Information Gain and Loss for a Quantum Maxwell’s Demon

We use continuous weak measurements of a driven superconducting qubit to experimentally study the information dynamics of a quantum Maxwell’s demon. We show how information gained by a demon who can track single quantum trajectories of the qubit can be converted into work using quantum coherent feedback. We verify the validity of a quantum fluctuation theorem with feedback by utilizing information obtained along single trajectories. We demonstrate, in particular, that quantum backaction can lead to a loss of information in imperfect measurements. We furthermore probe the transition between information gain and loss by varying the initial purity of the qubit.

Figure 1

Classical demon vs quantum demon. (a) In the classical situation, the dynamics can be seen as an evolution of the populations in the definite eigenstates, yet in the quantum case (b), the dynamics includes coherences and can no longer be understood as a classical mixture. (c) The experimental configuration consists of a quantum two-level system coupled to a cavity mode via a dispersive interaction, which allows for both weak and strong measurements of the qubit state populations. A resonant drive at the qubit transition frequency turns these relative populations into coherences and vice versa, leading to coherent quantum evolution.

Figure 2

Experimental sequence. (Step 1) The qubit is initialized in a thermal state at inverse temperature $\beta$. (Step 2) As the first step in determining the energy change through projective measurements, we perform a projective measurement (labeled “$X$”) in the energy basis. (Step 3) We drive the qubit with a coherent drive characterized by ${\mathrm{\Omega }}_R/2\pi =0.8\mathrm{MHz}$ while the quantum demon monitors the qubit evolution with a near-quantum-limited detector. The demon’s knowledge about the qubit state, which depends on density matrix evolution, can be expressed in terms of the expectation values $x\equiv ⟨{\sigma }_x⟩$ and $z\equiv ⟨{\sigma }_z⟩$ (solid lines) with corresponding tomographic validation (dashed lines) showing that the demon’s expectation values are verified with quantum state tomography [31, 32]. The information change is calculated using Eq. (4) for each run of the experiment. (Step 4) The demon uses the acquired information from the previous step to apply a rotation to bring the qubit to the ground state. (Step 5) We perform a projective measurement labeled “$Z$” as the second step in a two-point energy measurement.

Figure 3

Experimental test of the quantum fluctuation theorem. (a) Transition probabilities are calculated using the two projective measurements in steps 2 and 5. (b) By tracking a single quantum trajectory, we calculate the information change (4). The diagonal basis $z^{\prime }$ at time $t$ is indicated. (c) Combining the transition probabilities and the information change, we verify the quantum fluctuation theorem (6) (round markers); however if the information is ignored ($I=0$), the fluctuation theorem is not valid (square markers). (d) Information change $I$, Eq. (3), along single quantum trajectories; the dotted curve shows a typical information change obtained by Eq. (4). The background color shows the distribution of information change for many trajectories. The solid curve is the average information change $⟨I⟩$, Eq. (5). Two dashed lines show the Shannon entropy of initial state $H(0)$ (coarse dash) and $H(0)-\ln (2)$ (fine dash), which indicate the maximum and minimum limit for information change for a given initial state. (e) Red (blue) dots show the qubit state distribution obtained by trajectories at $t=2\mu \mathrm{s}$ right before (after) the feedback rotation and the average of this distribution indicated by the red (green) circle. The cross indicates the reconstruction of the state of the qubit after the feedback using projective measurements.

Figure 4

Transition from information gain to information loss. (a) Classical and quantum noise corresponding to the unknown detector configuration randomly shifts the probe tone [shown here as a coherent state in the quadrature ($I–Q$) space of the electromagnetic field] and degrades the total efficiency of the measurement. The inefficient unraveling of the SME is the statistical average of possible unravelings corresponding to different detector configurations. (b) Transition from information gain to information loss by changing the purity of the initial state of the system controlled by its temperature via $z_0=\tanh (\beta /2)$. The solid line indicates the Shannon entropy of the initial state, which is obtainable only with projective measurements corresponding to the Lanford-Robinson bound [50].