Entanglement-induced state ordering under local operations

We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form $\mathcal{E}\otimes \mathcal{I}$. We show that a set of maximally entangled states is by the action of $\mathcal{E}\otimes \mathcal{I}$ 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 ${\varrho }_1^{\left(\mathrm{in}\right)}$ and ${\varrho }_2^{\left(\mathrm{in}\right)}$ such that the entanglement $E\left[{\varrho }_1^{\left(\mathrm{in}\right)}\right]$ of the first state is larger than the entanglement $E\left[{\varrho }_2^{\left(\mathrm{in}\right)}\right]$ of the second state are transformed into states ${\varrho }_1^{\left(\mathrm{out}\right)}$ and ${\varrho }_2^{\left(\mathrm{out}\right)}$ such that $E\left[{\varrho }_2^{\left(\mathrm{out}\right)}\right]>E\left[{\varrho }_1^{\left(\mathrm{out}\right)}\right]$.

Figure 1

(Color online) The input versus output concurrence diagram for both families of states ${\varrho }_1,{\varrho }_2$ and for the depolarizing channel with $p=1∕2$. The states from the counterexample 1 discussed in the text are displayed and the change in the ordering is visible. The region under the line $C^{\mathrm{out}}=C^{\mathrm{in}}$ represents the allowed region that is achievable by local channels. Concurrence is measured in dimensionless units.