Coherence orders, decoherence, and quantum metrology

Since the dawn of quantum theory, coherence has been attributed as a key to understand the weirdness of fundamental concepts such as, e.g., the wave-particle duality and the Stern-Gerlach experiment. Recently, based on a resource theory approach, the notion of quantum coherence was revisited and a plethora of coherence quantifiers was proposed. In this work, we address such issues employing the language of coherence orders developed by the NMR community. This allowed us to investigate the role played by different subspaces of the Hilbert-Schmidt space into physical processes and quantum protocols. We found some links between decoherence and each coherence order. Moreover, we propose a sufficient and straightforward method to testify to the usefulness of a given state for quantum enhanced phase estimation, relying on a minimal set of elements belonging to the density matrix.

Figure 1

Depiction of all possible transitions of a three-qubit system and their relation to each coherence order. The third coherence order in red (thicker), second in blue, first in black, and zeroth in green (thinnest). The arrows indicate the direction of a positive order. The dashed lines separate the transitions associated with each line of Fig. 2.

Figure 2

Coherence orders for (a) $N=1$, (b) $N=2$, and (c) $N=3$. Note the pattern as $N$ increases.

Figure 3

Open-system dynamics for each coherence order for a maximally coherent three-qubit state, $|+++\rangle$, coupled to Ornstein-Uhlenbeck environments, in the slow correlation time regime. In particular, we set $\mathrm{\Gamma }/\left(2{\gamma }^2\right)\approx 1$, with $\mathrm{\Gamma }=100$ ${(\text{rad}/\text{s})}^2, \tau =1/\gamma =0.1$ s, and ${\lambda }_l=\{1,0.8,0.2\}$. (a) Case of a common environment. All coherence orders decay with a single rate. The higher the order, the faster is the decay. The zeroth order remains invariant under dephasing. (b) Case of three independent environments. Notice that both ${\tilde{C}}_0$ and ${\tilde{C}}_2$ coincide along the range $0\le t\left(s\right)\le 1$. There is a multicomponent decay for all coherence orders but the third one. Moreover, there is no direct connection between the specific coherence order and how fast or slow is the decay due to dephasing. In both cases (a) and (b) we plot the normalized coherence orders ${\left\{{\tilde{C}}_m\right\}}_{m=0,1,2,3}$, with ${\tilde{C}}_m:={\mathcal{C}}_{\left|m\right|}^{{\ell }_1}(\rho \left(t\right))/{\mathcal{C}}_{\left|m\right|}^{{\ell }_1}(\rho \left(0\right))$.

Figure 4

Plot of quantum Fisher information $F_Q\left({\rho }_{\tau }\right)$ (red solid line), squared speed ${\mathcal{S}}_{\tau }({\rho }_0,H)$ (blue dashed line), ${\mathcal{B}}_{\tau ,m_{\text{max}}}({\rho }_0,H)$ (black dot dashed line), $F_I({\rho }_0,H)$ (brown loosely dotted line), and $F_I^{m_{\text{max}}}({\rho }_0,H)$ (magenta loosely dot dashed line) related to the evolved state ${\rho }_{\tau }=U_{\tau }{\rho }_0{U}_{\tau }^{†}$, where ${\rho }_0=\left(\frac{1-p}{8}\right)\mathbb{I}+p|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }|$ and $|\mathrm{\Psi }\rangle =cos\varphi |{\text{GHZ}}_3\rangle +sin\varphi |001\rangle$ ($m_{\text{max}}$), for (a) $\varphi =0$, (b) $\varphi =\pi /6$, (c) $\varphi =\pi /4$, and (d) $\varphi =\pi /3$. Notice that the unitary evolution is generated by the Hamiltonian $H={\sum }_{l=1}^3{\mathbb{I}}^{\otimes l-1}\otimes I_l^z\otimes {\mathbb{I}}^{\otimes 3-l}$. Here we choose $\tau =\pi /6$. The gray dotted line represents the QFI related to the three-qubits uniform superposition initial state ${\rho }_0^{\text{class}}=|+++\rangle \langle +++|$ which undergoes such unitary dynamics.

