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Quantum Science量子力学量子力学（quantum mechanics）是物理学的分支学科。它描述原子尺度及原子尺度以下的自然行为和规律。 它是所有量子物理学的基础，包括量子化学、量子场论、量子技术和量子信息科学。 量子力学与相对论一起被认为是现代物理学的两大基本支柱。19世纪末，人们发现旧有的经典理论无法解释微观系统，于是经由物理学家的努力，在20世纪初创立量子力学，解释了这些现象。量子力学从根本上改变人类对物质结构及其相互作用的理解。除了透过广义相对论描写的引力外，迄今所有基本相互作用均可以在量子力学的框架内描述（量子场论）。 量子理论的重要应用包括宇宙学、量子化学、量子光学、量子计算、超导磁体、发光二极管、激光器、晶体管和半导体如微处理器等。 爱因斯坦可能是在科学文献中最先给出术语“量子力学”的物理学者。 量子力学逐渐从理论中兴起，用来解释与经典物理学不相符的观测结果，例如马克斯·普朗克在1900年解决黑体辐射问题，以及阿尔伯特·爱因斯坦1905年论文中能量与频率的对应关系，该论文解释了光电效应影响。 这些理解微观现象的早期尝试，现在被称为“旧量子论”，导致尼尔斯·玻尔、欧文·薛定谔、维尔纳·海森堡、马克斯·玻恩、保罗·狄拉克等人在1920年代中期全面发展了量子力学。 现代理论是用各种专门发展的数学形式体系来表达的。 其中之一，称为波函数的数学实体以概率幅的形式提供有关粒子能量、动量和其他物理特性的测量结果的信息。
Quantum mechanics, also known as quantum physics, is the fundamental physical theory that describes the behavior of matter and of light; the behaviors it models typically occur at and below the scale of atoms, and have often been described as counterintuitive. Its concepts and methods have been applied across many disciplines, including quantum chemistry, quantum biology, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and (optical) microscopic) scale; however, it is insufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have bound states that are quantized to discrete values of energy, momentum, angular momentum, and other quantities, in contrast to classical systems where these quantities can be measured continuously.
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View content license ↗ 在量子信息学中，量子比特（英语：quantum bit），又称Q比特（qubit）是量子信息的计量单位。传统电脑使用的是0和1，量子电脑虽然也是使用0跟1，但不同的是，量子电脑的0与1可以同时计算。在经典系统中，一个比特在同一时间，只有0或1，只存在一种状态，但量子比特可以同时是1和0，两种状态同时存在，这种效果叫量子叠加。这是量子电脑计算目前独有的特性。
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information, the quantum version of the classic binary bit. A qubit can be physically realized with a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of multiple states simultaneously, a property that is fundamental to quantum mechanics and quantum computing. A qubit can be generalized as a dimension d=2 qudit or a binary qudit.
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View content license ↗ 线性组合（英语：Linear combination）是线性代数中具有如下形式的表达式。其中 v i {\displaystyle v_{i}} 为任意类型的项， a i {\displaystyle a_{i}} 为标量。这些标量称为线性组合的系数或权。 w = a 1 v 1 + a 2 v 2 + a 3 v 3 + ⋯ + a n v n {\displaystyle w=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+\cdots +a_{n}v_{n}}
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
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View content license ↗ 在量子力学中，薛定谔方程（Schrödinger equation）是描述物理系统的量子态随时间演化的偏微分方程，为量子力学的基础方程之一，其以发表者奥地利物理学家埃尔温·薛定谔而命名。关于量子态与薛定谔方程的概念涵盖于基础量子力学假说里，无法从其它任何原理推导而出。 在经典力学里，人们使用牛顿第二定律描述物体运动。而在量子力学里，类似的运动方程为薛定谔方程。薛定谔方程的解完备地描述物理系统里，微观尺寸粒子的量子行为；这包括分子系统、原子系统、亚原子系统；另外，薛定谔方程的解还可完备地描述宏观系统，可能乃至整个宇宙。 薛定谔方程可以分为“含时薛定谔方程”与“不含时薛定谔方程”两种。含时薛定谔方程与时间有关，描述量子系统的波函数怎样随着时间而演化。不含时薛定谔方程则与时间无关，描述了定态量子系统的物理性质；该方程的解就是定态量子系统的波函数。量子事件发生的概率可以用波函数来计算，其概率幅的绝对值平方就是量子事件发生的概率密度。 薛定谔方程所属的波动力学可以数学变换为维尔纳·海森堡的矩阵力学，或理察·费曼的路径积分表述。薛定谔方程是个非相对论性方程，不适用于相对论性理论；对于相对论性微观系统，必须改使用狄拉克方程或克莱因-戈尔登方程等。
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.
