Generating steady quantum coherence and magic through an autonomous thermodynamic machine by utilizing a spin bath

The generation of steady quantumness in the presence of an environment is of utmost importance if we are to build practical quantum devices. We propose a scheme of generating steady coherence and magic in a qubit system attached to a heat bath at a certain temperature through its interaction with another qubit system attached to a spin bath. Coherence generation in the reduced qubit is always possible in this model. The steady coherence in the reduced qubit attached to the heat bath may be used to enhance the subsequent transient performance of a quantum absorption refrigerator. For the case of generation of magic, which is the quantum resource responsible for implementation of gates which are not simulatable via stabilizer computation, we show that there exists a critical temperature of the heat bath beyond which it is not possible to create magic in the reduced qubit attached to the heat bath. Below the critical temperature, the strength of interaction between the qubits must lie within a certain region for the creation of magic. We further note that by increasing the strength of coupling of the second qubit to the spin bath, typified by the reset probability, keeping the coupling strength of the first qubit to the heat bath, it is possible to increase the critical temperature of the heat bath for the creation of magic.

Figure 1

Schematic diagram of model used in the work.

Figure 2

(left) Dependence of the steady-state coherence of the first qubit on the heat bath temperature $T_1$ and (right) the reset probability for the spin bath $p_2$.

Figure 3

If a qubit state equilibrates on the $z$ axis of the Bloch sphere (blue blob) or on the $x$ axis of the Bloch sphere (golden blob), the state lies within the stabilizer polytope. The first scenario is associated with the thermal state of the heat bath, the second with the equilibrium state of the angular-momentum bath depicted in our model.

Figure 4

Values of linear functions of Bloch vector [cf. Eq. (8)] of the reduced qubit attached to the heat bath vs the reset probability $p_1=p_2=p$ (left) and the heat bath temperature $T_1$ (right). Any one of the curves falling in the pale yellow region indicates the presence of magic. If all three lines lie in the light gray region, then the state is within the stabilizer polytope.

Figure 5

Magic creation in the limit of low spin bath temperature $T_2$. (left) Dependence of critical temperature $T_{\text{crit}}$ on the reset probability $p_1=p_2=p$. Creation of magic is possible in the pale yellow region and impossible in the light gray region. (right) Allowed interval for interaction strength $g$ with respect to heat bath temperature $T_1$ for creation of magic. Creation of magic is possible in the pale yellow region and impossible in the dark brown region.

Figure 6

Condition for creation of magic for high spin bath temperature $T_2$. (left) Dependence of critical temperature $T_{\text{crit}}$ on the reset probability $p_1=p_2=p$. Creation of magic is possible in the pale yellow region and impossible in the light gray region. (right) Allowed interval for interaction strength $g$ with respect to heat bath temperature $T_1$ for creation of magic. Creation of magic is possible in the pale yellow region and impossible in the dark brown region.

Figure 7

Effect of asymmetry $\mu$ between the reset probabilities $p_1$ and $p_2=\mu p_1$ on the critical temperature $T_{\text{crit}}$ for creation of magic in the low spin bath temperature $T_2$ limit (left) and high spin bath temperature $T_2$ limit (right). Creation of magic is possible in the pale yellow region and impossible in the light gray region.