arXiv subjects

A. S. Holevo

Publications and source records attributed to A. S. Holevo.

At least 37 records · Page 2Linked to original sources

arXiv2016

Lower semicontinuity of the entropic disturbance and its applications in quantum information theory

We prove that for any infinite-dimensional quantum channel the entropic disturbance (defined as difference between the $χ$-quantity of a generalized ensemble and that of the image of the ensemble under the channel) is lower semicontinuous on the natural set of its definition. We establish a number of useful corollaries of this property, in particular, we prove the continuity of the output $χ\textrm{-}$quantity and the existence of $χ$-optimal ensemble for any quantum channel under the energy-type input constraint.

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arXiv2016

Quantum signal+noise models: beyond i.i.d

Recently, the Gaussian optimizer conjecture in quantum information theory was confirmed for bosonic Gaussian gauge-covariant or contravariant channels. These results use the i.i.d. model of the quantum noise. In this paper we consider quantum Gaussian signal+noise model with time-continuous stationary coloured noise. A proof of the coding theorem for the classical capacity of quantum broadband gauge-covariant Gaussian channels is given. We also discuss and compare the "broadband" and the "bandpass" models of time-continuous time-continuous stationary coloured noise.

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arXiv2016

Majorization and additivity for multimode bosonic Gaussian channels

Recently, the longstanding Gaussian optimizer conjecture was proven for bosonic Gaussian gauge-covariant or contravariant channels in the work of Giovannetti, Holevo and Garcia-Patron [3]. In the paper of Mari, Giovannetti and Holevo [11] this result was strengthened for one-mode channels by establishing that the output for the vacuum or coherent input majorizes the output for any other input. In the present paper we give the multimode extension of the result of [11], including sufficient conditions under which the coherent states are the only optimizers. We also discuss direct implications of this multimode majorization result to the positive solution of the additivity problem for the Gaussian channels. In particular, we demonstrate the additivity of the output Renyi entropies of arbitrary order p>1. Finally, we present an alternative derivation of a majorization property of Glauber's coherent states by Lieb and Solovej [10], basing on the method of the work [3].

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arXiv2015

On the constrained classical capacity of infinite-dimensional covariant quantum channels

The additivity of the minimal output entropy and that of the $χ$-capacity are known to be equivalent for finite-dimensional irreducibly covariant channels. In this paper we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to Bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper \cite{ghg} to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particular, this implies a multimode generalization of the "threshold condition", obtained for single mode in \cite{schaefer}) and the proof of the fact that under this condition the classical "Gaussian capacity" resulting from optimization over Gaussian inputs is equal to the full classical capacity. We also investigate implications of the gauge-covariance condition for the single- and multimode Bosonic Gaussian channels.

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arXiv2015

Gaussian optimizers and the additivity problem in quantum information theory

We give a survey of the two remarkable analytical problems of quantum information theory. The main part is a detailed report of the recent (partial) solution of the quantum Gaussian optimizers problem which establishes an optimal property of Glauber's coherent states -- a particular instance of pure quantum Gaussian states. We elaborate on the notion of quantum Gaussian channel as a noncommutative generalization of Gaussian kernel to show that the coherent states, and under certain conditions only they, minimize a broad class of the concave functionals of the output of a Gaussian channel. Thus, the output states corresponding to the Gaussian input are "the least chaotic", majorizing all the other outputs. The solution, however, is essentially restricted to the gauge-invariant case where a distinguished complex structure plays a special role. We also comment on the related famous additivity conjecture, which was solved in principle in the negative some five years ago. This refers to the additivity or multiplicativity (with respect to tensor products of channels) of information quantities related to the classical capacity of quantum channel, such as $(1\rightarrow p)$-norms or the minimal von Neumann or Rényi output entropies. A remarkable corollary of the present solution of the quantum Gaussian optimizers problem is that these additivity properties, while not valid in general, do hold in the important and interesting class of the gauge-covariant Gaussian channels.

