Using Nonlocal Coherence to Quantify Quantum Correlation

Abstract

We reexamine quantum correlation from the fundamental perspective of its consanguineous quantum property, the coherence. We emphasize the importance of specifying the tensor product structure of the total state space before discussing quantum correlation. A measure of quantum correlation for arbitrary dimension bipartite states using nonlocal coherence is proposed, and it can be easily generalized to the multipartite case. The quantification of non-entangled component within quantum correlation is investigated for certain states.

Appendix: Proof for the Existence of Unitaries to Produce \(\mathcal{L}(\rho)=0\)

Consider an arbitrary local unitary operation U ₁ performing on the first subsystem ρ ₁ of the bipartite system ρ, it produces

(12)

On the other hand, perform arbitrary local unitary operations U ₁ and U ₂ on the two subsystems, respectively. Then tracing out the second subsystem will produce the reduced density matrix of the first subsystem as

(13)

Therefore, $\operatorname{tr}_{2}\{(U_{1}\otimes{U_{2}})\rho(U_{1}^{\dagger}\otimes {U_{2}^{\dagger}})\}=U_{1}\rho_{1}U_{1}^{\dag}$. Similarly, one can obtain

$$ \operatorname{tr}_a \bigl\{(U_1 \otimes{U_2})\rho\bigl(U_1^\dagger\otimes {U_2^\dagger }\bigr) \bigr\}=U_b \rho_b{U_b}^\dag,\quad{a}\neq{b}\in\{1,2\}. $$

(14)

There exists always unitary operation U _1d , U _2d to diagonalize the subsystem ρ ₁, ρ ₂, thus under these specific unitary operations Eq. (13) and Eq. (14) give

$$ \begin{array}{@{}l} \displaystyle\mathrm{tr}_2 \bigl \{(U_{1d}\otimes{U_{2d}})\rho\bigl(U_{1d}^\dagger \otimes {U_{2d}^\dagger}\bigr) \bigr\} =\rho_1^{(d)}= \sum_{ik}\rho_{ikik}^c\vert i \rangle_1{_1} \langle i\vert , \\[4mm] \displaystyle\mathrm{tr}_1 \bigl\{(U_{1d} \otimes{U_{2d}})\rho\bigl(U_{1d}^\dagger \otimes {U_{2d}^\dagger}\bigr) \bigr\} =\rho_2^{(d)}= \sum_{ik}\rho_{ikik}^c\vert k \rangle_2{_2} \langle k\vert , \end{array} $$

(15)

where $\rho_{ikik}^{c}$ is the diagonal element of subsystem $\rho _{1}^{(d)}$, $\rho_{2}^{(d)}$ transformed by the local unitary operations. Since $\rho_{1}^{(d)}$, $\rho_{2}^{(d)}$ is diagonal, the off-diagonal elements are vanishing, giving

$$ \sum_{i\neq{m}}\biggl|\sum_{j=n}\rho_{ijmn}^c\biggr| +\sum_{j\neq{n}}\biggl|\sum_{i=m}\rho_{ijmn}^c\biggr|\equiv\mathcal{L}(\rho)=0. $$

(16)

Therefore for bipartite system, there exist always local unitaries to produce $\mathcal{L}(\rho)=0$.