Optimal Cloning and Singlet Monogamy

The inability to produce two perfect copies of an unknown state is inherently linked with the inability to produce maximal entanglement between multiple spins. Despite this, there is no quantitative link between how much entanglement can be generated between spins, and how well an unknown state can be cloned. This situation is remedied by giving a set of sufficient conditions such that a Completely Positive map can be optimally implemented as a teleportation operation into a standard, reference, state. The case of arbitrary $1\to N$ asymmetric cloning of $d$-dimensional spins can then be solved exactly, yielding the concept of “singlet monogamy.” The utility of this relation is demonstrated by calculating properties of Heisenberg systems, and contrasting them with the results from standard monogamy arguments.

Figure 1

For a 3-qudit state which is maximally entangled between spin 0 and spins 1 and 2, the optimal trade-off between singlet fractions is plotted, as specified by monogamy (dashed line, qubits only) and singlet monogamy ($d=2$, 3, 4, 100), derived from asymmetric cloning.