exploring boundaries of quantum convex structures: special role of unitary processes

We address the question of finding the most effective convex decompositions into boundary elements (so-called boundariness) for sets of quantum states, observables and channels. First we show that in general convex sets the boundariness essentially coincides with the question of the most distinguishable element, thus, providing an operational meaning for this concept. Unexpectedly, we discovered that for any interior point of the set of channels the optimal decomposition necessarily contains a unitary channel. In other words, for any given channel the best distinguishable one is some unitary channel. Further, we prove that boundariness is sub-multiplicative under composition of systems and explicitly evaluate its maximal value that is attained only for the most mixed elements of the considered convex structures.

quant-ph↗

arXiv2015

Two projections in a synaptic algebra

We investigate P. Halmos' two projections theorem, (or two subspaces theorem) in the context of a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra).

math.FA↗

arXiv2014

Randomization theorems for quantum channels

The classical randomization criterion is an important result of statistical decision theory. Recently, a quantum analogue has been proposed, giving equivalent conditions for two sets of quantum states, ensuring existence of a quantum channel mapping one set close to the other, in $L_1$-distance. In the present paper, we extend these concepts in several ways. First, sets of states are replaced by channels and randomization is performed by either post- or pre-composition with another channel. The $L_1$-distance is replaced by the diamond norm. Secondly, the maps are not required to be completely positive, but positivity is given by an admissible family of convex cones. It is shown that the randomization theorems, generalizing both quantum and classical randomization criteria, can be proved in the framework of base section norms, including the diamond norm and its dual. The theory of such norms is developed in the Appendix.

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arXiv2013

Base norms and discrimination of generalized quantum channels

We introduce and study norms in the space of hermitian operators, obtained from base norms in positively generated subspaces. These norms are closely related to discrimination of so-called generalized quantum channels, including quantum states, channels and networks. We further introduce generalized quantum decision problems and show that the maximal average payoff of decision procedures is again given by these norms. We also study optimality of decision procedures, in particular, we obtain a necessary and sufficient condition under which an optimal 1-tester for discrimination of quantum channels exists, such that the input state is maximally entangled.

quant-ph↗

arXiv2013

Extremal generalized quantum measurements

A measurement on a section K of the set of states of a finite dimensional C*-algebra is defined as an affine map from K to a probability simplex. Special cases of such sections are used in description of quantum networks, in particular quantum channels. Measurements on a section correspond to equivalence classes of so-called generalized POVMs, which are called quantum testers in the case of networks. We find extremality conditions for measurements on K and characterize generalized POVMs such that the corresponding measurement is extremal. These results are applied to the set of channels. We find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1-tester such that the corresponding measurement is not extremal.

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arXiv2012

Comparison of quantum binary experiments

A quantum binary experiment consists of a pair of density operators on a finite dimensional Hilbert space. An experiment E is called ε-deficient with respect to another experiment F if, up to ε, its risk functions are not worse than the risk functions of F, with respect to all statistical decision problems. It is known in the theory of classical statistical experiments that 1. for pairs of probability distributions, one can restrict to testing problems in the definition of deficiency and 2. that 0-deficiency is a necessary and sufficient condition for existence of a stochastic mapping that maps one pair onto the other. We show that in the quantum case, the property 1. holds precisely if E consist of commuting densities. As for property 2., we show that if E is 0-deficient with respect to F, then there exists a completely positive mapping that maps E onto F, but it is not necessarily trace preserving.

quant-ph↗

arXiv2012

Reversibility conditions for quantum operations

We give a list of equivalent conditions for reversibility of the adjoint of a unital Schwarz map with respect to a set of quantum states. A large class of such conditions is given by preservation of distinguishability measures: f-divergences, L_1 -distance, quantum Chernoff and Hoeffding distances; here we summarize and extend the known results. Moreover, we prove a number of conditions in terms of the properties of a quantum Radon-Nikodym derivative and factorization of states in the given set. Finally, we show that reversibility is equivalent with preservation of a large class of quantum Fisher informations and χ^2-divergences.

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