Spin Quantum Heat Engine Quantified by Quantum Steering

Following the rising interest in quantum information science, the extension of a heat engine to the quantum regime by exploring microscopic quantum systems has seen a boon of interest in the last decade. Although quantum coherence in the quantum system of the working medium has been investigated to play a nontrivial role, a complete understanding of the intrinsic quantum advantage of quantum heat engines remains elusive. We experimentally demonstrate that the quantum correlation between the working medium and the thermal bath is critical for the quantum advantage of a quantum Szilárd engine, where quantum coherence in the working medium is naturally excluded. By quantifying the nonclassical correlation through quantum steering, we reveal that the heat engine is quantum when the demon can truly steer the working medium. The average work obtained by taking different ways of work extraction on the working medium can be used to verify the real quantum Szilárd engine.

Figure 1

Conventional and modified Szilárd engine. (a) For the conventional Szilárd engine, the working medium is a single atom that initially stayed in a thermal equilibrium state. A demon measures which half of the box the atom is in. If the atom is in the right half of the box, a movable shutter is put down in the middle. Then the shutter is hung to a load and extracts work from the atom. (b) For the modified Szilárd engine, both the working medium and the bath are resembled by a single spin qubit, respectively. Alice (the demon) prepares the initial state of the whole system, performs measurement $\widehat{M_i}$ on the bath qubit, and tells Bob the operations ${\widehat{U}}_i^{\pm }$ depending on the measurement outcomes $\pm 1$. Bob implements the operations ${\widehat{U}}_i^{\pm }$ on the working medium qubit to extract work from its internal energy.

Figure 2

Experimental circuit for implementing the modified Szilárd engine. (a) The experimental system: a nitrogen vacancy (NV) center in diamond. The bath and the working medium are resembled by the intrinsic electron spin and nitrogen nuclear spin of the NV center. (b) The quantum circuit for the experimental implementation. In the state preparation step, different circuits are used to prepare different types of correlated initial states. The circuit in the left dashed box is for the Werner states, and the right dashed box is for the Gibbs-invariant mixed states. ${\theta }_y^{0,\pm }$ and ${\varphi }_y$ are rotations along $y$ axis. The rotation angle and the dephasing processes are dependent on $\eta$ and $q$. In the measurement step, depending on the decomposition $D_{i=1,2,3}$, the electron spin is measured with ${\widehat{M}}_{1,2,3}={\sigma }_{z,y,x}$. This is equivalently realized by the operations labeled by $D_i$ (listed below), which rotates the measurement basis of $\widehat{M_i}$ to the ${\sigma }_z$ Pauli basis. Finally, the work is extracted by the controlled rotations ${\widehat{U}}_i^{\pm }$ on the nuclear spin, and the final energy of the nuclear spin is measured [56].

Figure 3

Difference between work extracted from classical and quantum global states. The quantum (classical) global state is a pure entangled state ${\rho }_1$ (separable state ${\rho }_2$). The blue (red) data points are the work difference of measurement ${\widehat{M}}_1={\sigma }_z$ (${\widehat{M}}_2={\sigma }_y$), with error bars representing one standard deviation. The solid lines are the theoretical predictions. For the ${\widehat{M}}_1$ measurement, work extraction from both the classical and quantum global states are optimal, yielding the same extracted work. While for the ${\widehat{M}}_2$ measurement, work extraction from the classical state is no longer optimal, and can extract less work than from the quantum global state.

Figure 4

Identification of quantumness with the work extraction inequality. (a) Violation of the work extraction inequality ${\overline{W}}_3$ for the Gibbs-invariant mixed states. The color indicates the difference between the extracted work and its local work extraction bound, with the white color indicates the boundary where the work extraction inequality is violated. The dashed black line is the theoretical prediction of the boundary. (b) Comparison between the extracted work and its local work extraction bound for the pure entangled states. The data points are experimental results of the extracted work, with the error bars representing one standard deviation, and the solid lines representing the theoretical predictions. The dashed lines are the local work extraction bound. The color of red represents the work extraction with $(c_1,c_2,c_3)=(1/3,1/3,1/3)$, and yellow represents the work extraction inequality proposed in [42]. From the insets, the extracted work of ${\overline{W}}_3$ violates its local work extraction bounds, whereas ${\overline{W}}_B$ doesn’t, showing that work extraction inequality ${\overline{W}}_3$ is more effective than ${\overline{W}}_B$. (c),(d) Violation of the work extraction inequality ${\overline{W}}_2$ and ${\overline{W}}_2$ for the Werner states. (e) Comparison between the extracted work and its local work extraction bound for the standard Werner states. For the standard Werner states ($\eta =0$), the work extraction inequalities can be violated when $q>1/\sqrt{2}$ for ${\overline{W}}_2$ and $q>1/\sqrt{3}$ for ${\overline{W}}_3$, as marked by the dashed black lines. This is consistent with the well-known linear steering inequalities [60]. (f),(g) Correspondence between quantum steering and work extraction inequality violation [61]. The experimental results in (a),(d) are plotted as the vertical axis of the data points, and the horizontal axis is the steering inequality violation calculated from the corresponding states. The clear positive correlation demonstrates the effectiveness of identifying quantum steering with work extraction inequalities.