Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection

TL;DR

The paper introduces the Triangle Criterion, a geometric and operational tool for detecting and distilling magic in multi-qubit quantum states, revealing new limitations and features of magic protocols.

Contribution

It presents the Triangle Criterion as a new mixed-state magic detection method, demonstrating its advantages and limitations in multi-qubit magic distillation.

Findings

Multi-qubit magic distillation protocols outperform single-qubit schemes.

Low-rank multi-qubit states generally cannot be distilled by single-qubit protocols.

An upper bound on the minimal purity of magic states is derived and supported by evidence.

Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) Criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.