  The equivalence of thermodynamic ensembles is at the heart of statistical
mechanics and central to our understanding of equilibrium states of matter.
Recently, it has been shown that there is a formal connection between the
dynamics of open quantum systems and the statistical mechanics in an extra
dimension. This is established through the fact that an open system dynamics
generates a Matrix Product state (MPS) which encodes the set of all possible
quantum jump trajectories and permits the construction of generating functions
in the spirit of thermodynamic partition functions. In the case of
continuous-time Markovian evolution, such as that 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. This establishes an
interesting analogy: The continuous and discrete MPS correspond to different
ensembles of quantum trajectories, one characterised 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.
