Detecting Heat Leaks with Trapped Ion Qubits

The concept of passivity has been conceived to set bounds on the evolution of microscopic systems initialized in thermal states. We experimentally demonstrate the utility of two frameworks, global passivity and passivity deformation, for the detection of coupling to a hidden environment. We employ a trapped-ion quantum processor, where system qubits undergoing unitary evolution may optionally be coupled to an unobserved environment qubit, resulting in a heat leak. Evaluating the measurement data from the system qubits only, we show that global passivity can verify the presence of a heat leak, which is not detectable by a microscopic equivalent of the second law of thermodynamics. Furthermore, we experimentally show that passivity deformation allows for even more sensitive detection of heat leaks, as compared to global passivity, and detect a heat leak with an error margin of 5.3 standard deviations, in a scenario where other tests fail.

Figure 1

(a) A number of system qubits and one environment qubit are initialized to thermal states. The system qubits undergo a “black box” evolution and are measured. In the presence of a heat leak, the system is coupled to an unobserved environment qubit. (b) Laser-driven operations couple the qubit states $|0⟩$ and $|1⟩$.

Figure 2

Quantum circuits used for demonstrating a violation of (a) the global passivity inequality Eq. (6), and (b) the passivity deformation inequality Eq. (11). In both cases, all qubits are initialized to thermal states [before (i)], the system qubits undergo unitary evolution [before (ii)], and one of the system qubits is optionally swapped with the environment qubit before measurement (iii). Initialization to thermal states is depicted by a flame, with the flame size indicating the spin temperature.

Figure 3

Results of the global passivity protocol, Fig. 2, 6700 repetitions: $\mathrm{\Delta }⟨{\widehat{B}}^{\alpha }⟩$ is shown as a function of $\alpha$. For the case with the final SWAP gate (brown, dashed line), the system qubits are coupled to the environment qubit, and we observe $\mathrm{\Delta }⟨{\widehat{B}}^{\alpha }⟩\le 0$ for $\alpha <0.5090(75)$. For the case where the SWAP gate is not performed (purple, dashed line), we observe no violation of the global passivity inequality [Eq. (8)]. The experimental data (dashed) are shown along with error channels (shaded, see text) and compared to theoretical values (green). The inset shows the measured populations for system qubits $c$, $h$, from which the experimental data including the SWAP gate were calculated.

Figure 4

Results of the passivity deformation protocol, Fig. 2, 3200 repetitions: For varying $\xi$, we compare the energy change of the cold qubit (blue) to the rescaled energy change of the hot qubit (red) (left- and right-hand sides of inequality (19). Panel (a) shows the scenario with the SWAP gate, where we find a violation of inequality (19) for ${\xi }_m\le \xi \le -0.880(1)$ (dotted arrow). Panel (b) shows the case without the final SWAP gate, where no violation of Eq. (19) is observed. Experimental data (dashed) are compared to theoretical values (solid, green). The measured populations for system qubits $h$, $c$, from which the data were calculated, are shown as insets.