| Fenchel--Rockafellar duality | statement | theorem C.3 | arxiv |
| 1W8E0W6GEG | statement | lemma A.3 | arxiv |
| Semiclassical limit of Gaussian $2$-QW barycenters | statement | theorem 2.7 | arxiv |
| 23KJZXG2W8 | statement | proposition 4.5 | arxiv |
| Gaussian channel induced by a vacuum ancilla | statement | lemma D.5 | arxiv |
| Compactness under additional marginal energy constraints | statement | corollary 4.3 | arxiv |
| 2NZF61SKQQ | statement | proposition D.1 | arxiv |
| Conjugate of the constrained additive-potential functional | statement | proposition C.12 | arxiv |
| Quadratic Schr\"odinger operators with multiplicity | statement | lemma B.1 | arxiv |
| Conjugate of the cost-constraint indicator | statement | proposition C.11 | arxiv |
| Reduction to the isotropic one-mode case | statement | proposition E.1 | arxiv |
| Duality with barycentric energy constraints | statement | theorem 2.2 | arxiv |
| Existence of QOT barycenters | statement | theorem 2.1 | arxiv |
| Confinement for the costs with compact resolvent | statement | proposition C.7 | arxiv |
| 2-QW distances between displaced thermal states | statement | proposition 5.28 | arxiv |
| Gaussian channel | definition | definition 5.18 | arxiv |
| Truncation and monotone convergence for positive observables | statement | proposition 3.3 | arxiv |
| Gaussian projection decreases the $2$-QW cost | statement | proposition 5.9 | arxiv |
| Gaussian reduction for $2$-QW barycenters | statement | theorem 2.5 | arxiv |
| Two-sided bounds | statement | proposition 4.4 | arxiv |
| Barycenter of pure inputs | statement | lemma 5.26 | arxiv |
| Dual formulation of the covariance matrix optimization problem | statement | proposition 5.14 | arxiv |
| Existence of a convergent subsequence in \(\Gamma(\rho, \sigma)\ | statement | theorem A.5 | arxiv |
| Covariance uniqueness for arbitrary barycenters and optimal couplings with faithful inputs | statement | corollary 5.23 | arxiv |
| \rev{Compact} resolvent | definition | definition 3.1 | arxiv |
| Partial trace | definition | definition 3.4 | arxiv |
| Unitary equivalence of the $\tau$-CCR and standard CCR representations | statement | proposition 3.13 | arxiv |
| Complementary slackness for the covariance dual | statement | corollary 5.15 | arxiv |
| Dichotomy for quadrature-generated costs | statement | proposition C.8 | arxiv |
| Injectivity and Gaussianity under covariance slackness | statement | proposition 5.21 | arxiv |
| Gaussian $2$-QW covariance reduction | statement | proposition 5.12 | arxiv |
| Second-marginal injectivity and Gaussianity under covariance slackness | statement | corollary 5.22 | arxiv |
| Legendre--Fenchel conjugate | definition | definition C.2 | arxiv |
| BQ89YEEAAN | statement | corollary 4.2 | arxiv |
| CFAQ0WP04X | statement | corollary 5.29 | arxiv |
| Uniform Gaussian-channel representation under covariance slackness | statement | proposition 5.20 | arxiv |
| Unitary--orthosymplectic identification | statement | proposition D.2 | arxiv |
| Williamson's theorem | statement | theorem 5.5 | arxiv |
| Covariance optimality for arbitrary barycenters | statement | corollary 5.13 | arxiv |
| Closedness of marginal energy sublevel sets | statement | proposition A.6 | arxiv |
| States with finite canonical second moments | definition | definition 5.1 | arxiv |
| Multimode extension | statement | corollary E.2 | arxiv |
| Purely discrete spectrum from compact resolvent | statement | proposition A.1 | arxiv |
| State reconstruction via a Gaussian channel with full-column-rank \rev{\mbox{phase-space}} transformation matrix | statement | lemma 5.19 | arxiv |
| Convexity and properness of the constrained additive functional | statement | proposition C.9 | arxiv |
| Global uniqueness with a faithful Gaussian input | statement | theorem 2.6 | arxiv |
| Uniqueness for displaced thermal inputs | statement | proposition 5.27 | arxiv |
| Identification of the Fenchel--Rockafellar right-hand side | statement | proposition C.13 | arxiv |
| Gaussian projection | definition | definition 5.4 | arxiv |
| Identification of the Fenchel--Rockafellar left-hand side | statement | proposition C.10 | arxiv |
| Quadratic Hamiltonians generated by a regular Weyl system | statement | proposition 3.8 | arxiv |
| Standard-CCR reduction of a faithful-input covariance-dual optimizer | statement | proposition 5.24 | arxiv |
| Nonnegative operators with compact resolvent | statement | corollary A.2 | arxiv |
| Finite energy of the barycenter for quadrature-generated costs | statement | proposition 2.4 | arxiv |
| Uniqueness of the barycenter covariance and faithful-input cross-covariances | statement | proposition 5.16 | arxiv |
| Nonuniqueness and non-Gaussianity for partially pure inputs | example | example 5.31 | arxiv |
| Wigner function of a Gaussian state | definition | definition 5.25 | arxiv |
| Unconstrained barycenter duality | statement | corollary 2.3 | arxiv |
| NRV6A1613R | statement | lemma C.4 | arxiv |
| P6NKGKZ9VW | statement | lemma C.6 | arxiv |
| Compactness of constrained energy sublevels | statement | proposition 4.1 | arxiv |
| Lower semicontinuity | statement | proposition A.4 | arxiv |
| Boundary family of barycenters | statement | lemma G.1 | arxiv |
| Compatibility with unbounded observables | statement | proposition 3.6 | arxiv |
| Gaussian optimal couplings | statement | theorem 5.10 | arxiv |
| Spectral integral against a \rev{positive trace-class operator | definition | definition 3.2 | arxiv |
| Tensorization of barycenters | statement | lemma 5.30 | arxiv |
| Unique $2$-QW barycenters for isotropic thermal states | statement | corollary 2.8 | arxiv |
| Symplectic normal form under covariance slackness | statement | lemma D.6 | arxiv |
| Gaussian states | definition | definition 5.2 | arxiv |
| VG06CP27HQ | statement | lemma C.5 | arxiv |
| Quantum couplings | definition | definition 3.7 | arxiv |
| VYXKCGQ35S | statement | proposition 4.6 | arxiv |
| Robertson--Schr\"odinger uncertainty relation | statement | theorem 5.3 | arxiv |
| Gaussian unitary | definition | definition 5.6 | arxiv |
| Reduction of moments and Gaussianity | statement | proposition 5.7 | arxiv |
| Effective domain | definition | definition C.1 | arxiv |
| Compact resolvent of the square-sum | statement | corollary 3.12 | arxiv |
| Vacuum-mode representation from a Williamson decomposition | statement | lemma D.4 | arxiv |
| Trace-norm continuity of partial traces | statement | lemma 3.5 | arxiv |
| Uniqueness among Gaussian barycenters | statement | corollary 5.17 | arxiv |
| Stone--von Neumann theorem with multiplicity | statement | theorem 3.10 | arxiv |
| Quadratic costs depend only on moments | statement | proposition 5.8 | arxiv |
| Gaussian projection of barycenters and optimal couplings | statement | proposition 5.11 | arxiv |
