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arXiv 2405.04438quant-phmath-phmath.MPmath.SP
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多项式高斯积分算子的正定性与纠缠性

Positivity and entanglement of polynomial Gaussian integral operators

Richárd Balka, András Csordás, Gábor Homa

备注：
Minor modifications on version 2; final version. 32 pages, 2 figures, 1 table
发表信息：
Progress of Theoretical and Experimental Physics, Volume 2024, Issue 10, October 2024, 103A01
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AI总结：

该研究探讨开放量子系统中多项式高斯积分算子的正定性与纠缠性，给出正定性的高斯部分必要条件、奇次多项式算子非半正定结论，引入高斯核预序以扩展正定性检验范围，并推导其对纠缠判定的作用。

摘要：

正定性保持是开放量子系统动力学中的一个重要问题：正定性的破缺始终标志着模型有效性的边界。我们研究自伴多项式高斯积分算子$\boldsymbol{\widehat{κ}_{PG}}$的正定性，即其多变量核$κ_{PG}$是多项式$P$与高斯核$κ_G$的乘积。这类算子频繁出现在开放量子系统中。\n 我们证明，仅当高斯部分为正定时，$\boldsymbol{\widehat{κ}_{PG}}$才可能为正定，这提供了一种强有效性且十分简便的正定性检验方法。这一结论对密度算子$\boldsymbol{\widehat{κ}_{PG}}$的两体纠缠有重要推论：如果高斯密度算子$\boldsymbol{\widehat{κ}_G}$不满足Peres-Horodecki判据，那么对应的多项式高斯密度算子$\boldsymbol{\widehat{κ}_{PG}}$对所有$P$都不满足该判据，因此它们全部是纠缠的。\n 我们证明，具有奇次多项式的多项式高斯算子不可能是半正定的。\n 我们在高斯核上引入了一种新的预序关系$\preceq$：若$κ_{G_0}\preceq κ_{G_1}$，则对所有多项式$P$，$\boldsymbol{\widehat{κ}_{PG_0}\geq 0}$可推出$\boldsymbol{\widehat{κ}_{PG_1}\geq 0}$。因此，确定一个多项式高斯算子的正定性，就能判定许多具有相同多项式因子的其他多项式高斯算子的正定性，这可以通过将检验应用于更大的算子集合来改进任意给定的正定性检验方法。我们将展示一个示例，证明这确实能让正定性检验变得更加灵敏和高效。该预序关系对纠缠问题也有启示意义。

英文摘要：

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.

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