Curator's Take
This article presents a significant advancement in detecting quantum entanglement with fewer experimental measurements, potentially making entanglement verification much more practical in real quantum systems. The researchers show that instead of needing many partial transpose moments to certify entanglement, you can often get away with measuring just three carefully chosen moments, which could dramatically reduce the experimental overhead in quantum labs. What's particularly exciting is their proof that for certain important quantum states like stabilizer states, measuring just the first five moments can give you the full power of the complete positive partial transpose test. This work bridges fundamental quantum theory with experimental reality, offering quantum experimentalists more efficient tools to verify the entanglement that's crucial for quantum computing and communication protocols.
— Mark Eatherly
Summary
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$[(ρ^Γ)^k]$, where $ρ^Γ$ denotes the partially transposed density matrix of a quantum state $ρ$. 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 $ρ^Γ$ 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.