For classical ergodic discrete Markov chains, we can bound the time taken to reach the stationary distribution to the spectral properties of the transition matrix. I will outline this bound below.


Let $\Omega$ be a state space and $P$ a transition matrix on $\Omega$. The stationary distribution $\pi$ is the row vector such that $\pi P = \pi$, i.e., the distribution doesn't change after evolution generated by $P$.

Given some tolerance $\epsilon$, we define the equilibration (or mixing) time $t_{\mathrm{eq}}(\epsilon)$ as

$$ t_{\mathrm{eq}}(\epsilon) = \min_{t \in \mathbb{N}} \left\{ t : \max_{x\in \Omega} \left\lVert \sum_{y \in \Omega} \left [P^{t} (x,y) \right]- \pi \right\rVert_{TV} \leq \epsilon \right\}, $$

where $\lVert \cdot \rVert_{TV}$ is the total variation distance. In other words, $t_{\mathrm{eq}}$ is the minimum time such that the distance from the stationary distribution is at most $\epsilon$.

We say an ergodic Markov chain is reversible if it satisfies detailed balance, i.e. $\pi(x) P(x,y) = \pi(y) P(y,x)$.

Define $\Delta$ as the spectral gap of the transition matrix $P$. That is, if we order the eigenvalues of $P$ as $1 = \lambda_{1} > \lambda_{2} \geq \cdots \geq \lambda_{n}$, we define $\Delta = \lambda_{1} - \lambda_{2}$.

Equilibration time bounds

Then, for a reversible ergodic discrete Markov chain, one can prove the following bound:

$$ \left(\dfrac{1}{\Delta} - 1\right) \log\left(\dfrac{1}{2\epsilon}\right) \leq t_{\mathrm{eq}}(\epsilon) \leq \dfrac{1}{\Delta} \log\left(\dfrac{1}{\epsilon \pi_{\mathrm{min}}}\right), $$

where $\pi_{\mathrm{min}} = \min_{x \in \Omega} \pi(x)$.


Can we make similar bounds for the equilibration time of an ergodic open quantum system which obeys a Markovian master equation? In other words, can we place any bounds on the time taken to reach thermal equilibrium?

  • 2
    $\begingroup$ Nice question. I couldn't attempt an answer but this observation may be relevant. Quantum master equations derived under the Born-Markov and secular approximation give rise to decoupled equations for populations and coherences in the Hamiltonian eigenbasis. The populations obey a classical Markov master equation for which the above-stated bounds presumably apply directly. The coherences can be upper-bounded in terms of the population dynamics using a recent result of Lostaglio, Korzekwa and Milne (see Theorem 2 in particular). $\endgroup$ – Mark Mitchison Aug 17 '18 at 15:13
  • $\begingroup$ Thanks for the very helpful comment, Mark! The paper you linked looks promising. It would be interesting to see whether one could use their Thm. 2 to say something about the density matrix as a whole, like the distance between the density matrix and the thermal state. $\endgroup$ – Oliver Lunt Aug 17 '18 at 21:07

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