How Many Bits Does It Take to Track an Open Quantum System?

A $D$-dimensional Markovian open quantum system will undergo quantum jumps between pure states, if we can monitor the bath to which it is coupled with sufficient precision. In general, these jumps, plus the between-jump evolution, create a trajectory which passes through infinitely many different pure states. Here we show that, for any ergodic master equation, one can expect to find an adaptive monitoring scheme on the bath that can confine the system state to jumping between only $K$ states, for some $K\ge (D-1{)}^2+1$. For $D=2$ we explicitly construct a two-state ensemble for any ergodic master equation, showing that one bit is always sufficient to track a qubit.

Figure 1

The average number of bits required to keep track of the pure state of a qubit described by Eq. (11), as a function of dimensionless driving power $ϵ={\Omega }^2/{\gamma }^2$. The solid line is the von Neumann entropy for ${\rho }_{\mathrm{ss}}$, a lower bound on the memory required. The other nine curves are for 11 different adaptive unravellings. Dashed lines are for two-state jumping. Dotted lines are for cyclic three-state jumping.

Figure 2

Solid arrows show Bloch vectors for two-state jumping (a)–(c), and three-state jumping (d). Shades (colors) match those of Fig. 1. The volume of the sphere at the tip of each arrow represents the probability that the qubit occupies the corresponding pure state. The dashed arrow is ${\overrightarrow{r}}_{\mathrm{ss}}$. For all plots, $\Omega /\gamma =0.2$ ($ϵ=0.04$).