Broadcasting Non-Gaussian Quantum States

• Broadcasting Non-Gaussian States is the study of distributing quantum states with non-Gaussian features, crucial for advanced quantum protocols.
• The research shows that Gaussian channels degrade non-Gaussianity, necessitating explicit non-Gaussian operations to preserve these quantum resources.
• Experimental and theoretical analyses highlight techniques for state preparation, certification, and the limitations in cloning non-Gaussian properties.

Broadcasting of non-Gaussian States refers to the distribution, sharing, or cloning-like propagation of quantum states possessing non-Gaussian features—those not representable by classical Gaussian states or mixtures—across multiple subsystems, modes, or spatially separated locations. In the framework of continuous-variable (CV) quantum information, non-Gaussianity is regarded as a resource essential for tasks that are unattainable with Gaussian states and operations alone, such as universal quantum computation, fault-tolerant quantum error correction, nonlocality tests, and certain forms of secure quantum communication. The theoretical and experimental foundations, operational principles, and limitations of broadcasting non-Gaussian states have been rigorously explored, especially within the resource theory context that distinguishes Gaussian operations (as "free") from genuinely non-Gaussian manipulations.

1. Theoretical Framework and Resource Theory of Non-Gaussianity

The resource theory of non-Gaussianity treats Gaussian states and Gaussian operations as the free set, while non-Gaussian states—those with a non-Gaussian characteristic function or Wigner function—serve as valuable resources for a variety of quantum protocols. The relative entropy of non-Gaussianity is a central quantitative measure, defined as

$\text{NG}(\rho) = S(\Gamma_{\rho}) - S(\rho)$

where $\Gamma_\rho$ is the reference Gaussian state having the same first and second moments as $\rho$, and $S(\cdot)$ is the von Neumann entropy (Chatterjee et al., 23 Oct 2025). This measure captures the deviation of an arbitrary state from Gaussianity in terms of informational content.

Broadcasting, within this theory, is distinct from simple state transfer—it asks whether the non-Gaussian resource present in one subsystem can be shared or "cloned" across several subsystems via allowed (typically Gaussian) operations. While analogous to the well-known no-cloning theorem, the no-broadcasting question for non-Gaussian resources specifically addresses to what extent the unique quantum features embodied in non-Gaussian statistics can be shared among multiple parties.

2. No-Go Theorem: Impossibility of Broadcasting Non-Gaussianity via Gaussian Operations

The rigorous no-go theorem of (Chatterjee et al., 23 Oct 2025) demonstrates that broadcasting of non-Gaussian states via Gaussian operations is impossible. The proof is not anchored in the relative entropy of non-Gaussianity alone, because this measure is not super-additive:

$$\Delta \mathrm{NG}_{\Phi_+} := \mathrm{NG}(\Phi_+^{AB}) - \mathrm{NG}(\Phi_+^A) - \mathrm{NG}(\Phi_+^B)$$

can be negative for certain entangled cat states ("$\Phi_+$"), ruling out monotonicity-based no-go results. The proof instead employs analysis of the fixed points of Gaussian channels under the transformation of the covariance matrix:

$\Lambda \rightarrow X^T \Lambda X + Y,$

where $X$ and $Y$ are determined by the channel dilation.

If the Schur radius $r(X) < 1$ (all eigenvalues $|λ_i| < 1$), then only Gaussian states can be fixed points. If $r(X) > 1$ no fixed point exists, and for $r(X) = 1$, further analysis shows that the complementary output of the channel is necessarily Gaussian. Therefore, starting from a non-Gaussian state, any attempt to broadcast it using Gaussian channels must degrade (not preserve) the non-Gaussianity in the output modes. The main theorem is thus:

Non-Gaussianity cannot be broadcast using Gaussian operations.

Furthermore, if two uncorrelated systems interact via Gaussian dynamics, an increase in non-Gaussianity in one subsystem is unavoidably accompanied by a decrease in the non-Gaussianity of the other, formalizing a resource-consistency or trade-off constraint (Chatterjee et al., 23 Oct 2025).

