Finite-rank non-Gaussian bosonic states admit a holomorphic description in the Bargmann representation: after a zero-free Gaussian factor is separated off, their non-Gaussian structure is encoded by a finite stellar divisor. This article introduces a topological refinement of stellar rank for regular parameterized families of such states. Rather than only counting the zeros of the stellar divisor, we follow their motion under deformations of the state and record the associated braid monodromy. In the finite-Fock chart, a regular degree-r stellar state is represented by a monic polynomial with r simple zeros. The regular stratum is biholomorphic to the unordered configuration space of r points in the complex plane, and its fundamental group is the Artin braid group on r strands. Thus braid monodromy is an intrinsic invariant of loops in the regular finite-rank stellar state space. We then extend the construction to admissible finite stellar divisors of the form E_tau,mu(z) P(z); the zero-free Gaussian parameters form a contractible fiber over the same configuration-space base. Experimentally motivated finite-Fock families, especially the cubic subspace spanned by the first four Fock states, provide concrete laboratories, while trinomial slices yield explicit discriminants and local half-twists. The resulting invariant is post-tomographic and applies to preparation loops and parameterized families; it complements Wigner negativity, stellar rank, approximate stellar rank, and other scalar diagnostics of non-Gaussianity.
Finite-rank non-Gaussian bosonic states admit a holomorphic description in the Bargmann representation: after a zero-free Gaussian factor is separated off, their non-Gaussian structure is encoded by a finite stellar divisor. This article introduces a topological refinement of stellar rank for regular parameterized families of such states. Rather than only counting the zeros of the stellar divisor, we follow their motion under deformations of the state and record the associated braid monodromy. In the finite-Fock chart, a regular degree-r stellar state is represented by a monic polynomial with r simple zeros. The regular stratum is biholomorphic to the unordered configuration space of r points in the complex plane, and its fundamental group is the Artin braid group on r strands. Thus braid monodromy is an intrinsic invariant of loops in the regular finite-rank stellar state space. We then extend the construction to admissible finite stellar divisors of the form E_tau,mu(z) P(z); the zero-free Gaussian parameters form a contractible fiber over the same configuration-space base. Experimentally motivated finite-Fock families, especially the cubic subspace spanned by the first four Fock states, provide concrete laboratories, while trinomial slices yield explicit discriminants and local half-twists. The resulting invariant is post-tomographic and applies to preparation loops and parameterized families; it complements Wigner negativity, stellar rank, approximate stellar rank, and other scalar diagnostics of non-Gaussianity.
We introduce a general framework for the analysis of non-Gaussian entanglement in bosonic states of finite stellar rank. The central result is the full characterization of their entanglement structure through the atomic decomposition of their stellar polynomial and its associated structural graph, whose connected components determine the mode-intrinsic entanglement content of the state and all partitions compatible with passive separability. An essential ingredient in this construction is the concept of essential variables, which identify the minimal number of effective modes involved in a core state, in direct correspondence with the symplectic rank. This reduction provides the foundation for decomposing stellar polynomials into atomic factors and for revealing the underlying entanglement structure. Building on this, we derive complete separability criteria for two-mode states, expressed through hyperplane decompositions of zero sets, and for stellar-rank-2 states across an arbitrary number of modes. Applications to several example states illustrate how the method isolates genuinely non-Gaussian resources and quantifies preparation complexity.
Under the contraction of SU(1,1) to the Heisenberg-Weyl group, the zeros of the Husimi function of a state redistribute according to a single scale, and we determine which of them survive. The stellar rank of a single-mode bosonic pure state, the number of these zeros, characterises the Gaussian states on the plane through the vanishing of the rank. On the Poincaré disk, where the same construction applies to the discrete series of SU(1,1), it does not, because the zero set leaves a zero-free factor undetermined. We therefore ask how the zero structure behaves in the contraction limit rather than how it should be classified at fixed Bargmann index. The entire difference between the two Hilbert spaces reduces to one weight sequence wk(m)=(2k)m/(2k)m, whose monotonicity in k drives every estimate below. The rescaled states form a normal family, every nonzero limit point lies in the Segal-Bargmann space, and those with finitely many zeros are exactly the finite-rank Fock states. The rank is upper semicontinuous, with equality precisely when all zeros scale as O(1/2k), the scale of the coherent-state labels. A two-sided Harnack estimate pins the profile on compact sets. Separately, a family of rank r degenerates if and only if the amplitudes on the lowest r+1 weight vectors tend to zero. Mass escapes at the edge of the rescaled disk instead, by a first-order transition of the semiclassical Husimi potential at an explicit critical coupling. Finally, the contraction limit retains exactly the lowest 2r+3 amplitudes, and this number cannot be lowered.
