All steerable quantum correlations can provide thermodynamic advantages in cooling

TL;DR

The paper investigates whether quantum steering can enhance cooling by using steerable assemblages to withdraw more heat from a reservoir than any unsteerable, classically correlated resource. It formalizes heat extraction via $Q_c$ and analyzes the performance gap between steerable and unsteerable resources, connecting the maximal cooling advantage $\xi_{\max}$ to the steering robustness $\mathcal{R}(\{\rho_{a|x}\}_{a,x})$ through a lower bound $\xi_{\max} \ge 1 + \mathcal{R}(\{\rho_{a|x}\}_{a,x})$. The central result provides an operational interpretation of steering robustness as a thermodynamic witness and shows that, for certain constructions such as MUBs and maximally entangled or isotropic states, the advantage can scale with dimension, yielding an unbounded quantum cooling advantage in suitable regimes. The findings have practical implications for quantum thermodynamics and scalable quantum technologies, suggesting experimental feasibility via quench and thermalization steps and motivating future work on multipartite extensions and steerability-based thermodynamic tasks.

Abstract

The removal of heat generated during computation poses a major challenge for both classical and quantum information processing. In particular, heat removal is directly linked to a fundamental requirement of quantum computation: the ability to reset a system to a pure state before computation. Efficient cooling is therefore crucial both for advancing our understanding of thermodynamics in the quantum regime and for enabling the development of modern quantum technologies. In this work, we devise a cooling task that exploits steerability, a fundamental form of quantum correlations, to demonstrate a provable quantum advantage over classically correlated scenarios in which steerability is absent. We quantify this advantage by the ratio between the heat removed using steerable quantum correlations and the heat removed using unsteerable classical correlations. Specifically, we show that steerable quantum correlations always provide a thermodynamic advantage in the cooling task. Remarkably, we further establish that the maximum achievable advantage is directly related to a geometric measure of steerability known as steerability robustness. Our results suggest that this thermodynamic advantage can serve as a witness of steerability. Finally, we present examples showing that the advantage can increase with the dimension of the underlying system.

Figures (3)

• Figure 1: In Panel A, we illustrate the preparation of resources for heat removal (cooling) from the bath. Alice and Bob share a quantum state $\hat{\rho}_{AB}$. Alice samples an input $x$ uniformly from the set $\mathcal{X} = \{0, \ldots, n-1\}$ and performs the measurement $M_x$ on her subsystem of the joint state $\hat{\rho}_{AB}$. After the measurement, Alice communicates the outcome $a$ and the input $x$ to Bob. As a result of Alice's measurement, Bob obtains the conditional state $\hat{\rho}_{a|x}$, given in Eq. \ref{['conditional_state']}, with probability $p(a|x)$, as given in Eq. \ref{['conditional_prob']}. This state then serves as a resource for withdrawing heat from the bath. Panel B illustrates the procedure for heat removal from a bath at inverse temperature $\beta$. Upon receiving the pair $(a,x)$ from Alice, Bob quenches his Hamiltonian from the initial trivial Hamiltonian $H=0$ to a final Hamiltonian $H_{a|x}$. This step involves no contact with the bath, so the entropy remains unchanged, as indicated by arrow (1) in the average energy–entropy diagram. In the next step, represented by arrow (2), Bob thermalizes his subsystem by placing it in contact with the thermal bath. During this process, heat is withdrawn from the bath and the entropy of the state changes. Finally, in the step indicated by arrow (3), Bob returns the Hamiltonian to its initial trivial form via another quench, again without contacting the bath. On average, the heat withdrawn from the bath is given by $Q_c = 1/n \sum_{a,x} p(a|x) \left[\Tr(\hat{\rho}_{a|x} H_{a|x}) - \Tr(\hat{\gamma}_{a|x} H_{a|x}) \right].$
• Figure 2: In this diagram, we present a parallel comparison between the Otto refrigerator and the cooling protocol assisted by correlations introduced in this article. Panel (A) illustrates a bidimensional representation of the Otto refrigeration cycle in the "phase space" defined by the system's state and Hamiltonian. Driven by the temperature difference between the two baths, the refrigerator withdraws heat $Q_c$ from the cold reservoir and releases heat $Q_h$ into the hot reservoir. To complete the cycle, an input work of total magnitude $Q_h + Q_c = W_{\mathrm{in}} + W_{\mathrm{out}}$ is required, where $W_{\mathrm{in}} = \mathrm{Tr}(H_B \gamma_h) - \mathrm{Tr}(H_A \gamma_h)$ and $W_{\mathrm{out}} = \mathrm{Tr}(H_A \gamma_c) - \mathrm{Tr}(H_B \gamma_c)$. In panel (B), we represent the cooling mechanism based on assemblages arising from unsteerable and steerable correlations. We compare the amount of heat $Q_c$ withdrawn when the refrigerator operates with assemblages generated by unsteerable correlations versus those produced by steerable correlations. The latter cool the bath more, providing a quantum thermodynamic advantage.
• Figure 3: We plot the lower bound of $\xi_{\max}\!\left(\left\{\rho_{a|x}^{\mathrm{isotropic}}(\eta)\right\}_{a,x}\right)$, given in Eq. \ref{['ximax_ineq']}, as a function of $d$ and $\eta$ (orange surface) for $2 \leq d \leq 500$ and $\left(\sqrt{d}+\frac{1}{d+\sqrt{d}+1}\right)^{-1} < \eta \leq 1$, as specified in Eq. \ref{['eta+condition']}. The plot shows that this surface lies strictly above the plane $z = 1$ (shown in blue), demonstrating that $\xi_{\max}\!\left(\left\{\rho_{a|x}^{\mathrm{isotropic}}(\eta)\right\}_{a,x}\right) > 1$. This confirms the presence of quantum advantage, which increases with the dimension $d$ throughout the parameter regime $\left(\sqrt{d}+\frac{1}{d+\sqrt{d}+1}\right)^{-1} < \eta \leq 1$.

Theorems & Definitions (1)

• Theorem 1