A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only
d+1 projective measurement bases to fully reconstruct an arbitrary
d-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for
d=6, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.
This paper develops a comprehensive framework for quantum information theory within the paradigm of pseudo-Hermitian quantum mechanics. We establish necessary and sufficient conditions for the existence of biorthogonal Riesz bases, a fundamental component in the formulation of pseudo-Hermitian quantum information. Additionally, we propose a spectral theorem for pseudo-Hermitian operators and explore their decomposition. We introduce key postulates of pseudo-Hermitian quantum mechanics and independently derive the time-evolution operator for pseudo-Hermitian systems. Furthermore, we extend the classical notion of quantum channels to pseudo-Hermitian quantum information theory and provide various characterizations, including the Kraus, Stinespring, and Choi representations. Our findings contribute to the mathematical foundations of non-Hermitian quantum mechanics and its applications in quantum information processing.
Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.