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View content license ↗ 在量子力学里，密度算符（英语：density operator）与其对应的密度矩阵（英语：density matrix）专门描述混合态量子系统的物理性质。纯态是一种可以直接用态矢量 | ψ ⟩ {\displaystyle |\psi \rangle } 来描述的量子态，混合态则是由几种纯态依照统计概率组成的量子态。假设一个量子系统处于纯态 | ψ 1 ⟩ {\displaystyle |\psi _{1}\rangle } 、 | ψ 2 ⟩ {\displaystyle |\psi _{2}\rangle } 、 | ψ 3 ⟩ {\displaystyle |\psi _{3}\rangle } 、……的概率分别为 w 1 {\displaystyle w_{1}} 、 w 2 {\displaystyle w_{2}} 、 w…
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles of states. These arise in quantum mechanics in two different situations: when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of the ensemble of possible preparations; and when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.
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View content license ↗ 在数学、尤其是泛函分析中，矢量空间 V {\displaystyle V} 上的自伴算子（self-adjoint operator）是一类特殊的线性算子（自同态），其伴随算子是其自身。根据不同的需要，可以讨论 V {\displaystyle V} 为拓扑矢量空间、赋范矢量空间、巴拿赫空间乃至希尔伯特空间的情况，使得伴随算子、自伴算子可以具有更丰富的性质，一个重要的例子是希尔伯特空间上自伴算子的谱定理。 若 V {\displaystyle V} 是具有规范正交基的有限维复向量空间，其上自伴算子在该基下的矩阵是埃尔米特矩阵——该矩阵等于自身的共轭转置。有限维的谱定理表明，对于一个算子 A {\displaystyle A} ，总能找到 V {\displaystyle V} 上的规范正交基使得 A {\displaystyle A} 在该基下的矩阵是一个对角矩阵，且这些对角元都是实数。 无穷维希尔伯特空间上的自伴算子的谱定理与此类似：一个算子是自伴的，当且仅当其酉等价于一个实值乘法算子。不过，黑林格-特普利茨定理表明了定义于全空间的自伴算子必然是有界的，从而无界算子至多只能定义在全空间的一个稠密子空间上，故对于无界算子须对定义域的问题多加注意。定义域的问题造成了对称算子和自伴算子的区分，而这区分对于谱定理等结论而言是至关重要的。 自伴算子在量子力学中也有重要地位。在量子力学公理的狄拉克-冯诺依曼表述中，位置、动量、角动量和自旋等物理可观测量是由希尔伯特空间上的自伴算子表示。在哈密顿算子的谱（能级）具有重要的物理意义的同时，哈密顿算子中的动能项通常由导数算子构成，而无穷维空间中的导数算子是典型的无界算子。
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is a linear map A {\displaystyle A} (from V {\displaystyle V} to itself) that is its own adjoint. That is, ⟨ A x , y ⟩ = ⟨ x , A y ⟩ {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle } for all x , y ∈ V {\displaystyle x,y\in V} .
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View content license ↗ 在泛函分析中，幺正算符（英语：unitary operator，或称酉算符）是定义在希尔伯特空间上的有界线性算符 U : H → H {\displaystyle U:H\rightarrow H} ，满足如下规律： U ∗ U = U U ∗ = I {\displaystyle U^{*}U=UU^{*}=I} 其中 U ∗ {\displaystyle U^{*}} 是 U {\displaystyle U} 的厄米转置, 而 I : H → H {\displaystyle I:H\rightarrow H} 是恒等算符。
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.
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View content license ↗ 在物理学里，特别是在量子力学里，处于某种状态的物理系统，它所具有的一些性质，可以经过一序列的物理运作过程而得知。这些可以得知的性质，称为可观察量（observable）。例如，物理运作可能涉及到施加电磁场于物理系统，然后使用实验仪器测量某物理量的数值。在经典力学的系统里，任何可以用实验测量获得的可观察量，都可以用定义于物理系统状态的实函数来表示。在量子力学里，物理系统的状态称为量子态，其与可观察量的关系更加微妙，必须使用线性代数来解释。根据量子力学的数学表述，量子态可以用存在于希尔伯特空间的态矢量来代表，量子态的可观察量可以用厄米算符来代表。
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is described by a linear operator. For example, these operators might represent submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.
