Detecting entanglement from few partial transpose moments and their decay via weight enumerators

The $p_3$-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments $p_k=$Tr$[(\rho^\Gamma)^k]$, where $\rho^\Gamma$ denotes the partially transposed density matrix of a quantum state $\rho$. While most of these generalizations are strictly more powerful than the $p_3$-PPT criterion, their $m$-th level versions usually rely on the availability of $p_k$ for all moment orders $k\le m$. Here, we show that one can alternatively compare any three PT moments of orders $k<l<m$, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where $x=(m-l)/(m-k)$. Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to $p_k$ for $k \le 5$ is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-$m$ criterion is as powerful as the PPT criterion whenever $\rho^\Gamma$ has no more than $(m+1)/2$ distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of $p_k$ under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.