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

arXiv:quant-ph/0201056 · Journal of Mathematical Physics

A lower bound on the quantum capacity of channels with correlated errors

Mitsuru Hamada

Published Sep 1, 2002• 34 citations
arXivHTMLPDF

Abstract

The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1−exp[−nE(R)+o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R0, which implies R0 is a lower bound on the quantum capacity. This work is an extension of the author’s previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002); quant-ph/0109114, LANL (2001); M. Hamada, quant-ph/0112103, LANL (2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this article to those in the previous works is similar to that of Markov chains to sequences of independent identically distributed random variables.

DOI
10.1063/1.1495537

Related papers

Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

M. Hamada

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.

arXiv:quant-ph/0112103•IEEE Transactions on Information Theory•Sep 1, 2002•24 citations

Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

Mitsuru Hamada

On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function $E(R).$ The $E(R)$ is positive below some threshold ${R}_{0},$ which implies ${R}_{0}$ is a lower bound on the quantum capacity.

arXiv:quant-ph/0109114•Physical Review A•Apr 15, 2002•19 citations

An exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

M. Hamada

On a quantum memoryless channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity of the channel.

IEEE International Symposium on Information Theory (ISIT)•Jun 25, 2003•1 citation

Quantum-channel capacity of very noisy channels

David Peter Divincenzo, Peter W. Shor, John A. Smolin

We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters $f$ when $f>0.80944$. Random coding has nonzero capacity only for $f>0.81071$; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.

arXiv:quant-ph/9706061•Physical Review A•Feb 1, 1998•297 citations

Information rates achievable with algebraic codes on discrete memoryless quantum channels

M. Hamada

The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic (stabilizer) codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have the desired performance are symplectic codes. On the depolarizing channel, this work's bound is actually the highest possible rate at which symplectic codes work reliably. The details of this work can be found in (M. Hamada, 2002).

IEEE International Symposium on Information Theory (ISIT)•Jan 1, 2003•1 citation

Information Rates Achievable With Algebraic Codes on Quantum Discrete Memoryless Channels

M. Hamada

The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably

arXiv:quant-ph/0207113•IEEE Transactions on Information Theory•Nov 22, 2005•30 citations
The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1−exp[−nE(R)+o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R0, which implies R0 is a lower bound on the quantum capacity. This work is an extension of the author’s previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002); quant-ph/0109114, LANL (2001); M. Hamada, quant-ph/0112103, LANL (2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this article to those in the previous works is similar to that of Markov chains to sequences of independent identically distributed random variables.
Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.
On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function E(R).E(R).E(R). The E(R)E(R)E(R) is positive below some threshold R0,{R}_{0},R0​, which implies R0{R}_{0}R0​ is a lower bound on the quantum capacity.
We present a family of additive quantum error-correcting codes whose capacities exceed those of quantum random coding (hashing) for very noisy channels. These codes provide nonzero capacity in a depolarizing channel for fidelity parameters fff when f>0.80944f>0.80944f>0.80944. Random coding has nonzero capacity only for f>0.81071f>0.81071f>0.81071; by analogy to the classical Shannon coding limit, this value had previously been conjectured to be a lower bound. We use the method introduced by Shor and Smolin of concatenating a nonrandom repetition (cat) code within a random code to obtain good codes. The cat code with block size five is shown to be optimal for single concatenation. The best known multiple-concatenated code we found has a block size of 25. We derive a general relation between the capacity attainable by these concatenation schemes and the coherent information of the inner code states.
The highest information rate at which quantum error-correction schemes work reliably on a channel, which is called the quantum capacity, is proven to be lower bounded by the limit of the quantity termed coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic (stabilizer) codes. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension, and the codes that are proven to have the desired performance are symplectic codes. On the depolarizing channel, this work's bound is actually the highest possible rate at which symplectic codes work reliably. The details of this work can be found in (M. Hamada, 2002).
The highest information rate at which quantum error-correction schemes work reliably on a channel is called the quantum capacity. Here this is proven to be lower-bounded by the limit of coherent information maximized over the set of input density operators which are proportional to the projections onto the code spaces of symplectic stabilizer codes. The quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a completely positive linear map on a Hilbert space of finite dimension. The codes that are proven to have the desired performance are symplectic stabilizer codes. On the depolarizing channel, the bound proven here is actually the highest possible rate at which symplectic stabilizer codes work reliably