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View content license ↗ 在数学上，特别是线性代数中，对于一个给定的方阵 A {\displaystyle A} ，它的特征向量（eigenvector，也译固有向量、本征向量） v {\displaystyle v} 经过这个线性变换之后，得到的新向量仍然与原来的 v {\displaystyle v} 保持在同一条直线上，但其长度或方向也许会改变。即 A v = λ v {\displaystyle Av=\lambda v} ， λ {\displaystyle \lambda } 为标量，即特征向量的长度在该线性变换下缩放的比例，称 λ {\displaystyle \lambda } 为其特征值（eigenvalue，也译固有值、本征值）。如果特征值为正，则表示 v {\displaystyle v} 在经过线性变换的作用后方向也不变；如果特征值为负，说明方向会反转；如果特征值为0，则是表示缩回零点。但无论怎样，仍在同一条直线上。图1给出了一个以油画《蒙娜丽莎》为题材的例子。在一定条件下（如其矩阵形式为实对称矩阵的线性变换），一个变换可以由其特征值和特征向量完全表述，也就是说：所有的特征向量组成了这向量空间的一组基底。一个特征空间（eigenspace）是具有相同特征值的特征向量与一个同维数的零向量的集合，可以证明该集合是一个线性子空间，比如 E λ = { u ∈ V ∣ A u…
In linear algebra, an eigenvector ( EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector v {\displaystyle \mathbf {v} } of a linear transformation T {\displaystyle T} is scaled by a constant factor λ {\displaystyle \lambda } when the linear transformation is applied to it: T v = λ v {\displaystyle T\mathbf {v} =\lambda \mathbf {v} } . The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor λ {\displaystyle \lambda } (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在量子力学里，量子态（英语：quantum state）指的是量子系统的状态。态矢量可以用来抽象地表示量子态。采用狄拉克标记，态矢量表示为右矢 | ψ ⟩ {\displaystyle \left\vert \psi \right\rangle } ；其中，在符号内部的希腊字母 ψ {\displaystyle \psi } 可以是任何符号，字母，数字，或单字。例如，在计算氢原子能谱时，能级与主量子数 n {\displaystyle n} 有关，所以，每个量子态的态矢量可以表示为 | n ⟩ {\displaystyle \left\vert n\right\rangle } 。 一般而言，量子态可以是纯态或混合态。上述案例是纯态。混合态是由很多纯态组成的概率混合。不同的组合可能会组成同样的混合态。当量子态是混合态时，可以用密度矩阵做数学描述，这密度矩阵实际给出的是概率，不是密度。纯态也可以用密度矩阵表示。
In quantum physics, a quantum state is a mathematical entity that represents a physical system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system. Quantum states are either pure or mixed, and have several possible representations. Pure quantum states are commonly represented as a vector in a Hilbert space. Mixed states are statistical mixtures of pure states and cannot be represented as vectors on that Hilbert space, and instead are commonly represented as density matrices. Common examples of quantum states are the wave functions describing position and momentum, finite-dimensional vectors describing spin such as the singlet, and states describing many-body quantum systems in a Fock space.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学和数学物理中，泡利矩阵是一组三个2×2的幺正厄米复矩阵，一般都以希腊字母σ来表示，但有时当他们在和同位旋的对称性做链接时，会被写成τ。他们在泡利表像（σz表像）可以写成： σ 1 = σ x = [ 0 1 1 0 ]…
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter σ {\displaystyle \sigma } (sigma), and occasionally by τ {\displaystyle \tau } (tau) when used in connection with isospin symmetries.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学中，张量积，记为 ⊗ {\displaystyle \otimes } ，可以应用于不同的上下文中如向量、矩阵、张量、向量空间、代数、拓扑向量空间和模。在各种情况下这个符号的意义是同样的:最一般的双线性运算。在某些上下文中也叫做外积。
In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) {\displaystyle (v,w)} , where v ∈ V , w ∈ W {\displaystyle v\in V,w\in W} , to an element of V ⊗ W {\displaystyle V\otimes W} denoted v ⊗ w {\displaystyle v\otimes w} .
