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arXiv:2608.21434

Maximal-Rank Squeezing from Rank-One Quenches: An Exact Spectral-Reachability Theorem for Gaussian Bosonic Systems

Kamran Ansari

quant-phSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

We prove an exact spectral-reachability theorem for multimode squeezing generated by finite-rank quadratic quenches in Gaussian bosonic systems. For a stable sign-definite stiffness quench $Ω^2 \to Ω^2+\varepsilon U U^T$, the kernel, range, and rank of the anomalous Bogoliubov block are determined exactly by the block-Krylov subspace generated by $Ω$ and $U$. In particular, $\operatorname{rank}(β_\varepsilon)=\sum_a \operatorname{rank}(P_a U)$, where $P_a$ projects onto each distinct free-frequency eigenspace. Consequently, the microscopic rank of a perturbation does not bound the number of squeezing channels it can activate: for a simple spectrum, a single rank-one element with nonzero overlap with every mode produces a full-rank anomalous Bogoliubov response at every nonzero stable coupling. A signed Lyapunov-Gramian path shows that spectrally reachable directions cannot disappear through finite-coupling cancellation. The theorem also gives exact dark sectors, degeneracy bottlenecks, and actuator-position rules without computing the full Bogoliubov transformation. These results provide a direct hardware-screening principle for multimode Gaussian systems: one or a few physical elements can provide simultaneous support across many canonical squeezing channels, although algebraic rank alone does not imply strong squeezing or independent tunability.

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12 pages, 1 figure, v1

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Passive transformations of an nnn-mode squeezed probe become locally unidentifiable when the quantum Fisher matrix loses rank. For pure zero-mean Gaussian probes, we show that the CnC_{n}Cn​ restricted-root system of Sp(2n,R)/U(n)\mathrm{Sp}(2n,\mathbb{R})/\mathrm{U}(n)Sp(2n,R)/U(n) governs this exact sloppiness. For squeezing magnitudes r1,…,rnr_{1},\ldots,r_{n}r1​,…,rn​ in the canonical basis, the roots 2rj2r_{j}2rj​ and rj±rkr_{j}\pm r_{k}rj​±rk​ label the local phase rotations and two beam-splitter quadratures, and their vanishing identifies every Fisher-null direction. The associated Fisher weights quantify the approach to each singular wall. At fixed positive mean photon number, we solve the E-optimal probe-design problem of maximizing the smallest passive Fisher eigenvalue in two explicit generator normalizations. In the canonical phase and beam-splitter angle convention, the unique optimum in the fundamental Weyl chamber is an arithmetic progression approaching the consecutive-odd-integer dual-Weyl direction at large resource. With an invariant generator norm, the smallest eigenvalue is independent of the passive frame and the optimum lies exactly on that ray at every resource. Uniform squeezing instead minimizes the local-phase A-optimal cost while leaving one beam-splitter quadrature unidentifiable for every mode pair. Finally, the two quadratures of each identifiable mode pair saturate the quantum-geometric incompatibility bound, and the largest compatible passive submodel has n(n+1)/2n(n+1)/2n(n+1)/2 parameters at a regular spectrum. The resulting classification separates local identifiability, sensitivity optimization, and simultaneous attainability.
We study how many Gaussian components a continuous-variable quantum state needs, adopting the approximate Gaussian rank framework of Hahn–Takagi–Ferrini–Yamasaki and the fidelity-ball relaxation of the stellar-rank literature; no novelty is claimed for the definitions. For mixed states we formulate the measure-theoretic ess-sup rank roof over generalized ensembles and prove its attainment and Gaussian-channel monotonicity — a partial answer, on the rank side, to the discrete-versus-continuous decomposition question the cited work leaves open. Our main contributions are three. First, for single-mode dictionaries of bounded common-squeezing Gaussian atoms, we prove zero-counting lower bounds: robust zeros of a target's Bargmann function survive fidelity-ball perturbation and are counted against an exponential-sum zero budget. The dictionary restriction is provably forced — counterexample families show no bound on the unrestricted rank can follow from raw disk zero counts alone. Second, an inequivalence theorem: the thermal lossy cat of our earlier preprint has Gaussian rank at most 2 yet no finite-rank pre-loss operator representation at any efficiency, separating two natural compressibility notions in one direction. Third, measured curves: family-constrained approximation residuals and a real-data rank proxy on public GKP homodyne data, and a robust-zero census for finite-energy GKP states in which all three computed configurations support the lower-bound premises for the untruncated comb. The census is numerically supported, not computer-certified (float64 without outward rounding; numerical root-finding); a machine-checked claim register binds every quoted number to committed artifacts. Multimode extension and a general-squeezing plateau conjecture remain open.
The power and limitations of classical simulation are central to understanding quantum computational advantages. A leading simulation paradigm is based on coherent decomposition into classically tractable free states where the decomposition rank determines the simulation complexity. Proving strong lower bounds on this number is a notoriously difficult and mathematically rich problem, as exemplified by the qubit stabilizer rank problem. Here, we study the Gaussian version of this problem in both bosonic and fermionic systems and establish robust exponential lower bounds on Gaussian rank. In particular, we prove that for every pure non-Gaussian state on finitely many modes, with definite parity in the fermionic case, the approximate border Gaussian rank of its tensor powers grows at least exponentially at any fixed norm error below one. Our proofs combine reduction to four modes with Majorana spectral bounds for fermions, and Gaussian postselection with an entropy-based rank bound for bosons. As concrete examples, we derive explicit exponential lower bounds for the four-mode fermionic GHZ state and the bosonic single-photon state. The bosonic results require no assumption on the energy of the target state. Our results show that non-Gaussianity universally entails exponential Gaussian decomposition complexity, setting fundamental limitations on decomposition-based classical simulation of bosonic and fermionic systems.