量子互文性

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"Contextuality" redirects here. For the principle of contextuality in Linguistics, see Context (language use).

Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured (the measurement context). More formally, the measurement result (assumed pre-existing) of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

量子互文性是量子力学现象学的一个特征，据此，对量子观测变量的测量不能简单地被认为是揭示了预先存在的值。在现实的隐变量理论中，任何试图这样做的尝试都会导致数值取决于同时被测量的其他（兼容的）观测变量的选择（测量背景）。更正式地说，一个量子观测物的测量结果（假定是预先存在的）取决于在同一测量集合中的其他互换观测物。


Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.[1][2] The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,[3] graph theory,[4] hypergraphs,[5] algebraic topology,[6] and probabilistic couplings.[7]

语境性首先被贝尔-科琴-斯佩克定理证明是量子现象学的一个特征。语境性的研究已经发展成为量子基础的一个主要兴趣点，因为这一现象具体化了量子理论的某些非经典和反直觉的方面。从舍理论、图论、超图、代数拓扑学和概率耦合学的角度，已经开发了一些强大的数学框架来研究和更好地理解语境性。


Nonlocality, in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from the Fine–Abramsky–Brandenburger theorem.[8][3]

在贝尔定理的意义上，非局域性可以被看作是更普遍的语境性现象的一个特例，在这种情况下，测量语境包含分布在空间上的分离区域的测量。这是由Fine-Abramsky-Brandenburger定理得出的。


Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.[9][10][11][12] Contemporary research has increasingly focused on exploring its utility as a computational resource.

量子语境性已被确定为量子计算速度提升和量子优势的来源。当代的研究越来越注重探索其作为计算资源的效用。


Kochen and Specker 科亨和斯佩克

Main article: Kochen–Specker theorem

The need for contextuality was discussed informally in 1935 by Grete Hermann,[13] but it was more than 30 years later when Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that realistic noncontextual hidden variable theories cannot reproduce the empirical predictions of quantum mechanics.[14] Such a theory would suppose the following.

1935年，格雷特-赫尔曼非正式地讨论了语境性的必要性，但30多年后，西蒙-B-科亨和恩斯特-斯佩克，以及约翰-贝尔分别构建了证明，即任何能够解释量子力学现象学的现实隐变量理论对于希尔伯特空间维数为3或更大的系统都是语境性的。恩斯特-斯佩克定理证明，现实的非情境性隐变量理论不能重现量子力学的经验预测。这样的理论会假设以下情况。


  1. All quantum-mechanical observables may be simultaneously assigned definite values (this is the realism postulate, which is false in standard quantum mechanics, since there are observables which are undefinite in every given quantum state). These global value assignments may deterministically depend on some 'hidden' classical variable which, in turn, may vary stochastically for some classical reason (as in statistical mechanics). The measured assignments of observables may therefore finally stochastically change. This stochasticity is however epistemic and not ontic as in the standard formulation of quantum mechanics.所有的量子力学观测变量可以同时被赋予确定的值（这是现实主义假设，在标准量子力学中是错误的，因为有一些观测变量在每个给定的量子状态下都是不确定的）。这些全局值的分配可能确定性地取决于一些 "隐藏的 "经典变量，而这些变量又可能由于一些经典的原因而随机变化（如在统计力学中）。因此，观察物的测量赋值最终可能会随机地变化。然而，这种随机性是认识上的，而不是像量子力学的标准表述那样是本体的。
  2. Value assignments pre-exist and are independent of the choice of any other observables which, in standard quantum mechanics, are described as commuting with the measured observable, and they are also measured.价值分配预先存在，并且独立于任何其他观察物的选择，在标准量子力学中，这些观察物被描述为与被测量的观察物互换，并且它们也被测量。
  3. Some functional constraints on the assignments of values for compatible observables are assumed (e.g., they are additive and multiplicative, there are however several versions of this functional requirement).假设对兼容的观测变量的赋值有一些功能约束（例如，它们是加法和乘法的，然而这种功能要求有几个版本）。