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在量子力学里，开放量子系统的量子相干性会因为与外在环境发生量子纠缠而随着时间逐渐丧失，这效应称为量子退相干（英语：Quantum decoherence），又称为量子去相干。量子退相干是量子系统与环境因量子纠缠而产生的后果。由于量子相干性而产生的干涉现象会因为量子退相干而变得消失无踪。量子退相干促使系统的量子行为变迁成为经典行为，这过程称为“量子至经典变迁”（quantum-to-classical transition）。德国物理学者汉斯·泽贺最先于1970年提出量子退相干的概念。自1980年以来，量子退相干已成为热门研究论题。 实际而言，不存在孤立系统，特别是不存在孤立宏观系统，通过某种方式，每个量子系统都会持续地与外在环境耦合，发生量子纠缠，从而形成纠缠态。因此，量子退相干可以视为存在于量子系统内部的相干性随着时间流易而退定域（delocalize）至量子系统与环境所组成的纠缠系统，换句话说，量子系统内部的几个成分彼此之间的相位关系，会逐渐地退定域至整个系统，也就是说，量子系统的相位信息会持续地泄露至环境，从而有效地促使伴随着相干性的干涉现象消失无踪。 量子退相干能够解释为什么不会观察到干涉现象，但是，量子退相干能否解释波函数坍缩的后果，这论题至今仍旧存在巨大争议，一个很重要的原因就是，很难将这论题跟量子力学的诠释做分割，而人们各自有各自青睐的诠释。量子退相干是一种标准量子力学效应，关于它是否能够解释波函数坍缩的后果，存在有很多种观点，大多数过于乐观或过于悲观的观点，皆可追溯至对于量子退相干运作范围的误解。 量子退相干不是一种量子力学诠释，而是利用量子力学分析获得的结果。它严格遵守量子力学，并没有对量子力学的基础表述做任何修改。很多完成的量子实验已证实量子退相干的存在与正确性。 在实现量子计算机方面，量子退相干是一种必须面对的挑战，因为量子计算机的运作倚赖维持量子相干态的演化不被环境搅扰。简言之，必需良好维持量子相干态与管控量子退相干，才能够实际进行量子运算。
Quantum decoherence is the loss of quantum coherence. It involves generally a loss of information of a system to its environment. Quantum decoherence has been studied to understand how quantum systems convert to systems that can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is one of the primary practical applications of the concept.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在概率论和信息论中，两个随机变量的互信息（mutual Information，MI）度量了两个变量之间相互依赖的程度。具体来说，对于两个随机变量，MI是一个随机变量由于已知另一个随机变量而减少的“信息量”（单位通常为比特）。互信息的概念与随机变量的熵紧密相关，熵是信息论中的基本概念，它量化的是随机变量中所包含的“信息量”。 MI不仅仅是度量实值随机变量和线性相关性(如相关系数)，它更为通用。互信息决定了随机变量 ( X , Y ) {\displaystyle {\displaystyle (X,Y)}} 的联合分布与 X {\displaystyle X} 和 Y {\displaystyle Y} 的边缘分布的乘积之间的差异。MI是点互信息（Pointwise Mutual Information，PMI）的期望。克劳德·香农在他的论文A Mathematical Theory of Communication中定义并分析了这个度量，但是当时他并没有将其称为“互信息”。这个词后来由罗伯特·法诺创造。互信息也称为信息增益。
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair ( X , Y ) {\displaystyle (X,Y)} is from the product of the marginal distributions of X {\displaystyle X} and Y {\displaystyle Y} .
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View content license ↗ 阿达马变换（Hadamard transform），或称沃尔什-阿达玛转换，是一种广义傅立叶变换（Fourier transforms），作为变换编码的一种在影片编码当中使用有很久的历史。在近来的影片编码标准中，阿达马变换多被用来计算SATD(一种影片残差信号大小的衡量)。 在数字信号处理大型集成电路算法的领域中，阿达马变换是一种简单且重要的算法之一，主要能针对频谱做快速的分析。
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on a tuple of 2m numbers. The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2. It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (French: [adamaʁ]), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh.
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View content license ↗ 量子线路或沿用经典称呼而称作量子电路，是在抽象概念下，对于量子信息存储单元（例如量子比特）进行操作的线路。组成包括了于量子信息存储单元、线路（时间线），以及各种逻辑门；最后常需要量子测量将结果读取出来。电路需要能够对量子比特执行以实现量子计算的最小操作集被称为迪文森佐判据（DiVincenzo's criteria）。 编写电路时，水平轴是时间，从左侧开始并在右侧结束。水平线是量子比特，双线代表经典比特。由这些线连接的项目是在量子比特上执行的操作，例如量子闸和测量。这些线定义了事件的顺序，通常不是实体电缆。 量子线路器件的图形描述是使用彭罗斯图形符号 (Penrose graphical notation)的变体来描述的。理查德·费曼 (Richard Feynman) 在1986年使用了量子线路符号的早期版本。
In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly other actions. The minimum set of actions that a circuit needs to be able to perform on the qubits to enable quantum computation is known as DiVincenzo's criteria. Circuits are written such that the horizontal axis is time, starting at the left hand side and ending at the right. Horizontal lines are qubits, doubled lines represent classical bits. The items that are connected by these lines are operations performed on the qubits, such as measurements or gates. These lines define the sequence of events, and are usually not physical cables. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation. Richard Feynman used an early version of the quantum circuit notation in 1986.