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arXiv2014

Numerical and analytical research of the impact of decoherence on quantum circuits

Three different levels of noisy quantum schemes modeling are considered: vectors, density matrices and Choi-Jamiolkowski related states. The implementations for personal computers and supercomputers are described, and the corresponding results are shown. For the level of density matrices, we present the technique of the fixed rank approximation and show some analytical estimates of the fidelity level.

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arXiv2014

Coding theorems for hybrid channels. II

The present work continues investigation of the capacities of measurement (quantum-classical) channels in the most general setting, initiated in~\cite{HCT}. The proof of coding theorems is given for the classical capacity and entanglement-assisted classical capacity of the measurement channel with arbitrary output alphabet, without assuming that the channel is given by a bounded operator-valued density.

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arXiv2014

A solution of the Gaussian optimizer conjecture

The long-standing conjectures of the optimality of Gaussian inputs for Gaussian channel and Gaussian additivity are solved for a broad class of covariant or contravariant Bosonic Gaussian channels (which includes in particular thermal, additive classical noise, and amplifier channels) restricting to the class of states with finite second moments. We show that the vacuum is the input state which minimizes the entropy at the output of such channels. This allows us to show also that the classical capacity of these channels (under the input energy constraint) is additive and is achieved by Gaussian encodings.

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arXiv2013

Ultimate communication capacity of quantum optical channels by solving the Gaussian minimum-entropy conjecture

Optical channels, such as fibers or free-space links, are ubiquitous in today's telecommunication networks. They rely on the electromagnetic field associated with photons to carry information from one point to another in space. As a result, a complete physical model of these channels must necessarily take quantum effects into account in order to determine their ultimate performances. Specifically, Gaussian photonic (or bosonic) quantum channels have been extensively studied over the past decades given their importance for practical purposes. In spite of this, a longstanding conjecture on the optimality of Gaussian encodings has yet prevented finding their communication capacity. Here, this conjecture is solved by proving that the vacuum state achieves the minimum output entropy of a generic Gaussian bosonic channel. This establishes the ultimate achievable bit rate under an energy constraint, as well as the long awaited proof that the single-letter classical capacity of these channels is additive. Beyond capacities, it also has broad consequences in quantum information sciences.

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arXiv2013

On the entanglement-assisted classical capacity of infinite-dimensional quantum channels

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 $χ$-capacity of constrained channels are obtained and conditions for their coincidence are given. Sufficient conditions for continuity of the entanglement-assisted classical capacity as a function of a channel are obtained. Some applications of the obtained results to analysis of Gaussian channels are considered. A general (continuous) version of the fundamental relation between the coherent information and the measure of privacy of classical information transmission by infinite-dimensional quantum channel is proved.

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arXiv2012

Gaussian classical-quantum channels: gain of entanglement-assistance

In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; the embedding of the classical input into quantum is always possible and therefore the classical entanglement-assisted capacity C_{ea} under appropriate input constraint is well defined. We prove a general property of entropy increase for weak complementary channel, that implies the equality C_{ea}=C (where C is the unassisted capacity) for certain class of c-q Gaussian channel under appropriate energy-type constraint. On the other hand, we show by explicit example that the inequality C<C_{ea} is not unusual for constrained c-q Gaussian channel.

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arXiv2012

Mathematical modeling of quantum noise and the quality of hardware components of quantum computers

In the present paper methods and algorithms of modeling quantum operations for quantum computer integrated circuits design are developed. We examine different ways of quantum operation descriptions, including operator-sums, unitary representations, Choi-Jamiolkowski state representations and the corresponding chi-matrices, as well as quantum system evolution operators. The results of modeling of practically important quantum gates: SQiSW (square root of i-SWAP gate), controlled-NOT (CNOT), and controlled Z-transform (CZ) subject to different decoherence mechanisms are presented. These mechanisms include analysis of depolarizing quantum noise and processes of amplitude and phase relaxation. Finally, we consider error correction of phase flip, and the tasks of creating and maintaining the entanglement, as well as its breaking for two- and multi-qubit realizations of quantum operations. Importance of the present analysis for the quality and efficiency of quantum information technologies in practical applications is discussed.

