Suppose we have two states of a system where I tell you that there is a probability $p_1$ of being in state $1$, and probability $p_2$ of being in state $2$. The total state can be written as a vector in $L^1$ normed space:

$$p=\begin{pmatrix}p_1 \\ p_2 \end{pmatrix}, ||p||=p_1+p_2=1$$

If we define a transition matrix for a Markov process:

$$T=\begin{pmatrix}t_{11}&t_{12} \\ t_{21}&t_{22}\end{pmatrix}$$

Then the next state would be:

$$p'=Tp=\begin{pmatrix}t_{11}p_1+t_{12}p_2 \\ t_{21}p_1+t_{22}p_2\end{pmatrix}$$

Now my understanding of density matrices and quantum mechanics is that it should contain classical probability theory in addition to strictly quantum phenomena.

Classical probabilities in the density matrix formalism are mapped as:

$$p=\begin{pmatrix}p_1 \\ p_2 \end{pmatrix} \rightarrow \rho=\begin{pmatrix}p_1&0 \\ 0&p_2 \end{pmatrix}$$

And I want to obtain:

$$p'=\begin{pmatrix}t_{11}p_1+t_{12}p_2 \\ t_{21}p_1+t_{22}p_2\end{pmatrix} \rightarrow \rho'=\begin{pmatrix}t_{11}p_1+t_{12}p_2&0 \\ 0&t_{21}p_1+t_{22}p_2\end{pmatrix}$$

My attempt:

Define an operator $U$ such that:

$$\rho'=U\rho U^\dagger$$ $$\implies \begin{pmatrix}t_{11}p_1+t_{12}p_2&0 \\ 0&t_{21}p_1+t_{22}p_2\end{pmatrix}=\begin{pmatrix}u_{11}&u_{12} \\ u_{21}&u_{22}\end{pmatrix}\begin{pmatrix}p_1&0 \\ 0&p_2\end{pmatrix}\begin{pmatrix}u_{11}^*&u_{21}^* \\ u_{12}^*&u_{22}^*\end{pmatrix}$$

$$=\begin{pmatrix}|u_{11}|^2p_1+|u_{12}|^2p_2&u_{11}u_{21}^*p_1+u_{12}u_{22}^*p_2 \\ u_{21}u_{11}^*p_1+u_{12}^*u_{22}p_2 & |u_{21}|^2p_1+|u_{22}|^2p_2 \end{pmatrix}$$

Evidently, $|u_{ij}|^2=t_{ij}$, but the off diagonal terms aren't easily made zero, (I've wrestled with the algebra and applied all the proper normalizations of probability theory).

What would be the correct way to apply a Markov process in the density matrix formalism? It seems really basic and something that this formalism should be able to naturally handle.

  • 1
    $\begingroup$ You may want consider asking this type of questions at quantumcomputing.stackexchange.com $\endgroup$ – Alexander Pozdneev Apr 23 '18 at 20:50
  • $\begingroup$ Didn't know that was a thing. Is there a quick way to repost or share it there? Edit: Nevermind, just copy-pasted $\endgroup$ – Connor Dolan Apr 23 '18 at 20:52
  • $\begingroup$ I suggest to copy-paste the question to QC and link to this one. Meanwhile, I'll flag this for a moderator as "on hold". $\endgroup$ – Alexander Pozdneev Apr 23 '18 at 20:55
  • $\begingroup$ This question has been reposted to another SE site. $\endgroup$ – rob Apr 23 '18 at 21:34
  • $\begingroup$ @rob: Why on earth would this not be about physics? It is about evolution of quantum systems! $\endgroup$ – Norbert Schuch Apr 23 '18 at 22:04