Symmetry and Asymmetry in Bosonic Gaussian Systems: A Resource-Theoretic Framework

TL;DR

This work develops a comprehensive resource theory of Gaussian asymmetry by pairing symmetry representations with bosonic Gaussian dynamics. It proves that every Gaussian covariant channel can be realized via symmetry-preserving Gaussian unitaries coupled to a symmetry-respecting Gaussian environment (Gaussian dilation), and it introduces tractable monotones that quantify Gaussian asymmetry via displacement and covariance data. A key technical advance is the extension of Williamson's theorem to simultaneous normal-mode decompositions with a conserved quadratic charge, enabling precise conservation laws for closed Gaussian dynamics and distinguishing Gaussian from non-Gaussian behavior. The framework yields concrete examples (e.g., SU(2) covariant amplifiers/attenuators) and provides a complete characterization of reachable first moments under Gaussian covariant dynamics. Together, these results offer a robust operational toolkit for analyzing symmetry constraints in CV quantum information tasks and open avenues for phase-space symmetry witnesses and generalized non-Gaussian resource theories.

Abstract

We study the interplay of symmetries and Gaussianity in bosonic systems, under closed and open dynamics, and develop a resource theory of Gaussian asymmetry. Specifically, we focus on Gaussian symmetry-respecting (covariant) operations, which serve as the free operations in this framework. We prove that any such operation can be realized via Gaussian Hamiltonians that respect the symmetry under consideration, coupled to an environment prepared in a symmetry-respecting pure Gaussian state. We further identify a family of tractable monotone functions of states that remain non-increasing under Gaussian symmetry-respecting dynamics, and are exactly conserved in closed systems. We demonstrate that these monotones are not generally respected under non-Gaussian symmetry-respecting dynamics. Along the way, we provide several technical results of independent interest to the quantum information and optics communities, including a new approach to the Stinespring dilation theorem, and an extension of Williamson's theorem for the simultaneous normal mode decomposition of Gaussian systems and conserved charges.

Figures (6)

