Rotation-Invariant Symmetries of Gaussian States

• Rotation-invariant Gaussian states are continuous-variable quantum states that remain unchanged under symplectic rotations preserving canonical commutation relations.
• They are characterized by symmetry groups and covariance matrix structures that enable efficient protocols for state discrimination, entanglement testing, and optimal measurement design.
• The study of these symmetries provides actionable insights for purification strategies, topological analysis, and the development of robust quantum information systems.

Rotation-invariant symmetries of Gaussian states refer to the invariance of these quantum states—specifically in continuous-variable (CV) systems—under rotations within phase space or, more generally, under orthogonal or symplectic transformations that preserve canonical commutation relations. These symmetries are foundational in characterizing the structure of Gaussian states, defining their symmetry groups, classifying their transformations, and providing operational tools in quantum information theory. They also underlie efficient state discrimination, optimal measurement design, entanglement theory, and tomography. This topic integrates deep results from representation theory, convex geometry, symplectic analysis, functional analysis, and experimental methodology.

1. Symmetry Groups of Gaussian States

In the Hilbert space $L^2(\mathbb{R}^n)$, the set $S_n$ of all Gaussian states is exactly invariant under a group $\mathcal{G}_n$ of unitary operators defined as those $U$ for which $U \rho U^* \in S_n$ for any Gaussian state $\rho$ (Parthasarathy, 2011). This group can be fully characterized as

$U = \lambda\, W(\boldsymbol{\alpha})\, \Gamma(L)$

where:

• $\lambda$ is a complex phase ($|\lambda| = 1$),
• $W(\boldsymbol{\alpha})$ is a Weyl displacement (phase space translation) operator,
• $L \in Sp(2n, \mathbb{R})$ is a real symplectic matrix,
• $\Gamma(L)$ is the (Bogoliubov) automorphism implementing $L$ at the operator level.

The symplectic group $Sp(2n, \mathbb{R})$ is defined as follows: $$Sp(2n, \mathbb{R}) = \{ L \in GL(2n, \mathbb{R}) : L^T J L = J \}$$ with

$$J = \begin{pmatrix} 0 & -I \ I & 0 \end{pmatrix}$$

where $I$ is the $n \times n$ identity. The transformations encoded in $Sp(2n, \mathbb{R})$ encompass all passive (rotational) and active (squeezing) canonical transforms that preserve commutation relations.

Every symmetry operation—rotation, squeezing, and translation—corresponds to an element of this group, with rotations as a prominent subset. Under such symplectic transformations, covariance matrices transform as $S \mapsto L^T S L$.

2. Covariance Matrices, Convex Geometry, and Purification

The convex set $K_n$ of admissible $2n \times 2n$ real symmetric covariance matrices $S$ for Gaussian states is constrained by the uncertainty relation: $2S - iJ \geq 0$ $K_n$ is closed and convex, with its extreme points exactly those matrices $S = \frac{1}{2} L^{T} L$, $L \in Sp(2n, \mathbb{R})$ (Parthasarathy, 2011). Pure Gaussian states correspond to these extreme covariance matrices, whereas any mixed Gaussian state has a covariance matrix expressible as

$S = \frac{1}{2}(L^T L + M^T M)$

for some $L, M \in Sp(2n, \mathbb{R})$. This structure immediately links covariance geometry with symplectic/rotational invariance.

A crucial structural property is that any Gaussian state, mixed or pure, can be purified into a (higher-dimensional) pure Gaussian state via extension to $L^2(\mathbb{R}^{2n})$ (i.e., $S_n$ to $S_{2n}$), in a way compatible with symplectic symmetry (Parthasarathy, 2011).

3. Unitary Implementations: Role of Weyl and Bogoliubov Operators

General rotation-invariant symmetry operations on Gaussian states factor into Weyl operators (implementing phase-space translations) and unitaries implementing Bogoliubov automorphisms (transformations of the canonical operators generated by linear symplectic maps).

The Weyl operators $W(a)$ act as

$$W(a) W(b) = e^{-i \mathrm{Im} \langle a, b \rangle} W(a + b)$$

preserving the spread/covariance structure of the state but shifting mean values. Bogoliubov transformations correspond to $I(L)$, with $L \in Sp(2n, \mathbb{R})$, transforming

$I(L) W(a) I(L)^* = W(La)$

rotating and squeezing the state in phase space. Both pieces together guarantee that all unitary symmetry operations preserving the Gaussian family arise from this construction, and thus all rotation invariance is encapsulated.

