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Single energy measurement Integral Fluctuation theorem and non-projective measurements

Daniel AlonsoAntonia Ruiz Garc\'ia
Dec 2022
We study a Jarzysnki type equality for work in systems that are monitoredusing non-projective unsharp measurements. The information acquired by theobserver from the outcome $f$ of an energy measurement, and the subsequentconditioned normalized state $\hat \rho(t,f)$ evolved up to a final time $t$are used to define work, as the difference between the final expectation valueof the energy and the result $f$ of the measurement. The Jarzynski equalityobtained depends on the coherences that the state develops during the process,the characteristics of the meter used to measure the energy, and the noise itinduces into the system. We analyze those contributions in some detail tounveil their role. We show that in very particular cases, but not in general,the effect of such noise gives a factor multiplying the result that would beobtained if projective measurements were used instead of non-projective ones.The unsharp character of the measurements used to monitor the energy of thesystem, which defines the resolution of the meter, leads to different scenariosof interest. In particular, if the distance between neighboring elements in theenergy spectrum is much larger than the resolution of the meter, then a similarresult to the projective measurement case is obtained, up to a multiplicativefactor that depends on the meter. A more subtle situation arises in theopposite case in which measurements may be non-informative, i.e. they may notcontribute to update the information about the system. In this case, acorrection to the relation obtained in the non-overlapping case appears. Weanalyze the conditions in which such a correction becomes negligible. We alsostudy the coherences, in terms of the relative entropy of coherence developedby the evolved post-measurement state. We illustrate the results by analyzing atwo-level system monitored by a simple meter.