Pseudo-Moment Certificate: Theory & Applications

• Pseudo-moment certificates are finite-degree obstructions built from truncated or pseudo-moments that distinguish representability from non-representability and certify properties like positivity and non-SOS behavior.
• They employ semidefinite programming, dual linear functionals, and PSD Hankel matrices to rigorously separate genuine moment data from pseudo-moment constructs in classical and quantum contexts.
• Applications span entanglement certification using partial-transpose moments, channel separability via Choi-state moments, and detection of temporal quantum correlations with pseudo-density matrices.

Searching arXiv for relevant papers on pseudo-moment certificates and related moment/SOS frameworks. A pseudo-moment certificate is a certificate built from truncated moments, pseudo-moments, or dual linear functionals that establishes either representability or non-representability of finite moment data, or certifies positivity, non-SOS behavior, entanglement, channel separability, or temporal quantum correlations without reconstructing the full underlying measure or spectrum. In the truncated moment problem, it appears as a polynomial strictly positive on a semialgebraic set $K$ with $L_y(p)=0$; in SOS duality, it appears as a functional in the dual cone $\Sigma_{n,2d}^*$ that is nonnegative on all squares but negative on a target form; in quantum-information settings, it appears as a finite-moment witness based on partial-transpose moments, Choi-state moments, or pseudo-density-matrix moments (Henrion et al., 2023, Henrion, 1 Sep 2025, Yu et al., 2021, Liu et al., 2024).

1. Truncated-moment origin

The truncated moment problem asks whether a finite-dimensional vector $y=(y_\alpha)_{|\alpha|\le d}$ is obtained by integrating monomials against a non-negative measure supported on a set $K$. For a basic semialgebraic set

$$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$

the order-$d$ moment cone is

$$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$

while the cone of nonnegative polynomials is

$$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$

The associated moment functional is

$$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$

For compact $L_y(p)=0$0, the truncated moment cone is the dual cone of $L_y(p)=0$1: $L_y(p)=0$2 This duality yields a sharp alternative: either $L_y(p)=0$3, or there exists a polynomial $L_y(p)=0$4 with $L_y(p)=0$5. Such a polynomial is an algebraic certificate of unrepresentability, because strict positivity of $L_y(p)=0$6 on $L_y(p)=0$7 is incompatible with a nonzero representing measure satisfying $L_y(p)=0$8 (Henrion et al., 2023).

In this setting, a pseudo-moment certificate is therefore a finite-degree obstruction showing that a truncated vector behaves like moment data only at a relaxed level. The point is not full reconstruction of a measure, but separation from the true moment cone by a polynomial witness. A closely related formulation appears in the mapping of quantum separability to truncated $L_y(p)=0$9-moment problems: coefficients of a multipartite density matrix can be written as

$\Sigma_{n,2d}^*$0

and separability is equivalent to the existence of a positive measure supported on a compact semialgebraic parameter space of local states. Moment matrices and localizing matrices then provide the finite-dimensional SDP constraints governing this representability question (Bohnet-Waldraff et al., 2017).

2. Pseudo-moments, SOS duality, and non-SOS certificates

In the Moment-SOS hierarchy, pseudo-moments are truncated linear functionals that satisfy positivity constraints on squares and localizing constraints, but need not come from an actual measure. For a polynomial optimization problem over

$\Sigma_{n,2d}^*$1

the dual cone

$\Sigma_{n,2d}^*$2

consists of truncated moment sequences whose moment and localizing matrices are PSD. These are precisely the pseudo-moments used by SOS relaxations. The Christoffel-function formulation makes this duality explicit: every degree-$\Sigma_{n,2d}^*$3 SOS polynomial in $\Sigma_{n,2d}^*$4 is the reciprocal of the Christoffel function of some linear functional $\Sigma_{n,2d}^*$5, and more generally interior elements of $\Sigma_{n,2d}^*$6 admit Christoffel representations built from pseudo-moments (Lasserre, 2023).

A more algebraic pseudo-moment formalism is developed through moment polynomials. Let

$\Sigma_{n,2d}^*$7

with formal integration map $\Sigma_{n,2d}^*$8. A pseudo-moment evaluation is a homomorphism $\Sigma_{n,2d}^*$9 such that

$y=(y_\alpha)_{|\alpha|\le d}$0

Equivalently, the symbolic Hankel matrices induced by $y=(y_\alpha)_{|\alpha|\le d}$1 are PSD. The paper proves a positive solution to Hilbert’s 17th problem for pseudo-moments: a moment polynomial is nonnegative under all pseudo-moment evaluations if and only if it is a quotient of elements of the preordering generated by sums of squares and formal moments of squares. This sharply separates pseudo-moment positivity from positivity on actual measures, which is weaker and in general only admits perturbative SOS certificates (Klep et al., 2023).