In addition, Kochen and Specker constructed an explicitly noncontextual hidden variable model for the two-dimensional qubit case in their paper on the subject,S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", Journal of Mathematics and Mechanics 17, 59–87 (1967) thereby completing the characterisation of the dimensionality of quantum systems that can demonstrate contextual behaviour. Bell's proof invoked a weaker version of Gleason's theorem, reinterpreting the theorem to show that quantum contextuality exists only in Hilbert space dimension greater than two.Gleason, A. M, "Measures on the closed subspaces of a Hilbert space", Journal of Mathematics and Mechanics 6, 885–893 (1957).

此外，科亨和斯佩克在他们的论文中为二维量子比特的情况构建了一个明确的非语境的隐性变量模型，即S. Kochen和E.P. Specker,"量子力学中的隐藏变量问题"，《数学与力学杂志》17，59-87（1967）从而完成了可以证明上下文行为的量子系统的维度特征。贝尔的证明引用了格里森定理的一个弱化版本，重新解释了该定理，以表明量子情境性只存在于希尔伯特空间维度大于2的地方。格里森，A. M，"希尔伯特空间封闭子空间上的度量"，《数学与力学杂志》6，885-893（1957）。


Frameworks for contextuality 语境性的框架

Sheaf-theoretic framework 片断理论框架

The sheaf-theoretic, or Abramsky–Brandenburger, approach to contextuality initiated by Samson Abramsky and Adam Brandenburger is theory-independent and can be applied beyond quantum theory to any situation in which empirical data arises in contexts. As well as being used to study forms of contextuality arising in quantum theory and other physical theories, it has also been used to study formally equivalent phenomena in logic,[15] relational databases,[16] natural language processing,[17] and constraint satisfaction.[18]

参孙 · 阿布拉姆斯基和亚当 · 布兰登伯格提出的语境性方法是一种独立于理论的方法，可以超越量子理论应用于经验数据产生于语境的任何情况。除了用于研究量子理论和其他物理理论中出现的情境性形式外，它还被用于研究逻辑学、关系数据库、自然语言处理和约束满足中的形式等效现象。


In essence, contextuality arises when empirical data is locally consistent but globally inconsistent. Analogies may be drawn with impossible figures like the Penrose staircase, which in a formal sense may also be said to exhibit a kind of contextuality.[1]

从本质上讲，当经验数据局部一致但全局不一致的时候，就会出现语境性。可以用彭罗斯阶梯这样的不可能数字进行类比，在形式上也可以说它表现出一种语境性。


This framework gives rise in a natural way to a qualitative hierarchy of contextuality.

这个框架以一种自然的方式产生了背景性的质的层次结构。

  • (Probabilistic) contextuality may be witnessed in measurement statistics, e.g. by the violation of an inequality. A representative example is the KCBS proof of contextuality.
  • Logical contextuality may be witnessed in the 'possibilistic' information about which outcome events are possible and which are not possible. A representative example is Hardy's nonlocality proof of nonlocality.
  • Strong contextuality is a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be reproduced by a mixture of global value assignments, strong contextuality arises when no global value assignment is even compatible with the possible outcome events. A representative example is the original Kochen–Specker proof of contextuality.
  • (Probabilistic) contextuality may be witnessed in measurement statistics, e.g. by the violation of an inequality. A representative example is the KCBS proof of contextuality.
  • Logical contextuality may be witnessed in the 'possibilistic' information about which outcome events are possible and which are not possible. A representative example is Hardy's nonlocality proof of nonlocality.
  • Strong contextuality is a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be reproduced by a mixture of global value assignments, strong contextuality arises when no global value assignment is even compatible with the possible outcome events. A representative example is the original Kochen–Specker proof of contextuality.