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View content license ↗ 量子力学中，以自旋物理与核磁共振专家费利克斯·布洛赫命名的布洛赫球面是一种对于双态系统中纯态空间的几何表示法。在讨论量子比特的场合上常常运用到。
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. Mathematically each quantum mechanical system is associated with a separable complex Hilbert space H {\displaystyle H} . A pure state of a quantum system is represented by a non-zero vector ψ {\displaystyle \psi } in H {\displaystyle H} . The vectors ψ {\displaystyle \psi } and λ ψ {\displaystyle \lambda \psi } (with λ {\displaystyle \lambda } a non-zero complex number) represent the same state. A system with n mutually orthogonal quantum states can be described by a Hilbert space of dimension n.
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View content license ↗ 量子门（或量子逻辑门）在量子计算和特别是量子线路的计算模型里面是一个基本的，操作一个小数量量子比特的量子线路。它是量子线路的基础，就像传统逻辑门跟一般数字线路之间的关系。 与多数传统逻辑门不同，量子逻辑门是可逆的。然而，传统的计算可以只使用可逆的门表示。举例来说，可逆的Toffoli门可以实做所有的布尔函数。这个门有一个直接等同的量子门，也因此代表量子线路可以模拟所有传统线路的操作。 量子逻辑门使用酉矩阵表示。就像传统的逻辑门一样，它们是针对一个或两个比特进行操作，常见的量子逻辑门也是针对一个或两个量子比特进行操作。这也代表这一些量子门可以使用 2 × 2 或者 4 × 4 的酉矩阵表示。
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. According to quantum mechanics, a quantum system can only either evolve unitarily according to the Schrödinger equation, or be measured (sometimes called "Observed"). Quantum gates describe these unitary transformations, that occur when the system is not being measured. The expression "quantum gate" appears in relation to quantum processors, and in this context they are the logical operations that the quantum computer at the assembly language-level of abstraction (e.g. OpenQASM) can perform on the quantum data (qubits or quantum states) that they process. They can also be whole algorithms (e.g. the Quantum Fourier transform) if such algorithms contain no measurement operations.
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View content license ↗ 贝尔态是量子资讯科学中使用到的概念，是两个量子位元众多的量子态中的一种，而且是最简单单纯的一种量子纠缠态。贝尔态是一种纠缠并且归一向量基底的形式。归一意思是指粒子在该状态的全域几率是 1: ⟨ Φ | Φ ⟩ = 1 {\displaystyle \langle \Phi |\Phi \rangle =1} 。纠缠态则是指基底独立的叠加态.。由于这样的叠加态，当测量这个量子位元的时候，测量行为会使该量子位元在给定的几率下收敛成基底之一的状态。然而因为处于纠缠态，测量其中之一的量子位元即等同于马上给定了另外一个量子位元的测定值，而此测定值会是该贝尔态中给定该量子位元的值。
In quantum information science, the Bell's states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell's states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particles being in one of the mentioned states is 1: ⟨ Φ | Φ ⟩ = 1 {\displaystyle \langle \Phi |\Phi \rangle =1} . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the GHZ state for three or more subsystems.
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View content license ↗ 量子统计力学中，冯纽曼熵（英语：von Neumann entropy）是经典体系吉布士熵概念的拓展延伸。体系的冯纽曼熵为 S = ˙ − T r ( ρ ln ρ ) , {\displaystyle S{\dot {=}}-\mathrm {Tr} (\rho \ln \rho ),} 其中Tr表示求迹， ρ {\displaystyle \rho } 是体系的密度矩阵。
In physics, the von Neumann entropy, named after John von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is S = − tr ( ρ ln ρ ) , {\displaystyle S=-\operatorname {tr} (\rho \ln \rho ),} where tr {\displaystyle \operatorname {tr} } denotes the trace and ln {\displaystyle \operatorname {ln} } denotes the matrix version of the natural logarithm.