The quantum theory of the electromagnetic field enables the description of multiphoton states exhibiting nonclassical statistical properties, often reflected in non-Gaussian phase-space distributions. While non-Gaussianity alone does not fully characterize quantum states, several classifications have been proposed to hierarchize non-Gaussian states according to physically or informationally relevant resources. Here, we provide a physical interpretation of non-Gaussianity and connect it to a computational perspective by showing how a prominent classification-the stellar rank-emerges as a limiting case of the roots of polynomials that univocally represent bosonic states defined with a quantized phase reference, namely the Majorana polynomials. A direct consequence of our results is a revised interpretation of both the stellar rank and non-Gaussianity itself: when superselection rules are properly taken into account, quadrature non-Gaussianity - and nonzero stellar rank - act as witnesses of particle entanglement, rather than being linked with photon addition to Gaussian states as previously assumed. In addition, we show that because the stellar rank depends on a specific choice of coherent states, its relation to computational resources and potential quantum advantage is inherently basis-dependent, being naturally tied to quadrature eigenstates as the computational basis. Motivated by this observation, we generalize the notion of stellar rank to arbitrary computational bases, thereby establishing it as a genuine witness of bosonic resources that may enable quantum advantage.
Bosonic quantum error correction (QEC) relies on non-Gaussian bosonic encodings whose preparation cost is a central practical constraint. In this work, we use stellar rank as a resource measure to design and benchmark bosonic codes under finite non-Gaussian resources. For fixed cat and Gottesman--Kitaev--Preskill (GKP) code families, we show that finite stellar rank creates a trade-off among state approximability, energy, and logical protection under photon loss and photon-number dephasing, evaluated with optimal recovery. This trade-off implies that codewords with better ideal error-correction properties need not be optimal once finite-rank preparation constraints are imposed. Going beyond fixed-target codewords, we directly optimize bosonic encodings at fixed stellar rank, revealing noise-adapted code structures and concrete resource thresholds. Grid-like encodings emerge under photon loss, whereas approximately rotation-symmetric encodings arise under dephasing. In the optimized search, stellar rank k=2 suffices to surpass break-even for all dephasing strengths considered, while under photon loss the required rank increases with the loss rate. These results establish stellar rank as an operationally meaningful resource measure for bosonic QEC under practical state-preparation constraints.
Quantum non-Gaussian states and operations serve as fundamental resources for universal quantum computation, error correction, and high-precision metrology, extending beyond the Gaussian limits. While the stellar rank provides a rigorous hierarchical measure of non-Gaussianity, it remains challenging to determine experimentally. Conversely, witnesses of non-Gaussian features, based on the expectation values and variances of measurable observables, offer an accessible method for certifying non-Gaussian behavior but lack a direct connection to stellar rank. In this work, we establish a quantitative connection between these witnesses and stellar rank, demonstrating that the former can provide certifiable lower bounds on stellar rank. We introduce normalized expectation value and variance-based quantifiers and show that these witnesses form a consistent hierarchy of thresholds corresponding to stellar rank. Our results bridge the gap between abstract hierarchical measures and experimentally accessible quantifiers, enabling scalable certification of non-Gaussian states.
Quantifying non-Gaussianity is essential for assessing the computational power of bosonic quantum systems. We introduce a framework to witness and quantify non-Gaussianity from displaced parity measurements, and we directly apply it to experimental data obtained with superconducting cavities. Our method estimates fidelities with target pure states from the displaced parity measurements, enabling certification of non-Gaussianity via the stellar rank. We further introduce refined witnesses computed by measuring the value of the Wigner function at a few points, providing efficient certification tools outperforming existing ones. Applying these techniques to Fock, binomial, cat, and GKP experimental data, we rigorously certify non-Gaussian resources and establish a stellar rank of 8 for a GKP state, setting a new benchmark across all quantum platforms. Our methods enable a rigorous quantification of non-Gaussian quantum resources directly applicable to superconducting devices.