Nonlocal advantage of quantum coherence in high-dimensional states

By local measurements on party $A$ of a system $\mathit{AB}$ and classical communication between its two parties, one can achieve a nonlocal advantage of quantum coherence (NAQC) on party $B$. For the $l_1$ norm of coherence and the relative entropy of coherence, we generalized the framework of NAQC for two qubits and derived the criteria which capture NAQC in the ($d\times d$)-dimensional states when $d$ is a power of a prime. We also presented a framework for formulating NAQC and showed through explicit examples its capacity for capturing the NAQC states. Moreover, we proved that any bipartite state with NAQC is quantum entangled; thus the obtained criteria can also be used as an entanglement witness.

Figure 1

Schematic picture of our “coherence steering game.” There is a one-to-one correspondence between Alice's measurement $A_i$ and Bob's measurement basis $A_{{\alpha }_i}$ ($\forall i$), and $\left\{{\alpha }_i\right\}$ are any possible permutations of the elements of $\left\{i\right\}$.

Figure 2

The NAQC for ${\rho }_I$ of Eq. (17) with (a) $d=3$ and (b) $d=5$. For every plot, the red, blue, green, and purple (from top to bottom) lines correspond to ${\tilde{C}}_{l_1}^{na}\left({\rho }_I\right), C_{l_1}^{na}\left({\rho }_I\right), {\tilde{C}}_{re}^{na}\left({\rho }_I\right)$, and $C_{re}^{na}\left({\rho }_I\right)$, while the two black horizontal lines represent the corresponding bounds $C_{l_1}^m$ (top) and $C_{re}^m$ (bottom), respectively.