  • (概率)情境可见于测量统计，例如。违反了不平等原则。一个具有代表性的例子就是 KCBS 的语境性证明。
  • 在「可能性」资料中，我们可以看到哪些结果事件是可能的，哪些是不可能的。一个典型的例子是哈代的非定域性证明。
  • 强语境性是语境性的最大形式。当测量统计数据不能被全局值分配的混合物复制时，(概率)语境性就出现了，当全局值分配甚至不能与可能的结果事件兼容时，强语境性就出现了。一个具有代表性的例子是 Kochen-Specker 关于语境性的原始证明。

Each level in this hierarchy strictly includes the next. An important intermediate level that lies strictly between the logical and strong contextuality classes is all-versus-nothing contextuality,[15] a representative example of which is the Greenberger–Horne–Zeilinger proof of nonlocality.

Each level in this hierarchy strictly includes the next. An important intermediate level that lies strictly between the logical and strong contextuality classes is all-versus-nothing contextuality, a representative example of which is the Greenberger–Horne–Zeilinger proof of nonlocality.

此层次结构中的每个级别都严格包括下一个级别。严格介于逻辑上下文类和强上下文类之间的一个重要的中间层次是全有对全无上下文，Greenberger-Horne-Zeilinger 非定域性证明就是一个代表性的例子。

Graph and hypergraph frameworks

Adán Cabello, Simone Severini, and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories.[19] Within this framework experimental scenarios are described by graphs, and certain invariants of these graphs were shown have particular physical significance. One way in which contextuality may be witnessed in measurement statistics is through the violation of noncontextuality inequalities (also known as generalized Bell inequalities). With respect to certain appropriately normalised inequalities, the independence number, Lovász number, and fractional packing number of the graph of an experimental scenario provide tight upper bounds on the degree to which classical theories, quantum theory, and generalised probabilistic theories, respectively, may exhibit contextuality in an experiment of that kind. A more refined framework based on hypergraphs rather than graphs is also used.[5]

Adán Cabello, Simone Severini, and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories.A. Cabello, S. Severini, A. Winter, Graph-Theoretic Approach to Quantum Correlations", Physical Review Letters 112 (2014) 040401. Within this framework experimental scenarios are described by graphs, and certain invariants of these graphs were shown have particular physical significance. One way in which contextuality may be witnessed in measurement statistics is through the violation of noncontextuality inequalities (also known as generalized Bell inequalities). With respect to certain appropriately normalised inequalities, the independence number, Lovász number, and fractional packing number of the graph of an experimental scenario provide tight upper bounds on the degree to which classical theories, quantum theory, and generalised probabilistic theories, respectively, may exhibit contextuality in an experiment of that kind. A more refined framework based on hypergraphs rather than graphs is also used.

= = = Adán Cabello，Simone Severini，and Andreas Winter 提出了一个通用的图论框架来研究不同物理理论的语境性。Cabello, S. Severini, A.温特，量子关联的图论方法”，物理评论快报112(2014)040401。在这个框架下，实验场景用图形来描述，图形中的某些不变量具有特定的物理意义。在测量统计中可以看到语境性的一种方式是通过违反非语境不等式(也称为广义 Bell 不等式)。对于某些适当规范化的不等式，一个实验场景图的独立数、洛瓦兹数和分数包装数分别提供了经典理论、量子理论和广义概率理论在此类实验中展现情境性程度的紧上界。还使用了基于超图而不是图的更精确的框架。

