Entanglement Detection Beyond Measuring Fidelities

• Main Authors: Weilenmann, Mirjam, Dive, Benjamin, Trillo, David, Aguilar, Edgar A, Navascués, Miguel
• Format: Journal Article
• Language: English
• Published: 13-08-2020
• Subjects: Physics - Quantum Physics
• DOI: 10.48550/arxiv.1912.10056

Phys. Rev. Lett. 124, 200502 (2020) 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.