Maximum Entropy with Quartic Moments in Continuous-Variable Quantum Systems: Existence, Nonexistence, and Dual Geometry

We study the quantum exponential family of n bosonic modes generated by the phase-space polynomials of degree at most four, and the maximization of the von Neumann entropy under prescribed moments up to order four, the quantum analogue of a classical problem of Junk. We prove that the family is an analytic dually flat manifold with the Bogoliubov–Kubo–Mori metric and that strong duality holds for moment data in the interior of the realizable region. For such data, and under a density hypothesis also in the relative interior of faces, we characterize nonexistence completely: the maximizer fails to exist exactly when the data are the moments of a family member plus an excess fourth-order tensor supported on the null directions of its quartic part. For one mode, the kernel of a nonzero positive semidefinite polynomial operator of degree at most four has dimension at most two, so that maximizers also exist on the boundary. For two or more modes the admissible set is strictly larger than the classical one (companion paper); faces of infinite dimension occur, and nonexistence of Junk type appears inside faces. We compute the curvature of two-parameter photon-number families at their stratum b = 0, which equals −(1 + 3v₂)/2 for quadratic variance functions and −2 for thermal states. Near the Gaussian boundary the log-partition function stays bounded along regular approaches (in a one-dimensional classical model and for one mode) and diverges in the classical model when the squared cubic coefficient dominates the quartic one. Files: paper1_jmp.pdf (manuscript, 22 pages); verify_paper1.py (Python 3 script that reproduces the numerical values reported in the paper; requires numpy, scipy and sympy; about 20 seconds; comments are in Japanese). The following article has been submitted to the Journal of Mathematical Physics. After it is published, it will be found at https://pubs.aip.org/aip/jmp.