On Quantum Fluctuations Relations with Generalized Energy Measurements

Abstract

Quantum fluctuations relations are typically derived with projective measurements of energy at the beginning and the end of the protocol. Though projective measurements are easy to treat theoretically, they may be difficult to implement in experiments. We show that recent results on the force protocol with generalized measurements remain valid with more general forms of quantum evolution. In the case considered, states of an open quantum system are transformed according to an arbitrary trace-preserving completely positive map. The energy measurements used in the protocol are prescribed as a superposition of projective ones. Such measurements can be parameterized by a continuous variable substituted into acceptance functions of the apparatuses. The role of limited measurement precision is also clarified here. As is known, the standard Jarzynski equality and the Tasaki–Crooks theorem remain valid for the case of unital quantum channels. Using characteristic functions, we recast standard relations for the scenario with generalized measurements. With respect to the Jarzynski equality, generalized measurements lead to a negative shift in observed values of the difference of free energies. A deviation of the actual quantum channel from unital ones may also generate some shift with an arbitrary sign.