Bipartite entanglement-annihilating maps: Necessary and sufficient conditions

We fully characterize bipartite entanglement-annihilating (EA) channels that destroy entanglement of any state shared by subsystems and, thus, should be avoided in any entanglement-enabled experiment. Our approach relies on extending the problem to EA positive maps, the cone of which remains invariant under concatenation with partially positive maps. Due to this invariancy, positive EA maps adopt a well characterization and their intersection with completely positive trace-preserving maps results in the set of EA channels. In addition to a general description, we also provide sufficient operational criteria revealing EA channels. They have a clear physical meaning since the processes involved contain stages of classical information transfer for subsystems. We demonstrate the applicability of derived criteria for local and global depolarizing noises, and specify corresponding noise levels beyond which any initial state becomes disentangled after passing the channel. The robustness of some entangled states is discussed.

Figure 1

(a) Quantum channel as an input-output device. (b) Physical interpretation of the diagonal-sum representation. (c) Structure of EB channels.

Figure 2

Venn diagram of linear bipartite maps ${\Phi }^{\mathcal{A}\mathcal{B}}$. Convex figures correspond to convex sets.

Figure 3

PEA maps and physical meaning of Proposition 2 (positive maps $\left\{{\Lambda }_k\right\}$ are followed by one-sided EB operations).

Figure 4

Area of parameters $(q_1,q_2)$ where the local-depolarizing channel ${\Phi }_{q_1}^{\mathcal{A}}\otimes {\Phi }_{q_2}^{\mathcal{B}}$ is EA: (a) two-qubit system, $d^{\mathcal{A}}=d^{\mathcal{B}}=2$; (b) qutrit-qubit system, $d^{\mathcal{A}}=3$, $d^{\mathcal{B}}=2$. The validity of Proposition 2 is justified inside the dashed region, which provides all EA channels in case (a) and a subset of EA channels in case (b). Being applied to case (a), Corollary 2 detects EA behavior inside the green hatching.

Figure 5

Local (a) and global (b) depolarizing channels that surely annihilate or preserve entanglement of $d\times d$ systems. Entanglement of the state $|\gamma \rangle =\frac{1}{\sqrt{2}}\left(\right|11\rangle +|dd\rangle )$ is more robust than that of the maximally entangled state $|{\Psi }_{+}\rangle$ (MES).