Approximate Coherent State Rank

• Approximate Coherent State Rank is a fidelity-smoothed measure indicating the minimal number of coherent states needed to accurately approximate a bosonic quantum state.
• Constructive decompositions offer explicit simulation algorithms for both Fock and squeezed states, while Hankel matrix methods provide rigorous lower bounds.
• This concept connects continuous-variable resource theory with classical simulation complexity, impacting Boson Sampling and optical computation strategies.

Searching arXiv for papers on approximate coherent state rank and closely related notions. arXiv search: "approximate coherent state rank" Approximate coherent state rank is a fidelity-smoothed notion of coherent-state complexity for bosonic quantum states. For a target state, it asks how many coherent states must be superposed to approximate that state to a prescribed accuracy; in its limiting form, it asks for the minimal size of a coherent-state superposition that can approximate the state arbitrarily well. Recent work places it at the intersection of continuous-variable resource theory, classical simulation of quantum optics, and algebraic complexity: coherent-state decompositions yield constructive simulation algorithms, while lower-bound techniques based on Hankel matrices and the permanent show that small decompositions are impossible for important state families such as Boson Sampling inputs (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

1. Formal definitions and variants

For a single bosonic mode, the coherent state of complex amplitude $\alpha\in\mathbb C$ is

$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$

For $m$ modes, an $m$-mode coherent state is a tensor product

$$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$

A coherent-state superposition (CSS) of rank $k$ is a state of the form

$\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$

with distinct coherent states $\ket{\bm\alpha_j}$ and coefficients $c_j\in\mathbb C$. The exact coherent-state rank of a pure state is the smallest $k$ for which such an equality holds; if no finite such $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$0 exists, the rank is $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$1. In the older continuous-variable literature this is also the “degree of non-classicality” (Cottier et al., 1 Apr 2026).

Two closely related approximate notions appear in the literature. A fidelity-dependent quantity, used explicitly by Marshall and Anand, is the $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$2-approximate coherent state rank: $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$3 A limiting quantity is the approximate coherent state rank

$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$4

which is either a positive integer or $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$5 (Cottier et al., 1 Apr 2026). A conceptually equivalent formulation was already used in the coherent-state decomposition framework of 2023: the approximate coherent rank of $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$6 is the smallest integer $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$7 such that for any $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$8, there exists a coherent rank-$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$9 state $m$0 with $m$1 (Marshall et al., 2023).

This distinction matters. A family of states may admit finite-rank approximants at every fixed target fidelity while still having $m$2, because the required rank can grow as $m$3 (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

2. Constructive decompositions and upper bounds

The main constructive framework starts from finite Fock support. If

$m$4

is a single-mode state with support on at most $m$5 photons, then its approximate coherent rank is at most $m$6. An explicit approximation is built from $m$7 coherent states placed on a small circle in phase space,

$m$8

with coefficients chosen by a discrete Fourier transform so that the amplitudes $m$9 agree exactly with the target for $m$0, while leakage to higher Fock levels is suppressed as $m$1 (Marshall et al., 2023).

For a pure Fock state $m$2, this yields an explicit coherent expansion with exactly $m$3 coherent terms. The case $m$4 reduces to an odd cat state,

$m$5

so the approximate coherent rank of $m$6 is exactly $m$7 in this construction (Marshall et al., 2023).

The same logic extends tensorially to multimode Fock states. For $m$8, one obtains

$m$9

Two special cases are central. For the standard Boson Sampling input $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$0, each occupied mode contributes rank $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$1, so

$$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$2

For the bunched configuration $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$3, one gets only

$$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$4

These upper bounds directly determine simulation costs under linear optics (Marshall et al., 2023).

The same paper treats squeezing by truncating the Fock expansion and then applying the finite-support construction. Thus any squeezed vacuum can be approximated to arbitrary fixed accuracy with finite rank, but the required rank grows with squeezing strength and with the demanded fidelity. Numerically, for squeezing parameter $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$5 and mean photon number $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$6, a superposition of $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$7 coherent states yields fidelity $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$8, while for lower target fidelity around $$\ket{\bm\alpha}=\ket{\alpha_1}\otimes\cdots\otimes\ket{\alpha_m},\qquad \bm\alpha\in\mathbb C^m.$$9, as few as $k$0 coherent states suffice (Marshall et al., 2023).

