[Paper Review] Quantum Tomography From Few Full-Rank Observables

Vladislav Voroninski|arXiv (Cornell University)|Sep 29, 2013

Advanced X-ray Imaging Techniques5 references15 citations

TL;DR

This paper establishes that quantum state tomography can be successfully performed using PhaseLift with only O(n) measurements when observables are drawn from the Haar distribution on unitary or orthogonal groups. It proves exact recovery of pure states up to global phase with high probability, extending PhaseLift's guarantees beyond Gaussian measurements to physically realizable, structured random observables.

ABSTRACT

We establish that the PhaseLift algorithm recovers pure states from a constant number of full-rank observables with high probability.

Motivation & Objective

• To extend PhaseLift's exact recovery guarantees from Gaussian to structured, physically realizable measurement models in quantum tomography.
• To establish that O(n) measurements from Haar-distributed unitary or orthogonal matrices suffice for exact recovery of pure quantum states.
• To provide a theoretical foundation for quantum tomography using full-rank observables, addressing the gap between idealized Gaussian models and real-world implementations.
• To adapt the dual certificate technique to structured random ensembles, reducing measurement complexity from O(n log n) to O(n).
• To connect the results to Wright’s conjecture on the minimal number of observables needed to determine pure states, resolving a related variant.

Proposed method

• Uses the PhaseLift convex relaxation framework to recover density matrices from magnitude-only measurements.
• Employs a dual certificate construction based on the measurement operator A, leveraging unitary/orthogonal invariance of the Haar measure.
• Applies Talagrand’s concentration inequality to control the deviation of measurement operators on rank-2 Hermitian matrices.
• Establishes a Restricted Isometry Property of type 1 (RIP-1) for rank-2 matrices under Haar-distributed observables.
• Adapts the dual certificate technique from Gaussian settings to structured random matrices via a covering argument and Lipschitz analysis.
• Uses a modified convex program to extend results to noisy settings and uniform recovery over all signals.

Experimental results

Research questions

• RQ1 Can PhaseLift recover pure quantum states exactly using a sub-Gaussian number of full-rank observables drawn from a structured random ensemble?
• RQ2 Does the Haar-distributed measurement model—physically realizable in quantum tomography—support exact recovery with O(n) measurements?
• RQ3 Can the dual certificate approach used in Gaussian settings be adapted to unitary/orthogonal random ensembles to achieve O(n) measurement complexity?
• RQ4 What is the minimal number of full-rank observables required to uniquely determine any pure quantum state, and how does this relate to Wright’s conjecture?
• RQ5 How does the stability of PhaseLift under noise extend to structured random measurement models?

Key findings

• PhaseLift recovers any pure state x ∈ ℂⁿ up to global phase with high probability using m = O(n) measurements from Haar-distributed unitary or orthogonal matrices.
• The measurement operator A satisfies RIP-1 for rank-2 matrices with high probability when r = O(1) observables are used, ensuring exact recovery via the dual certificate method.
• The number of measurements is reduced from O(n log n) to O(n) by adapting the dual certificate construction, matching the best-known bounds for Gaussian measurements.
• The dual certificate Y exists in the range of A* with Y_T^⊥ ≺ 0 and ‖Y_T − e₁e₁*‖₂ ≤ 1/5, which guarantees uniqueness of the solution.
• The proof establishes concentration of measure via Talagrand’s inequality, showing exponential tail bounds for the operator norm deviation.
• The results extend to rank-k states with O(kn) measurements by generalizing the RIP-1 property to rank-2k matrices and using a modified dual certificate.