Contextuality-by-Default (CbD) framework

In the CbD approach,[20][21][22] developed by Ehtibar Dzhafarov, Janne Kujala, and colleagues, (non)contextuality is treated as a property of any system of random variables, defined as a set [math]\displaystyle{ \mathcal{R}=\left\{ R_{q}^{c}:q\in Q,q\prec c,c\in C\right\} }[/math] in which each random variable [math]\displaystyle{ R_{q}^{c} }[/math] is labeled by its content [math]\displaystyle{ q }[/math], the property it measures, and its context [math]\displaystyle{ c }[/math], the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); [math]\displaystyle{ q\prec c }[/math] stands for "[math]\displaystyle{ q }[/math] is measured in [math]\displaystyle{ c }[/math]". The variables within a context are jointly distributed, but variables from different contexts are stochastically unrelated, defined on different sample spaces. A (probabilistic) coupling of the system [math]\displaystyle{ \mathcal{R} }[/math] is defined as a system [math]\displaystyle{ S }[/math] in which all variables are jointly distributed and, in any context [math]\displaystyle{ c }[/math], [math]\displaystyle{ R^{c}=\left\{ R_{q}^{c}:q\in Q,q\prec c\right\} }[/math] and [math]\displaystyle{ S^{c}=\left\{ S_{q}^{c}:q\in Q,q\prec c\right\} }[/math] are identically distributed. The system  is considered noncontextual if it has a coupling [math]\displaystyle{ S }[/math] such that the probabilities [math]\displaystyle{ \Pr\left[S_{q}^{c}=S_{q}^{c'}\right] }[/math] are maximal possible for all contexts [math]\displaystyle{ c,c' }[/math] and contents [math]\displaystyle{ q }[/math] such that [math]\displaystyle{ q\prec c,c' }[/math]. If such a coupling does not exist, the system is contextual. For the important class of cyclic systems of dichotomous ([math]\displaystyle{ \pm1 }[/math]) random variables,  [math]\displaystyle{ \mathcal{C}_{n}=\left\{ \left(R_{1}^{1},R_{2}^{1}\right),\left(R_{2}^{2},R_{3}^{2}\right),\ldots,\left(R_{n}^{n},R_{1}^{n}\right)\right\} }[/math] ([math]\displaystyle{ n\geq2 }[/math]), it has been shown[23][24] that such a system is noncontextual if and only if

In the CbD approach, developed by Ehtibar Dzhafarov, Janne Kujala, and colleagues, (non)contextuality is treated as a property of any system of random variables, defined as a set \mathcal{R}=\left\{ R_{q}^{c}:q\in Q,q\prec c,c\in C\right\} in which each random variable R_{q}^{c} is labeled by its content q, the property it measures, and its context c, the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); q\prec c stands for "q is measured in c". The variables within a context are jointly distributed, but variables from different contexts are stochastically unrelated, defined on different sample spaces. A (probabilistic) coupling of the system \mathcal{R} is defined as a system S in which all variables are jointly distributed and, in any context c, R^{c}=\left\{ R_{q}^{c}:q\in Q,q\prec c\right\}  and S^{c}=\left\{ S_{q}^{c}:q\in Q,q\prec c\right\}  are identically distributed. The system  is considered noncontextual if it has a coupling S such that the probabilities \Pr\left[S_{q}^{c}=S_{q}^{c'}\right] are maximal possible for all contexts c,c' and contents q such that q\prec c,c'. If such a coupling does not exist, the system is contextual. For the important class of cyclic systems of dichotomous (\pm1) random variables,  \mathcal{C}_{n}=\left\{ \left(R_{1}^{1},R_{2}^{1}\right),\left(R_{2}^{2},R_{3}^{2}\right),\ldots,\left(R_{n}^{n},R_{1}^{n}\right)\right\} (n\geq2), it has been shown that such a system is noncontextual if and only if

