M. E. Shirokov

M. E. Shirokov
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

About

142
Publications
4,534
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1,166
Citations
Additional affiliations
January 2004 - present
Russian Academy of Sciences
Position
  • Leading Researcher
January 1999 - present
Moscow Institute of Physics and Technology
Position
  • Professor (Full)

Publications

Publications (142)
Preprint
We consider a quasi-classical version of the Alicki-Fannes-Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called "quasi-c...
Preprint
Necessary and sufficient conditions for convergence (local continuity) of the quantum relative entropy are obtained. Some applications of these conditions are considered. In particular, the preservation of local continuity of the quantum relative entropy under completely positive linear maps is established.
Preprint
We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows to expand the scope of possible applications of the method. We p...
Preprint
General methods of quantitative and qualitative continuity analysis of characteristics of composite quantum systems are described. Several modifications of the Alicki-Fannes-Winter method are considered, which make it applicable to a wide class of characteristics in both finite-dimensional and infinite-dimensional cases. A new approximation method...
Article
We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods, we obtain continuity bounds for several important characteristics of a multipartite quantum state:...
Preprint
Full-text available
Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative concave functions. Applications of this result to some problems of mathematical physics are indicated. Several o...
Preprint
Special approximation technique for analysis of different characteristics of states of multipartite infinite-dimensional quantum systems is proposed and applied to study of the relative entropy of entanglement and its regularisation. We prove several results about analytical properties of the multipartite relative entropy of entanglement and its re...
Article
We consider a family of equivalent norms (called operator -norms) on the algebra of all bounded operators on a separable Hilbert space induced by a positive densely defined operator on . By choosing different generating operators we can obtain the operator -norms producing different topologies, in particular, the strong operator topology on bounded...
Article
Рассмотрено семейство эквивалентных норм (названных операторными $E$-нормами) на алгебре $\mathfrak{B}(\mathscr{H})$ всех ограниченных операторов в сепарабельном гильбертовом пространстве $\mathscr{H}$, индуцированных положительным плотно определенным оператором $G$ в $\mathscr{H}$. Выбирая разные операторы $G$, можно получить операторные $E$-нормы...
Article
It is shown that a sequence {Φn} of quantum channels strongly converges to a quantum channel Φ0 if and only if there exist a common environment for all the channels and a corresponding sequence {Vn} of Stinespring isometries strongly converging to a Stinespring isometry V0 of the channel Φ0. A quantitative description of the above characterization...
Preprint
Full-text available
We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay a special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods we obtain continuity bounds for several important characteristics of a multipartite quantum state:...
Article
Full-text available
We describe an universal method for quantitative continuity analysis of entropic characteristics of energy-constrained quantum systems and channels. It gives asymptotically tight continuity bounds for basic characteristics of quantum systems of wide class (including multi-mode quantum oscillators) and channels between such systems under the energy...
Preprint
Full-text available
We describe the class (semigroup) of quantum channels mapping states with finite entropy into states with finite entropy. We show, in particular, that this class is naturally decomposed into three convex subclasses, two of them are closed under concatenations and tensor products. We obtain asymptotically tight universal continuity bounds for the ou...
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We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to t...
Preprint
Full-text available
We show that for any energy observable every extreme point of the set of quantum states with bounded energy is a pure state. This allows us to write every state with bounded energy in terms of a continuous convex combination of pure states of bounded energy. Furthermore, we prove that any quantum state with finite energy can be represented as a con...
Preprint
Full-text available
It is well known that the quantum mutual information and its conditional version do not increase under local channels. I this paper we show that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual info...
Preprint
Full-text available
It is proved that the energy-constrained Bures distance between arbitrary infinite-dimensional quantum channels is equal to the operator E-norm distance from any given Stinespring isometry of one channel to the set of all Stinespring isometries of another channel with the same environment. The same result is shown to be valid for arbitrary quantum...
Article
For a given positive operator G we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with \(\sqrt G \)-bounded and \(\sqrt G \)-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by G (...
Article
In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigro...
Article
Full-text available
By using estimates for the variation of quantum mutual information and the relative entropy of entanglement, we obtain ε-exact lower estimates for distances from a given quantum channels to sets of degradable, antidegradable, and entanglement-breaking channels. As an auxiliary result, we obtain ε-exact lower estimates for the distance from a given...
