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.
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.
Continuous-variable quantum-state tomography usually reconstructs a truncated Fock-basis density matrix. We study a compact alternative: finite mixtures of displaced, squeezed Gaussian kets whose ρ = BB† construction is positive semidefinite by construction, fitted by per-sample homodyne likelihood with closed-form gradients, and composed with physical loss and noise channels. On the public propagating-light GKP dataset of Konno et al. (Science 2024), a rank-4 model with 92 real parameters matches the empirical full-rank maximum-likelihood frontier (255 parameters) at confidence-interval resolution on held-out likelihood. On a synthetic thermal-noise target that we prove lies outside the model family — no detection efficiency and no finite rank reproduces it exactly — the channel-composed model fitted blind exceeds a full-rank MLE run under the pre-declared 900-second baseline budget, a verdict that holds across all five pre-declared seed and noise configurations. We do not claim a universally superior method: comparisons on real data reuse observations across splits, the strongest baseline is test-selected, and the blind result covers one target class. The contribution is a compact, physically constrained model family together with a fully falsification-first research record — negative results, superseded scorings, and pre-declared protocols are all preserved in the accompanying repository.
The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows.
(i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states.
(ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states.
(iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states.
(iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the n-mode Fock state ∣1⟩⊗n, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009].
Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.
We introduce the stellar decomposition , a novel method for characterizing non-Gaussian states produced by photon-counting measurements on Gaussian states. Given an ( m + n ) -mode Gaussian state G , we express it as an ( m + n ) -mode "Gaussian core state" G c o r e followed by an m -mode Gaussian transformation T that only acts on the first m modes. The defining property of the Gaussian core state G c o r e is that measuring the last n of its modes in the photon-number basis leaves the first m modes on a finite Fock support, i.e. a core state. Since T is measurement-independent and G c o r e has an exact and finite Fock representation, this decomposition exactly describes all non-Gaussian states obtainable by projecting n modes of G onto the Fock basis. For pure states we prove that a physical pair ( G c o r e , T ) always exists with G c o r e pure and T unitary. For mixed states, we establish necessary and sufficient conditions for ( G c o r e , T ) to be a Gaussian mixed state and a Gaussian channel. We also develop a semidefinite program to extract the "largest" possible Gaussian channel when these conditions fail. Finally, we present a formal stellar decomposition for generic operators, which is useful in simulations where the only requirement is that the two parts contract back to the original operator. The stellar decomposition leads to practical bounds on achievable state quality in photonic circuits and for GKP state generation in particular. Our results are based on a new characterization of Gaussian completely positive maps in the Bargmann picture, which may be of independent interest.
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 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.
Recent algorithms for classical simulation of quantum mechanics have runtime whose superpolynomial component is given by a linear dependence on the number of terms required to write large tensor products of certain "magic" states as superpositions of "free" states (which may be stabilizer states, fermionic Gaussian states or others). Surprisingly little is known about the number of terms in such decompositions, called ranks (e.g. the stabilizer rank, fermionic-Gaussian rank etc.). For complexity theoretic reasons they are expected to grow exponentially in the number of tensor factors but, while exponential upper bounds are known for both the stabilizer and fermionic-Gaussian rank of magic states; the best known lower bounds on the stabiliser rank are quadratic, and no bounds on the fermionic-Gaussian rank are known beyond fixed constants. In this work we prove that any fermionic Gaussian decomposition of ∣M⟩⊗k consists of Ω(1.4k) terms. Here ∣M⟩ is the most standard magic state for fermionic linear optics: the 4 qubit state that may be consumed to implement a swap gate. We also prove essentially matching bounds on the δ-approximate rank of the same state, lower bounding it by the same quantity that bounds the exact rank, multiplied by a factor of 1−δ2. Our results on exact fermionic Gaussian rank apply directly to any product of k fixed parity non-Gaussian states, although the same is not true of the approximate rank. Finally, we prove that the coherent state border rank of an n-mode bosonic Fock state with mj bosons in mode j is exactly ∏j(1+mj), answering a conjecture of Ref. [1] and obtain lower bounds on the approximate coherent state rank given by the same quantity multiplied by a function of the fidelity of the approximation, emphasizing the broad applicability of the method we employ.