Certification of mixed states with few single-qubit measurements

Develop a certification procedure analogous to the shadow-overlap method for target mixed states and ascertain whether almost all approximately low-rank mixed states can be certified using only polynomially many single-qubit measurements, despite known worst-case lower bounds for arbitrary mixed states.

Background

For arbitrary mixed states, existing lower bounds (e.g., for certifying the maximally mixed state) rule out protocols with polynomial sample complexity even with entangled measurements. Nonetheless, instance-optimal results suggest efficient certification might be possible for structured classes such as approximately low-rank mixed states, albeit typically with entangled measurements.

Extending the single-qubit measurement framework to mixed states, and determining whether almost all approximately low-rank mixed states are certifiable with few single-qubit measurements, would significantly broaden the practical impact of the proposed certification approach.