For homogeneous forms, the dual cone to the SOS cone is

$y=(y_\alpha)_{|\alpha|\le d}$2

A pseudo-moment certificate that a form $y=(y_\alpha)_{|\alpha|\le d}$3 is not SOS is a linear functional $y=(y_\alpha)_{|\alpha|\le d}$4 such that $y=(y_\alpha)_{|\alpha|\le d}$5. In matrix form, $y=(y_\alpha)_{|\alpha|\le d}$6 is encoded by a moment matrix $y=(y_\alpha)_{|\alpha|\le d}$7. Exact arithmetic constructions show that positive non-SOS forms with rational or integer coefficients admit rational or integer pseudo-moment certificates, and symmetry reduction yields explicit certificates for the Motzkin sextic, Robinson sextic, Reznick octic, and Choi–Lam quartic. In the ternary sextic case, extreme rays of $y=(y_\alpha)_{|\alpha|\le d}$8 have rank $y=(y_\alpha)_{|\alpha|\le d}$9; in the quaternary quartic case, extreme rays of $K$0 have rank $K$1 (Henrion, 1 Sep 2025).

A common misconception is that PSD moment matrices already imply representability by a measure. The literature is explicit that this is false in general: pseudo-moments satisfy the quadratic positivity constraints used by SOS, but may still fail to correspond to any genuine measure. The certificate is therefore dual and separation-based rather than reconstructive unless additional flatness or realizability conditions are available (Klep et al., 2023, Henrion, 1 Sep 2025).

3. Entanglement certification from finite moments of the partial transpose

For a bipartite state $K$2, the partial transpose with respect to subsystem $K$3 is

$K$4

and the PPT criterion states

$K$5

The moments of the partially transposed state are

$K$6

The central finite-data question is the PT-moment problem: given $K$7, is there a separable state consistent with these moments? In this context, a pseudo-moment certificate is a finite list of inequalities or matrix-positivity conditions involving $K$8 such that violation proves that no PPT state is compatible with the measured moments (Yu et al., 2021).

The first construction uses Hankel matrices. For a sequence $K$9, define

$$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$0

Specializing to PT moments yields the necessary PPT condition

$$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$1

For $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$2, this reduces to

$$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$3

which is exactly the $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$4-PPT criterion. Higher-order Hankel conditions are strictly stronger than earlier two-moment scalar inequalities because they exploit the full matrix structure of the finite moment sequence (Yu et al., 2021).

The second construction is optimal. Writing the spectrum of $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$5 as $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$6, one has

$$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$7

The PT-moment problem becomes the feasibility problem $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$8, where $$K=S(g):=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_k(x)\ge 0\},$$9 is the set of moment tuples generated by nonnegative spectra. This leads to polynomial optimization over spectra and bounds

$d$0

For $d$1, the resulting $d$2-OPPT criterion is necessary and sufficient for compatibility with a PPT spectrum and is strictly stronger than $d$3. More generally, the OPPT conditions are logically optimal for the available finite moment data (Yu et al., 2021).

A complementary route maps separability itself to a truncated $d$4-moment problem. The coefficients of a multipartite density matrix become truncated moments, moment and localizing matrices impose the semialgebraic constraints defining local state spaces, SDP infeasibility is a certificate of entanglement, and a flat extension yields a finitely atomic representing measure and hence an explicit separable decomposition. The algorithm always gives a certificate of entanglement if the state is entangled; if the state is separable, typically a certificate of separability is obtained in a finite number of steps and an explicit decomposition into separable pure states can be extracted (Bohnet-Waldraff et al., 2017).

These two lines address different finite-data regimes. PT-moment certificates are “finite, experimentally accessible shadows” of the PPT test; truncated-moment SDPs are global separability certificates. Neither escapes the basic PPT limitation: PT-moment criteria detect only NPT entanglement, so PPT entangled states remain undetected (Yu et al., 2021).

4. Channel separability and Choi-state moment certificates

For a quantum channel $d$5, the Choi matrix is

$d$6

The separability of $d$7 across different system–ancilla cuts defines three channel classes treated by the truncated-moment method: SEP channels correspond to separability across $d$8, entanglement-breaking channels correspond to separability across $d$9, and fully separable channels require separability across all bipartite cuts of $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$0. In each case, the coordinates of the Choi state in a fixed basis are mapped to a truncated moment sequence $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$1, turning channel separability into a $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$2-truncated moment problem (Milazzo et al., 2020).

The moment matrix and localizing matrices have the standard form

$$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$3

where the polynomials $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$4 encode positivity of the local factor operators. A feasible extension of the given truncated data to higher order with

$$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$5

together with a flatness condition

$$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$6

implies the existence of an $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$7-atomic representing measure supported on the appropriate semialgebraic set $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$8. That measure yields a separable decomposition of the Choi matrix, so the SDP solution is a separability certificate in the constructive sense (Milazzo et al., 2020).

This framework is exact when flatness is detected, but practically hierarchical. A feasible SDP without flatness is a pseudo-moment object consistent with separability constraints up to the chosen order, while a flat extension upgrades that pseudo-moment certificate to an actual representing measure. The computational complexity and the performance depend on the number of variables $$M(K)_d:=\Big\{y:\exists\,\mu\ge 0,\ \mathrm{supp}(\mu)\subset K,\ y_\alpha=\int_K x^\alpha\,d\mu\Big\},$$9 in the truncated moment sequence and on the size of the moment matrix $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$0. The method was applied to families of 2-qubit and single-qutrit channels, and in the qutrit case it provided an answer for examples previously explored through the negativity $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$1, a criterion that remains inconclusive for Choi matrices with $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$2 (Milazzo et al., 2020).

