Jarzynski equality for quantum stochastic maps

Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are applied for exemplary systems subjected to quantum channels acting on a finite-dimensional Hilbert space.

Figure 1

One-qubit quantum channels acting on the Bloch ball. Correction term in the Jarzynski equality (43) depends on the product $\tau ·\mathbf{B}$ and vanishes in the unital case (a) and in the case (d), for which both vectors are perpendicular. Correction term is maximal in the cases (b) and (f), for which these vectors are parallel.