The Gram Matrix of a PT-symmetric Quantum System

Quantumness of pure-state ensembles via coherence of Gram matrix based on generalized $α$-$z$-relative Rényi entropy

The Gram matrix of a set of quantum pure states plays key roles in quantum information theory. It has been highlighted that the Gram matrix of a pure-state ensemble can be viewed as a quantum state, and the quantumness of a pure-state ensemble can thus be quantified by the coherence of the Gram matrix [Europhys. Lett. \textbf{134} 30003]. Instead of the $l_1$-norm of coherence and the relative entropy of coherence, we utilize the generalized $α$-$z$-relative Rényi entropy of coherence of the Gram matrix to quantify the quantumness of a pure-state ensemble and explore its properties. We show th…

A Gramian Approach to Entanglement in Bipartite Finite Dimensional Systems: The case of pure states

It has been observed that the reduced density matrices of bipartite qudit pure states possess a Gram matrix structure. This observation has opened a possibility of analysing the entanglement in such systems from the purely geometrical point of view. In particular, a new quantitative measure of an entanglement of the geometrical nature, has been proposed. Using the invented Gram matrix approach, a version of a non-linear purification of mixed states describing the system analysed has been presented.

实 Gram 矩阵 Hafnian 与对称高斯 Hafnian 的平移反集中性

Shifted Anticoncentration for Real Gram Hafnians and Symmetric Gaussian Hafnians

本文证明实高斯 Gram 矩阵 hafnian 的平移反集中定理，给出密度界与多项式增长系数，并推广至对称高斯情形。

Self-consistent tomography of the state-measurement Gram matrix

State and measurement tomography make assumptions about the experimental states or measurements. These assumptions are often not justified because state preparation and measurement errors are unavoidable in practice. Here we describe how the Gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so called globally completable. This is for instance the case if the unknown measurements are known to be projective and non-degenerate. The computed Gram matrix determines the states, and the measurement ope…

Quantum dynamics and Gram's matrix

We propose to analyse the statistical properties of a sequence of vectors using the spectrum of the associated Gram matrix. Such sequences arise e.g. by the repeated action of a deterministic kicked quantum dynamics on an initial condition or by a random process. We argue that, when the number of time-steps, suitably scaled with respect to $\hbar$, increases, the limiting eigenvalue distribution of the Gram matrix reflects the possible quantum chaoticity of the original system as it tends to its classical limit. This idea is subsequently applied to study the long-time properties of sequences o…

量子卡尔曼分解：一种格拉姆矩阵方法

The Quantum Kalman Decomposition: A Gramian Matrix Approach

本文提出一种基于格拉姆矩阵的替代方法，推导量子线性系统的卡尔曼规范形式，并通过实例验证其有效性。