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:1402.4120

Random-Unitary Depolarization Ensures the Correctability of All Quantum Channels

Samuel R. Hedemann

Published Feb 17, 2014• 0 citations
arXivHTMLPDF

Abstract

We prove that if any error channel has a Kraus decomposition that is simultaneously correctable and Hilbert-Schmidt (HS) complete, then the existence of Kraus sets with these properties guarantees the correctability of all quantum channels. As a proof of the existence of such Kraus sets, the $n$-level depolarization channel is shown to have a random-unitary (RU) decomposition that is both HS complete and correctable due its RU nature, thereby proving that all quantum channels are correctable. As an application, conditions for universal error-correction operations are presented.

DOI
10.48550/arxiv.1402.4120

Related papers

Restoration of a quantum state in a dephasing channel via environment-assisted error correction

Xinyu Zhao, Samuel R. Hedemann, Ting Yu

In this paper, we propose an environment-assisted error correction scheme to fully recover a multiple-qubit state subject to a phase damping noise. We establish the theoretical framework and the operational procedure to restore an unknown initial quantum state for an $N$-qubit model interacting with either individual baths or a common bath. We give an explicit construction of the random unitary (RU) Kraus decomposition for an $N$-qubit model interacting with a common bath. We also demonstrate how to use only one unitary reversal operation to restore an arbitrary state with phase damping noise. In principle, the initial state can always be recovered with a success probability of 1. Interestingly, we found that non-RU decomposition can also be used to restore some particular entangled states. This may open a new path to restore a quantum entangled state beyond the standard RU scheme.

arXiv:1305.4627•Physical Review A•Aug 19, 2013•43 citations

Decoherence-Free Communication over Multiaccess Quantum Channels

Maciej Demianowicz

In this paper we consider decoherence-free communication over multiple access and k-user quantum channels. First, we concentrate on a hermitian unitary noise model U for a two-access bi-unitary channel and show that in this case a decoherence-free code exists if the space of Schmidt matrices of an eigensubspace of U exhibits certain properties of decomposability. Then, we show that our technique is also applicable for generic random unitary two-access channels. Finally, we consider the applicability of the result to the case of a larger number of senders and general Kraus operators.

arXiv:1209.0120•Open Systems & Information Dynamics•Jun 1, 2013•1 citation

A Polar Decomposition for Quantum Channels: Theory and Applications

Matthew Alexander

Every potential benefit of quantum computers would be lost without methods to protect the computations from error. Quantum channels provide a framework for understanding error in quantum systems. In general, for a system of size d, a quantum channel may require as many as ~d^4 parameters to characterize it. We introduce the leading Kraus approximation, a simplification which reduces the number of parameters to ~d^2, while accurately approximating two figures of merit important to the experimentalist: the unitarity and average process fidelity. Additionally, applying the leading Kraus approximation declutters investigations into the set of quantum channels. When applied, a natural decomposition of channels arises, separating behaviour into coherent and decoherent contributions. We find that eliminating the coherent contribution provides the greatest increase to the fidelity, while decoherent processes provide a generalization of depolarizing channels.

•Aug 23, 2019•0 citations

Unambiguous unitary quantum channels

Shengjun Wu, X.-M. Chen

Unambiguous unitary maps and unambiguous unitary quantum channels are introduced and some of their properties are derived. These properties ensures certain simple form for the measurements involved in realizing an unambiguous unitary quantum channel. Error correction and unambiguous error correction with nonzero probability are discussed in terms of unambiguous unitary quantum channels. We not only re-derive the well-known condition for a set of errors to be correctable with certainty, but also obtain a necessary and sufficient condition for the errors caused by a noisy channel to be correctable with any nonzero probability. Dense coding with a partially entangled state can also be viewed as an unambiguous unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained.

arXiv:quant-ph/0509212•Quantum Information and Computation•Nov 1, 2007•0 citations

Universal recovery in approximate quantum error correction

Dor Elimelech, Victor V. Albert, Alexander Barg

Universal recovery -- the existence of a single recovery map that corrects an entire family of error channels -- is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set. In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill--Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.

