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arXiv:quant-ph/0611253 · Journal of Physics A Mathematical and General

Bounds on action of local quantum channels

Peter Štelmachovič, Vladimír Bužek

Published Jun 15, 2005• 0 citations
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Abstract

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.

DOI
10.1088/0305-4470/38/26/014

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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 $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.
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 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.
Quantum superchannels are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps we motivate and define superchannels on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.
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.
One of the fundamental concepts of quantum information theory is that of entanglement purification; that is, the transformation of a partially entangled state into a smaller-dimensional, more completely entangled state. Of particular interest are protocols for entanglement purification (EPPs) that alternate purely local operations with one- or two-way classical communication. In the present work, we consider a more general, but simpler, class of transformations, called separable superoperators. Since every EPP is a separable superoperator, bounds on separable superoperators apply as well to EPPs; we use this fact to give a new upper bound on the rate of EPPs on Bell-diagonal states, and thus on the capacity of Bell-diagonal channels.