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View content license ↗ 量子力学中，林德布拉德方程（英语：Lindblad equation）是最常用的主方程之一，其常用来描述密度矩阵的含时演化（通常是非幺正的）。 薛定谔方程是林德布拉德方程在特殊情况的推论。薛定谔方程展现的是系统的态矢量随时间的演化，只能处理纯态演化，而林德布拉德方程所展现的是系统的密度矩阵随时间的演化（密度矩阵可以表征系统的混态），所以林德布拉德方程比薛定谔方程更加一般。
In quantum mechanics, the Franke–Gorini–Kossakowski–Sudarshan–Lindblad (FGKSL) master equation (named after Valentin Franke, Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation, master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contact with their surroundings. The resulting dynamics are no longer unitary, but still satisfy the property of being trace-preserving and completely positive for any initial condition. The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation.
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View content license ↗ 在数学的文献中，线性标准转换（英语：Linear Canonical Transform，简称LCT）也称作线性正则变换、ABCD转换、广义Fresnel变换等。在汉米尔顿力学中，线性标准转换是积分变换的一个代表家族，并且能够将许多经典的转换进行广义化，例如傅立叶变换、分数傅立叶变换、拉普拉斯变换、菲涅尔转换(电磁波在空气中传播)、高斯-魏尔斯特拉斯转换、包格曼转换等等。此转换提供了这些最常使用的线性转换一个统一框架，并且在光学、信号转换以及系统响应领域中都提供一般化的概念。尤其从系统工程的角度看来，线性标准转换提供一个强大的光学系统设计和分析的工具。 此转换有四维变数的线性积分转换和一个限制条件，因此实际上是一个三维自由度的积分变换的家族。 在群论中，线性标准转换属于特殊线性群(SR(2))在时频域上的一个作用群。
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(C) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered.
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View content license ↗ 科恩－沈吕九方程（英语：Kohn–Sham equation，简称科恩－沈方程）在密度泛函理论里面指的是与真实体系相关的虚拟体系所满足的薛定谔方程。该虚拟体系中的粒子（通常是电子）在无相互作用的有效势场中运动，粒子密度在空间各点均与真实系统相同。科恩－沈吕九方程中的有效势通常用 v s ( r ) {\displaystyle v_{\rm {s}}(\mathbf {r} )} 或 v e f f ( r ) {\displaystyle v_{\rm {eff}}(\mathbf {r} )} ) 来表示，称为科恩－沈势。虚拟系统中的粒子是彼此无相互作用的费米子，因此科恩－沈方程的精确解为单个斯莱特行列式，行列式中的轨道则称为科恩－沈轨道，每一个科恩－沈轨道都可以表示为原子轨道的线性组合，也可以按照基函数展开。科恩－沈方程的形式如下： ( − ℏ 2 2 m ∇…
The Kohn-Sham equations are a set of mathematical equations used in quantum mechanics to simplify the complex problem of understanding how electrons behave in atoms and molecules. They introduce fictitious non-interacting electrons and use them to find the most stable arrangement of electrons, which helps scientists understand and predict the properties of matter at the atomic and molecular scale.
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View content license ↗ 朗道量子化是指均匀磁场中带电粒子的回旋轨道发生的量子化。这些带电粒子能量在一系列分立的数值中取值，形成朗道能级。朗道能级是简并的，每一能级上电子的电子数量与外加磁场的强度成正比。由朗道量子化可以得出外磁场会导致材料中电子性质的振荡。这一理论是由苏联物理学家列夫·朗道于1930年提出的。
In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels. These levels are degenerate, with the number of electrons per level directly proportional to the strength of the applied magnetic field. It is named after the Soviet physicist Lev Landau who developed the theory in 1930. Landau quantization contributes towards magnetic susceptibility of metals, known as Landau diamagnetism. Under strong magnetic fields, Landau quantization leads to oscillations in electronic properties of materials as a function of the applied magnetic field known as the De Haas–Van Alphen and Shubnikov–de Haas effects. Landau quantization is a key ingredient in explanation of the integer quantum Hall effect.