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arXiv2012

Quantum channels and their entropic characteristics

One of the major achievements of the recently emerged quantum information theory is the introduction and thorough investigation of the notion of quantum channel which is a basic building block of any data-transmitting or data-processing system. This development resulted in an elaborated structural theory and was accompanied by the discovery of a whole spectrum of entropic quantities, notably the channel capacities, characterizing information-processing performance of the channels. This paper gives a survey of the main properties of quantum channels and of their entropic characterization, with a variety of examples for finite dimensional quantum systems. We also touch upon the "continuous-variables" case, which provides an arena for quantum Gaussian systems. Most of the practical realizations of quantum information processing were implemented in such systems, in particular based on principles of quantum optics. Several important entropic quantities are introduced and used to describe the basic channel capacity formulas. The remarkable role of the specific quantum correlations - entanglement - as a novel communication resource, is stressed.

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arXiv2011

On extreme Bosonic linear channels

The set of all channels with fixed input and output is convex. We first give a convenient formulation of necessary and sufficient condition for a channel to be extreme point of this set in terms of complementary channel, a notion of big importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra due to Arveson. We then apply this formulation to prove the main result of this note: under certain nondegeneracy conditions, purity of the environment is necessary and sufficient for extremality of Bosonic linear (quasi-free) channel. It follows that Gaussian channel between finite-mode Bosonic systems is extreme if and only if it has minimal noise.

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arXiv2011

Information capacity of quantum observable

In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give the formulas for unassisted and entanglement-assisted classical capacities $C,C_{ea}$ and consider some explicitly solvable cases which give simple examples of entanglement-breaking channels with $C<C_{ea}.$ We also elaborate on the ensemble-observable duality to show that $C_{ea}$ for the measurement channel is related to the $χ$-quantity for the dual ensemble in the same way as $C$ is related to the accessible information. This provides both accessible information and the $χ$-quantity for the quantum ensembles dual to our examples.

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arXiv2011

On the Choi-Jamiolkowski Correspondence in Infinite Dimensions

We give a mathematical formulation for the Choi-Jamiolkowski (CJ) correspondence in the infinite-dimensional case in the form close to one used in quantum information theory. We show that there is no need to use a limiting procedure to define "unnormalized maximally entangled state" and the corresponding analog of the Choi matrix since they can be defined rigorously as, in general, nonclosable forms on an appropriate dense subspace. The properties of these forms are discussed in Sec. 2. An important question is: when the CJ form is given by a bounded operator. This is the case for entanglement-breaking channels: we prove this in Sec. 3 along with a version of a result of Wolf et al. characterizing CJ operators which correspond to such channels by giving precise definitions of a separable operator and a relevant integral. In Sec. 4 we obtain explicit expressions for CJ forms and operators defining a general Bosonic Gaussian channel. In Sec. 5 we give a decomposition of the Gaussian CJ form into product of the four principal types and a necessary and sufficient condition for existence of the bounded CJ operator.

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arXiv2010

Mutual and coherent informations for infinite-dimensional quantum channels

The work is devoted to study of quantum mutual information and coherent information -- the two important characteristics of 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 quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of quantum mutual information and coherent information following from the above identities is observed. An upper bound for the coherent information is obtained.

math-ph↗

arXiv2010

Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results

A formulation of the generalized minimal output entropy conjecture for Gaussian channels is presented. It asserts that, for states with fixed input entropy, the minimal value of the output entropy of the channel (i.e. the minimal output entropy increment for fixed input entropy) is achieved by Gaussian states. In the case of centered channels (i.e. channels which do not add squeezing to the input state) this implies that the minimum is obtained by thermal (Gibbs) inputs. The conjecture is proved to be valid in some special cases.

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