quant-pharXiv:2608.28962•Sep 1, 2026•0 citations

Probabilistic unitary quantum channels

Shengjun Wu

Probabilistic unitary maps and probabilistic unitary quantum channels are introduced, many quantum information applications, including (unambiguous) teleportation, can be described as probabilistic unitary quantum channels. Some properties of probabilistic unitary maps and probabilistic unitary quantum channels are derived. The property of a probabilistic unitary quantum channel ensures certain simple form for the measurements involved in realizing the probabilistic unitary quantum channel. The relation between a probabilistic unitary quantum channel and a uniformly entangled state is established. The combined operation of a noisy quantum channel and the error correction operation is a probabilistic unitary channel, from this point of view the condition for the errors to be correctable is easily derived. The condition for the errors caused by a noisy channel to be correctable with any nonzero probability is also obtained. Dense coding with a partially entangled state can also be viewed as a probabilistic unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained. 1

•Sep 29, 2005•0 citations
We prove that if any error channel has a Kraus decomposition that is simultaneously correctable and Hilbert-Schmidt (HS) complete, then the existence of Kraus sets with these properties guarantees the correctability of all quantum channels. As a proof of the existence of such Kraus sets, the nnn-level depolarization channel is shown to have a random-unitary (RU) decomposition that is both HS complete and correctable due its RU nature, thereby proving that all quantum channels are correctable. As an application, conditions for universal error-correction operations are presented.
In this paper, we propose an environment-assisted error correction scheme to fully recover a multiple-qubit state subject to a phase damping noise. We establish the theoretical framework and the operational procedure to restore an unknown initial quantum state for an NNN-qubit model interacting with either individual baths or a common bath. We give an explicit construction of the random unitary (RU) Kraus decomposition for an NNN-qubit model interacting with a common bath. We also demonstrate how to use only one unitary reversal operation to restore an arbitrary state with phase damping noise. In principle, the initial state can always be recovered with a success probability of 1. Interestingly, we found that non-RU decomposition can also be used to restore some particular entangled states. This may open a new path to restore a quantum entangled state beyond the standard RU scheme.
In this paper we consider decoherence-free communication over multiple access and k-user quantum channels. First, we concentrate on a hermitian unitary noise model U for a two-access bi-unitary channel and show that in this case a decoherence-free code exists if the space of Schmidt matrices of an eigensubspace of U exhibits certain properties of decomposability. Then, we show that our technique is also applicable for generic random unitary two-access channels. Finally, we consider the applicability of the result to the case of a larger number of senders and general Kraus operators.
Every potential benefit of quantum computers would be lost without methods to protect the computations from error. Quantum channels provide a framework for understanding error in quantum systems. In general, for a system of size d, a quantum channel may require as many as ~d^4 parameters to characterize it. We introduce the leading Kraus approximation, a simplification which reduces the number of parameters to ~d^2, while accurately approximating two figures of merit important to the experimentalist: the unitarity and average process fidelity. Additionally, applying the leading Kraus approximation declutters investigations into the set of quantum channels. When applied, a natural decomposition of channels arises, separating behaviour into coherent and decoherent contributions. We find that eliminating the coherent contribution provides the greatest increase to the fidelity, while decoherent processes provide a generalization of depolarizing channels.
Unambiguous unitary maps and unambiguous unitary quantum channels are introduced and some of their properties are derived. These properties ensures certain simple form for the measurements involved in realizing an unambiguous unitary quantum channel. Error correction and unambiguous error correction with nonzero probability are discussed in terms of unambiguous unitary quantum channels. We not only re-derive the well-known condition for a set of errors to be correctable with certainty, but also obtain a necessary and sufficient condition for the errors caused by a noisy channel to be correctable with any nonzero probability. Dense coding with a partially entangled state can also be viewed as an unambiguous unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained.
Universal recovery -- the existence of a single recovery map that corrects an entire family of error channels -- is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set. In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill--Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.
Probabilistic unitary maps and probabilistic unitary quantum channels are introduced, many quantum information applications, including (unambiguous) teleportation, can be described as probabilistic unitary quantum channels. Some properties of probabilistic unitary maps and probabilistic unitary quantum channels are derived. The property of a probabilistic unitary quantum channel ensures certain simple form for the measurements involved in realizing the probabilistic unitary quantum channel. The relation between a probabilistic unitary quantum channel and a uniformly entangled state is established. The combined operation of a noisy quantum channel and the error correction operation is a probabilistic unitary channel, from this point of view the condition for the errors to be correctable is easily derived. The condition for the errors caused by a noisy channel to be correctable with any nonzero probability is also obtained. Dense coding with a partially entangled state can also be viewed as a probabilistic unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained. 1