© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:quant-ph/0010038 · Physical Review A

Entanglement purification of unknown quantum states

Todd A. Brun, Carlton M. Caves, Rüdiger Schack

Published Mar 21, 2001• 27 citations

Abstract

A concern has been expressed that ``the Jaynes principle can produce fake entanglement'' [R. Horodecki et al., Phys. Rev. A 59, 1799 (1999)]. In this paper we discuss the general problem of distilling maximally entangled states from N copies of a bipartite quantum system about which only partial information is known, for instance, in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principle of maximum entropy to more than one copy of a system, but the nature of this problem is classical and was discussed extensively by Jaynes. Under the additional assumption that the state ${\ensuremath{\rho}}^{(N)}$ of the N copies of the quantum system is exchangeable, one can write down a simple general expression for ${\ensuremath{\rho}}^{(N)}.$ By measuring one or more of the subsystems, one can gain information and update the state estimate for the remaining subsystems with the quantum version of the Bayes rule. Using this rule, we show how to modify two standard entanglement purification protocols, one-way hashing and recurrence, so that they can be applied to exchangeable states. We thus give an explicit algorithm for distilling entanglement from an unknown or partially known quantum state.

DOI: 10.1103/physreva.63.042309

Related papers

Quantum entanglement and the maximum-entropy states from the Jaynes principle

A. K. Rajagopal

We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is shown in a particular example considered recently by Horodecki et al. [Phys. Rev. A 59, 1799 (1999)], by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a ``thermodynamiclike'' description of the entanglement arising out of the maximum entropy principle.

arXiv:quant-ph/9903083•Physical Review A•Dec 1, 1999•14 citations

Necessary and sufficient condition for distillability with unit fidelity from finite copies of a mixed state: The most efficient purification protocol

Ping‐Xing Chen, Lin-Mei Liang, Cheng-Zu Li +1 more

It is well known that any entangled mixed state in $2\ensuremath{\bigotimes}2$ systems can be purified via infinite copies of the mixed state. But, can one distill a pure maximally entangled state from finite copies of a mixed state in any bipartite system by local operation and classical communication? This is more meaningful in practical application. We give a necessary and sufficient condition for this distillability. This condition requires that there exist distillable subspaces. According to this condition, one can judge easily whether a mixed state is distillable or not. We also analyze some properties of distillable subspaces, and discuss the most efficient purification protocol. Finally, we discuss the distillable enanglement of a two-qubit system for the case of finite copies.

arXiv:quant-ph/0110045•Physical Review A•Aug 21, 2002•12 citations

Distilling multipartite pure states from a finite number of copies of multipartite mixed states

Ping‐Xing Chen, Cheng-Zu Li

This paper addresses the question of the distillation of entanglement from a finite number of multipartite mixed states. It is shown that, if one can distill a pure entangled state from n copies of a mixed state ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}},$ there must be at least a subspace in the whole Hilbert space of all the copies such that the projection of ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}}^{\ensuremath{\bigotimes}n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general positive operator valued measurements on a single particle, in principle. Finally, we discuss experimental realizability of the entanglement purification.

arXiv:quant-ph/0311095•Physical Review A•Jan 14, 2004•3 citations

Entanglement processing and statistical inference: The Jaynes principle can produce fake entanglement

Ryszard Horodecki, Michał Horodecki, Paweł Horodecki

We show, by explicit examples, that the Jaynes principle when applied to compound quantum systems may produce the entangled maximum entropy states compatible with data coming from nonentangled (separable) states. It means that the Jaynes statistical inference scheme may lead to a wrong conclusion about entanglement, which is a crucial parameter in quantum information theory. We suggest that in all the processes where entanglement is needed, the proper inference scheme should involve minimization of entanglement. Examples illustrating the proposed scheme are provided.

Physical Review A•Mar 1, 1999•70 citations

Entanglement purification of hypergraph states

Lina Vandré, Otfried Gühne

Entanglement purification describes a primitive in quantum information processing, where several copies of noisy quantum states are distilled into a few copies of nearly pure states of high quality via local operations and classical communication. Especially in the multiparticle case, the task of entanglement purification is complicated, as many inequivalent forms of pure state entanglement exist and purification protocols need to be tailored for different target states. In this paper we present optimized protocols for the purification of hypergraph states, which form a family of multiqubit states that are relevant from several perspectives. We start by reformulating an existing purification protocol in a graphical language. This allows for systematical optimization and we present improvements in three directions. First, one can optimize the sequences of the protocol with respect to the ordering of the parties. Second, one can use adaptive schemes, where the measurement results obtained within the protocol are used to modify the protocols. Finally, one can improve the protocol with respect to the efficiency, requiring fewer copies of noisy states to reach a certain target state.

arXiv:2301.11341•Physical Review A•Dec 14, 2023•7 citations

The Purification and Distillation of Entanglement

Cheng-Zu Li

Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed state.

