Markovian evolution of quantum coherence under symmetric dynamics

Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work, we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalization to higher-dimensional systems of the relation $T_2\le 2T_1$ for qubit decoherence and relaxation times. It also enables us to witness and assess the role of non-Markovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indefinitely sustain a superposition of different energy states. Finally, we establish a formal connection between the resource-theoretic and the master equation approaches to thermodynamics, with the former being a non-Markovian generalization of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications.

Figure 1

Elementary example. The evolution of the initial qubit state $\rho \left(0\right)$ towards the stationary state $\rho \left(\infty \right)$ presented on the Bloch sphere. During the evolution the difference between the current population and the stationary population, $\mathrm{\Delta }p\left(t\right)=\left|p\right(t)-\pi |$, must decrease. Due to a constraint linking relaxation and decoherence processes, at any time the ratio between the current and initial coherence $\left|c\right(t\left)\right|/\left|c\right(0\left)\right|$ is bounded by $\sqrt{\mathrm{\Delta }p\left(t\right)/\mathrm{\Delta }p\left(0\right)}$.

Figure 2

Optimal coherence mixing. Within a mode $\mathrm{\Omega }$ (consisting of coherence elements ${\rho }_{10}$ and ${\rho }_{32}$), the optimal mixing of coherence elements can be achieved via covariant Markovian dynamics: the initial total coherence within the mode $|{\rho }_{10}\left(0\right)|+|{\rho }_{32}\left(0\right)|$ is equal to the final total coherence $|{\rho }_{10}\left(t\right)|+|{\rho }_{32}\left(t\right)|$, and for $t\to \infty$ we obtain $|{\rho }_{10}\left(t\right)|=|{\rho }_{32}\left(t\right)|$.

Figure 3

(a) Coherence transfer within a mode $\mathrm{\Omega }$ between overlapping coherence elements $\left|1\right.〉\left.〈0\right|$ and $\left|2\right.〉\left.〈1\right|$. (b) Coherence transfer within a mode $\mathrm{\Omega }$ between nonoverlapping coherence elements $\left|1\right.〉\left.〈0\right|$ and $\left|3\right.〉\left.〈2\right|$.

Figure 4

Qubit covariant dynamics: Markovian vs non-Markovian. (a) Initial state with $p\left(0\right)=\frac{1}{6}$ and $c\left(0\right)=\sqrt{5}/6$, and $\rho \left(\infty \right)$ such that $\mathbf{\pi }=(\frac{1}{2},\frac{1}{2})$. (b) Initial state with $p\left(0\right)=\frac{1}{4}$ and $c\left(0\right)=\frac{1}{4}$, and $\rho \left(\infty \right)$ such that $\mathbf{\pi }=(\frac{3}{4},\frac{1}{4})$. The dashed lines show the maximum coherence preservation possible with Markovian covariant dynamics with a given fixed point $\rho \left(\infty \right)$; the solid lines show the maximum coherence preservation possible for general covariant operations with a fixed state given by $\rho \left(\infty \right)$.

Figure 5

Eigenvalues of stochastic matrices. The eigenvalues of a $d\times d$ stochastic matrix all lie within the unit circle on the complex plane, independently of $d$. For a given $d$, points corresponding to the eigenvalues of a stochastic $d\times d$ matrix are given by the Karpelevič theorem [44, 45] and are depicted in dark gray. Points that correspond to the eigenvalues of an embeddable stochastic $d\times d$ matrix, specified by Theorem 5, are depicted in light gray.

Figure 6

Families of thermodynamic quantum channels. By incorporating assumptions of Markovianity and quantum detailed balance into the set of generalized thermal operations, one obtains the set of Davies maps. These are precisely those which achieve optimal coherence transformation in Theorem 1.

Figure 7

The block-diagonal structure of the Choi state. An example of the Choi state $J\left[\mathcal{E}\right]$ of a covariant map $\mathcal{E}$ for a qutrit system described by an equidistant Hamiltonian, i.e., $H=\hslash \mathrm{\Omega }(\left|1\right.〉\left.〈1\right|+2\left|2\right.〉\left.〈2\right|)$. (a) The diagonal terms of $J\left[\mathcal{E}\right]$ (in blue) describe the evolution of populations, i.e., the transition rates $P_{x^{\prime }|x}$ between energy eigenstates $|x\rangle \left.〈x\right|$ and $\left|x^{\prime }\right.〉\left.〈x^{\prime }\right|$. (b) The off-diagonal terms of $J\left[\mathcal{E}\right]$ describe the preserved “fraction” $C_{y|y}^{x|x}$ of coherence term $|x\rangle \langle y|$ (in red), and the amount $C_{y^{\prime }|y}^{x^{\prime }|x}$ of coherence transferred (in yellow) between coherence terms $|x\rangle \langle y|$ and $\left|x^{\prime }\right.〉\left.〈y^{\prime }\right|$.

Figure 8

Choice of phases for overlapping elements of a mode. The choice of phases ${\phi }_{x^{\prime }x}$ that saturates the bound of Theorem 2 for all elements of a mode. The phases are given for the case of three overlapping elements: ${\sigma }_{10}$, ${\sigma }_{21}$, and ${\sigma }_{32}$ (hence $x_i=0,...,3$), but the table can be easily extended noticing the structure of each diagonal. The phases in the corners of the matrix, in this case ${\phi }_{30}$ and ${\phi }_{03}$, do not need to be fixed to saturate the bound.