Preprint
Full-text available
We describe an universal method for quantitative continuity analysis of entropic characteristics of energy-constrained quantum systems and channels. It gives asymptotically tight continuity bounds for different characteristics of a multi-mode quantum oscillator and quantum channels acting on this system under the energy constraint. The main applica...
Article
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tigh...
Preprint
We analyse possibility to extend a quantum operation (sub-unital normal CP linear map on the algebra $B(H)$ of bounded operators on a separable Hilbert space $H$) to the space of all operators on $H$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $\,\Phi\,$ can be uniquely extended to a bounded linea...
Preprint
In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigro...
Preprint
In this brief note we describe relations between the well known notion of a relatively bounded operator and the operator E-norms considered in [arXiv:1806.05668]. We show that the set of all $\sqrt{G}$-bounded operators equipped with the E-norm induced by a positive operator $G$ is the Banach space of all operators with finite E-norm and that the $...
Preprint
The completion of the cone of CP linear maps between Banach spaces of trace class operators w.r.t. the metric induced by the energy-constrained diamond norm are described. This completion consists of linear maps defined on the linear span of states with finite energy and characterized by the Stinespring-like representation with operators (unbounded...
Article
Full-text available
We consider energy-constrained infinite-dimensional quantum channels from a given system (satisfying a certain condition) to any other systems. We show that dealing with basic capacities of these channels we may assume (accepting arbitrarily small error $\epsilon$) that all channels have the same finite-dimensional input space -- the subspace corre...
Preprint
We consider a family of norms (called operator E-norms) on the algebra $B(H)$ of all bounded operators on a separable Hilbert space $H$ induced by a positive densely defined operator $G$ on $H$. Each norm of this family produces the same topology on $B(H)$ depending on $G$. By choosing different generating operator $G$ one can obtain operator E-nor...
Article
Full-text available
In [arXiv:1712.03219] the existence of a strongly (pointwise) converging sequence of quantum channels that can not be represented as a reduction of a sequence of unitary channels strongly converging to a unitary channel is shown. In this work we give a simple characterization of sequences of quantum channels that have the above representation. The...
Article
With the rapid growth of quantum technologies, knowing the fundamental characteristics of quantum systems and protocols is essential for their effective implementation. A particular communication setting that has received increased focus is related to quantum key distribution and distributed quantum computation. In this setting, a quantum channel c...
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Full-text available
We consider a family of energy-constrained diamond norms on the set of Hermitian- preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-di...
Article
Full-text available
We present a characterization of the strong (pointwise) convergence of quantum channels in terms of their Stinespring representations. It states that a sequence $\{\Phi_n\}$ of channels strongly converges to a channel $\Phi_0$ if and only if there exist a common environment for all the channels and a corresponding sequence $\{V_n\}$ of Stinespring...
Article
Full-text available
We consider an important characteristic of a quantum channel called the entropic disturbance. It is defined as the difference between the -quantity of a generalized ensemble and that of the image of the ensemble under the channel. We prove the lower semicontinuity of the entropic disturbance for any infinite- dimensional quantum channel on its natu...
Article
Full-text available
We start with Fannes’ type and Winter’s type tight (uniform) continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quanti...
Article
We consider the family of energy-constrained diamond seminorms on the set of all completely bounded linear maps between Banach spaces of trace class operators. We prove that this family generates the strong (pointwise) convergence topology on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional c...
Article
Full-text available
The tight, in a sense, lower estimates of diamond-norm distance from a given quantum channel to the sets of degradable, antidegradable and entanglement-breaking channels are obtained via the tight continuity bounds for quantum mutual information and for relative entropy of entanglement in finite-dimensional case. As an auxiliary result there are es...
Article
Full-text available
We show that a positive linear map preserves local continuity (convergence) of the entropy if and only if it preserves finiteness of the entropy, i.e. transforms operators with finite entropy to operators with finite entropy. The last property is equivalent to the boundedness of the output entropy of a map on the set of pure states. This criterion...
Article
Full-text available
We present a family of easily computable upper bounds for the Holevo quantity of ensemble of quantum states depending on a reference state as a free parameter. These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that appropriate choice of the reference state gives tight upper bounds for the...
