Gaussian Steering-Breaking Channels in CV Systems

• Gaussian steering-breaking channels are quantum channels for CV systems that destroy EPR steering by satisfying a specific matrix inequality.
• They are characterized by gain and noise matrices, with thresholds (e.g., η ≤ 1/2 for pure-loss channels) defining when steering is broken.
• These channels play a pivotal role in quantum resource theory and secure protocols, particularly impacting one-sided device-independent QKD.

Gaussian steering-breaking channels are a class of quantum channels acting on continuous-variable (CV) systems which universally destroy the ability to demonstrate Einstein-Podolsky-Rosen (EPR) steering via any input bipartite Gaussian state. Analogous to entanglement-breaking channels, steering-breaking channels mark an operational boundary between those quantum processes that preserve nonclassical correlations and those that do not. Their characterization involves matrix inequalities on covariance matrices and plays a foundational role in quantum information theory, quantum resource theory, and secure quantum communication protocols.

1. Definitions and Structural Properties

A Gaussian steering-breaking channel is defined, for $N$-mode CV systems, via its action on covariance matrices. Any such channel $\Lambda$ is determined by matrices $M\in\mathbb{R}^{2N\times2N}$ (gain) and $N\in\mathbb{R}^{2N\times2N}$ (noise), with its effect on a covariance matrix $V$ being $\Lambda[V]=MVM^T+N$. The canonical symplectic form is $\Omega_N=\bigoplus_{j=1}^N\begin{pmatrix}0&1\-1&0\end{pmatrix}$.

Steering-breaking channels are those for which, for any bipartite Gaussian input state $\rho_{AB}$ (covariance $V_{AB}$), the output state $(\Lambda\otimes\text{id})(\rho_{AB})$ is unsteerable according to the Wiseman–Jones–Doherty criterion:

$$V_{AB}' + (0_A \oplus i\Omega_N) \succeq 0, \quad \text{for all } V_{AB},$$

where $V_{AB}'$ is the transformed covariance matrix. This condition must be verified for all $V_{AB}$, indicating complete destruction of $A \rightarrow B$ steering (Kiukas et al., 2017, Ma et al., 7 Nov 2025, Yan et al., 2 Sep 2024, Heinosaari et al., 2015).

2. Characterization Theorem and Matrix Inequality

A central result is a necessary and sufficient matrix inequality: a Gaussian channel $\Lambda$ with parameters $(M,N)$ is steering-breaking if and only if

$N - iM^T\Omega_N M \succeq 0.$

This Hermitian matrix must be positive semidefinite. The proof leverages the covariance-matrix formalism, the behavior of the Choi–Jamiołkowski correspondence for CV systems, and the extremal properties of infinitely-squeezed two-mode vacuum states (Kiukas et al., 2017, Ma et al., 7 Nov 2025, Heinosaari et al., 2015).

The sufficiency follows by considering the Schur complement structure after channel action, and necessity is established via considering steering criteria on maximally entangled Gaussian inputs. This criterion also underpins the convexity of the set of steering-breaking channels and relates them closely to incompatibility-breaking and entanglement-breaking maps.

3. Relation to Measurement Incompatibility and State-Channel Duality

Fundamental to the theory is the duality between quantum steering and measurement incompatibility. Through a generalized Choi–Jamiołkowski correspondence, steerability of $\rho_{AB}$ by Alice's measurements is equivalent to incompatibility of the measurement images under the channel adjoint $\Lambda^*$.

Specifically, a channel is Gaussian incompatibility-breaking (and hence steering-breaking) if it maps all sets of measurements into jointly Gaussian measurable sets—this is formalized by the same matrix inequality above. Thus, the fine structure of steering-breaking channels can be read as the continuous-variable generalization of resource-destroying maps for both steering and measurement incompatibility (Kiukas et al., 2017, Heinosaari et al., 2015).

4. Examples of Steering-Breaking Channels

Key parametric families include:

Channel Type	Matrix Parameters	Steering-Breaking Threshold
Pure-loss (Attenuator)	$M = \sqrt{\eta}\,I$, $N = (1-\eta)I$	$\eta \le 1/2$
Additive Noise	$M = I$, $N = nI$	$n \ge 1$
Amplifier	$M = \sqrt{G}I$, $N = (G-1)I$	Never steering-breaking at quantum limit

For a pure-loss channel, steering is destroyed for transmission $\eta \le 1/2$. For additive noise, steering is broken when noise $n \ge 1$. Ideal amplifiers do not break steering unless additional noise is introduced. These analytic thresholds have clear operational meaning in quantum communication, such as bounding the transmissivity or tolerable noise for key distribution protocols relying on steering (Kiukas et al., 2017, Ma et al., 7 Nov 2025, Heinosaari et al., 2015).

5. Steering-Breaking Channels and Resource Theory

Within the resource theory of quantum steering, steering-breaking channels constitute the set of free operations: they always map Gaussian unsteerable states to Gaussian unsteerable states. The matrix inequality

$N - iM^T\Omega_N M \geq 0$

or its block-wise generalization serves as the master test to identify free maps. Maximal Gaussian unsteerable channels are the superset of channels that always output unsteerable states, though closed-form tests are not always available for this class (Yan et al., 2 Sep 2024).

These channels are closed under composition and contain local Gaussian unitaries and noise processes. Notably, all entanglement-breaking Gaussian channels are steering-breaking, but the converse does not hold: some steering-breaking channels preserve certain forms of entanglement (Heinosaari et al., 2015).

6. Measurement Class Dependence and Non-Gaussian Criteria

Gaussian steering-breaking channels are defined relative to Gaussian measurements. However, non-Gaussian measurement schemes, such as truncated local orthogonal observables (TLOOs), can detect steering that Gaussian criteria miss. For instance, lossy channels with transmissivity $1/3 \lesssim \eta \le 1/2$ can still permit steering detected by non-Gaussian tests for moderate squeezing values (Ji et al., 2015).

The true "steering-breaking" region, defined as the set of parameters for which no steering is detectable by any measurement (Gaussian or non-Gaussian), may be strictly smaller than that obtained from Gaussian tests alone. Systematic identification of these regions requires evaluating more general LHS models and non-Gaussian steering inequalities. Experimental results confirm the possibility of steering "sudden death" and "revival" via engineering correlated noise channels, highlighting directionality and asymmetry in the phenomenon (Deng et al., 2021).

7. Operational Applications and Physical Significance

Gaussian steering-breaking channels delimit the boundaries of practical quantum protocols that depend on steering as a resource. In one-sided device-independent quantum key distribution (QKD) and quantum teleportation, security and operational fidelity rely on the preservation of EPR steering; steering-breaking channels on the trusted party's subsystem render such tasks infeasible (Kiukas et al., 2017, Ma et al., 7 Nov 2025). Their matrix-inequality characterization yields direct criteria for channel engineering in experimental CV quantum communication.

Additionally, steering-breaking maps provide a unified language for the analysis of quantum resource destroyers and the interplay between entanglement, steering, and measurement incompatibility in both theoretical and applied CV quantum information science.