An operator version of the same program shows how rank grows under non-Gaussian transformations. A single creation operator $k$1 has operator coherent rank $k$2; more generally, a degree-$k$3 polynomial in $k$4 increases rank by at most a factor $k$5, and a truncated squeezing expansion of order $k$6 increases rank by at most $k$7 (Marshall et al., 2023).

3. Lower bounds and exact characterization

A systematic lower-bound theory was introduced in 2026 through Hankel matrices built from Fock amplitudes. For a single-mode state

$k$8

define rescaled coefficients $k$9 and the $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$0 Hankel matrix

$\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$1

If $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$2 is a CSS of rank $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$3, then $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$4 admits a Vandermonde factorization and therefore

$\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$5

for all $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$6. Combining this with the Young–Eckart–Mirsky theorem and an inequality relating Frobenius distance of Hankel matrices to Hilbert-space distance yields a generic lower bound: if

$\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$7

then

$\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$8

A rescaled version with a free parameter $\ket\phi=\sum_{j=1}^k c_j\ket{\bm\alpha_j},$9 strengthens the bound numerically by replacing $\ket{\bm\alpha_j}$0 with a weighted Hankel matrix $\ket{\bm\alpha_j}$1 (Cottier et al., 1 Apr 2026).

The same framework leads to a complete single-mode characterization. Let $\ket{\bm\alpha_j}$2. Then $\ket{\bm\alpha_j}$3 if and only if the sequence $\ket{\bm\alpha_j}$4 satisfies a linear recurrence relation of order $\ket{\bm\alpha_j}$5. Equivalently, $\ket{\bm\alpha_j}$6 can be written as a finite superposition of displaced core states,

$\ket{\bm\alpha_j}$7

where the $\ket{\bm\alpha_j}$8 are pairwise distinct, each $\ket{\bm\alpha_j}$9 is a finite Fock superposition with highest Fock number $c_j\in\mathbb C$0, and

$c_j\in\mathbb C$1

This theorem sharply separates finite from infinite approximate coherent state rank (Cottier et al., 1 Apr 2026).

Several consequences are immediate. If $c_j\in\mathbb C$2 is a single-mode core state with highest photon number $c_j\in\mathbb C$3, then

$c_j\in\mathbb C$4

For the pure Fock state $c_j\in\mathbb C$5, one moreover has the fine-grained stability statement

$c_j\in\mathbb C$6

By contrast, every nontrivial squeezed state $c_j\in\mathbb C$7 has

$c_j\in\mathbb C$8

The proof uses the fact that the rescaled Fock coefficients of a squeezed vacuum do not satisfy any finite-order linear recurrence with constant coefficients unless $c_j\in\mathbb C$9 (Cottier et al., 1 Apr 2026).

4. Multimode structure, simulation complexity, and Boson Sampling

The multimode theory has both constructive and hardness components. For finite multimode Fock superpositions,

$k$0

if the total photon number is at most $k$1 and the $k$2-photon component is nonzero, then

$k$3

The proof compresses photons into a single mode by a passive linear unitary and then postselects the remaining modes onto vacuum, reducing the problem to a single-mode core state (Cottier et al., 1 Apr 2026).

The state $k$4 is qualitatively harder. A linear bound $k$5 follows from the multimode core-state theorem, but a much stronger result is known: $k$6 This super-polynomial lower bound is obtained by showing that an $k$7-approximate CSS for $k$8 induces a family of multilinear arithmetic formulas approximating the permanent uniformly on unitary matrices, and then extending Raz’s lower bound for exact multilinear formulas to the border, or approximate, setting (Cottier et al., 1 Apr 2026).

These lower bounds are directly relevant to simulation complexity. In the constructive framework, an $k$9-mode rank-$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$00 state

$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$01

can be stored with $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$02 complex numbers; a full $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$03-mode linear-optical unitary can be applied in $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$04 time; and a fixed Fock-basis amplitude can be computed in $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$05 operations. For standard Boson Sampling, the decomposition $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$06 gives time complexity $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$07 and space complexity $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$08. For the bunched input $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$09, the complexity drops to $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$10 time and $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$11 space (Marshall et al., 2023).

