Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The additivity problem and constrained quantum channels

Alexander Semenovich Holevo

Published Jun 1, 2005• 1 citation
Publisher

Abstract

We give formulations of the famous additivity conjecture for several important quantities characterizing quantum channel and prove their global equivalence to the additivity of the classical capacity of a channel under input constrains (like mean energy constrain).
DOI
10.1117/12.620489

Related papers

The additivity problem for constrained quantum channels

Александр Семенович Холево, Максим Евгеньевич Широков

One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:

Russian Mathematical Surveys•Apr 30, 2004•0 citations

On some additivity problems in quantum information theory

Grigori Gennadievich Amosov, Alexander Semenovich Holevo, Reinhard F. Werner

A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture.

arXiv:math-ph/0003002•Mar 4, 2000•105 citations

Additivity/Multiplicativity Communication Channels

Grigori Gennadievich Amosov, Alexander Semenovich Holevo, Reinhard F. Werner

A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the “classical capacity”, and the “maximal output purity”. All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Kluwer Academic Publishers eBooks•Feb 24, 2006•0 citations

On the multiplicativity conjecture for quantum channels

Grigori Gennadievich Amosov, Alexander Semenovich Holevo

A multiplicativity conjecture for quantum communication channels is formulated, validity of which for the values of parameter $p$ close to 1 is related to the solution of the fundamental problem of additivity of the channel capacity in quantum information theory. The proof of the conjecture is given for the case of natural numbers $p$.

arXiv:math-ph/0103015•Mar 12, 2001•31 citations

Entanglement-assisted capacity of constrained channels

A. S. Holevo

In this paper we fil a gap in previous work by proving the conjectured formula for the antanglement-assisted capacity of quantum channel with additive input constraint (such as Bosonic Gaussian channel). The main tools are the coding theorem for classical-quantum constrained channels and a finite dimensional approximation of the input density operators for the entanglement-assisted capacity.

arXiv:quant-ph/0211170•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jul 23, 2003•24 citations

Novel geometrical solution to additivity problem of classical quantum channel capacity

László Gyöngyösi, Sándor Imre

Quantum channel additivity is currently an active area of research in Quantum Information Theory. The additivity conjectures have emerged from an attempt to find a closed form expression for the capacity of a noisy quantum channel. To this day, strict additivity for quantum channel capacity has been conjectured, but not proven. The open questions related to additivity of quantum channel capacities can be discussed by our novel approach. We show an efficient computational geometric method to analyze the additivity property, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as distance measure.

IEEE Sarnoff Symposium•Apr 1, 2010•8 citations
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the "classical capacity", and the "maximal output purity". All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture.
A class of problems in quantum information theory, having an elementary formulation but still resisting solution, concerns the additivity properties of various quantities characterizing quantum channels, notably the “classical capacity”, and the “maximal output purity”. All known results, including extensive numerical work, are consistent with the conjecture that these quantities are indeed additive (resp. multiplicative) with respect to tensor products of channels. A proof of this conjecture would have important consequences in quantum information theory. In particular, according to this conjecture, the classical capacity or the maximal purity of outputs cannot be increased by using entangled inputs of the channel. In this paper we state the additivity/multiplicativity problems, give some relations between them, and prove some new partial results, which also support the conjecture. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A multiplicativity conjecture for quantum communication channels is formulated, validity of which for the values of parameter ppp close to 1 is related to the solution of the fundamental problem of additivity of the channel capacity in quantum information theory. The proof of the conjecture is given for the case of natural numbers ppp.
Quantum channel additivity is currently an active area of research in Quantum Information Theory. The additivity conjectures have emerged from an attempt to find a closed form expression for the capacity of a noisy quantum channel. To this day, strict additivity for quantum channel capacity has been conjectured, but not proven. The open questions related to additivity of quantum channel capacities can be discussed by our novel approach. We show an efficient computational geometric method to analyze the additivity property, using quantum Delaunay tessellation on the Bloch ball and quantum relative entropy as distance measure.