= = = Contextuality-by-Default (CbD) framework = = = 在由 Ehtibar Dzhafarov，Janne Kujala 和他的同事们开发的 CbD 方法中，(non) contextuality 被视为任何随机变量系统的一个性质，定义为集合数学{ r } = left { r { q }{ q }{ q }{ c } : q In q,其中每个随机变量 r _ { q } ^ { c }被标记为其内容 q，它测量的性质，以及它的上下文 c，记录它的记录环境的集合(包括但不限于它被记录在一起的其他随机变量) ;Q prec c 代表“ q 用 c 来衡量”。上下文中的变量是联合分布的，但来自不同上下文的变量是随机无关的，定义在不同的样本空间。系统数学{ r }的(概率)耦合定义为系统 s 中所有变量联合分布，在任意上下文 c 中，r ^ { c } = 左{ r _ { q } ^ { c } : q in q，q prec 右}和 s ^ { c } = 左{ s _ { q } ^ { c } : q in q，q prec 右}均匀分布。如果系统有一个耦合 s，使得左[ s _ { q } ^ { c } = s _ { q } ^ { c’}右]的概率对于所有上下文 c，c’和内容 q 都是最大的，则系统被认为是非上下文的。如果这种耦合不存在，则系统是上下文的。对于二值(pm1)随机变量循环系统的重要类，数学{ c }{ n } = 左{左(r _ {1} ^ {1} ，r _ {2} ^ {1}右) ，左(r _ {2} ^ {2} ，r _ {3} ^ {2}右) ，ldots，左(r _ { n } ^ { n } ，r _ {1}{ n }右)(n geq2) ，证明了这样的系统是非当且仅当

[math]\displaystyle{ D\left(\mathcal{C}_{n}\right)\leq\Delta\left(\mathcal{C}_{n}\right), }[/math]

D\left(\mathcal{C}_{n}\right)\leq\Delta\left(\mathcal{C}_{n}\right),

D left (mathcal { c }{ n } right) leq Delta left (mathcal { c }{ n } right) ,

where

where

在哪里

[math]\displaystyle{ \Delta\left(\mathcal{C}_{n}\right)=\left(n-2\right)+\left|R_{1}^{1}-R_{1}^{n}\right|+\left|R_{2}^{1}-R_{2}^{2}\right|+\ldots\left|R_{n}^{n-1}-R_{n}^{n}\right|, }[/math]

\Delta\left(\mathcal{C}_{n}\right)=\left(n-2\right)+\left|R_{1}^{1}-R_{1}^{n}\right|+\left|R_{2}^{1}-R_{2}^{2}\right|+\ldots\left|R_{n}^{n-1}-R_{n}^{n}\right|,

Delta left (mathcal { c }{ n } right) = left (n-2 right) + left | r _ {1} ^ {1}-r _ {1}{ n } ^ {右 | + 左 | r _ {2} ^ {2}{2}{右 | + ldots 左 | r _ { n }{ n-1}{ n }{ n } ^ { n }右 | ,

and

and

及

[math]\displaystyle{ D\left(\mathcal{C}_{n}\right)=\max\left(\lambda_{1}\left\langle R_{1}^{1}R_{2}^{1}\right\rangle +\lambda_{2}\left\langle R_{2}^{2}R_{3}^{2}\right\rangle +\ldots+\lambda_{n}\left\langle R_{n}^{n},R_{1}^{n}\right\rangle \right), }[/math]

D\left(\mathcal{C}_{n}\right)=\max\left(\lambda_{1}\left\langle R_{1}^{1}R_{2}^{1}\right\rangle +\lambda_{2}\left\langle R_{2}^{2}R_{3}^{2}\right\rangle +\ldots+\lambda_{n}\left\langle R_{n}^{n},R_{1}^{n}\right\rangle \right),

D left (mathcal { c }{ n } right) = max left (λ _ {1} left langle r _ {1} ^ {1} r _ {2} ^ {1}{1} right rangle + λ _ {2} left langle r _ {2} ^ {2} r _ {3} ^ {2} right rangle + ldots + λ _ { n } left langle r _ { n } ^ { n } ，r _ {1}{ n } right rangle right right) ,

