Quantum work and the thermodynamic cost of quantum measurements

Quantum work is usually determined from two projective measurements of the energy at the beginning and at the end of a thermodynamic process. However, this paradigm cannot be considered thermodynamically consistent as it does not account for the thermodynamic cost of these measurements. To remedy this conceptual inconsistency we introduce a paradigm that relies only on the expected change of the average energy given the initial energy eigenbasis. In particular, we completely omit quantum measurements in the definition of quantum work, and hence quantum work is identified as a thermodynamic quantity of only the system. As main results we derive a modified quantum Jarzynski equality and a sharpened maximum work theorem in terms of the information free energy. A comparison of our results with the standard approach allows one to quantify the informational cost of projective measurements.

Figure 1

Average work $〈W〉$ [Eq. (21)] (blue, solid line), together with the change of equilibrium free energy $\mathrm{\Delta }F$ [Eq. (22)] (red, dotted line) and the informational maximum work theorem, $\beta \mathrm{\Delta }F+S\left({\tilde{\rho }}_{\tau }\right|\left|{\rho }_{\tau }^{\mathrm{eq}}\right)$ [Eq. (17)] (purple, dashed line) for the parametric harmonic oscillator (20) with $\hslash =1, \beta =1, {\omega }_0=1$, and ${\omega }_{\tau }=2$.

Figure 2

Average work $〈W〉$ [Eq. (21)] (blue, solid line), together with the change of equilibrium free energy $\mathrm{\Delta }F$ [Eq. (22)] (red, dotted line) and the informational maximum work theorem, $\beta \mathrm{\Delta }F+S\left({\tilde{\rho }}_{\tau }\right|\left|{\rho }_{\tau }^{\mathrm{eq}}\right)$ [Eq. (17)] (purple, dashed line) for the parametric harmonic oscillator (20) with $\hslash =1, \beta =1, {\omega }_0=2$, and ${\omega }_{\tau }=1$.