3. Non-Gaussian State Preparation and Measurement for Broadcasting

A rich variety of protocols exist for preparing and potentially broadcasting non-Gaussian states, all requiring explicitly non-Gaussian operations or measurements:

• Photon addition and subtraction on Gaussian states, producing, e.g., photon-subtracted squeezed states with Wigner negativity.
• Engineering eigenstates of quadratic Hamiltonians involving EPR-like operators, yielding hierarchies from the Gaussian two-mode squeezed vacuum (TMSV, the ground state) to the non-Gaussian "entangled number states" (ENS) as excited states (0906.1659):

  $$A_\xi = \frac{a - \xi b^\dagger}{\sqrt{1-\xi^2}},\qquad \Omega(\xi) = (\xi^{-1} - \xi)\left(2 A_\xi^\dagger A_\xi + 1\right),$$

  with excitations generated by repeated action of $A_\xi^\dagger$.

• Postselected von Neumann (projective) measurements coupling a two-level system to a Gaussian pointer, enabling generation of non-Gaussian superposition states (e.g., squeezed cat states) dependent on tunable weak values and interaction strengths (Yao et al., 29 Sep 2025).
• Quantum switches controlling the causal order of Gaussian operations. Deterministically engineered non-Gaussian states arise from interference between distinct operation sequences, leveraging the non-convexity of the set of Gaussian states and operations (Koudia et al., 2021).
• Mode filtering and high-pass optical filtering in photon-subtraction setups to enhance the purity and robustness of non-Gaussian states during teleportation or transfer (Takeda et al., 2012).
• Hardware-efficient environment-assisted protocols directly emit propagating non-Gaussian wavepackets (e.g., grid states, multi-component cat states) via dissipative stabilization in engineered superconducting circuits with multi-photon jumps (Khanahmadi et al., 6 Apr 2025).

A key experimental consideration is that only explicitly non-Gaussian operations (such as engineered measurements, nonlinear drives, or multi-photon dissipation) are capable of generating and preserving non-Gaussianity for broadcasting—Gaussian operations always fail this task.

4. Criteria and Verification of Non-Gaussianity under Transmission

The identification and certification of non-Gaussianity in broadcast or distributed quantum states is facilitated by operational criteria:

• Quantum non-Gaussianity can be certified using the relationship between the vacuum probabilities of the state before and after passing through a lossy channel of transmittance $T$ (Fiurášek et al., 2021):

  $$p_0 = \langle 0 | \rho | 0 \rangle, \quad q_0(T) = \sum_{n=0}^\infty p_n (1-T)^n,$$

  where $p_n$ is the photon-number distribution. For Gaussian states, $q_0(T)$ is bounded by a threshold for any given $p_0$, and exceeding this bound certifies quantum non-Gaussianity.

• Certification via mean photon number: For a given vacuum probability $p_0$, an upper bound $n_{\text{th}}$ for the mean photon number $n$ exists such that $n < n_{\text{th}}$ implies genuine non-Gaussianity.
• The EPR-steering inequality can witness non-Gaussianity for pure bipartite states by establishing that for Gaussian states

  $$\Delta^2(q_2|q_1) \cdot \Delta^2(x_2|x_1) \leq 1/4,$$

  and exceeding this value certifies non-Gaussian character (Gómez et al., 2015).

These criteria are experimentally robust—even after significant attenuation and in multimode settings—because Gaussianity is preserved under loss, so demonstration of non-Gaussianity in an output state indicates that the prepared and transmitted state retained its quantum resource nature.