with the maximum taken over all [math]\displaystyle{ \lambda_{i}=\pm1 }[/math] whose product is [math]\displaystyle{ -1 }[/math]. If [math]\displaystyle{ R_{q}^{c} }[/math] and [math]\displaystyle{ R_{q}^{c'} }[/math], measuring the same content in different context, are always identically distributed, the system is called consistently connected (satisfying "no-disturbance" or "no-signaling" principle). Except for certain logical issues,[7][21] in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to [math]\displaystyle{ D\left(\mathcal{C}_{n}\right)\leq n-2, }[/math]which includes the Bell/CHSH inequality ([math]\displaystyle{ n=4 }[/math]), KCBS inequality ([math]\displaystyle{ n=5 }[/math]), and other famous inequalities.[25] That nonlocality is a special case of contextuality follows in CbD from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable (this generalizes Arthur Fine's analysis of Bell's theorem). CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is strongly consistently connected, which means that the joint distributions of [math]\displaystyle{ \left\{ R_{q_{1}}^{c},\ldots,R_{q_{k}}^{c}\right\} }[/math] and [math]\displaystyle{ \left\{ R_{q_{1}}^{c'},\ldots,R_{q_{k}}^{c'}\right\} }[/math] coincide whenever [math]\displaystyle{ q_{1},\ldots,q_{k} }[/math] are measured in contexts [math]\displaystyle{ c,c' }[/math]. However, unlike most approaches to contextuality, CbD allows for inconsistent connectedness, with [math]\displaystyle{ R_{q}^{c} }[/math] and [math]\displaystyle{ R_{q}^{c'} }[/math] differently distributed. This makes CbD applicable to physics experiments in which no-disturbance condition is violated,[24][26] as well as to human behavior where this condition is violated as a rule.[27] In particular, Vctor Cervantes, Ehtibar Dzhafarov, and colleagues have demonstrated that random variables describing certain paradigms of simple decision making form contextual systems,[28][29][30] whereas many other decision-making systems are noncontextual once their inconsistent connectedness is properly taken into account.[27]

with the maximum taken over all \lambda_{i}=\pm1  whose product is -1. If R_{q}^{c} and R_{q}^{c'}, measuring the same content in different context, are always identically distributed, the system is called consistently connected (satisfying "no-disturbance" or "no-signaling" principle). Except for certain logical issues, in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to D\left(\mathcal{C}_{n}\right)\leq n-2,which includes the Bell/CHSH inequality (n=4), KCBS inequality (n=5), and other famous inequalities. That nonlocality is a special case of contextuality follows in CbD from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable (this generalizes Arthur Fine's analysis of Bell's theorem). CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is strongly consistently connected, which means that the joint distributions of \left\{ R_{q_{1}}^{c},\ldots,R_{q_{k}}^{c}\right\}  and \left\{ R_{q_{1}}^{c'},\ldots,R_{q_{k}}^{c'}\right\}  coincide whenever q_{1},\ldots,q_{k} are measured in contexts c,c'. However, unlike most approaches to contextuality, CbD allows for inconsistent connectedness, with R_{q}^{c} and R_{q}^{c'} differently distributed. This makes CbD applicable to physics experiments in which no-disturbance condition is violated, as well as to human behavior where this condition is violated as a rule. In particular, Vctor Cervantes, Ehtibar Dzhafarov, and colleagues have demonstrated that random variables describing certain paradigms of simple decision making form contextual systems, whereas many other decision-making systems are noncontextual once their inconsistent connectedness is properly taken into account.

