Equivalence of matrix product ensembles of trajectories in open quantum systems

The equivalence of thermodynamic ensembles is at the heart of statistical mechanics and central to our understanding of equilibrium states of matter. Recently, a formal connection has been established between the dynamics of open quantum systems and statistical mechanics in an extra dimension: an open system dynamics generates a matrix product state (MPS) encoding all possible quantum jump trajectories which allows to construct generating functions akin to partition functions. For dynamics generated by a Lindblad master equation, the corresponding MPS is a so-called continuous MPS which encodes the set of continuous measurement records terminated at some fixed total observation time. Here, we show that if one instead terminates trajectories after a fixed total number of quantum jumps, e.g., emission events into the environment, the associated MPS is discrete. The continuous and discrete MPS correspond to different ensembles of quantum trajectories, one characterized by total time, the other by total number of quantum jumps. Hence, they give rise to quantum versions of different thermodynamic ensembles, akin to “grand canonical” and “isobaric,” but for trajectories. Here, we prove that these trajectory ensembles are equivalent in a suitable limit of long time or large number of jumps. This is in direct analogy to equilibrium statistical mechanics where equivalence between ensembles is only strictly established in the thermodynamic limit. An intrinsic quantum feature is that the equivalence holds only for all observables that commute with the number of quantum jumps.

Figure 1

Sketch of trajectory ensembles. Time runs from left to right. Wiggly lines indicate quantum jumps, and straight ones deterministic evolution between jumps under $H_{\mathrm{eff}}$. (a) The $s$ ensemble. All trajectories in the ensemble are of a total fixed time $\tau$ but can have any number of quantum jumps. Furthermore, between the time of the last jump and the final time there may be a period of no-jump evolution. (b) The $x$ ensemble. All trajectories in the ensemble have the same number of quantum jumps $K$, but their time extension can fluctuate. In this case, trajectories terminate after the $K\mathrm{th}$ jump.

Figure 2

Sketch of concentration of measure. The $s$ ensemble (in blue) corresponds to all trajectories with fixed overall time ${\tau }^{*}$ and any number of quantum jumps $K$, and where the time of the last jump $t_K$ is in general smaller than $\tau$. A subset of this set is that of trajectories where the number of jumps is exactly $K^{*}$. The $x$ ensemble (in red), in contrast, is that of trajectories all with fixed number of jumps $K^{*}$ but any time extent $\tau$ where trajectories terminate with the last jump ${\tau }^{*}=t_{K^{*}}$. The subset of trajectories with $\tau$ exactly ${\tau }^{*}$ (in purple) also belongs to the $s$ ensemble. The two ensembles are equivalent when concentrated to this intersection if the fields $s$ and $x$ defining the ensembles are such that ${\langle K\rangle }_s=K^{*}$ and ${\langle \tau \rangle }_x={\tau }^{*}$ (see Sec. 4b).