Taken together, these results yield a precise picture. Coherent-state decompositions provide an explicit simulation algorithm whose cost is linear in the decomposition rank, but for $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$12 that rank cannot remain polynomial in $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$13. The consequence is an unconditional barrier to efficient classical simulation of Boson Sampling via coherent-state decompositions (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

5. Relation to neighboring rank notions

Several nearby notions use similar language but refer to different mathematical objects.

Notion	Defining object	Operational role
Approximate coherent state rank	Minimal CSS size achieving target fidelity	Classical simulation cost of bosonic computations
Approximate stellar rank	Minimal stellar rank within target fidelity	Bounds approximate Gaussian state conversion
Approximate stabilizer rank	Minimal stabilizer decomposition size within target norm error	Classical simulation cost of Clifford-based computations
Approximate coherence rank	Nearest rank class of an optical coherence matrix in eigenvalue space	Scattering-immune optical communications

Approximate stellar rank is defined for continuous-variable states by

$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$14

where the exact stellar rank is based on the factorization of the stellar function into a polynomial times a Gaussian entire function. It is a faithful monotone under Gaussian protocols and is used to derive no-go theorems for approximate Gaussian state conversion (Hahn et al., 2024). This is not a coherent-state decomposition rank, but it is closely related: the 2023 coherent-state decomposition framework explicitly connects coherent-state rank bounds to stellar rank, with non-classicality decomposing into contributions from single-photon additions and squeezing (Marshall et al., 2023).

Approximate stabilizer rank plays an analogous role for qubit magic states. It is defined by minimizing stabilizer rank within a prescribed Hilbert-space error ball, and it controls the classical simulation cost of Clifford circuits. The stabilizer-rank literature provides examples where exact rank is exponential but approximate rank is constant, illustrating how sharply exact and approximate decompositional complexity can diverge (Lovitz et al., 2021).

A separate source of confusion is the optical-communications notion of approximate coherence rank. In that setting the object is not a bosonic state vector but a classical optical coherence matrix $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$15; “coherence rank” is the number of non-zero eigenvalues of $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$16, and the practical approximation rule assigns a measured state to the nearest ideal eigenvalue vertex in Euclidean distance (Harling et al., 25 Jul 2025). Despite the similar terminology, this is a different invariant.

6. Conceptual status and open problems

The present theory is defined by a tension between constructive upper bounds and rigorous lower bounds. Constructively, many important states admit useful finite-rank approximants at any fixed target accuracy. Rigorously, only a special class of single-mode states has finite limiting rank $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$17: precisely those whose rescaled Fock amplitudes satisfy a finite-order linear recurrence, or equivalently those that are finite superpositions of displaced core states (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

This distinction resolves an apparent ambiguity around squeezed states. The decomposition framework of 2023 shows that squeezed vacua can be approximated to arbitrary fixed accuracy by finite CSS rank, with the required rank depending on the fidelity target. The lower-bound theory of 2026 proves that nontrivial squeezed states nevertheless satisfy

$$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$18

The two statements concern different quantities: finite $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$19 for each fixed $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$20 does not imply finite $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$21 (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

Several technical questions remain open. The 2023 framework leaves open whether the $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$22 upper bound for the Fock state $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$23 is optimal for all $$\ket{\alpha} = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\ket n.$$24, how to compute minimal coherent-state rank for arbitrary states, and how to fully characterize the trade-off between coherent-state rank, squeezing, and stellar rank. The 2026 lower-bound work describes itself as initiating a systematic study, and it identifies the extension of Hankel-matrix methods to Gaussian rank as a nontrivial problem (Marshall et al., 2023, Cottier et al., 1 Apr 2026).

The resulting picture is already clear in broad outline. Approximate coherent state rank is simultaneously a non-classicality measure, a one-shot approximation parameter, and a simulation-complexity invariant. Its constructive side is governed by explicit coherent-state decompositions of Fock-supported and mildly non-Gaussian states; its obstructive side is governed by spectral properties of Hankel matrices and by deep lower bounds for the permanent. In continuous-variable quantum information, it is therefore both a representation-theoretic notion and a complexity-theoretic one (Marshall et al., 2023, Cottier et al., 1 Apr 2026).