Characterizing coherence with quantum observables

We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence. Our measure allows one to quantify coherence without having to perform tomography of the quantum state and can be directly calculated from measurement expectation values. This definition of coherence allows the decomposition into contributions corresponding to the nonclassical correlations between the subsystems and localized on each subsystem. The method can also be applied to cases where the full set of measurement operators is unavailable. An estimator using the truncated measurement operators can be used to obtain lower bound to the genuine value of coherence. We illustrate the method for several bipartite systems and show the singular behavior of the coherence measure in a spin-1 chain, characteristic of a quantum phase transition.

Figure 1

The coherence $C$ of various bipartite quantum states including their contributions from the local coherence $C_{\text{L}}$ and global correlation $\delta$. The states and measurement operators are: (a) $|{\mathrm{\Psi }}_S\rangle =|+\rangle |+\rangle$ and $|{\mathrm{\Psi }}_E\rangle =\left(\right|0\rangle |0\rangle +|1\rangle |1\rangle )/\sqrt{2}$ in (13) with Pauli matrices ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in \{I,{\sigma }^x,{\sigma }^y,{\sigma }^z\}/\sqrt{2}$. (b) $|{\mathrm{\Psi }}_S\rangle =\left(\right|0\rangle +|1\rangle +|2\rangle )\left(\right|0\rangle +|1\rangle +|2\rangle )/3, |{\mathrm{\Psi }}_E\rangle =\left(\right|0\rangle |0\rangle +|1\rangle |1\rangle +|2\rangle |2\rangle )/\sqrt{3}$ in (13). Coherence measured with Gell-Mann matrices ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{G}}_{\text{SU}\left(3\right)}=\{\sqrt{2/3}I,{\lambda }^{\left(1\right)},\cdots ,{\lambda }^{\left(8\right)}\}/\sqrt{2}$ (solid lines), truncated spin basis ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{L}}_{n=2}$ (dashed lines) where ${\mathcal{L}}_n=\{\sqrt{n(2+n)/3}I,S^x,S^y,S^z\}/\sqrt{n(1+n)(2+n)/3}$ are the orthonormal spin-$n/2$ operator set. (c) Total coherence of spin squeezed state evolved under Markovian dephasing with rate $\mathrm{\Gamma }=1$ for $n$ qubit spin ensembles. Coherence measured with Gell-Mann matrices ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{G}}_{\text{SU}\left(n\right)}$ (solid lines) and truncated operators ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{L}}_n$ (dashed lines). (d) Total coherence of the ground state of the generalized AKLT model between two sites separated by $r$ sites. Coherence measured using ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{G}}_{\text{SU}\left(3\right)}$. For the truncated operators ${\mathcal{A}}_l,{\mathcal{B}}_{l^{\prime }}\in {\mathcal{L}}_{n=2}$, using estimators coherences are lower for $g\ge 0$ but zero for $g\le 0$.