Quantum effects improve the energy efficiency of feedback control

The laws of thermodynamics apply equally well to quantum systems as to classical systems, and because of this, quantum effects do not change the fundamental thermodynamic efficiency of isothermal refrigerators or engines. We show that, despite this fact, quantum mechanics permits measurement-based feedback control protocols that are more thermodynamically efficient than their classical counterparts. As part of our analysis, we perform a detailed accounting of the thermodynamics of unitary feedback control and elucidate the sources of inefficiency in measurement-based and coherent feedback.

Figure 1

Illustration on the Bloch sphere of the conditional rotations of the auxiliary initially in a mixed state with Bloch vector $-\lambda \widehat{z}$. When the system is initially prepared in the $|+\vec{m}\rangle$ ($|-\vec{m}\rangle$) eigenstate, the auxiliary is conditionally rotated under $U_{\mathrm{meas}}^{\vec{m}}$ by an angle $\theta$ counterclockwise (clockwise) in the $xz$ plane.

Figure 2

(a) Depiction of the conditional evolution of $U_{\pm }^x$ on the post-measurement state given an $x$-measurement outcome $|+\rangle$ (right) or $|-\rangle$ (left). The gray dotted circle denotes the initial length of the Bloch vector, $\alpha$. (b) Depiction of the conditional evolution of $U_{\pm }^z$ after a $z$ measurement.

Figure 3

Plot of the efficiency of the measurement-based feedback protocol using and explicit $x$ measurement, ${\varepsilon }_x^{\mathrm{mb}}$ (dashed) [Eq. (21)], the same measurement-based protocol performed coherently, ${\varepsilon }_x^{\mathrm{coh}}$ (solid) [Eq. (22)], and the measurement-based protocol that uses a $z$ measurement, ${\varepsilon }_z$ (gray) [Eq. (23)], as functions of the strength parameter $\theta$ with $\alpha =0.4$ and $\lambda =0.8$. Note that at $\theta =0$ the efficiency is not defined, since there is no measurement, and as a result the protocol does not change the entropy of the system.

Figure 4

Time evolution of the mutual information for the $x$ measurement (black) and $z$ measurement (gray) during the course of measurement and feedback for a measurement strength $\theta =\pi /4\approx 0.78$, with an $x$-measurement feedback rotation of $\varphi \approx 1.89$ [Eq. (18)].