arXiv subjects

Anna Jenčová

Publications and source records attributed to Anna Jenčová.

25 records · Page 2Linked to original sources

arXiv2020

Rényi relative entropies and noncommutative $L_p$-spaces II

We study an extension of the sandwiched Rényi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $α\in [1/2,1)$. This work is intended as a continuation of [A. Jenčová, Ann. Henri Poincaré 19, 2513-2542, 2018], where the values $α>1$ were studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri Poincaré 9, 1843-1867, 2018] and treat them in the framework of Kosaki's noncommutative $L_p$-spaces. Using the variational formula, recently obtained by F. Hiai, for $α\in [1/2,1)$, we prove the data processing inequality with respect to positive trace preserving maps and show that for $α\in (1/2,1)$, equality characterizes sufficiency (reversibility) for any 2-positive trace preserving map.

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arXiv2019

On the properties of spectral effect algebras

The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We also provide some results on the structure of spectral effect algebras and their state spaces and investigate the direct products and direct convex sums of spectral effect algebras. In the case of spectral effect algebras with sharply determining state space, stronger properties can be proved: the spectral decompositions are essentially unique, the algebra is sharply dominating and the set of its sharp elements is an orthomodular lattice. The article also contains a list of open questions that might provide interesting future research directions.

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arXiv2019

On period, cycles and fixed points of a quantum channel

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.

math.OA↗

arXiv2018

Incompatible measurements in a class of general probabilistic theories

We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite dimensional general probabilistic theories (GPT).The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the maximum incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPT. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the super-quantum non-classical effects that were observed on these spaces.

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arXiv2017

Conditions on the existence of maximally incompatible two-outcome measurements in General Probabilistic Theory

We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite dimensional General Probabilistic Theory. The conditions are on the geometry of the state space, they require existence of two pairs of parallel exposed faces with additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet non-existing) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.

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arXiv2017

On monoids in the category of sets and relations

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ is a poset with respect to inclusion. We examine the 2-category of monoids $\mathbf{RelMon}$ in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

math.RA↗

arXiv2016

Conditions for optimal input states for discrimination of quantum channels

We find optimality conditions for testers in discrimination of quantum channels. These conditions are obtained using semidefinite programming and are similar to optimality conditions for POVMs obtained by Holevo for ensembles of quantum states. We get a simple condition for existence of an optimal tester with any given input state with maximal Schmidt rank, in particular with a maximally entangled input state and we show the pitfalls of using input states with not maximal Schmidt rank. In case when maximally entangled state is not the optimal input state an error estimate is obtained. The results for maximally entangled input state are applied to covariant channels, qubit channels, unitary channels and simple projective measurements.

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