最大乘以所有 lambda { i } = pm1，其乘积为 -1。如果在不同上下文中测量相同内容的 r _ { q } ^ { c }和 r _ { q } ^ { c’}总是同分布的，则称系统为一致连通(满足“无干扰”或“无信令”原则)。除了某些逻辑问题，在这种情况下，CbD 专门处理量子物理学中的语境问题。特别地，对于一致连通循环系统，上述非上下文准则退化为 d 左(数学{ c } _ { n }右) leq n-2，其中包括 Bell/CHSH 不等式(n = 4)、 KCBS 不等式(n = 5)和其他著名的不等式。非局部性是上下文关系在 CbD 中的一种特殊情况，它源于这样一个事实，即对于随机变量的联合分布等价于是同一个随机变量的可测函数(这推广了 Arthur Fine 对 Bell 定理的分析)。如果系统是强一致连通的，则 CbD 本质上与 Abramsky 的层理论方法的概率部分相一致，这意味着左{ r { q {1} ^ { c } ，ldots，r { q { k } ^ { c }右}和左{ r { q {1} ^ { c’} ，ldots，r { q { k } ^ { c’}右}的联合分布在 q {1} ，l } ，q { k }的联合分布在 c，c’中测量。然而，与大多数上下文方法不同的是，CbD 允许不一致的连接，而 r _ { q } ^ { c }和 r _ { q }是以不同的方式分布的。这使得生物多样性公约适用于违反无扰动条件的物理实验，以及违反这一条件的人类行为。特别是，Vctor Cervantes，Ehtibar Dzhafarov 和他的同事们已经证明，描述简单决策的某些范例的随机变量形成了语境系统，而其他许多决策系统一旦恰当地考虑到它们不一致的连通性，就是非语境的。

Operational framework

An extended notion of contextuality due to Robert Spekkens applies to preparations and transformations as well as to measurements, within a general framework of operational physical theories.[31] With respect to measurements, it removes the assumption of determinism of value assignments that is present in standard definitions of contextuality. This breaks the interpretation of nonlocality as a special case of contextuality, and does not treat irreducible randomness as nonclassical. Nevertheless, it recovers the usual notion of contextuality when outcome determinism is imposed.

An extended notion of contextuality due to Robert Spekkens applies to preparations and transformations as well as to measurements, within a general framework of operational physical theories. With respect to measurements, it removes the assumption of determinism of value assignments that is present in standard definitions of contextuality. This breaks the interpretation of nonlocality as a special case of contextuality, and does not treat irreducible randomness as nonclassical. Nevertheless, it recovers the usual notion of contextuality when outcome determinism is imposed.

由于 Robert Spekkens 而产生的语境的扩展概念适用于准备和转换，也适用于测量，在可操作的物理理论的一般框架内。关于度量，它移除了上下文性的标准定义中存在的值赋值确定性的假设。这打破了非局部性作为上下文特例的解释，也没有把不可约的随机性当作非经典随机性处理。然而，当结果决定论被强加时，它恢复了语境的通常概念。

Spekkens' contextuality can be motivated using Leibniz's law of the identity of indiscernibles. The law applied to physical systems in this framework mirrors the entended definition of noncontextuality. This was further explored by Simmons et al,[32] who demonstrated that other notions of contextuality could also be motivated by Leibnizian principles, and could be thought of as tools enabling ontological conclusions from operational statistics.

Spekkens' contextuality can be motivated using Leibniz's law of the identity of indiscernibles. The law applied to physical systems in this framework mirrors the entended definition of noncontextuality. This was further explored by Simmons et al,A.W. Simmons, Joel J. Wallman, H. Pashayan, S. D. Bartlett, T. Rudolph, "Contextuality under weak assumptions", New J. Phys. 19 033030, (2017). who demonstrated that other notions of contextuality could also be motivated by Leibnizian principles, and could be thought of as tools enabling ontological conclusions from operational statistics.

斯佩肯斯的语境性可以用莱布尼茨的不可区别的等同原则定律来推动。在这个框架中，适用于物理系统的定律反映了非上下文的纠缠定义。西蒙斯等人对此进行了进一步的探索。西蒙斯，乔尔 · j · 沃尔曼，h · 帕沙扬，s · d · 巴特利特，t · 鲁道夫，“弱假设下的语境”，新 j · 菲斯。19 033030, (2017).他证明，其他关于情境性的概念也可以受到莱布尼兹原则的激励，并可以被认为是能够从操作统计中得出本体论结论的工具。

