Quantifying mixed-state entanglement via partial transpose and realignment moments

TL;DR

This paper introduces efficient, measurable entanglement witnesses based on partial transpose and realignment moments, enabling quantification and certification of entanglement in mixed quantum states and many-body systems, even under noise.

Contribution

The authors develop new entanglement witnesses from partial transpose and realignment moments that are efficiently measurable and applicable to large, noisy quantum systems, advancing entanglement quantification methods.

Findings

Efficient algorithms for testing entanglement in mixed states with bounded entropy.

Measurement protocols for Schmidt and operator Schmidt ranks using few copies.

Robust certification of quantum circuit depth in noisy environments.

Abstract

Entanglement plays a crucial role in quantum information science and many-body physics, yet quantifying it in mixed quantum many-body systems has remained a notoriously difficult problem. Here, we introduce families of quantitative entanglement witnesses, constructed from partial transpose and realignment moments, which provide rigorous bounds on entanglement monotones. Our witnesses can be efficiently measured using SWAP tests or variants of Bell measurements, thus making them directly implementable on current hardware. Leveraging our witnesses, we present several novel results on entanglement properties of mixed states, both in quantum information and many-body physics. We develop efficient algorithms to test whether mixed states with bounded entropy have low or high entanglement, which previously was only possible for pure states. We also provide an efficient algorithm to test the Schmidt rank using only two-copy measurements, and to test the operator Schmidt rank using four-copy measurements. Further, our witnesses enable robust certification of quantum circuit depth even in the presence of noise, a task which so far has been limited to noiseless circuits only. Finally, we show that the entanglement phase diagram of Haar random states, quantified by the partial transpose negativity, can be fully established solely by computing our witness, a result that also applies to any state 4-design. Our witnesses can also be efficiently computed for matrix product states, thus enabling the characterization of entanglement in extensive many-body systems. Finally, we make progress on the entanglement required for quantum cryptography, establishing rigorous limits on pseudoentanglement and pseudorandom density matrices with bounded entropy. Our work opens new avenues for quantifying entanglement in large and noisy quantum systems.