Cloning quantum entanglement in arbitrary dimensions

We have found a quantum cloning machine that optimally duplicates the entanglement of a pair of $d$-dimensional quantum systems prepared in an arbitrary isotropic state. It maximizes the entanglement of formation contained in the two copies of any maximally entangled input state, while preserving the separability of unentangled input states. Moreover, it cannot increase the entanglement of formation of isotropic states. For large $d$, the entanglement of formation of each clone tends to one-half the entanglement of the input state, which corresponds to a classical behavior. Finally, we investigate a local entanglement cloner, which yields entangled clones with one-fourth the input entanglement in the large-$d$ limit.

Figure 1

(i) Optimal $d\times d$ entanglement cloner; (ii) “local” entanglement cloner, as defined in Sec. 4. Here, $A$ and $B$ stand for Alice’s and Bob’s part of the bipartite state, while $a$ and $b$ stand for two clones. Entanglement is indicated by double loops.

Figure 2

Entanglement of formation, $E_F$, of the clone of a maximally entangled input state versus the fidelity $F$ of the clone for various dimensions $d=2–20$ (the lowest curve corresponds to $d=2$, while the highest curve corresponds to $d=20$). The circles show the maximum achievable fidelity and the corresponding entanglement of formation. The crosses mark the crossover between the expression of the fidelity corresponding to the second and third lines of Eq. (26).

Figure 3

Entanglement of formation, $E_F$, of the clones of a maximally entangled state obtained by the optimal (nonlocal) cloner (엯) and the local cloner (+) versus the entanglement of the input state, $E_{\mathrm{in}}$, for various dimensions $d=2–200$. The apparent “discontinuity” in both curves is due to the crossover from the second to the third line of Eq. (26) for $d⩾7$ (optimal cloner) and $d⩾13$ (local cloner). Solid lines represent the asymptotics of $E_F$ for large $d$ in both cases.