On PSI-complete and PSIR-complete measurements

Abstract

I construct a POVM which has 2d rank-one elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (quant-ph/0404137). I show that if a rank-one POVM is required to be informationally complete for all pure states in d dimensions, it must have at least 3d-2 elements. I also show that, in a POVM which is informationally complete for all pure states in d dimensions, for any vector there must be at least 2d-1 POVM elements which do not annihilate that vector.

9 pages, LaTeX, no figures v2: Added comment about MUBs; corrected typos

References in corpus (3)

• On SIC-POVMs and MUBs in Dimension 6
• Minimal Informationally Complete Measurements for Pure States
• Quantum measurements and finite geometry

Cited by in corpus (36)

• SIC-POVMs and the Extended Clifford Group
• Quantum Tomography under Prior Information
• Uniqueness of Quantum States Compatible with Given Measurement Results
• On SIC-POVMs in Prime Dimensions
• Strictly-complete measurements for bounded-rank quantum-state tomography
• The power of being positive: Robust state estimation made possible by quantum mechanics
• Informational completeness in bounded-rank quantum-state tomography
• Pure State Tomography with Pauli Measurements
• Fisher-symmetric informationally complete measurements for pure states
• Random unitary maps for quantum state reconstruction
• Experimental study of optimal measurements for quantum state tomography
• PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
• Universal compressive characterization of quantum dynamics
• Scalable Quantum Tomography with Fidelity Estimation
• How many orthonormal bases are needed to distinguish all pure quantum states?
• Reconstruction of Signals from Magnitudes of Redundant Representations
• Stable phase retrieval with low-redundancy frames
• Invertibility and Robustness of Phaseless Reconstruction
• The Role of Topology in Quantum Tomography
• A set of $4d-3$ observables to determine any pure qudit state
• Modern compressive tomography for quantum information science
• Pure state tomography with parallel unentangled measurements
• Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case
• Constrained Quantum Tomography of Semi-Algebraic Sets with Applications to Low-Rank Matrix Recovery
• Phase Retrieval from 4N-4 Measurements
• Obtaining conclusive information from incomplete experimental quantum tomography
• Phaseless Signal Recovery in Infinite Dimensional Spaces using Structured Modulations
• Phase Retrieval using Lipschitz Continuous Maps
• Pure State Tomography with Fourier Transformation
• On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem
• Minimal orthonormal bases for pure quantum state estimation
• The minimal measurement number problem in phase retrieval: a review of recent developments
• Minimal covariant observables identifying all pure states
• Almost everywhere injectivity conditions for the matrix recovery problem
• PhaseLiftOff: an Accurate and Stable Phase Retrieval Method Based on Difference of Trace and Frobenius Norms
• Almost Everywhere Generalized Phase Retrieval