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

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Abstract In light of the rapid experimental development of quantum devices, tools for estimating quantum properties, particularly entanglement, must be improved to keep pace. 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 ⩽ 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 x p m 1 − x must be entangled, where x = ( m − l ) / ( m − k ) . Using the example of locally depolarized Greenberger–Horne–Zeilinger states, a family of states that has been used as a test for the quantum character of large-scale states prepared on quantum computers, 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

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