Dimensional advantage in network cooling with hybrid oscillator-qudit systems

TL;DR

This work establishes a fundamental no-go theorem for cooling a CV oscillator with a qubit regulator under a measurement-based JC-type protocol, and proves a twofold dimensional advantage when using higher-dimensional regulators. By analyzing single- and multi-oscillator networks, the authors show that increasing regulator dimension reduces the required cooling cycles and expands feasibility to higher initial energies, with linear networks achieving near-unit fidelity while star networks falter. The study extends to hybrid CV–DV systems, where a fourth- or higher-dimensional regulator yields near-unit cooling and shorter cycle counts, enabling efficient generation of non-Gaussian resources such as CAT codes and N00N states. Together, these results suggest scalable, dimension-aware cooling protocols that can serve as practical primitives for preparing low-entropy resources in quantum technologies.

Abstract

We examine the cooling of networks of oscillators through repeated unitary evolution followed by conditional measurement on a finite-dimensional auxiliary system, coupled via Jaynes-Cummings type interaction. We prove that near-perfect cooling of the oscillator to vacuum is fundamentally impossible when the auxiliary system is a qubit, establishing a no-cooling theorem for a two-level regulator. Moving beyond this limitation, we reveal a twofold dimensional advantage of higher-dimensional auxiliaries - reducing the number of required cycles, and enabling the efficient cooling of oscillators with higher initial energies. We further show that, while extending the network leads to a saturation of this dimensional advantage at moderate auxiliary dimensions, near-perfect cooling remains achievable for linear network configurations but fails for star networks. Moreover, we highlight the adaptability of the proposed protocol by demonstrating efficient cooling of hybrid continuous- and discrete-variable systems that naturally support the generation of non-Gaussian and entangled quantum resources.

Figures (4)

• Figure 1: Schematic diagram illustrating different components of the cooling protocol.$(a).$ Network of resonators, each with level spacing $\omega_f$, arranged in a linear configuration with the first resonator on the right coupled with strength $\lambda$ to the $d$-level regulator, with level spacing $\omega_a$. The resonators interact with themselves with coupling strength $\tilde{\lambda}$. $(b).$ A star network of resonators, each interacting with the qudit regulator via strength $\lambda$, while there is no coupling between the resonators themselves. $(c).$ The cooling protocol for a collection of resonators, $\rho_{\mathcal{V}}$, with the help of a $d$-level regulator initialized in the energy state, $\ket{k}$. The entire system is evolved through the unitary, $\hat{U}(t_{\text{opt}}^k)$, over the optimal time, $t_{\text{opt}}^k$, following which the regulator is post-selected on the eigenstate $\ket{k}$. The evolution-measurement process is repeated several times in order to bring $\rho_{\mathcal{V}}$ to its ground state, $\ketbra{\mathbf{0}}{\mathbf{0}}$, with $\ket{\mathbf{0}}$ denoting that all the resonators are in the vacuum state, $\ket{0} \otimes \ket{0} \otimes \dots$.
• Figure 2: Energy of the resonator that admits near-perfect cooling with qudit auxiliary regulators. Required number of evolution-measurement cycles, $N$ (ordinate), against the average resonator energy, $\langle \hat{\mathcal{N}}_e \rangle$ (abscissa), for different regulator dimensions, $d = 4$ (red dashed line), $d = 5$ (green dotted line), and $d = 6$ (black dash-dotted line). The inset shows the same for $d = 3$ (blue solid line), for which $N$ is considerably higher even for a low-energetic resonator. Both axes are dimensionless.
• Figure 3: Minimum number of cycles required for cooling different network sizes and topologies: Number of evolution--measurement rounds, $N$ (ordinate), as a function of the regulator dimension, $d$ (abscissa), for cooling a network of $M=2$ (dotted line) and $M=3$ (solid line) oscillators in the linear (hollow circles) and star (solid stars) interaction topologies. Increasing the regulator dimension produces a clear dimensional advantage, reducing the required number of rounds and leading to saturation. The linear network enables near-perfect cooling with substantially fewer rounds, while the star configuration saturates at lower fidelities but with a higher success probability. The values cited are correct up to the third decimal place, and both the axes are dimensionless.
• Figure 4: Hybrid cooling of a displaced squeezed thermal resonator and qudit system coupled to a finite-dimensional auxiliary regulator:$(a)$ Number of evolution–measurement rounds, $N$ (ordinate), required to reach the saturation fidelity, against the regulator dimension, $d$ (abscissa), for post-selection on the ground, $k=0$ (hollow squares), and excited, $k=1$ (stars), states. Excited-state measurement requires more rounds but yields significantly higher final fidelity. $(b)$ Dependence of $N$ (ordinate) on the system dimension $d_s$ (abscissa) for fixed regulator dimension, $d=4$ under excited-state post-selection ($\ket{k=1}_R$). The inset corresponds to the success probability $P^{1}_{4,N}$ (ordinate) with $d_s$ (abscissa). Increasing the auxiliary dimension reduces the required rounds while maintaining near-unit fidelity, demonstrating a dimensional advantage. All the axes are dimensionless.

Theorems & Definitions (3)

• proof
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