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View content license ↗ 卡皮查-狄拉克效应（英语：Kapitsa–Dirac effect，简称KDE）是一种量子现象，即拥有良好空间相干性的粒子束（一般情况下为电子束）通过光场驻波时发生衍射，它是受激康普顿散射的一种特殊情况。该效应由保罗·狄拉克与彼得·列昂尼多维奇·卡皮察于1933年首次提出。 KDE可以通过德布罗意1924年的波粒二象性理论得到解释。因为粒子具有波的性质，粒子束会被以驻波形式存在的电磁场的空间周期性结构所散射，散射波又会与自身发生干涉（粒子束强度随空间位置而变，就像一般的光学衍射一样，产生极大与极小峰）。 实验上观测KDE要求高度相干光束，这在激光发明前是不可能实现的。2001年进行的实验证明了所猜想的衍射峰。
The Kapitza–Dirac effect is a quantum mechanical effect consisting of the diffraction of matter by a standing wave of light, in complete analogy to the diffraction of light by a periodic grating, but with the role of matter and light reversed. The effect was first predicted as the diffraction of electrons from a standing wave of light by Paul Dirac and Pyotr Kapitsa (or Peter Kapitza) in 1933. The effect relies on the wave–particle duality of matter as stated by the de Broglie hypothesis in 1924. The matter-wave diffraction by a standing wave of light was first observed using a beam of neutral atoms. Later, the Kapitza-Dirac effect as originally proposed was observed in 2001.
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View content license ↗ 数学上，卡迪森-辛格问题（英语：Kadison–Singer problem）于1959年提出，有关泛函分析，问某个特定C*-代数上的任意线性泛函，延拓到另一个较大的C*-代数时，是仅有唯一的可能，抑或可以有多个不同的延拓。2013年，问题得到解决，答案为肯定（即唯一）。 问题源出1940年代保罗·狄拉克对量子力学理论基础的研究。1959年，理查德·卡迪森与艾沙道尔·辛格给出严格的问题叙述。此后，发现纯数学、应用数学、工程学、计算机科学等学科的多个未解问题，皆与卡迪森-辛格问题等价。卡迪森、辛格，以及日后多个作者，都相信问题答案为否定（即不唯一），然而于2013年，亚当·马库斯、丹尼尔·斯皮尔曼、尼基·斯里瓦斯塔瓦合著论文给出肯定的答案。翌年，三人因此获SIAM颁发波利亚奖。 马-斯-斯三氏皆为计算机科学家，本来并非研究C*-代数。马库斯甚至称自己在解决该问题后，“仍无法用C*-代数的语言来描述它”。解决问题的转捩点，是乔尔·安德森（Joel Anderson）将其重写成不牵涉C*-代数理论的等价形式。安德森于1979年证明，其“铺砌猜想”（英语：paving conjecture）与卡迪森-辛格问题等价。该猜想仅牵涉有限维希尔伯特空间的算子，而相比之下，原问题的空间则是无穷维。此后，亦有其他学者，如尼克·威佛（Nik Weaver），在有限维空间中，给出其他等价问法。威佛的版本吸引了马-斯-斯三氏研究。而此版本用交织多项式族（英语：interlacing family）获解决。
In mathematics, the Kadison–Singer problem, posed in 1959, was a problem in functional analysis about whether certain extensions of certain linear functionals on certain C*-algebras were unique. The uniqueness was proved in 2013. The statement arose from work on the foundations of quantum mechanics done by Paul Dirac in the 1940s and was formalized in 1959 by Richard Kadison and Isadore Singer. The problem was subsequently shown to be equivalent to numerous open problems in pure mathematics, applied mathematics, engineering and computer science. Kadison, Singer, and most later authors believed the statement to be false, but, in 2013, it was proven true by Adam Marcus, Daniel Spielman and Nikhil Srivastava, who received the 2014 Pólya Prize for the achievement. The solution was made possible by a reformulation provided by Joel Anderson, who showed in 1979 that his "paving conjecture", which only involves operators on finite-dimensional Hilbert spaces, is equivalent to the Kadison–Singer problem.
