Detection of nonabsolute separability in quantum states and channels through moments

In quantum information and computation, the generation of entanglement through unitary gates remains a significant and active area of research. However, there are states termed as absolutely separable, from which entanglement cannot be created through any non-local unitary action. Thus, from a resource-theoretic perspective, non-absolutely separable states are useful as they can be turned into entangled states using some appropriate unitary gates. In this work, we propose an efficient method to detect non-absolutely separable states. Our approach relies on evaluating moments that can bypass the need for full state tomography, thereby enhancing its practical applicability. We then present several examples in support of our detection scheme. We also address a closely related problem concerning states whose partial transpose remains positive under any arbitrary non-local unitary action. Furthermore, we examine the effectiveness of our moment-based approach in the detection of quantum channels that are not absolutely separating, which entails the detection of resource preserving channels. Finally, we demonstrate the operational significance of non-absolutely separable states by proving that every such state can provide an advantage in a quantum-channel discrimination task.

💡 Research Summary

The paper addresses the problem of identifying quantum states and channels that are not absolutely separable (AS) or absolutely PPT (APPT) without resorting to full state tomography. Absolutely separable states remain separable under any global unitary transformation and thus constitute a “free” resource in the resource‑theoretic framework of entanglement. In contrast, non‑absolutely separable states can be turned into entangled states by a suitable global unitary and therefore act as a useful resource. Existing detection methods rely on eigenvalue‑based criteria or geometric balls, both of which require complete knowledge of the density matrix and become infeasible as the system dimension grows.

To overcome this limitation, the authors propose a moment‑based approach. They focus on the partial‑transpose moments
(p_n = \operatorname{Tr}