Article
We show that a positive linear map preserves local continuity (convergence) of the entropy if and only if it preserves finiteness of the entropy, i.e. transforms operators with finite entropy to operators with finite entropy. The last property is equivalent to the boundedness of the output entropy of a map on the set of pure states. This criterion...
Article
We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (when it is finite) and on the maximal level of input energy (when the input dimension is infinite). First we prove continuity bounds for the output conditional mutual information for a single channel and for $n$ copies of a chan...
Article
We describe a modification of the Alicki-Fannes-Winter method (used for proving uniform continuity of functions on the set of quantum states). It allows to show uniform continuity on the set of states with bounded energy of any approximately affine function having limited growth with increasing energy. Some applications in quantum information theor...
Article
Full-text available
We consider a new entropic characteristic of a quantum channel -- the $\Delta_\chi$-quantity, defined as difference between the $\chi$-quantity of a generalized ensemble and that of the image of the ensemble under the channel. We prove that it is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries...
Article
We show how to use properties of the quantum conditional mutual information to obtain continuity bounds for information characteristics of quantum channels depending on their input dimension. First we prove tight estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. Then...
Article
Quantitative analysis of discontinuity of basic characteristics of quantum states and channels is presented. First we consider general estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mea...
Article
Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of...
Article
One of the central results of the theory of probability measures on metric spaces is Prokhorov’s theorem on weak compactness of a subset of probability measures. In the present paper this theorem is used to obtain a criterion of weak compactness for families of generalized quantum ensembles, i.e., Borel probability measures on the set of quantum st...
Article
A quantitative analysis of continuity of the quantum mutual information and of the Holevo quantity is presented. First we obtain Fannes type and Winter type tight continuity bounds for the quantum mutual information and their specifications for qc-states. Then we show that the Holevo quantity is continuous on the set of all ensembles of m states if...
Article
Full-text available
We analyze the squashed entanglement of a state of an infinite-dimensional bipartite system (defined by direct translation of the finite-dimensional definition). We show the validity of all basic properties of an entanglement measure (with the continuity replaced by the lower semicontinuity) on the set of states having at least one finite marginal...
Article
Full-text available
Several important measures of quantum correlations of a state of a finite-dimensional composite system are defined as linear combinations of marginal entropies of this state. This paper is devoted to the infinite-dimensional generalizations of such quantities and to the analysis of their properties. We introduce the notion of faithful extension of...
Article
Full-text available
A generalization of the superactivation of quantum channel capacities to the case of n>2 channels is considered. An explicit example of such superactivation for the 1-shot quantum zero-error capacity is constructed for n=3. Some implications of this example and its reformulation on terms of quantum measurements are described.
Article
We show that unbounded number of channel uses may be necessary for perfect transmission of a quantum state. For any n we explicitly construct low-dimensional quantum channels (d A =4, d E =2 or 4) whose quantum zero-error capacity is positive but the same n-shot capacity is zero. We give estimates for quantum zero-error capacity of such channels (a...
Article
Full-text available
We obtain a simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel. This shows that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that...
Article
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We begin with a detailed description of a low dimensional quantum channel ($d_A=4, d_E=3$) demonstrating the symmetric form of superactivation of one-shot zero-error quantum capacity. This means appearance of a noiseless (perfectly reversible) subchannel in the tensor square of a channel having no noiseless subchannels. Then we show that the supera...
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We consider examples of low dimensional quantum channels demonstrating different forms of superactivation of one-shot zero-error capacities, in particular, the extreme superactivation (this complements the result of T.Cubitt and G.Smith [4]). We also describe classes of quantum channels whose zero-error classical and quantum capacities cannot be su...
Article
Full-text available
We propose examples of low dimensional quantum channels demonstrating different forms of superactivation of one-shot zero-error capacities, in particular, the extreme superactivation (this complements the recent result of T.S.Cubitt and G.Smith). We also describe classes of quantum channels whose zero-error classical and quantum capacities cannot b...
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Full-text available
Properties of Bosonic linear (quasi-free) channels, in particular, Bosonic Gaussian channels with two types of degeneracy are considered. The first type of degeneracy can be interpreted as existence of noise-free canonical variables (for Gaussian channels it means that $\det\alpha=0$). It is shown that this degeneracy implies existence of (infinite...
