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:0707.4401

Entanglement measures: state ordering vs local operations

Mário Ziman, Vladimír Bužek

Published Jul 30, 2007• 2 citations
arXivHTMLPDF

Abstract

A set of all states of a bi-partite quantum system can be divided into subsets each of which contains states with the same degree of entanglement. In this paper we address a question whether local operations (without classical communication) affect the entanglement-induced state ordering. We show that arbitrary unilocal channel (i.e., a channel that acts on one sub-system of a bi-partite system only) might change the ordering for an arbitrary nontrivial measure of entanglement. A slightly weaker result holds for the maximally entangled states. In particular, the maximally entangled states might not remain the most entangled ones at the output of a unilocal noise channel.

DOI
10.48550/arxiv.0707.4401

Related papers

Entanglement-induced state ordering under local operations

Liscie udolie Quniverse

We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form ExI. We show that a set of maximally entangled states is by the action of ExI transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective of what the input state of the composite system is. We show that within this bound the entanglement-induced state ordering is ''relative'' and can be changed by the action of local channels. That is, two states {rho}{sub 1}{sup (in)} and {rho}{sub 2}{sup (in)} such that the entanglement E[{rho}{sub 1}{sup (in)}] of the first state is larger than the entanglement E[{rho}{sub 2}{sup (in)}] of the second state are transformed into states {rho}{sub 1}{sup (out)} and {rho}{sub 2}{sup (out)} such that E[{rho}{sub 2}{sup (out)}]>E[{rho}{sub 1}{sup (out)}].

Physical Review A•Jan 15, 2006•0 citations

Entanglement-induced state ordering under local operations

Mário Ziman, Vladimír Bužek

We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$. We show that a set of maximally entangled states is by the action of $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$ transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective of what the input state of the composite system is. We show that within this bound the entanglement-induced state ordering is ``relative'' and can be changed by the action of local channels. That is, two states ${\ensuremath{\varrho}}_{1}^{(\mathrm{in})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{in})}$ such that the entanglement $E[{\ensuremath{\varrho}}_{1}^{(\mathrm{in})}]$ of the first state is larger than the entanglement $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{in})}]$ of the second state are transformed into states ${\ensuremath{\varrho}}_{1}^{(\mathrm{out})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{out})}$ such that $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.

arXiv:quant-ph/0510017•Physical Review A•Jan 1, 2006•24 citations

Hierarchies of localizable entanglement due to spatial distribution of local noise

Ratul Banerjee, Amit Kumar Pal, Aditi Sen

Complete characterization of a noisy multipartite quantum state in terms of entanglement requires full knowledge of how the entanglement content in the state is affected by the spatial distribution of noise in the state. Specifically, we find that if the measurement basis in the protocol of computing localizable entanglement and the basis of the Kraus operator representing the local noisy channel do not commute, the information regarding the noise is retained in the system even after the qubit is traced out after measurement. Using this result and the basic properties of entanglement under noise, we present a set of hierarchies that localizable entanglement over a specific subsystem in a multiqubit state can obey when local noise acts on the subparts or on all the qubits of the whole system. In particular, we propose two types of hierarchies---one tailored according to the number of noisy unmeasured qubits, and the other one that depends additionally on the cardinality of the set of noisy measured qubits, leading to the classification of quantum states. We report the percentage of states satisfying the proposed hierarchies in the case of random three- and four-qubit systems and show, using both analytical methods and numerical simulations, that in almost all the cases, anticipated hierarchies tend to hold with the variation of the strength of noise.

arXiv:2003.02175•Physical Review Research•Apr 12, 2022•6 citations

Maximally Entangled Set of Multipartite Quantum States

Julio I. de Vicente, Cornelia Spee, Barbara Kraus

Entanglement is a resource in quantum information theory when state manipulation is restricted to local operations assisted by classical communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.

arXiv:1305.7398•Physical Review Letters•Sep 10, 2013•145 citations

Operational Multipartite Entanglement Measures

Katharina Schwaiger, David Sauerwein, Martí Cuquet +2 more

We introduce two operational entanglement measures that are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication and the other characterizes the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the Greenberger-Horne-Zeilinger and W state as the most powerful pure three-qubit states regarding state manipulation.

arXiv:1503.00615•Physical Review Letters•Oct 7, 2015•68 citations

Unextendible Entanglement of Quantum Channels

Vishal Singh, Mark M. Wilde

Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.

arXiv:2407.15944•IEEE Transactions on Information Theory•May 7, 2025•1 citation
A set of all states of a bi-partite quantum system can be divided into subsets each of which contains states with the same degree of entanglement. In this paper we address a question whether local operations (without classical communication) affect the entanglement-induced state ordering. We show that arbitrary unilocal channel (i.e., a channel that acts on one sub-system of a bi-partite system only) might change the ordering for an arbitrary nontrivial measure of entanglement. A slightly weaker result holds for the maximally entangled states. In particular, the maximally entangled states might not remain the most entangled ones at the output of a unilocal noise channel.
We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form ExI. We show that a set of maximally entangled states is by the action of ExI transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective of what the input state of the composite system is. We show that within this bound the entanglement-induced state ordering is ''relative'' and can be changed by the action of local channels. That is, two states {rho}{sub 1}{sup (in)} and {rho}{sub 2}{sup (in)} such that the entanglement E[{rho}{sub 1}{sup (in)}] of the first state is larger than the entanglement E[{rho}{sub 2}{sup (in)}] of the second state are transformed into states {rho}{sub 1}{sup (out)} and {rho}{sub 2}{sup (out)} such that E[{rho}{sub 2}{sup (out)}]>E[{rho}{sub 1}{sup (out)}].
We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$. We show that a set of maximally entangled states is by the action of $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$ transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective of what the input state of the composite system is. We show that within this bound the entanglement-induced state ordering is ``relative'' and can be changed by the action of local channels. That is, two states ${\ensuremath{\varrho}}_{1}^{(\mathrm{in})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{in})}$ such that the entanglement $E[{\ensuremath{\varrho}}_{1}^{(\mathrm{in})}]$ of the first state is larger than the entanglement $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{in})}]$ of the second state are transformed into states ${\ensuremath{\varrho}}_{1}^{(\mathrm{out})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{out})}$ such that $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.
Complete characterization of a noisy multipartite quantum state in terms of entanglement requires full knowledge of how the entanglement content in the state is affected by the spatial distribution of noise in the state. Specifically, we find that if the measurement basis in the protocol of computing localizable entanglement and the basis of the Kraus operator representing the local noisy channel do not commute, the information regarding the noise is retained in the system even after the qubit is traced out after measurement. Using this result and the basic properties of entanglement under noise, we present a set of hierarchies that localizable entanglement over a specific subsystem in a multiqubit state can obey when local noise acts on the subparts or on all the qubits of the whole system. In particular, we propose two types of hierarchies---one tailored according to the number of noisy unmeasured qubits, and the other one that depends additionally on the cardinality of the set of noisy measured qubits, leading to the classification of quantum states. We report the percentage of states satisfying the proposed hierarchies in the case of random three- and four-qubit systems and show, using both analytical methods and numerical simulations, that in almost all the cases, anticipated hierarchies tend to hold with the variation of the strength of noise.
Entanglement is a resource in quantum information theory when state manipulation is restricted to local operations assisted by classical communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.
We introduce two operational entanglement measures that are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication and the other characterizes the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the Greenberger-Horne-Zeilinger and W state as the most powerful pure three-qubit states regarding state manipulation.
Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea ofk-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the “unextendible entanglement of a channel”, yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error.We then generalize the formalism ofk-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.