This use of pseudo-moments is therefore both algebraic and operational. It does not merely test PPT or spectral negativity; it reformulates channel separability as the existence of a representing measure over local operator parameters.

5. Temporal pseudo-moments and pseudo-density matrices

In the temporal setting, the pseudo-density matrix $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$3 extends the density matrix formalism to correlations between different times. It is Hermitian, has unit trace, and may have negative eigenvalues. Those negative eigenvalues are the key indicators of quantum temporal correlations. The paper on temporal certification focuses on the first and second moments of $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$4, especially the second-order moment

$$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$5

A central observation is: $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$6 Hence the inequality $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$7 is a pseudo-moment certificate of temporal quantum correlations. The maximal possible value for an $$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$8-qubit system with pure input and unitary channel is

$$P(K)_d:=\{f\in\mathbb{R}[x]_{\le d}: f(x)\ge 0,\ \forall x\in K\}.$$9

and $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$0 is a lower bound on $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$1 (Liu et al., 2024).

The protocol avoids full pseudo-density-matrix tomography by virtually preparing $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$2 within a single time slice. With a control qubit, a maximally mixed ancilla, a controlled-SWAP gate, the physical channel $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$3, and a final measurement on the control qubit, one obtains

$$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$4

Randomized measurements are then applied to this virtual PDM, and an unbiased estimator $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$5 for $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$6 is constructed from a unitary 2-design; variance bounds use a 4-design. The sample-complexity analysis gives

$$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$7

where $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$8 is the number of distinct random unitaries and $$L_y\Big(\sum_\alpha p_\alpha x^\alpha\Big):=\sum_\alpha p_\alpha y_\alpha.$$9 the number of measurements per unitary. In ensemble-average platforms such as NMR, the effective runtime scales with $L_y(p)=0$00 and is independent of system size in measurement-basis complexity (Liu et al., 2024).

The criterion is sufficient but not necessary. The paper explicitly shows a case where $L_y(p)=0$01 has negative eigenvalues but $L_y(p)=0$02, so the second moment misses existing negativity. A plausible implication is that higher-order temporal moments could play the same strengthening role that higher partial-transpose moments play in entanglement detection, but that extension is left as future work (Liu et al., 2024).

6. Cone geometry, extraction algorithms, and structural certificates

The pseudo-moment cone also has a precise convex-geometric realization. For homogeneous degree $L_y(p)=0$03, the dual SOS cone $L_y(p)=0$04 is the cone of linear functionals nonnegative on all squares. In matrix form, its realization is the cone of PSD generalized multivariate Hankel matrices. For fixed $L_y(p)=0$05, if a moment matrix is formed by $L_y(p)=0$06 generically chosen weighted atoms, then its minimal face in the matrix realization of the pseudo-moment cone is simplicial and generated by the planted rank-one atoms. This simplicial regularizability implies that, in that regime, a pseudo-moment certificate is locally indistinguishable from an actual atomic moment decomposition (Kang et al., 7 May 2026).

Based on this geometry, a Carathéodory-type extreme-ray decomposition algorithm for spectrahedral cones produces

$L_y(p)=0$07

where each $L_y(p)=0$08 is an extreme ray. Specialized to the pseudo-moment cone, it yields an efficient atomic decomposition method for generically generated moment matrices in the same $L_y(p)=0$09 regime and exactly recovers the planted atoms and weights. Outside the guaranteed regime, the stabilized numerical implementation demonstrates strong recovery performance and suggests that the algorithm may serve as a practical sampler of high-rank extreme rays. This distinction matters conceptually: rank-one rays correspond to evaluation atoms and genuine measures, whereas higher-rank extreme rays correspond to purely pseudo-moment certificates (Kang et al., 7 May 2026).

A different structural certificate appears in semidefinite relaxations for positive-dimensional real varieties. There, truncated moment matrices are used to compute a Pommaret basis of an ideal $L_y(p)=0$10 nested between the input ideal $L_y(p)=0$11 and its real radical. The stopping certificate is a corank condition,

$L_y(p)=0$12

where $L_y(p)=0$13 counts degree-$L_y(p)=0$14 basis elements of Pommaret class $L_y(p)=0$15. When this holds, the reduced basis of $L_y(p)=0$16 is a weak Pommaret basis of $L_y(p)=0$17. This is a moment-based certificate of involutive completeness rather than a measure-extraction result, and it is explicitly designed for the positive-dimensional case, where flatness is generally unavailable (Ma et al., 2012).

These developments delimit two broad regimes. In one regime, pseudo-moment certificates are exactable: flat extension, simplicial faces, or atomic decompositions turn them into genuine measures or explicit decompositions. In the other, they remain intrinsically dual objects—separating functionals, PSD Hankel matrices, or finite-moment witnesses that certify impossibility, non-SOSness, non-representability, or nonclassicality without ever becoming honest moments. That tension is not a defect but the organizing principle of the subject.