No-broadcasting of non-Gaussian states

TL;DR

The paper investigates whether non-Gaussianity can be broadcast in continuous-variable quantum systems using only Gaussian operations. It combines a resource-theoretic insight—NG is not generally super-additive—with a fixed-point analysis of Gaussian channels (via Lyapunov stability) to establish a no-broadcasting theorem for non-Gaussianity, including a treatment of the complementary channel and degradability. The main contributions are the demonstration that exact broadcasting is impossible under Gaussian dynamics, the classification of Gaussian-channel fixed points through the spectral radius $r(X)$, and the argument that non-Gaussian resources like $Wigner$ negativity cannot be shared without introducing non-Gaussian operations. This result highlights a fundamental limitation of Gaussian processing in distributing non-classical features and guides the design of CV quantum technologies that rely on non-Gaussian resources.

Abstract

Gaussian states are of fundamental importance in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and an elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In terms of quantum resource theory, we can then recognize non-Gaussian states as resources and Gaussian operations and states as free, which can be used and prepared easily. Given such a structure of resource theory, the task of broadcasting the resource is to determine if the resource content of a state can be cloned in a meaningful way, which, if possible, provides a strong operation for manipulation of the resource. In this work, we prove that broadcasting of non-Gaussian states via Gaussian operations is not possible. For this, we first show that the relative entropy of non- Gaussianity is not super-additive, which rules it out as a prime candidate in the analysis of such no-go results. Our proof is then based on understanding fixed points of Gaussian operations and relates to the theory of control systems. The no-go theorem also states that if two initially uncorrelated systems interact by Gaussian dynamics and non-Gaussianity is created at one subsystem, then the non-Gaussianity of the other subsystem must be reduced. Further, keeping the set of free operations fixed to Gaussian operations, we can also comment on the broadcasting of Wigner negativity and genuine quantum non-Gaussianity.

Figures (3)

• Figure 1: Broadcasting protocol. Everything in blue is Gaussian (including the unitary), and states in pink need to be non-Gaussian for a successful broadcasting protocol.
• Figure 2: Variation of $\Delta NG_{\phi_+}$ as given by (\ref{['eqr1']}). The negativity of the values shows that for those $\alpha$ the state violates super-additivity. Refer to the text for more details.
• Figure 3: The setting for Theorem-\ref{['thrm3']}. If, using a Gaussian operation $\mathcal{R}$, the state $\rho_B$ can be recovered from $\sigma_B$, then the effective local operation $\mathcal{G}$ is Gaussian and can therefore be implemented without having access to $\rho_B$. The figure shows the contradiction that we can broadcast $\rho_B$ if this is not the case.

Theorems & Definitions (17)

• definition 1
• proposition 1
• proof
• lemma 1
• theorem 1
• lemma 2
• theorem 2
• proof
• lemma 3
• theorem 3