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View content license ↗ 多体问题为一大类物理问题的通称。那些问题与大量粒子构成的微观系统有关，且粒子之间有相互作用。要精确描述这些微观系统，将会用到量子力学。三体以上的系统即被视为多体系统，不过因为三体和四体可以用特定的方法处理，有时会被归类为少体系统。在这样的量子系统中，粒子之间不断相互作用，产生量子相关性以及纠缠。因此，系统的波函数很复杂，并含有大量信息，常常无法进行精确或可分析的计算。所以，多体理论物理学常常必须依赖针对问题的一组近似，并且是最多计算的科学领域之一。
The quantum many-body problem is a general name for a vast category of physical problems pertaining to deriving the behavior of multi-particle systems using fundamental quantum-mechanical principles. The goal of many-body physics is to find new principles to describe macroscopic systems, using principles that pertain to microscopic systems.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 分子电子跃迁表示分子中价电子从一个能级因为吸收能量时，跃迁到一个更高的能级；或者释放能量，跃迁到更低的能级的过程。如果起始能级的能量比最终能级的能量高，原子便会释放能量（通常以电磁波的形式发放）。相反，如果起始能级的能量较低，原子便会吸收能量。释放与吸收的能量等于这两个能级的能量之差。 在此过程中的能量变化提供了分子结构的信息，并决定了许多分子性质如颜色。有关电子跃迁的能量和辐射频率的关系由普朗克定律决定。 一般，我们应用电子跃迁来说明单个原子。当讨论多原子分子时，我们应用分子轨道理论。也可以视单个原子为单原子分子，将各种情况的电子跃迁统一到分子电子跃迁的框架下来。这里的能级是基于分子轨道理论提出的。 有机化合物中的电子跃迁在电磁频谱的紫外区或可见光区发生，可以由UV/VIS光谱测得。在HOMO σ带处的电子可被激发到 LUMO 的σ带。这个过程被写作σ → σ*跃迁。同样有电子从π键轨道激发至反π键轨道π*，写作π → π*跃迁。助色基团的自由电子对被写为孤对电子n，孤电子对有自己的跃迁，如芳香π键跃迁。下列是已存在的分子电子跃迁： σ → σ* π → π* n → σ* n → π* 芳香 π → 芳香 π*
In theoretical chemistry, molecular electronic transitions take place when electrons in a molecule are excited from one energy level to a higher energy level. The energy change associated with this transition provides information on the structure of the molecule and determines many of its properties, such as colour. The relationship between the energy involved in the electronic transition and the frequency of radiation is given by Planck's relation.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 介观物理学（mesoscopic physics）是凝聚态物理学的一个分支学科，研究的是尺寸范围从纳米级到微米级之间的材料 。“介观（mesoscopic）”这个词汇，是由Van Kampen于1976年所创，指的是介乎于微观和宏观之间的尺度。介观物理学的研究对宏观物体的小型化（比如微电子器件）有重要要意义。介观尺度和纳米材料的典型尺度（10-9~10-7m）有很大重合，因此介观物理学与纳米技术领域有着密切的联系。这一领域的研究常称为“介观物理和纳米科技”。 从微观尺度上看，介观材料和宏观材料一样含有大量原子，但它和宏观材料具有很不一样的性质：宏观物体遵循经典力学定律，样本的性质由构成材料的平均值给出；而介观物体的电子行为通常需要在量子力学层面进行建模，其样本性质会受到涨落的影响而偏离平均值 。
Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in size between the nanoscale for a quantity of atoms (such as a molecule) and of materials measuring micrometres. The lower limit can also be defined as being the size of individual atoms. At the microscopic scale are bulk materials. Both mesoscopic and macroscopic objects contain many atoms. Whereas average properties derived from constituent materials describe macroscopic objects, as they usually obey the laws of classical mechanics, a mesoscopic object, by contrast, is affected by thermal fluctuations around the average, and its electronic behavior may require modeling at the level of quantum mechanics. A macroscopic electronic device, when scaled down to a meso-size, starts revealing quantum mechanical properties. For example, at the macroscopic level the conductance of a wire increases continuously with its diameter.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 位置空间与动量空间是物理学中一对联系紧密的矢量空间。 位置空间（或称实空间、坐标空间）是空间中所有物体的位矢r的集合。这个空间通常是三维的。位矢定义了空间中的一个点。如果位矢随时间会发生变化的话，那么它就可以描绘出一个路径或一个面，如粒子的运动轨迹。 动量空间是空间中所有物体的动量矢量的集合。这个空间通常也是三维的。一个物体的动量可以反映它的运动情况。无论在经典力学还是在量子力学中，动量都是非常重要的一个概念。然而，依据量子力学的德布罗意关系，p = ħk，一个自由粒子的动量正比于波矢。系统的所有波矢的集合构成波矢空间。在不严格区分动量与波矢时，这两个概念可以混用。但在晶体中，德布罗意关系并不成立。 位置与动量间的对偶性是庞特里亚金对偶性的一个例子。 位矢r的量纲为[L]，动量p的量纲为[M][L][T]−1，波矢k的量纲为[L]−1，因而类比于角频率ω之于时间t，k可以视为系统空间上的频率。一个系统的物理现象既可以用位矢描述，也可以用动量描述。两种描述方式所提供的系统信息是等价的。通常利用r描述更为直观，但在固体物理学中，k更为常用。
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with dimension of mass⋅length⋅time−1. Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
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