Finite-resolution ancilla-assisted measurements of quantum work distributions

Work is an observable quantity associated with a process; however, there is no Hermitian operator associated with its measurement. We consider an ancilla-assisted protocol measuring the work done on a quantum system driven by a time-dependent Hamiltonian via two von Neumann measurements of the system's energy carried out by a measuring apparatus modeled as a free particle of finite localization and interaction time with the system. We consider system Hamiltonians which both commute and do not commute at different times, finding corrections to fluctuation relations like the Jarzynski equality and the Crooks relation. This measurement model allows us to quantify the effect that measuring has on the estimated work distribution and associated average work done on the system and average heat exchanged with the measuring apparatus.

Figure 1

The measurement model described in the text is depicted as a quantum circuit. The system and apparatus are prepared at time $t_p$ in the states ${\rho }_S\left(t_p\right)$ and ${\rho }_A\left(t_p\right)$, respectively, and their free evolution is described by $U_S$ and $U_A$. The apparatus and system interact at times $t_i$ and $t_f$ via the interaction unitaries $U_i$ and $U_f$, and a final measurement of the apparatus takes place at time $t_m$ and is described by the POVM ${\mathrm{\Pi }}_x$.

Figure 2

The three measurement model layers outlined in Sec. 2c, qualitatively illustrating their associated averages ${\langle W\rangle }_S$, ${\langle \tilde{W}\rangle }_S$, and ${\langle W\rangle }_{\mathrm{dist}}$. The latter two definitions reduce to the former upon taking the ideal measurement limit.

Figure 3

For a system prepared in an equally weighted superposition of the two energy eigenstates, which corresponds to $\alpha =\frac{1}{\sqrt{2}}$ at the time of the first measurement $\kappa t_i=2$, we show the work distribution at $\kappa t_m=4$ for different values of $\theta$. The mass of the measuring apparatus and the duration of the interaction are taken to be $\frac{\kappa m}{{\sigma }_p^2}=1000$ and $\kappa \mathrm{\Delta }=0.2$.

Figure 4

For an initially excited two-level system, corresponding to $\alpha =0$ at the time of the first measurement $\kappa t_i=2$, the differences in average work $\frac{\mathrm{\Delta }{W}_{\mathrm{int}}}{\kappa }$ and $\frac{\mathrm{\Delta }{W}_{\mathrm{POVM}}}{\kappa }$, as defined in Eqs. (10) and (13), are plotted at the time of measurement $\kappa t_m=4$ for $\theta \in [0,\pi ]$. In general, these differences are nonzero, illustrating a discrepancy between the average estimated work and modification due to the act of measuring the work distribution, with the average work defined in Eq. (8) corresponding to the work done in the absence of measurement. The mass of the measuring apparatus and the duration of the interaction are taken to be $\frac{\kappa m}{{\sigma }_p^2}=1000$ and $\kappa \mathrm{\Delta }=0.2$.

Figure 5

For an initial equally weighted classical mixture, corresponding to ${\rho }_S\left(t_i\right)=\frac{1}{2}(|0\rangle \langle 0|+|1\rangle \langle 1|)$ at the time of the first measurement $\kappa t_i=2$, we show the work distribution at $\kappa t_m=4$ for the same values of $\theta$ displayed in Fig. 3. The mass of the measuring apparatus and the duration of the interaction are taken to be $\frac{\kappa m}{{\sigma }_p^2}=1000$ and $\kappa \mathrm{\Delta }=0.2$.