Figure 5

Plot of quantum Fisher information $F_Q\left({\rho }_{\tau }\right)$ (red solid line), squared speed ${\mathcal{S}}_{\tau }({\rho }_0,H)$ (blue dashed line), ${\mathcal{B}}_{\tau ,m_{\text{max}}}({\rho }_0,H)$ (black dot dashed line), $F_I({\rho }_0,H)$ (brown loosely dotted line), and $F_I^{m_{\text{max}}}({\rho }_0,H)$ (magenta loosely dot dashed line) related to the evolved state ${\rho }_{\tau }=U_{\tau }{\rho }_0{U}_{\tau }^{†}$, where (a) example 1: ${\rho }_0=|\psi \rangle \langle \psi |$, with $|\psi \rangle =cos\varphi |001\rangle +sin\varphi |{\text{GHZ}}_3\rangle$ ($m_{\text{max}}=3$), and (b) example 2: ${\rho }_0=|\psi \rangle \langle \psi |$, with $|\psi \rangle =cos\varphi |000\rangle +sin\varphi |\text{W}\rangle$ ($m_{\text{max}}=2$). In both cases such unitary evolution is generated by the Hamiltonian $H$ in Eq. (A3) and $\tau =\pi /6$. The gray dotted line represents the QFI related to the three-qubits uniform superposition initial state ${\rho }_0^{\text{class}}=|+++\rangle \langle +++|$ which undergoes the unitary dynamics discussed before.

Figure 6

Example 3: Plot of quantum Fisher information $F_Q\left({\rho }_{\tau }\right)$ (red solid line), squared speed ${\mathcal{S}}_{\tau }({\rho }_0,H)$ (blue dashed line), ${\mathcal{B}}_{\tau ,m_{\text{max}}}({\rho }_0,H)$ (black dot dashed line), $F_I({\rho }_0,H)$ (brown loosely dotted line), and $F_I^{m_{\text{max}}}({\rho }_0,H)$ (magenta loosely dot dashed line) related to the evolved state ${\rho }_{\tau }=U_{\tau }{\rho }_0{U}_{\tau }^{†}$, where ${\rho }_0=\left(\frac{1-p}{8}\right)\mathbb{I}+p|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }|$ and $|\mathrm{\Psi }\rangle =cos\varphi |{\text{GHZ}}_3\rangle +sin\varphi |001\rangle$ ($m_{\text{max}}$), for (a) $\varphi =0$, (b) $\varphi =\pi /6$, (c) $\varphi =\pi /4$, and (d) $\varphi =\pi /3$. Notice that the unitary evolution is generated by the Hamiltonian $H$ in Eq. (A3) and $\tau =\pi /6$. The gray dotted line represents the QFI related to the three-qubits uniform superposition initial state ${\rho }_0^{\text{class}}=|+++\rangle \langle +++|$ which undergoes such unitary dynamics.

Figure 7

Plot of quantum Fisher information $F_Q\left({\rho }_{\tau }\right)$ (red solid line), squared speed ${\mathcal{S}}_{\tau }({\rho }_0,H)$ (blue dashed line), ${\mathcal{B}}_{\tau ,m_{\text{max}}}({\rho }_0,H)$ (black dot dashed line), $F_I({\rho }_0,H)$ (brown loosely dotted line), and $F_I^{m_{\text{max}}}({\rho }_0,H)$ (magenta loosely dot dashed line) related to the evolved state ${\rho }_{\tau }=U_{\tau }{\rho }_0{U}_{\tau }^{†}$, where (a) example 4: ${\rho }_0=(1-p)|+00\rangle \langle +00|+p|{\text{GHZ}}_3\rangle \langle {\text{GHZ}}_3|$ ($m_{\text{max}}=3$) and (b) example 5: ${\rho }_0=(1-p)|+++\rangle \langle +++|+p|{\text{GHZ}}_3\rangle \langle {\text{GHZ}}_3|$ ($m_{\text{max}}=3$). In both cases such unitary evolution is generated by the Hamiltonian $H$ in Eq. (A3) and $\tau =\pi /6$. The gray dotted line represents the QFI related to the three-qubits uniform superposition initial state ${\rho }_0^{\text{class}}=|+++\rangle \langle +++|$, which undergoes the unitary dynamics discussed before.