Maximal Gaussian Unsteerable Superchannels

• The paper demonstrates that maximal Gaussian unsteerable superchannels are those that factor through two maximal unsteerable Gaussian channels, ensuring no steering resource is generated.
• It establishes necessary and sufficient matrix inequalities that precisely characterize when a superchannel preserves the maximal unsteerable set in continuous-variable systems.
• The framework offers structural insights and tight monotonicity bounds, clarifying the role of free operations in continuous-variable quantum steering resource theories.

A maximal Gaussian unsteerable superchannel is a higher-order map in the resource theory of Gaussian continuous-variable (CV) quantum channels, defined by its property of preserving the set of maximal unsteerable Gaussian channels under composition. These objects provide a key structural tool for quantifying and bounding quantum steering in CV settings, with applications to resource theory, quantum information protocols, and channel manipulation. The framework relies on precise matrix inequalities and factorization results that sharply characterize which superchannels are “free” with respect to Gaussian channel steering.

1. Structure of Gaussian Superchannels in the Continuous-Variable Setting

Consider two parties $A$ and $B$ possessing $N$-mode bosonic quantum systems with phase-space Hilbert spaces $H_A, H_B$. A superchannel $\Phi$ is a map that sends any CPTP (completely positive trace-preserving) map $\phi$ on $H_A\otimes H_B$ to another such map. A Gaussian superchannel specifically preserves the Gaussianity of Gaussian channels.

The structure theorem for Gaussian superchannels states:

$\Phi(\phi) = \chi_2 \circ \phi \circ \chi_1$

Here, the pre-processing $\chi_1$ and post-processing $\chi_2$ are Gaussian channels defined by

$$\begin{aligned} \chi_1(\rho) &= \phi(\Sigma E^T \Sigma, 0, 0)\left(\rho\right) \ \chi_2(\rho) &= \phi(A, Y, \nu)\left(\rho\right) \end{aligned}$$

where $A, E, Y$ are real matrices in $M_{2(m+n)}(\mathbb{R})$, $Y = Y^T$, $\nu \in \mathbb{R}^{2(m+n)}$, and

$$\Sigma = \bigoplus_{k=1}^{m+n} \mathrm{diag}(1, -1)$$

$$\Omega = \bigoplus_{k=1}^{m+n}\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}$$

The complete positivity constraints are: $\begin{aligned} Y + i\Omega - iA\Omega A^T &\ge 0 \tag{1a} \ i\Omega - iE\Omega E^T &\ge 0 \tag{1b} \end{aligned}$

For any Gaussian channel $\phi(K, M, d)$, its image under $\Phi$ is given by

$$K' = A K \Sigma E^T \Sigma, \quad M' = A M A^T + Y, \quad d' = A d + \nu$$

This decomposition enables the classification of superchannels via their constituent Gaussian channels.

2. Definitions: Unsteerable and Maximal Unsteerable Superchannels

Let $\mathcal{GC}_{US}$ denote the set of Gaussian unsteerable channels, defined by the matrix inequality: $$M + (0_A \oplus i\Omega_B) - K (0_A \oplus i\Omega_B) K^T \ge 0 \tag{2}$$ A Gaussian channel lies in $\mathcal{GC}_{US}$ if it never induces steerability from $A$ to $B$ on any input Gaussian state.

Maximal unsteerable Gaussian channels, $\mathcal{GC}_{MUS}$, are defined as those that preserve the entire set of unsteerable Gaussian states: $$\phi \in \mathcal{GC}_{MUS} \iff \phi(\mathcal{GS}_{US}) \subseteq \mathcal{GS}_{US}$$

At the superchannel level, the following definitions are made:

• $\mathcal{SGC}_{US}$: Gaussian superchannels that map $\mathcal{GC}_{US}$ into itself.
• $\mathcal{SGC}_{MUS}$: Maximal Gaussian unsteerable superchannels, mapping $\mathcal{GC}_{MUS}$ into itself.

3. Characterization: Necessary and Sufficient Conditions

For a Gaussian superchannel $\Phi = \Phi(A, E, Y, \nu)$:

Unsteerable Superchannels ($\mathcal{SGC}_{US}$)

$$\Phi \in \mathcal{SGC}_{US} \iff \begin{cases} Y + (0 \oplus i\Omega) - A (0 \oplus i\Omega) A^T \ge 0 \ (0 \oplus i\Omega) - E (0 \oplus i\Omega) E^T \ge 0 \end{cases} \tag{3}$$

That is, both $\chi_2(A, Y, \nu)$ and $\chi_1(\Sigma E^T \Sigma, 0, 0)$ must be Gaussian unsteerable channels individually.

