Fundamental Limitation on the Detectability of Entanglement

TL;DR

This paper reveals a fundamental trade-off in entanglement detection, showing that efficient criteria often require exponential resources and that multi-copy measurements can exponentially enhance detection capabilities, highlighting inherent limitations and quantum advantages.

Contribution

It formalizes the fundamental limitations in entanglement detectability and introduces a systematic method to evaluate the detection capability of various criteria.

Findings

Single-copy detection requires exponentially many observables.

Detection capability decays double-exponentially without multi-copy measurements.

Multi-copy measurements can exponentially improve detection effectiveness.

Abstract

Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponential resources, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to theoretically evaluate the detection capability of entanglement criteria. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double-exponentially. Furthermore, if multi-copy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems.