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Positivity and entanglement of polynomial Gaussian integral operators

Richárd Balka, András Csordás, Gábor Homa

TL;DR

The paper addresses positivity and entanglement for polynomial Gaussian density operators with kernels $\kappa_{PG}(x,y)=P(x,y)\kappa_G(x,y)$. The authors prove that positivity of the Gaussian part $\widehat{\kappa}_G$ is necessary for $\widehat{\kappa}_{PG}$ to be positive, derive that odd-degree polynomials cannot yield positive semidefinite operators, and introduce a preorder $\preceq$ on Gaussian kernels that propagates positivity to a larger family of operators. They further connect these results to entanglement via the Peres–Horodecki criterion, showing that if the Gaussian part is entangled (NPT), then the polynomial-Gaussian extension is entangled as well, at least in key bipartite configurations. The approach relies on symplectic (Williamson) decomposition and the Wigner–Weyl framework to reduce multidimensional problems to tractable one-dimensional analyses, offering practical tests and guidance for positivity checks and entanglement detection under quantum operations. Overall, the work provides new tools for certificate-based positivity analysis and yields practical consequences for open quantum systems and non-Gaussian state entanglement.

Abstract

Positivity preservation is an important issue in the dynamics of open quantum systems: positivity violations always mark the border of validity of the model. We investigate the positivity of self-adjoint polynomial Gaussian integral operators $\widehatκ_{PG}$, that is, the multivariable kernel $κ_{PG}$ is a product of a polynomial $P$ and a Gaussian kernel $κ_G$. These operators frequently appear in open quantum systems. We show that $\widehatκ_{PG}$ can be only positive if the Gaussian part is positive, which yields a strong and quite easy test for positivity. This has an important corollary for the bipartite entanglement of the density operators $\widehatκ_{PG}$: if the Gaussian density operator $\widehatκ_G$ fails the Peres-Horodecki criterion, then the corresponding polynomial Gaussian density operators $\widehatκ_{PG}$ also fail the criterion for all $P$, hence they are all entangled. We prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite. We introduce a new preorder $\preceq$ on Gaussian kernels such that if $κ_{G_0}\preceq κ_{G_1}$ then $\widehatκ_{PG_0}\geq 0$ implies $\widehatκ_{PG_1}\geq 0$ for all polynomials $P$. Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of another polynomial Gaussian operators having the same polynomial factor, which might improve any given positivity test by carrying it out on a much larger set of operators. We will show an example that this really can make positivity tests much more sensitive and efficient. This preorder has implication for the entanglement problem, too.

Positivity and entanglement of polynomial Gaussian integral operators

TL;DR

The paper addresses positivity and entanglement for polynomial Gaussian density operators with kernels . The authors prove that positivity of the Gaussian part is necessary for to be positive, derive that odd-degree polynomials cannot yield positive semidefinite operators, and introduce a preorder on Gaussian kernels that propagates positivity to a larger family of operators. They further connect these results to entanglement via the Peres–Horodecki criterion, showing that if the Gaussian part is entangled (NPT), then the polynomial-Gaussian extension is entangled as well, at least in key bipartite configurations. The approach relies on symplectic (Williamson) decomposition and the Wigner–Weyl framework to reduce multidimensional problems to tractable one-dimensional analyses, offering practical tests and guidance for positivity checks and entanglement detection under quantum operations. Overall, the work provides new tools for certificate-based positivity analysis and yields practical consequences for open quantum systems and non-Gaussian state entanglement.

Abstract

Positivity preservation is an important issue in the dynamics of open quantum systems: positivity violations always mark the border of validity of the model. We investigate the positivity of self-adjoint polynomial Gaussian integral operators , that is, the multivariable kernel is a product of a polynomial and a Gaussian kernel . These operators frequently appear in open quantum systems. We show that can be only positive if the Gaussian part is positive, which yields a strong and quite easy test for positivity. This has an important corollary for the bipartite entanglement of the density operators : if the Gaussian density operator fails the Peres-Horodecki criterion, then the corresponding polynomial Gaussian density operators also fail the criterion for all , hence they are all entangled. We prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite. We introduce a new preorder on Gaussian kernels such that if then implies for all polynomials . Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of another polynomial Gaussian operators having the same polynomial factor, which might improve any given positivity test by carrying it out on a much larger set of operators. We will show an example that this really can make positivity tests much more sensitive and efficient. This preorder has implication for the entanglement problem, too.
Paper Structure (21 sections, 18 theorems, 117 equations, 2 figures, 1 table)

This paper contains 21 sections, 18 theorems, 117 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. History
  3. Motivation and physical applications
  4. Positivity violations of the density operator in the Caldeira--Leggett master equation
  5. Entanglement between two oscillators in the presence of heat bath
  6. Positivity checks and entanglement detection after applying quantum operations
  7. Theoretical background
  8. Structure of the paper
  9. Statement of results
  10. When the Gaussian is not positive
  11. When the polynomial is of odd degree
  12. A useful preorder on Gaussian kernels
  13. Application of our preorder: an example
  14. Entanglement
  15. When the Gaussian is not positive
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $\kappa\in L^2(\mathbb{R}^{2n})$ is a continuous kernel. Then $\widehat{\kappa}$ is positive semidefinite if and only if for all $x_1,\dots, x_k \in \mathbb{R}^n$ and $c_1,\dots, c_k\in \mathbb{C}$ we have

Figures (2)

  • Figure 1: Plot of $4\sigma_{RS}(\tau)/\hbar^2$ as a function of a dimensionless time parameter $\tau=\omega t$ on a semi-logarithmic scale coming from the Robertson--Schrödinger uncertainty relation \ref{['SRI']}. The parameters and the initial states are the same as in Figure 1(b) of Erratum2019, and also $\hbar$ is reinstated. The three different curves are plotted for the initial states given in \ref{['hos']}: $n=0$ (green); $n=1$ (red); and $n=2$ (black). If a curve goes below $1$, positivity is violated.
  • Figure 2: The quantities $e_3(\delta,\gamma)$ as a function of $\gamma$ for the parameters $\delta=0,100,1000$ scaled appropriately. As $\delta$ is larger, the root $\mathcal{Z}_3(\delta)$ becomes smaller, so we obtain a larger set of parameters $\gamma$ where positivity fails.

Theorems & Definitions (45)

  • Theorem 1.1: Mercer
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Claim 2.10
  • Example 2.11
  • ...and 35 more