Acta Sinica Quantum Optica•Jan 1, 2003•0 citations

A concern has been expressed that ``the Jaynes principle can produce fake entanglement'' [R. Horodecki et al., Phys. Rev. A 59, 1799 (1999)]. In this paper we discuss the general problem of distilling maximally entangled states from N copies of a bipartite quantum system about which only partial information is known, for instance, in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principle of maximum entropy to more than one copy of a system, but the nature of this problem is classical and was discussed extensively by Jaynes. Under the additional assumption that the state

{\ensuremath{\rho}}^{(N)}

of the N copies of the quantum system is exchangeable, one can write down a simple general expression for

{\ensuremath{\rho}}^{(N)}.

By measuring one or more of the subsystems, one can gain information and update the state estimate for the remaining subsystems with the quantum version of the Bayes rule. Using this rule, we show how to modify two standard entanglement purification protocols, one-way hashing and recurrence, so that they can be applied to exchangeable states. We thus give an explicit algorithm for distilling entanglement from an unknown or partially known quantum state.

We show that the Jaynes principle is indeed a proper inference scheme when applied to compound systems and will correctly produce the entangled maximum entropy states compatible with appropriate data. This is shown in a particular example considered recently by Horodecki et al. [Phys. Rev. A 59, 1799 (1999)], by including the dispersion of the entanglement along with its mean value as constraints and an application of the uncertainty principle. We also construct a ``thermodynamiclike'' description of the entanglement arising out of the maximum entropy principle.

It is well known that any entangled mixed state in $2\ensuremath{\bigotimes}2$ systems can be purified via infinite copies of the mixed state. But, can one distill a pure maximally entangled state from finite copies of a mixed state in any bipartite system by local operation and classical communication? This is more meaningful in practical application. We give a necessary and sufficient condition for this distillability. This condition requires that there exist distillable subspaces. According to this condition, one can judge easily whether a mixed state is distillable or not. We also analyze some properties of distillable subspaces, and discuss the most efficient purification protocol. Finally, we discuss the distillable enanglement of a two-qubit system for the case of finite copies.

This paper addresses the question of the distillation of entanglement from a finite number of multipartite mixed states. It is shown that, if one can distill a pure entangled state from n copies of a mixed state ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}},$ there must be at least a subspace in the whole Hilbert space of all the copies such that the projection of ${\ensuremath{\sigma}}_{\mathrm{ABC}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}}^{\ensuremath{\bigotimes}n}$ onto the subspace is a pure entangled state. We also show that the purification of entanglement or distillation of entanglement can be carried out by local joint projective measurements with the help of classical communication and local general positive operator valued measurements on a single particle, in principle. Finally, we discuss experimental realizability of the entanglement purification.

We show, by explicit examples, that the Jaynes principle when applied to compound quantum systems may produce the entangled maximum entropy states compatible with data coming from nonentangled (separable) states. It means that the Jaynes statistical inference scheme may lead to a wrong conclusion about entanglement, which is a crucial parameter in quantum information theory. We suggest that in all the processes where entanglement is needed, the proper inference scheme should involve minimization of entanglement. Examples illustrating the proposed scheme are provided.

Entanglement purification describes a primitive in quantum information processing, where several copies of noisy quantum states are distilled into a few copies of nearly pure states of high quality via local operations and classical communication. Especially in the multiparticle case, the task of entanglement purification is complicated, as many inequivalent forms of pure state entanglement exist and purification protocols need to be tailored for different target states. In this paper we present optimized protocols for the purification of hypergraph states, which form a family of multiqubit states that are relevant from several perspectives. We start by reformulating an existing purification protocol in a graphical language. This allows for systematical optimization and we present improvements in three directions. First, one can optimize the sequences of the protocol with respect to the ordering of the parties. Second, one can use adaptive schemes, where the measurement results obtained within the protocol are used to modify the protocols. Finally, one can improve the protocol with respect to the efficiency, requiring fewer copies of noisy states to reach a certain target state.

Distillation and purification of entanglement is an important field of quantum information both in theory and in experiment. This paper presents some main conclusions in this field, and discusses mainly the distillation of entanglement from finite copies of a mixed state. By using the conceptions of quasiseparable state and distillable subspace, we show a necessary condition and a sufficient and necessary condition of the distillability from finite copies of a mixed state. We also get the most efficient protocols of entanglement distillation for the case of finite copies. Finally we consider the generalization of these results to the case of infinite copies of a mixed state.