Selective entanglement breaking

We discuss the cases where local decoherence selectively degrades one type of entanglement more than other types. A typical case is called state ordering change, in which two input states with different amounts of entanglement undergoes a local decoherence and the state with the larger entanglement results in an output state with less entanglement than the other output state. We are also interested in a special case where the state with the larger entanglement evolves to a separable state while the other output state is still entangled, which we call selective entanglement breaking. For three-level or larger systems, it is easy to find examples of the state ordering change and the selective entanglement breaking, but for two-level systems it is not trivial whether such situations exist. We present a general strategy to construct examples of two-qubit states exhibiting the selective entanglement breaking regardless of entanglement measure. We also give a more striking example of the selective entanglement breaking in which the less entangled input state has only an infinitesimal amount of entanglement.

Figure 1

Strategy to find examples of the selective entanglement breaking. We use negativity $N\left(\widehat{\rho }\right)$ as an entanglement measure, since it is easy to calculate. Slight decrease in the degree of entanglement (the arrow) is done by LOCC, and hence the order of the two input states is measure independent. The order of the two output states is also measure independent, since one of them is separable.

Figure 2

Example of the selective entanglement breaking. The negativities of ${\widehat{\rho }}_1^{\mathrm{out}}$, ${\widehat{\rho }}_2^{\mathrm{out}}$, ${\widehat{\rho }}_3^{\mathrm{out}}$ $(t=1∕3)$ are indicated by the dotted curve, the dashed curve, and the solid curve, respectively.