Entanglement Detection Beyond Measuring Fidelities

TL;DR

This paper reveals that many entangled quantum states cannot be detected or quantified using fidelity measurements, introduces the concept of unfaithful states, and explores their properties and limitations in entanglement detection.

Contribution

It identifies unfaithful states that evade fidelity-based detection, introduces D-unfaithful states, and develops a hierarchy of semidefinite programs for better entanglement characterization.

Findings

Most bipartite states are entangled and unfaithful in low dimensions

Adding white noise can make pure entangled states unfaithful

Tensor powers can activate faithfulness in unfaithful states

Abstract

One of the most widespread methods to determine if a quantum state is entangled, or to quantify its entanglement dimensionality, is by measuring its fidelity with respect to a pure state. In this Letter we find a large class of states whose entanglement cannot be detected in this manner; we call them unfaithful. We find that unfaithful states are ubiquitous in information theory. For small dimensions, we check numerically that most bipartite states are both entangled and unfaithful. Similarly, numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added. We also find that faithfulness can be self-activated, i.e., there exist instances of unfaithful states whose tensor powers are faithful. To explore how the fidelity approach limits the quantification of entanglement dimensionality, we generalize the notion of an unfaithful state to that of a D-unfaithful state, one that cannot be certified as D-dimensionally entangled by measuring its fidelity with respect to a pure state. For describing such states, we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of Schmidt rank at most D.