Article
Full-text available
A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown tha...
Article
Full-text available
A coding theorem for entanglement-assisted communication via an infinite-dimensional quantum channel with linear constraints is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and χ-capacity of constrained channels are obtained, and conditions for their coincidence are given. Sufficient con...
Article
Full-text available
The method of complementary channel for analysis of reversibility (sufficiency) of a quantum channel with respect to families of input states (pure states for the most part) are considered and applied to Bosonic linear (quasi-free) channels, in particular, to Bosonic Gaussian channels. The obtained reversibility conditions for Bosonic linear channe...
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Full-text available
The coding theorem for the entanglement-assisted communication via infinite-dimensional quantum channel with linear constraint is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and the $\chi$-capacity of constrained channels are obtained and conditions for their coincidence are given. Suff...
Article
We briefly review the results related to the notion of stability of convex sets and consider some of their applications. We prove a corollary of the stability property which enables us to develop an approximation technique for concave functions on a wide class of convex sets. This technique yields necessary and sufficient conditions for the local c...
Article
Full-text available
Several relations between the Holevo capacity and entanglement-assisted classical capacity of a quantum channel are proved; necessary and sufficient conditions for their coincidence are obtained. In particular, it is shown that these capacities coincide if (respectively, only if) the channel (respectively, the χ-essential part of the channel) belon...
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A description of all quantum channels reversible with respect to a given complete family of pure states is obtained. Some applications in quantum information theory are considered.
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A necessary condition for reversibility (sufficiency) of a quantum channel with respect to complete families of states with bounded rank is obtained. A full description (up to isometrical equivalence) of all quantum channels reversible with respect to orthogonal and nonorthogonal complete families of pure states is given. Some applications in quant...
Article
Full-text available
It is easy to show coincidence of the entanglement-assisted classical capacity and the Holevo capacity for any c-q channel between finite dimensional quantum systems. In this paper we prove the converse assertion: coincidence of the above-mentioned capacities of a quantum channel implies that the $\chi$-essential part of this channel is a c-q chann...
Article
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It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian $H$ of the system satisfies the condition $\Tr\exp(-cH)<+\infty$ for any $c>0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is co...
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Global and local continuity conditions for the output von Neumann entropy for positive maps between Banach spaces of trace-class operators in separable Hilbert spaces are obtained. Special attention is paid to completely positive maps: infinite dimensional quantum channels and operations. It is shown that as a result of some specific properties of...
Article
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A condition for reversibility (sufficiency) of a channel with respect to a given countable family of states with bounded rank is obtained. This condition shows that a quantum channel preserving the Holevo quantity of at least one (discrete or continuous) ensemble of states with rank $\leq r$ has the r-partially entanglement-breaking complementary c...
Article
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Several relations between the Holevo capacity and the entanglement-assisted classical capacity of a quantum channel are proved, necessary and sufficient conditions for their coincidence are obtained. In particular, it is shown that these capacities coincide if (correspondingly, only if) the channel (correspondingly, the $\chi$--essential part of th...
Article
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The Schmidt number of a state of an infinite-dimensional composite quantum system is defined and several properties of the corresponding Schmidt classes are considered. It is shown that there are states with given Schmidt number such that any of their countable convex decompositions does not contain pure states of finite Schmidt rank. The classes o...
Article
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It is observed that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a q-c channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. This observa...
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A method of proving local continuity of concave functions on convex set possessing the $\mu$-compactness property is presented. This method is based on a special approximation of these functions. The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which par...
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The paper is devoted to the study of quantum mutual information and coherent information, two important characteristics of a quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given, and their properties are studied in detail. Basic identities relating the quantum mutual information and c...
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For a fixed convex domain in a linear metric space the problems of the continuity of convex envelopes (hulls) of continuous concave functions (the CE-property) and of convex envelopes (hulls) of arbitrary continuous functions (the strong CE-property) arise naturally. In the case of compact domains a comprehensive solution was elaborated in the 1970...
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Continuity properties of the output entropy of positive linear maps between Banach spaces of trace class operators are investigated with the special attention to the classes of quantum channels and operations. It is shown that finiteness of the output entropy of a positive map on the whole input state space implies its continuity. Sufficient condit...
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A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy...

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