Equivalence of matrix product ensembles of trajectories in open quantum systems

Abstract

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.

Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

“…This approach is a classical version of the known connection between continuous MPSs and open quantum dynamics [11][12][13]. It allows to describe in a compact way conditioned trajectory ensembles and demonstrate ensemble equivalences, in analogy with what can be done in the quantum case [15,16]. The key property of cMPSs is that of gauge invariance from which the equivalences follow.…”

“…This way of proving ensemble equivalence in terms of the distance between the corresponding cMPS vectors is the classical analog, for real vectors in an L1-space, of that of Ref. [16] for the case of quantum stochastic dynamics where the cMPS vectors are complex and belong to an L2-space (the systemoutput Hilbert space). A more detailed form of equivalence, termed operational equivalence in Ref.…”

“…A more detailed form of equivalence, termed operational equivalence in Ref. [16], can be proved in the following way. The idea is to consider a conditioned ensemble where the conditioning is on the whole time extension t, with t large, but focus only on the portion of trajectories up to time t 0 t. We now ask the question: how does the distribution of sub-trajectories up to time t 0 , from an ensemble conditioned microcanonically by some timeextensive quantity A for all t, compare to that for the same sub-trajectories of an ensemble conditioned canonically by s. This can be answered by comparing the "reduced" cMPSs where all information after t 0 is traced over.…”

“…The first and simplest one is the equivalence of concentration [5,16]: for long times, the canonical and microcanonical trajectory measures concentrate exponentially on the same region of their support, as long as K and s are related by Eq. (30).…”

“…It is possible in this way to represent the whole ensemble of quantum trajectories of the open system as a cMPS. This MPS approach can be then extended for the application of large-deviation methods to describe and characterise open quantum dynamics [15,16].…”