Maximal Unsteerable Superchannels ($\mathcal{SGC}_{MUS}$)

$$\Phi \in \mathcal{SGC}_{MUS} \iff \chi_1, \chi_2 \in \mathcal{GC}_{MUS}$$

This requires that both the pre- and post-processing channels are maximal Gaussian unsteerable, i.e., each satisfies the maximal-unsteerability matrix inequalities (see Theorem 3 of [Yan et al., PRA 110, 052427 (2024)]).

A maximal unsteerable Gaussian channel $\chi_i(K_i, M_i, 0)$ must satisfy: $$\forall w \in \mathbb{C}^{2(m+n)}:\quad w^* M_i w + |w^* K_i (0 \oplus \Omega) K_i^T w| \ge |w^* (0 \oplus \Omega) w|$$ This ensures maximal preservation of unsteerability under all compositions with unsteerable channels.

4. Resource-Theoretic Role and Implications

Within the resource theory of Gaussian channel steering:

• Resources are Gaussian channels capable of generating steerability from $A$ to $B$.
• Free channels ($\mathcal{GC}_{US}$) never generate steering.
• Free superchannels ($\mathcal{SGC}_{US}$) are those mapping free channels to free channels.
• A resource monotone $\mathcal{R}(\phi)$ satisfies $\mathcal{R}(\phi) \ge 0$, $\mathcal{R}(\phi) = 0$ if and only if $\phi \in \mathcal{GC}_{US}$, and: $$\mathcal{R}(\Phi(\phi)) \le \mathcal{R}(\phi) \qquad \forall\,\Phi \in \mathcal{SGC}_{US}$$

Maximal unsteerable superchannels ($\mathcal{SGC}_{MUS}$) form a subclass of free operations that do not enable an increase in the “resourceful” hull of channels. This property provides tight upper bounds for steering production under arbitrary Gaussian pre- and post-processing. A plausible implication is that these superchannels serve as the most general resource-non-generating transformations within the hull of maximal unsteerable Gaussian channels.

5. Examples and Explicit Constructions

Two principal classes of constructions follow directly from the characterization theorems:

• Composition Construction: If $\chi_1, \chi_2 \in \mathcal{GC}_{MUS}$, then any superchannel of the form $\Phi(\phi) = \chi_2 \circ \phi \circ \chi_1$ lies in $\mathcal{SGC}_{MUS}$. This requires checking the maximal-unsteerability inequalities for each composed channel.
• Explicit (2+2)-Mode Example: The paper presents a superchannel $\Phi(A, E, Y, 0)$ with $A, Y \in M_4(\mathbb{R})$, $E = I_4$, where both $(A, Y)$ and $(\Sigma E^T \Sigma, 0)$ fulfill the maximal-unsteerability inequalities, but the simpler unsteerability inequalities for $\mathcal{SGC}_{US}$ are violated. This explicitly realizes a superchannel in $\mathcal{SGC}_{MUS} \setminus \mathcal{SGC}_{US}$.

6. Structural Consequences and Closure Properties

Several structural relations hold:

• Every steering-annihilating superchannel is maximal unsteerable.
• There exist superchannels in $\mathcal{SGC}_{MUS}$ that are not in $\mathcal{SGC}_{US}$, establishing that maximal unsteerability is a strictly weaker requirement than unsteerability preservation.
• Both $\mathcal{SGC}_{US}$ and $\mathcal{SGC}_{MUS}$ are closed under concatenation, maintaining their respective properties under composition.
• The inclusion relations are strict: $$\mathcal{SGC}_{US} \subsetneq \mathcal{SGC}_{MUS} \subseteq \mathcal{SGC}$$ and for channels

$$\mathcal{GC}_{US} \subsetneq \mathcal{GC}_{MUS} \subseteq \mathcal{GC}$$

This hierarchy reflects the varying degrees of “freeness” with respect to steering preservation in the space of Gaussian superchannels.

7. Summary and Significance

Maximal Gaussian unsteerable superchannels are precisely the Gaussian superchannels that factor through two maximal Gaussian unsteerable channels, thereby preserving the maximal-unsteerable hull under arbitrary pre- and post-processing. Their explicit matrix characterizations offer necessary and sufficient conditions for identifying “free” superchannels in resource-theoretic analyses of Gaussian steering. These results enable the determination of tight monotonicity bounds for any quantitative measure of steering applied to Gaussian channels, providing clear structural underpinnings for resource-theoretic investigations in continuous-variable quantum information.