Quantum operation, quantum Fourier transform and semi-definite programming

Revisiting the quantum harmonic oscillator via unilateral Fourier transforms

The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integrable eigenfunctions.

设备依赖猜测概率的半定规划公式

A semi-definite programming formulation of the device-dependent guessing probability

针对已知量子态和测量装置但存在不可信对手的情况，提出半定规划方法计算最大猜测概率，并应用于多种场景确定可证随机性。

Optimal Unambiguous Discrimination of Quantum States

Unambiguously distinguishing between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In this work, an exact analytic solution to an optimum measurement problem involving an arbitrary number of pure linearly independent quantum states is presented. To this end, the relevant semi-definite programming task is reduced to a linear programming one with a feasible region of polygon type which can be solved via simplex method. The strength of the method is illustrated through some explicit examples. Also using the close connection betwe…

The Classical Limit of Quantum Mechanics and the Fejer Sum of the Fourier Series Expansion of a Classical Quantity

In quantum mechanics, the expectation value of a quantity on a quantum state, provided that the state itself gives in the classical limit a motion of a particle in a definite path, in classical limit goes over to Fourier series form of the classical quantity. In contrast to this widely accepted point of view, a rigorous calculation shows that the expectation value on such a state in classical limit exactly gives the Fejér's arithmetic mean of the partial sums of the Fourier series.

Moment 简介：一款基于矩矩阵的半定规划工具包

Introducing Moment: A toolkit for semi-definite programming with moment matrices

针对非交换多项式优化中矩矩阵构造计算成本高的问题，本文提出采用C++编写、提供MATLAB接口的Moment工具包，其构造矩矩阵松弛的速度较同类软件最高快四个数量级。

量子马尔可夫决策过程：用于最优解的动态规划与半定规划

Quantum Markov Decision Processes: Dynamic and Semi-Definite Programs for Optimal Solutions

针对 q-MDP，论文利用动态规划与半定规划的对偶性证明开环及保持经典状态闭环策略下存在平稳最优策略，并给出价值函数计算与双线性规划求解方法。

Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier

In quantum mechanics some properties are maximally incompatible, such as the position and momentum of a particle or the vertical and horizontal projections of a 2-level spin. Given any definite state of one property the other property is completely random, or unbiased. For N-level systems, the 6-level ones are the smallest for which a tomographically efficient set of N+1 mutually unbiased bases (MUBs) has not been found. To facilitate the search, we numerically extend the classification of unbiased bases, or Hadamards, by incrementally adjusting relative phases in a standard basis. We consider…

Applications of semi-definite optimization in quantum information protocols

This work is concerned with the issue of applications of the semi-definite programming (SDP) in the field of quantum information science. Our results of the analysis of certain quantum information protocols using this optimization technique are presented, and an implementation of a relevant numerical tool is introduced. The key method used is NPA discovered by Navascues et al. [Phys. Rev. Lett. 98, 010401 (2007)]. In chapter 1 a brief overview of mathematical methods used in this work is presented. In chapter 2 an introduction to quantum information science is given. Chapter 3 concerns the dev…

On some meaningful inner product for real Klein-Gordon fields with positive semi-definite norm

A simple derivation of a meaningful, manifestly covariant inner product for real Klein-Gordon (KG) fields with positive semi-definite norm is provided which turns out - assuming a symmetric bilinear form - to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December, 2003, and February, 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fouri…

An Introduction to Quantum Programming in Quipper

Quipper is a recently developed programming language for expressing quantum computations. This paper gives a brief tutorial introduction to the language, through a demonstration of how to make use of some of its key features. We illustrate many of Quipper's language features by developing a few well known examples of Quantum computation, including quantum teleportation, the quantum Fourier transform, and a quantum circuit for addition.

量子态与信道测量相对熵的半定优化

Semi-definite optimization of the measured relative entropies of quantum states and channels

本文利用变分公式和半定表示，通过半定规划高效计算量子态与信道的测量相对熵，并给出最优策略的数值表征，用于设计假设检验协议。