Entanglement-induced state ordering under local operations

Abstract

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 $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.