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Physical Review A

Entanglement-induced state ordering under local operations

Liscie udolie Quniverse

Published Jan 15, 2006• 0 citations
OpenAlex

Abstract

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)}].

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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

Entanglement measures: state ordering vs local operations

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Bounds on action of local quantum channels

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We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on multi-partite quantum states. We assume that the individual channels $Λ_j$ affect single-particle states so, that for an arbitrary input $ρ_j$, the distance $D_j (Λ_j [ ρ_j ], ρ_j)$ between the input $ρ_j$ and the output $Λ_j [ ρ_j ]$ of the channel is less than $ε$. Given this assumption we show that for an arbitrary {\em separable} two-partite state $ρ_{12}$ the distance between the input $ρ_{12}$ and the output $Λ_1\otimesΛ_2[ρ_{12} ]$ fulfills the bound $D_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq \sqrt{2+ 2 \sqrt{(1-1/d_1)(1-1/d_2)}} ε$ where $d_1$ and $d_2$ are dimensions of first and second quantum system respectively. On the contrary, entangled states are transformed in such a way, that the bound on the action of the local channels is $D_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq 2 \sqrt{2 - 1/d} \: ε$, where $d$ is the dimension of the smaller of the two quantum systems passing through the channels. Our results show that the fundamental distinction between the set of separable and the set of entangled states results into two different bounds which in turn can be exploited for a discrimination between the two sets of states. We generalize our results to multi-partite channels.

arXiv:quant-ph/0611253•Journal of Physics A Mathematical and General•Jun 15, 2005•0 citations

Concurrence versus purity: Influence of local channels on Bell states of two qubits

Liscie udolie Quniverse

We analyze how a maximally entangled state of two qubits (e.g., the singlet {psi}{sub s}) is affected by the action of local channels described by completely positive maps E. We analyze the concurrence and the purity of states {rho}{sub E}=ExI[{psi}{sub s}]. Using the concurrence-versus-purity phase diagram we characterize local channels E by their action on the singlet state {psi}{sub s}. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.

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Concurrence versus purity: Influence of local channels on Bell states of two qubits

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We analyze how a maximally entangled state of two qubits (e.g., the singlet ${\ensuremath{\psi}}_{s}$) is affected by the action of local channels described by completely positive maps $\mathcal{E}$. We analyze the concurrence and the purity of states ${\ensuremath{\varrho}}_{\mathcal{E}}=\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}[{\ensuremath{\psi}}_{s}]$. Using the concurrence-versus-purity phase diagram we characterize local channels $\mathcal{E}$ by their action on the singlet state ${\ensuremath{\psi}}_{s}$. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.

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Local State Discrimination and Ordering of Multipartite Entangled States

Sumit Rout, Ananda G. Maity, Amit Mukherjee +2 more

Local operations assisted with classical communication (LOCC) and/or its stochastic generalization induce an ordering relation among bipartite entangled states based on their inter-convertibility. This, in multipartite scenario, leads to different classes of entanglement where states within a class can be inter-converted but conversion among different classes is not possible, and thus results in incomparable multipartite entangled resources. Here we show that a bona-fide ordering relation, under the operational paradigm of LOCC, can be established between two such incomparable states based on their performance in some operational task. Entanglement assisted local state discrimination provides one such platform for resource comparison and the notion of genuinely nonlocal product states becomes quite pertinent in this regard. We provide the detailed framework along with explicit examples exhibiting how incomparable resources can be put into well defined ordered relation.

•Oct 31, 2019•1 citation
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 $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.
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 derive an upper bound on the action of a direct product of two quantum maps (channels) acting on multi-partite quantum states. We assume that the individual channels ΛjΛ_jΛj​ affect single-particle states so, that for an arbitrary input ρjρ_jρj​, the distance Dj(Λj[ρj],ρj)D_j (Λ_j [ ρ_j ], ρ_j)Dj​(Λj​[ρj​],ρj​) between the input ρjρ_jρj​ and the output Λj[ρj]Λ_j [ ρ_j ]Λj​[ρj​] of the channel is less than εεε. Given this assumption we show that for an arbitrary {\em separable} two-partite state ρ12ρ_{12}ρ12​ the distance between the input ρ12ρ_{12}ρ12​ and the output Λ1⊗Λ2[ρ12]Λ_1\otimesΛ_2[ρ_{12} ]Λ1​⊗Λ2​[ρ12​] fulfills the bound D12(Λ1⊗Λ2[ρ12],ρ12)≤2+2(1−1/d1)(1−1/d2)εD_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq \sqrt{2+ 2 \sqrt{(1-1/d_1)(1-1/d_2)}} εD12​(Λ1​⊗Λ2​[ρ12​],ρ12​)≤2+2(1−1/d1​)(1−1/d2​)​​ε where d1d_1d1​ and d2d_2d2​ are dimensions of first and second quantum system respectively. On the contrary, entangled states are transformed in such a way, that the bound on the action of the local channels is D12(Λ1⊗Λ2[ρ12],ρ12)≤22−1/d εD_{12} (Λ_1 \otimes Λ_2 [ ρ_{12} ], ρ_{12}) \leq 2 \sqrt{2 - 1/d} \: εD12​(Λ1​⊗Λ2​[ρ12​],ρ12​)≤22−1/d​ε, where ddd is the dimension of the smaller of the two quantum systems passing through the channels. Our results show that the fundamental distinction between the set of separable and the set of entangled states results into two different bounds which in turn can be exploited for a discrimination between the two sets of states. We generalize our results to multi-partite channels.
We analyze how a maximally entangled state of two qubits (e.g., the singlet {psi}{sub s}) is affected by the action of local channels described by completely positive maps E. We analyze the concurrence and the purity of states {rho}{sub E}=ExI[{psi}{sub s}]. Using the concurrence-versus-purity phase diagram we characterize local channels E by their action on the singlet state {psi}{sub s}. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.
We analyze how a maximally entangled state of two qubits (e.g., the singlet ${\ensuremath{\psi}}_{s}$) is affected by the action of local channels described by completely positive maps E\mathcal{E}E. We analyze the concurrence and the purity of states ${\ensuremath{\varrho}}_{\mathcal{E}}=\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}[{\ensuremath{\psi}}_{s}]$. Using the concurrence-versus-purity phase diagram we characterize local channels E\mathcal{E}E by their action on the singlet state ${\ensuremath{\psi}}_{s}$. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.
Local operations assisted with classical communication (LOCC) and/or its stochastic generalization induce an ordering relation among bipartite entangled states based on their inter-convertibility. This, in multipartite scenario, leads to different classes of entanglement where states within a class can be inter-converted but conversion among different classes is not possible, and thus results in incomparable multipartite entangled resources. Here we show that a bona-fide ordering relation, under the operational paradigm of LOCC, can be established between two such incomparable states based on their performance in some operational task. Entanglement assisted local state discrimination provides one such platform for resource comparison and the notion of genuinely nonlocal product states becomes quite pertinent in this regard. We provide the detailed framework along with explicit examples exhibiting how incomparable resources can be put into well defined ordered relation.