Determination of all pure quantum states from a minimal number of observables

Abstract

We show that for any positive integer $n$, the maps $x\in {\mathbb{C}}^n\mapsto \{{\left|⟨x,z_i⟩\right|}^2{\}}_{i=1}^{4n}\in {\mathbb{R}}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of $\mathbb{C}{P}^{n-1}$ into ${\mathbb{R}}^{4n-4}$. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for $n\ge 6$. To complement this information-theoretic study, we establish in a companion paper that the PhaseLift algorithm yields efficient phase retrieval from quadratic measurements with $O(1)$ unitary matrices, with high probability, where the unitaries are iid according to Haar measure.

References in corpus (4)

• Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
• An algebraic characterization of injectivity in phase retrieval
• On signal reconstruction without noisy phase
• Quantum Tomography From Few Full-Rank Observables

Cited by in corpus (11)

• A Partial Derandomization of PhaseLift using Spherical Designs
• Efficient Pure State Quantum Tomography from Five Orthonormal Bases
• How many orthonormal bases are needed to distinguish all pure quantum states?
• Quantum Tomography From Few Full-Rank Observables
• The Role of Topology in Quantum Tomography
• Dynamical Quantum Tomography
• Quantum tomography meets dynamical systems and bifurcations theory
• On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem
• Phase Retrieval using Lipschitz Continuous Maps
• Determination of All Unknown Pure Quantum States with Two Observables
• Phaseless Reconstruction from Space-Time Samples