Information-to-work conversion by Maxwell's demon in a superconducting circuit quantum electrodynamical system

Abstract

Information thermodynamics bridges information theory and statistical physics by connecting information content and entropy production through measurement and feedback control. Maxwell's demon is a hypothetical character that uses information about a system to reduce its entropy. Here we realize a Maxwell's demon acting on a superconducting quantum circuit. We implement quantum non-demolition projective measurement and feedback operation of a qubit and verify the generalized integral fluctuation theorem. We also evaluate the conversion efficiency from information gain to work in the feedback protocol. Our experiment constitutes a step toward experimental studies of quantum information thermodynamics in artificially made quantum machines.

Fig. 1

Maxwell’s demon and absolute irreversibility. a Concept of the experiment. The system initially prepared in a canonical distribution ${\widehat{\rho }}_{\mathrm{ini}}$ evolves in time. A projective measurement [with outcome x(=g or e)] by the demon projects the system onto a quantum state. The demon gains the stochastic Shannon entropy I_Sh and converts it into work W via a feedback operation $Û$ ($\mathrm{=}{Û}_{\mathrm{g}}$ or ${Û}_{\mathrm{e}}$). The forward process ends up in the final distribution ${\widehat{\rho }}_{\mathrm{fin}}$. The time-reversed reference process starts from a reference state ${\widehat{\rho }}_{\mathrm{r}}$, which we choose to be equal to ${\widehat{\rho }}_{\mathrm{ini}}$. The absolute irreversibility is quantified with λ_fb, the probability of those events in the time-reversed process whose counterparts in the original process do not exist (red dashed arrows). b Schematic of the feedback-controlled system in the experiment. The right panel shows a photograph of the qubit-resonator coupled system. The cavity resonator is disassembled to show its internal structure

Fig. 2

Generalized integral fluctuation theorem under feedback control. a Pulse sequence used in the experiment. The qubit is initialized with a projective measurement and postselection, followed by a resonant pulse excitation which prepares the qubit in a superposition state as an input. The two-point measurement protocol (TPM) involves two quantum non-demolition projective readout pulses. Depending on the outcome x of the first readout (x = g or e corresponding to the ground or the excited state of the qubit), a π-pulse for the feedback control is or is not applied. The π-pulse flips the qubit state to the ground state and extracts energy. The second readout with outcome z completes the protocol. See the Supplementary Note 2 for details. b Experimentally obtained statistical averages ${⟨{\mathrm{e}}^{\beta W-I_{\mathrm{Sh}}}⟩}_{\mathrm{PM}}$ (blue circles) and ${⟨W⟩}_{\mathrm{PM}}\mathrm{∕}\hslash {\omega }_{\mathrm{q}}$ (magenta circles) vs. the inverse initial qubit temperature 1/T. The blue solid curve (gray solid line) is the theoretical value of the probability 1 − λ_fb in the presence (absence) of absolute irreversibility. The magenta solid curve is the expectation value of the normalized extracted work. The corresponding blue and magenta dashed curves are obtained by a master equation, which takes into account the qubit relaxation during the pulse sequence

Fig. 3

Effects of the feedback error on the fluctuation theorem and the second law of thermodynamics. a Pulse sequence. Two readout pulses are inserted between the two TPM pulses in Fig. 2a. The outcome k( = g or e) obtained by the readout with a variable pulse amplitude is used for the feedback control. The feedback error probability ${ϵ}_{\mathrm{fb}}$ is a function of the measurement strength. The subsequent readout with outcome y projects the qubit state before the feedback control. See Supplementary Note 2 for details. b Experimentally determined $⟨{\mathrm{e}}^{\beta W-I_{\mathrm{QC}}}⟩$ (blue circles) and $⟨{\mathrm{e}}^{\beta W}⟩$ (red squares) vs. the feedback error probability ${ϵ}_{\mathrm{fb}}$. c $⟨I_{\mathrm{QC}}⟩$ (blue circles) and $⟨\beta W⟩$ (red squares) vs. ${ϵ}_{\mathrm{fb}}$. The black dashed line represents the Shannon entropy ${⟨I_{\mathrm{Sh}}⟩}_{\mathrm{PM}}$ of the qubit initial state, which is prepared at the effective temperature T = 0.14 K with the excited state occupancy of 0.097. Line-connected black dots in b, c show the simulated results incorporating the effect of qubit relaxation (Supplementary Note 2). Inset in c: information-to-work conversion efficiency η (green circles) and the simulated result (line-connected black dots). The gray dashed line indicates the value for the efficiency in the limit of the projective measurement due to the absolute irreversibility