• Figure 1: Gaussian systems and symmetry representations. Each system under consideration is a collection of bosonic modes carrying a representation of a symmetry group $G$, which, in general, can differ between subsystems. In this example, systems $A$, $B$, and $C$ have 3, 3, and 2 bosonic modes, respectively, and carry an arbitrary representation of group $G$ with dimension equal to the number of modes of the system. Since all Gaussian Hamiltonians are quadratic, interactions in this framework can only couple pairs of systems, i.e. they are 2--local. The symmetry further restricts these 2--local interactions (see Appendix \ref{['app:examples']} for further discussion and examples including the group $\operatorname{SU}(2)$ and the permutation group $\operatorname{S}_3$).
• Figure 2: Squeezed passive representations. We sketch the action of squeezed passive representations of U(1), $\theta \mapsto S_r(\theta) \coloneqq V_{\rm 1sq}(r)\exp(-\theta \Omega)V_{\rm 1sq}(-r)$ for different squeezing parameters $r$, on the displaced vacuum state $\hat{D}_{(3,0)^{\rm T}}\ket{0}$. These transformations move phase space points anti-clockwise on an ellipsoid, with the passive representation ($r = 0$), corresponding to a circle. Each element $S_r(\theta_0)$ for fixed $\theta_0$ commutes with representation $S_{r'}$ if and only if $r'=r$, whereas its transpose $S_r(\theta_0)^{\rm T} = S_{-r}(-\theta_0)$ commutes with $S_{r'}$ if and only if $r'=-r$. We discuss when a $(1,1)$--tensor and its transpose simultaneously commute with the same representation in Appendix \ref{['app:passive']}. The complex conjugate representations $\theta \mapsto S_r(-\theta)$ move points clockwise instead. We also depict the state's covariance matrix under the action of the representation with $r = 0.1$. The covariance matrix picks up squeezing along its orbit, but returns to the same state after a $\pi$ rotation, i.e. it respects the $\mathbb{Z}_2$ symmetry. We generalize this observation for any representation of $\operatorname{U}(1)$ in Appendix \ref{['app:u1breaking']}, where we show that a Gaussian operator which breaks the $\operatorname{U}(1)$ symmetry always breaks the symmetry of a finite $\operatorname{U}(1)$ subgroup of sufficiently high order. Instead, the state $V_{\rm 1sq}(r)\ket{0}$ is an invariant state of representation $S_r$ for all $r$.
• Figure 3: Relative entropy of asymmetry for 1--mode states. We consider 1--mode $\operatorname{U}(1)$ symmetry and calculate the Holevo asymmetry $\Gamma$ of coherent states $\ket{\alpha}$ and squeezed vacuum states $\ket{r}$ as a function of mean photon number $\langle n \rangle$. The maximum mean photon number $\langle n \rangle = 50$ in the plot corresponds to $|\alpha| = \sqrt{50} \approx 7.07$ and $r = \sinh^{-1}\left(\sqrt{50}\right) \approx 2.65$. The dashed lines represent the asymptotic behavior of the monotone as $r, |\alpha| \rightarrow \infty$. For $\langle n \rangle \lesssim 8.54$, coherent states have a larger relative entropy of asymmetry, whereas squeezed states exhibit the dominant asymptotic behavior.
• Figure 4: Type--1 asymmetry. Suppose a $G$--covariant Gaussian channel $\mathcal{E}$ sends an input state $\hat{\rho}_1$ to an output state $\hat{\rho}_2$. Then, there exists a $G$--covariant Gaussian channel $\mathcal{E}'$ that maps $\hat{\rho}'_1 = \mathcal{D}^{(1)}_{\boldsymbol{d}}(\hat{\rho}_1)$ to $\hat{\rho}'_2 = \mathcal{D}^{(1)}_{\boldsymbol{d}}(\hat{\rho}_2)$.
• Figure 5: Circuit for purification. We consider an invariant Gaussian state $\hat{\rho}_A$ with covariance matrix $\sigma = VDV^{\rm T}$, for symplectic $V$ and diagonal $D \ge I$. Registers $A$ and $\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu$ both comprise $n$ modes. The circuit operates in three steps: (i) preparation of the vacuum state on both registers, (ii) coupling of the two registers via $n$ 2--mode squeezers, each depending on a symplectic eigenvalue in $D$ and depicted by an arched line inside the box, and (iii) symplectic rotation on each of the registers $A$ and $\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu$ ($\hat{V}$ and $\hat{V}'$ are associated with symplectic matrices $V$ and $V\Omega$, respectively). Overall this circuit realizes the purification $\ket{\psi}$ with covariance matrix $\sigma_{\psi} = (V \oplus V\Omega) V_{\rm 2sq}(\tfrac{1}{2}\cosh^{-1}D)V_{\rm 2sq}(\tfrac{1}{2}\cosh^{-1}D)^{\rm T} (V \oplus V\Omega)^{\rm T}$, as given in Eq.(\ref{['eq:purif_cov']}). Lemma \ref{['lem:inv_gauss_purification']} proves that this is an invariant Gaussian purification of $\hat{\rho}$ and guarantees that it is unique up to an invariant Gaussian unitary $\hat{U}$ applied on the ancillary register $\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu$ according to Eq.(\ref{['eq:psi12U']}). We note that steps (i), (ii) and (iii) are generally not individually invariant, but Proposition \ref{['prop:pure_interconversion']} gives a $G$--invariant Gaussian unitary that prepares $\sigma_{\psi}$ from a $G$--invariant state that is uncorrelated across $A$ and $\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu$ which, however, requires the implementation of the globally correlated symplectic transformation $\sqrt{\sigma_{\psi}}$.

Theorems & Definitions (63)

• Proposition 0
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• Definition 3
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