Fluctuation theorems for quantum processes

We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that for a specific class of generalized measurements, which include projective measurements, unitality replaces microreversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.

Figure 1

The forward process protocol. A quantum state $\rho$ is prepared, measured ($\mathcal{P}$), evolved via a CPTP map $\mathcal{E}$, and measured again ($\mathcal{Q}$).

Figure 2

The reverse process protocol. A quantum state $\tilde{\rho }$ is prepared, measured ($\tilde{\mathcal{Q}}$), evolved via the dual CPTP map ${\mathcal{E}}^{*}$, and measured again ($\tilde{\mathcal{P}}$). The final state ${\tilde{\rho }}_f$ thus obtained is in general different from the state ${\rho }_p$.

Figure 3

Distance-entropy diagram illustrating the generalized second law of thermodynamics (see text for details).

Figure 4

The feedback protocol associated with the forward process with an intermediate measurement whose result introduces the conditional CP map ${\mathcal{E}}_j$.

Figure 5

Experimental results (blue dots with error bars) for $\langle v\rangle$ [from Eq. (45a)] and the best fit using the adiabatic Markovian master equation (red $\times$) with the extracted value $\kappa =2.34\times {10}^{-3}$. Main panel: as a function of $J$, with $t_f=5\mu$s in both the numerical simulations and the experiment. Inset: as a function of $t_f$, with $J=\frac{1}{2}$ in both the numerical simulations and the experiment.

Figure 6

A schematic of the architecture of the $D$-Wave One Rainier chip. The qubits (the labeled circles) are arranged in $4\times 4$ unit cells, with 8 qubits per unit cell. The allowed couplings, shown by lines connecting qubits, are programmable inductive couplers. Only green qubits corresponding to calibrated qubits are used in the experiments.

Figure 7

The annealing schedules $A\left(t\right)$ and $B\left(t\right)$ used in the system Hamiltonian in the experiment and in our numerical simulations.

Figure 8

Independent calculation of $\langle e^{-\beta (\Delta E-\Delta F)}\rangle$ (blue solid line) and $\gamma$ (red dashed line) as a function of the system-bath coupling strength using the adiabatic Markovian master equation with $t_f=5\mu s$, $J=\frac{1}{2}$, $T=17$ mK, and ${\omega }_c=8\pi$. Inset: magnification of the behavior near the top of the curves. The two curves overlap to within numerical accuracy.

Figure 9

Simulation results for $\langle \beta (\Delta E-\Delta F)\rangle$ (solid blue line) and the right-hand side of Eq. (13a) (red dashed line) using the Markovian master equation and $t_f=5\mu s$, $J=\frac{1}{2}$, $T=17$ mK, and ${\omega }_c=8\pi$. The two curves overlap to within numerical accuracy.