5. Experimental Demonstrations and Feasibility in Real-World Channels

Experimental implementations confirm the principle challenges and possibilities of broadcasting non-Gaussian states:

• Explicit transmission and full quantum homodyne tomography of photon-subtracted squeezed states ("Schrödinger kitten" states) over deployed telecommunication fiber (∼300 m) demonstrate survival of Wigner negativity (indicative of non-Gaussianity) even after 22% channel loss, with Wigner function minima such as $W(0,0)\approx -0.028$ after loss correction (Breum et al., 22 Sep 2025).
• Integrated platform proposals enable deterministic, single-shot emission of multi-component cat, grid, and entangled pair-cat states in superconducting circuits, with output wavepackets designed for direct use in quantum networking (Khanahmadi et al., 6 Apr 2025).
• Broadcast of engineered non-Gaussian states in optical channels leverages heralded photon subtraction, mode filtering, and postselected measurement to preserve Wigner negativity, essential for tests of Bell inequalities, quantum steering, and continuous-variable error correction.
• Protocols for communication and cryptography (e.g., quantum key distribution) utilizing non-Gaussian signal states (such as photon-added then subtracted coherent states, PASCS) offer enhanced key rates and robustness against attacks, directly benefiting from the anti-correlation properties and Wigner negativity of these states (Borelli et al., 2014).

However, the practical limitations of broadcasting are underscored by the probabilistic nature and low success rates of non-Gaussian state preparation schemes, as well as the impossibility of using only Gaussian operations for resource distribution.

6. Implications, Limitations, and Applications in Quantum Networks

The impossibility of broadcasting non-Gaussianity via Gaussian operations necessitates the use of explicit non-Gaussian processes for any task requiring distribution, sharing, or verification of non-Gaussian resources in a network. This restriction has deep implications:

• Universal quantum computation with continuous variables requires the distribution and manipulation of non-Gaussian states as canonical resources.
• Error correction codes with bosonic modes (such as cat and grid codes) rely on the robust broadcast and detection of non-Gaussian features for fault tolerance.
• Quantum networks must incorporate non-Gaussian operations (such as measurements, nonlinear optics, or dissipative engineering) at the modules or the interfaces for deployment or routing of resource states.
• Resource theory arguments and Lyapunov-type stability analysis from control systems theory underpin the non-generativity of non-Gaussian resources under free (Gaussian) operations, formalizing fundamental constraints on the architecture of quantum information protocols.
• Trade-offs resulting from the non-super-additivity of relative entropy measures and the fixed-point properties of Gaussian channels impose limits on resource sharing, redistribution, and the design of multi-user quantum systems.

A direct consequence is that any proposal purporting to distribute or broadcast non-Gaussian states within a scalable architecture must explicitly engineer non-Gaussian processes, eschewing purely Gaussian operation approaches.

7. Summary Table: Core Principles and Results

Topic	Core Principle/Result	Reference(s)
Resource Theory	Gaussian ops/states: free; non-Gaussian: valuable resource	(Chatterjee et al., 23 Oct 2025)
No-go Theorem	Broadcasting non-Gaussianity via Gaussian ops is impossible	(Chatterjee et al., 23 Oct 2025)
Non-Gaussian State Preparation	Photon subtraction/addition, projective measurement, quantum switch	(0906.1659, Yao et al., 29 Sep 2025, Koudia et al., 2021)
Certification under Loss	Vacuum probability and mean photon number criteria	(Fiurášek et al., 2021)
Experimental Demonstration	Transmission of Wigner-negative states in real-world fiber	(Breum et al., 22 Sep 2025)
Implications for Quantum Networks	Explicit non-Gaussian operations required for networked applications	(Khanahmadi et al., 6 Apr 2025, Borelli et al., 2014)
Trade-off Constraints	Increase in non-Gaussianity in one subsystem necessarily reduces it in the other under Gaussian interaction	(Chatterjee et al., 23 Oct 2025)

The broadcasting of non-Gaussian quantum states fundamentally shapes the possibilities and architecture of future quantum networks, quantum information processing, and quantum communication systems. Its study informs both the limitations imposed by physical law and the operational requirements for achieving universal, scalable, and secure quantum technologies.