The two-mode relative phase associated with Gaussian states plays an important role in quantum information processes in optical, atomic, and electronic systems. In this work, the origin and structure of the two-mode relative phase in pure Gaussian states is studied in terms of its dependences on the quadratures of the modes. This is done by constructing local canonical transformations to an associated two-mode squeezed state. The results are illustrated by studying the time dependence of the phase under a nonlocal unitary model evolution containing correlations between the modes. In a more general context, this approach may allow the two-mode phase to be studied in situations sensitive to different physical parameters within experimental configurations relevant to quantum information processing tasks.
We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of
2-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the
2-quantum Wasserstein distances of Caglioti--Golse--Mouhot--Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma--Trevisan~\cite{DPaTre19}.
The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices.
The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.
We introduce a canonical decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into two additive pieces: an even part that captures changes in the symplectic spectrum and an odd part associated with correlation-generating dynamics. On the pure-state manifold, the even contribution vanishes identically, while the odd contribution coincides with the QFI derived from the natural metric on the Siegel upper half-space, revealing a direct geometric underpinning of pure-Gaussian metrology. This also provides a link between the graphical representation of pure Gaussian states and an explicit expression for the QFI in terms of graphical parameters. For evolutions completely generated by passive Gaussian unitaries (orthogonal symplectics), the odd QFI vanishes, while thermometric parameters contribute purely to the even sector with a simple spectral form; we also derive a state-dependent lower bound on the even QFI in terms of the purity-change rate. We extend the construction to the full QFI matrix, obtaining an additive even odd sector decomposition that clarifies when cross-parameter information vanishes. Applications to unitary sensing (beam splitter versus two-mode squeezing) and to Gaussian channels (loss and phase-insensitive amplification), including joint phase loss estimation, demonstrate how the decomposition cleanly separates resources associated with spectrum versus correlations. The framework supplies practical design rules for continuous-variable sensors and provides a geometric lens for benchmarking probes and channels in Gaussian quantum metrology.
We show that given experimental data obtained from joint position and momentum measurements one can construct an optimal pure state approximating the observed quantum state. For that purpose we use a tool from multivariate statistical analysis (the MVE method), which relies on the existence of a minimum volume ellipsoid (the John--Löwner ellipsoid) containing a given convex set. This method allows us to determine the shape of the covariance ellipsoid, which is thereafter calibrated using the expression of the uncertainty principle in terms of the notion of symplectic capacity. We finally use the Wigner formalism to produce the best approximating Gaussian state.