Entanglement cost for steering assemblages

In this paper we answer the question of how one can quantify the amount of entanglement necessary to create a given steering assemblage when the original state and measurements are unknown. To do this, we extend the concepts of entanglement cost and entanglement of formation to steering assemblages and provide easy-to-calculate upper and lower bounds. We prove that the entanglement of formation for assemblages is not generally continuous and not a flat roof extension (in contrast with the state-based quantity), and we use numerical analysis to illustrate these properties. Finally, we discuss the consequences of these results for assemblage-to-assemblage conversion and suggest potential applications.

Figure 1

Bounds for the entanglement cost of assemblages created by measuring an entangled Werner state $W_p=p|{\mathrm{\Phi }}^{+}\rangle \langle {\mathrm{\Phi }}^{+}|+(1-p)\mathbb{I}/4$, with $1/3<p\le 1$, in the Pauli-$X$ and Pauli-$Z$ bases. Also plotted is the regularized relative entropy of entanglement with respect to PPT states ($E_{\mathrm{PPT}}^{\infty }$) for Werner states [48]. Because $E_{\mathrm{PPT}}^{\infty }$ is a lower bound to the entanglement cost, we can see that Werner states cannot be optimal for $p<0.85406$.

Figure 2

An example of an extremal assemblage: the weights of the subnormalized states are given by the relative length to the crossing point, which represents the marginal ${\rho }_{\mathit{\sigma }}$. Only one crossing of the two lines is possible, and the individual points cannot be decomposed; these two properties makes the assemblage extremal.

Figure 3

Comparison of decompositions for ${\tilde{\mathit{x}}}_1=(0,x_1,0,z_1)$ assemblages. Inside the red region equal-value decompositions give a better upper bound; outside the region the steering weight gives the better bound.

Figure 4

Results of numerical decomposition optimization for ${\tilde{\mathit{x}}}_1=(0,x_1,0,z_1)$ assemblages. We find there is a region where neither the equal-value nor steering-weight decompositions are optimal.