4. Rotation-Invariant Symmetries in State and Measurement Design

Rotation-invariant symmetries are central to designing measurement and communication protocols:

• Geometrically Uniform Symmetry (GUS): A set of states $\{|\gamma_j\rangle\}$ where each is generated by applying a unitary symmetry (typically rotation) to a reference, i.e., $|\gamma_j\rangle = Q^j |\gamma_0\rangle$ with $Q^K = I$ for some $K$ (Cariolaro et al., 2014). For $N$-mode Gaussian states, rotations leave the squeezed-displaced state class invariant, i.e.,

$$R(\varphi) D(\alpha) S(z) |0\rangle = D(e^{i\varphi}\alpha) S(e^{i\varphi} z e^{i\varphi^T}) |0\rangle$$

• Optimal Quantum Measurements: For geometrically uniform constellations, the Gram matrix of inner products is circulant, allowing diagonalization via discrete Fourier transform, which simplifies square-root measurement (SRM) constructions and yields closed-form optimal detection rules for quantum communications (e.g., in pulse position modulation).

5. Invariant Measures and Typicality

Several natural invariant measures exist on the manifold of Gaussian states—most notably, those inherited from the Haar (unitary) measure on the Gaussian symmetry group. For multimode states, the invariant measure can be decomposed over local and nonlocal degrees of freedom using the symplectic eigenvalues of the covariance matrix, with the nonlocal (entanglement-relevant) part given by

$$\prod_{j<k} (v_j^2 - v_k^2)^2 \prod_j (v_j^2 - 1)^{n_B - n_A} dv_j$$

where $v_j$ are symplectic eigenvalues (Lupo et al., 2012). When an energy constraint is imposed (e.g. fixing $\mathrm{tr}(S)$), the measure becomes finite and permits computation of statistical properties ("typical entanglement," etc.).

The Hilbert–Schmidt and Fisher–Rao measures (Link et al., 2015, Sohr et al., 2018), as well as measures corresponding to reductions of pure states, coincide on level sets of fixed purity, further enforcing the unitarily/symplectically invariant nature of typicality for rotation-invariant settings.

6. Characterization and Testing of Rotation-Invariant Gaussian States

A recent operational criterion based on representation theory provides a characterization: a pure (zero-mean) state is Gaussian if and only if its $k$-fold tensor power is invariant under all orthogonal transformations (rotations) among the copies for $k \geq 2$ (Girardi et al., 8 Oct 2025). This rotation-invariant symmetry underpins efficient protocols for certifying Gaussianity: for pure states, constant-copy rotation tests suffice (requiring only 2 or 3 copies), relying on projective measurement onto the rotation-invariant subspace.

For mixed states, no efficient test exists—the sample complexity scales exponentially with system size, indicating that the rotation-invariant property is tightly linked to the structure of pure Gaussian states but cannot, in practice, be used to efficiently test mixed-state Gaussianity in large systems.

7. Transformation, Disentanglement, and Topological Implications

Symplectic rotations (elements of $Sp(2n, \mathbb{R}) \cap O(2n)$) can be leveraged to disentangle any bipartite Gaussian state via conjugation by the corresponding metaplectic operator (Gosson, 2020). This action rotates the covariance matrix into a form compatible with the Werner–Wolf criterion for separability. The result generalizes the principle that entanglement of Gaussian states is, in principle, always removable by a passive, rotation-invariant operation, provided one can implement the appropriate symplectic transformation.

In more abstract settings, e.g., over general locally compact abelian groups, rotation-invariant symmetries manifest as invariance under the action of isotropic subgroups and are tied to the Fourier-analytic or group-theoretic properties of the Weyl system and associated 2-cocycles (Beny et al., 2022). Topological obstructions to generating entanglement in non-Euclidean settings are described through these symmetries.

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In summary, rotation-invariant symmetries of Gaussian states are realized as the invariance under symplectic transformations (especially in $Sp(2n, \mathbb{R})$), whose physical and operational significance ranges from defining the full automorphism group of the Gaussian sector to underpinning efficient state certification protocols, measurement design, entanglement characterizations, and topological properties of quantum states over abstract phase spaces. Their mathematical and physical import is central to the architecture of continuous-variable quantum theory and its applications in quantum information science.