Quantum Refrigeration with Indefinite Causal Order

We propose a thermodynamic refrigeration cycle which uses indefinite causal orders to achieve nonclassical cooling. The cycle cools a cold reservoir while consuming purity in a control qubit. We first show that the application to an input state of two identical thermalizing channels of temperature $T$ in an indefinite causal order can result in an output state with a temperature not equal to $T$. We investigate the properties of the refrigeration cycle and show that thermodynamically, the result is compatible with unitary quantum mechanics in the circuit model but could not be achieved classically. We believe that this cycle could be implemented experimentally using tabletop photonics. Our result suggests the development of a new class of thermodynamic resource theories in which operations are allowed to be performed in an indefinite causal order.

Figure 1

(a) and (b) illustrate channels ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ placed in a definite order, corresponding to the control qubit being in state $|1⟩⟨1|$ and $|0⟩⟨0|$, respectively. In (c) the quantum SWITCH places the channels in a superposition of causal orders. It entangles the order of the two channels with the state of the control qubit, in this case $|+⟩⟨+|$.

Figure 2

A circuit with an output equivalent to that of the quantum SWITCH of thermalizing channels.

Figure 3

The three steps of the refrigeration cycle of the ICO refrigerator. The black dot represents the working system, and the color of the outline indicates the temperature of the last reservoir(s) with which it has interacted. The dotted lines in step (i) represent the operation in the event of a measurement of $|+{⟩}_c$ (the undesired outcome) for the state of the control system.