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

2002), Entanglement and perfect quantum error correction

Benjamin W. Schumacher, Michael D. Westmorel

Published Dec 4, 2012• 4 citations
OpenAlex

Abstract

The entanglement of formation gives a necessary and sufficient condition for the existence of a perfect quantum error correction procedure. 1 Quantum error correction Suppose a composite quantum system RQ is initially in a pure joint input state ∣ΨRQ 〉. The subsystem Q undergoes a dynamical evolution described by a trace-preserving, completely positive (CP) map E. The joint output state is therefore ρ RQ′

Related papers

Entanglement and perfect quantum error correction

Benjamin Schumacher, Michael D. Westmoreland

The entanglement of formation gives a necessary and sufficient condition for the existence of a perfect quantum error correction procedure.

arXiv:quant-ph/0201061•Journal of Mathematical Physics•Sep 1, 2002•6 citations

Quantum Error Correction in the Presence of Initial System-Environment Correlations

Aysenur Turkmen, Abdullah Verçin

Quantum error correction is studied in a framework consisting of an open quantum system and its environment, jointly subjected to a unitary action, and an interaction-free reference system. It has been shown that coherent information between the initially correlated open system and reference system is conserved in the transmission stage of any quantum communication process, provided that the tripartite input is any pure Markov state and the overall evolution preserves its form. This conservation constitutes the necessary and sufficient condition for accomplishment of perfect error correction by a recovery channel even in the presence of initial system-environment correlations. Explicit expressions of the recovery operators and examples of the joint unitary evolution preserving the form of inputs are given for all classes of pure Markov states.

Open Systems & Information Dynamics•Jun 1, 2020•0 citations

Theory of quantum error-correcting codes

Emanuel H. Knill, Raymond Laflamme

Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.

Physical Review A•Feb 1, 1997•1,593 citations

A New Error Correction Intuitive Approach in Quantum Communications Protocols

Skander Aris, Merabtine Nadjim, Malek Benslama

Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.

American Journal of Applied Sciences•Aug 1, 2007•3 citations

Assignment maps for arbitrary reduced quantum dynamics

James Fullwood, Weixiang Ye

Given an open quantum system $S$ with environment $E$, we establish a one-to-one correspondence between completely positive trace-preserving (CPTP) maps from $S$ to $E$ and assignment maps initializing the composite system $SE$ in a possibly correlated state prior to unitary evolution. We show that every possible joint state $ρ_{SE}$ can be realized by such assignment maps, which are inherently Hermiticity-preserving and trace-preserving (HPTP). While these maps sacrifice global positivity, we derive a simple necessary and sufficient condition for any local state of $S$ to belong to their domain of validity. Furthermore, we prove that the geometry of this domain is governed by the joint state assigned to the maximally mixed input, ensuring that a non-zero volume of admissible local states can readily coexist with initial system-environment entanglement. Finally, we provide an operational protocol for witnessing such limits of validity.

quant-pharXiv:2610.27338•Oct 21, 2026•0 citations

Entanglement measures and quantum channels

Michael D. Westmoreland, Benjamin W. Schumacher

Like the coherent information, the entanglement of formation may be used as a measure of the quantum information sent over a channel. These measures are related, and share the property of giving necessary and sufficient conditions for perfect quantum error correction.

Optical Fiber Communication Conference•Jan 1, 2001•0 citations
The entanglement of formation gives a necessary and sufficient condition for the existence of a perfect quantum error correction procedure. 1 Quantum error correction Suppose a composite quantum system RQ is initially in a pure joint input state ∣ΨRQ 〉. The subsystem Q undergoes a dynamical evolution described by a trace-preserving, completely positive (CP) map E. The joint output state is therefore ρ RQ′
Quantum error correction is studied in a framework consisting of an open quantum system and its environment, jointly subjected to a unitary action, and an interaction-free reference system. It has been shown that coherent information between the initially correlated open system and reference system is conserved in the transmission stage of any quantum communication process, provided that the tripartite input is any pure Markov state and the overall evolution preserves its form. This conservation constitutes the necessary and sufficient condition for accomplishment of perfect error correction by a recovery channel even in the presence of initial system-environment correlations. Explicit expressions of the recovery operators and examples of the joint unitary evolution preserving the form of inputs are given for all classes of pure Markov states.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.
Given an open quantum system SSS with environment EEE, we establish a one-to-one correspondence between completely positive trace-preserving (CPTP) maps from SSS to EEE and assignment maps initializing the composite system SESESE in a possibly correlated state prior to unitary evolution. We show that every possible joint state ρSEρ_{SE}ρSE​ can be realized by such assignment maps, which are inherently Hermiticity-preserving and trace-preserving (HPTP). While these maps sacrifice global positivity, we derive a simple necessary and sufficient condition for any local state of SSS to belong to their domain of validity. Furthermore, we prove that the geometry of this domain is governed by the joint state assigned to the maximally mixed input, ensuring that a non-zero volume of admissible local states can readily coexist with initial system-environment entanglement. Finally, we provide an operational protocol for witnessing such limits of validity.