Quantum process inference for a single-qubit Maxwell demon

Xingrui Song, Mahdi Naghiloo, and Kater Murch

Phys. Rev. A 104, 022211 – Published 20 August 2021

Abstract

While quantum measurement theories are built around density matrices and observables, the laws of thermodynamics are based on processes such as the ones used in heat engines and refrigerators. The study of quantum thermodynamics fuses these two distinct paradigms. In this article, we highlight the usage of quantum process matrices as a unified language for describing thermodynamic processes in the quantum regime. We experimentally demonstrate this in the context of a quantum Maxwell demon, where two major quantities are commonly investigated: the average work extraction $〈 W 〉$ and the efficacy $γ$ , which measures how efficiently the feedback operation uses the obtained information. Using the tool of quantum process matrices, we develop optimal feedback protocols for these two quantities and experimentally investigate them in a superconducting circuit QED setup.

Received 31 January 2021
Revised 17 June 2021
Accepted 3 August 2021

DOI:https://doi.org/10.1103/PhysRevA.104.022211

Physics Subject Headings (PhySH)

Quantum thermodynamics

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Xingrui Song ¹, Mahdi Naghiloo^1,2, and Kater Murch ^1,*

¹Department of Physics, Washington University in St. Louis, St. Louis, Missouri 63130, USA
²Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*murch@physics.wustl.edu

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Vol. 104, Iss. 2 — August 2021

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Images

Figure 1
Quantum process matrix and quantum trajectory descriptions of the qubit evolution. (a) The signal $r$ obtained from continuous weak measurement. (b) The diagonal and off-diagonal matrix elements of the quantum process matrix $χ_{j k}$ . The colors of the shaded areas represent the phases of the off-diagonal elements. (c) The matrix elements of the quantum process matrix at $t = 0.94 μ s$ . The length and color of each bar represent the norm and phase of the corresponding matrix elements of $χ_{j k}$ , respectively. The quantum process matrix of a (rescaled) ideal Hadamard gate (dashed boxes). (d) The quantum trajectory for $ρ_{r}$ calculated from a weak measurement record with the quantum process matrix ( $x_{χ}, z_{χ}$ ; solid lines with symbols) and with the SME ( $x_{ρ}, z_{ρ}$ ; solid lines without symbols). These trajectories are verified with tomographic validation ( $x_{tom}, z_{tom}$ ; dotted lines). In this example, the qubit is initialized in the ground state.
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Figure 2
Demon utilizing quantum process inference. The qubit is prepared in a high-temperature initial state. We apply an initial measurement to determine $E_{i}$ . The system undergoes Rabi drive and continuous weak measurement. The demon infers the quantum process matrix $χ_{j k}$ from the measurement record and performs a feedback rotation to extract work. At the end, we apply a final measurement to determine $E_{f}$ , concluding the TPM protocol.
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Figure 3
Two different feedback protocols. (a) The feedback protocol that maximizes the work extraction: an ensemble of $ρ_{r}$ , obtained by tracking single trajectories via the SME from an initial thermal state $(β = 1.3)$ , is represented as points in the $X - Z$ plane. As shown in the plot, the feedback rotation angle is directly related to $arctan (z / x)$ returning the state to the $+ Z$ direction. Experimentally, this feedback rotation is approximated by 20 discrete angles separated by $π / 10$ . (b) The feedback protocol that maximizes the efficacy: the map is more complicated because the optimal feedback rotation (color) is determined by ${\tilde{ρ}}_{r}$ rather than $ρ_{r}$ (points on the $X - Z$ plane). (c) The efficacy of the two feedback protocols measured at different evolution times (circles) for $β = 2.5$ . The generalized Jarzynski equality is verified for the $ρ$ demon (squares). (d) Work advantage due to the two feedback protocols for $β = 2.5$ .
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Figure 4
Efficacy and work extraction advantages of the two protocols. (a) The efficacy gain for different initial temperatures and different quantum efficiencies at three evolution times. The negative values (black) are caused by statistical fluctuations. Efficiency lower than the experimental realization is obtained by adding Gaussian random numbers as discussed in Appendix pp6. (b) The corresponding work gain for the same parameters.
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Figure 5
Two approaches to calculate the work distribution. The left and right panels display the work distribution before and after the feedback rotation, respectively. (a) The TPM work distribution measured at $β = 0.5$ . The dashed lines indicate the average work. (b) The corresponding conditional work distribution viewed by the demon. $〈 W_{r} 〉$ is in fair agreement with $〈 W_{TPM} 〉$ .
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