Limits on the decay rate of quantum coherence and correlation

The study and control of coherence in quantum systems is one of the most exciting recent developments in physics. Quantum coherence plays a crucial role in emerging quantum technologies as well as fundamental experiments. A major obstacle to the utilization of quantum effects is decoherence, primarily in the form of dephasing that destroys quantum coherence, and leads to effective classical behavior. We show that there are universal relationships governing dephasing that constrain the relative rates at which quantum correlations can disappear. These lead effectively to speed limits which become especially important in multipartite systems.

Figure 1

Convex cone of allowed dephasing rates for $N=3$ and real dephasing, that is, $\Delta {\omega }_{23}=0$. The axes are $x=({\Gamma }_{12}+{\Gamma }_{23}+{\Gamma }_{13})/\sqrt{3}$, $y=({\Gamma }_{13}-{\Gamma }_{23})/\sqrt{2}$ and $z=\sqrt{\frac{2}{3}}({\Gamma }_{12}-\frac{1}{\sqrt{2}}({\Gamma }_{13}+{\Gamma }_{23}))$ and the constraint equation becomes $x^2/2\ge y^2+z^2$, which defines a circular cone in the positive octant of in the parameter space of $\{{\Gamma }_{12},{\Gamma }_{23},{\Gamma }_{13}\}$, tangential to the ${\Gamma }_{12}{\Gamma }_{23}$, ${\Gamma }_{23}{\Gamma }_{13}$, and ${\Gamma }_{13}{\Gamma }_{12}$ planes.

Figure 2

Constraint violation leads to nonphysical states. A plot of the minimum eigenvalue of $\rho \left(t\right)$ starting with $\rho \left(0\right)=|{\Psi }_{\mathrm{CS}}\rangle \langle {\Psi }_{\mathrm{CS}}|$ subject to pure dephasing $(H=0)$ with ${\Gamma }_{23}={\Gamma }_{14}=\mu \Gamma$ for different values of $\mu$ shows the emergence of negative eigenvalues for $\mu >2$.

Figure 3

Constraint violation map for three qubits with local dephasing rates $\Gamma$, two-qubit dephasing rates ${\mu }_1\Gamma$, and three-qubit dephasing rates ${\mu }_2\Gamma$. For $0\le {\mu }_1\le 2$ the first four of the eight constraints are satisfied but additional constraints may be violated; for example, in the yellow region, constraint 5 is violated. Additional constraints further restrict the set of allowed rates.