Marginal second-order moments do not suffice for entanglement detection

TL;DR

This paper demonstrates that knowing only the global and marginal second-order moments of a quantum state is insufficient for reliably detecting entanglement, highlighting the need for more comprehensive criteria.

Contribution

The authors derive conditions on local-unitary transformations and construct examples showing second-order moments alone cannot distinguish entangled states from separable ones.

Findings

Second-order moments do not uniquely determine entanglement status.

Constructed state pairs with identical moments but different entanglement.

Entanglement detection requires more than second-order moment information.

Abstract

The complete knowledge of the global and marginal second-order moments of a quantum state is in general insufficient for entanglement detection. By deriving the conditions on local-unitary (LU) transformations through the second-order moments, we construct pairs of separable and entangled states that are not LU equivalent, contain different amount of entanglement, and are not equivalent under stochastic local operations and classical communication, even though the states share identical global and marginal second-order moments.