Other frameworks and extensions

Other frameworks and extensions

= = 其他框架和扩展 = =

  • A form of contextuality that may present in the dynamics of a quantum system was introduced by Shane Mansfield and Elham Kashefi, and has been shown to relate to computational quantum advantages.[33] As a notion of contextuality that applies to transformations it is inequivalent to that of Spekkens. Examples explored to date rely on additional memory constraints which have a more computational than foundational motivation. Contextuality may be traded-off against Landauer erasure to obtain equivalent advantages.[34]
  • A form of contextuality that may present in the dynamics of a quantum system was introduced by Shane Mansfield and Elham Kashefi, and has been shown to relate to computational quantum advantages. As a notion of contextuality that applies to transformations it is inequivalent to that of Spekkens. Examples explored to date rely on additional memory constraints which have a more computational than foundational motivation. Contextuality may be traded-off against Landauer erasure to obtain equivalent advantages.


  • Shane Mansfield 和 Elham Kashefi 介绍了一种可能出现在量子系统动力学中的语境形式，并且已被证明与计算量子优势有关。作为一种适用于转换的语境性概念，它与斯佩肯斯的语境性概念是不等同的。迄今为止探讨的例子依赖于额外的内存约束，这些约束比基本动机更具计算性。情境性可以与消除兰道尔相互抵消，以获得同等的优势。

Fine–Abramsky–Brandenburger theorem

The Kochen–Specker theorem proves that quantum mechanics is incompatible with realistic noncontextual hidden variable models. On the other hand Bell's theorem proves that quantum mechanics is incompatible with factorisable hidden variable models in an experiment in which measurements are performed at distinct spacelike separated locations. Arthur Fine showed that in the experimental scenario in which the famous CHSH inequalities and proof of nonlocality apply, a factorisable hidden variable model exists if and only if an noncontextual hidden variable model exists.[8] This equivalence was proven to hold more generally in any experimental scenario by Samson Abramsky and Adam Brandenburger.[3] It is for this reason that we may consider nonlocality to be a special case of contextuality.

The Kochen–Specker theorem proves that quantum mechanics is incompatible with realistic noncontextual hidden variable models. On the other hand Bell's theorem proves that quantum mechanics is incompatible with factorisable hidden variable models in an experiment in which measurements are performed at distinct spacelike separated locations. Arthur Fine showed that in the experimental scenario in which the famous CHSH inequalities and proof of nonlocality apply, a factorisable hidden variable model exists if and only if an noncontextual hidden variable model exists. This equivalence was proven to hold more generally in any experimental scenario by Samson Abramsky and Adam Brandenburger. It is for this reason that we may consider nonlocality to be a special case of contextuality.

= = 精细-Abramsky-Brandenburger 定理 = = Kochen-Specker 定理证明了量子力学与现实的非上下文隐变量模型是不相容的。另一方面，Bell 定理在一个实验中证明了量子力学模型与因子分解的隐变量模型不兼容，在这个实验中，测量是在不同的类空分离位置进行的。表明，在著名的 CHSH 不等式和非局部性证明应用的实验场景中，当且仅当存在非上下文隐变量模型时，才存在因子分解隐变量模型。萨姆森 · 阿布拉姆斯基和亚当 · 布兰登伯格在任何实验场景中都证明了这种等价性。正是由于这个原因，我们可以认为非局部性是语境性的一种特殊情况。

Measures of contextuality

Measures of contextuality

= = 上下文度量 =

Contextual fraction

A number of methods exist for quantifying contextuality. One approach is by measuring the degree to which some particular noncontextuality inequality is violated, e.g. the KCBS inequality, the Yu–Oh inequality,[35] or some Bell inequality. A more general measure of contextuality is the contextual fraction.[11]

A number of methods exist for quantifying contextuality. One approach is by measuring the degree to which some particular noncontextuality inequality is violated, e.g. the KCBS inequality, the Yu–Oh inequality, or some Bell inequality. A more general measure of contextuality is the contextual fraction.

= = = 上下文分数 = = = 有许多方法可以量化语境。一种方法是通过测量某些特定的非上下文不平等被破坏的程度，例如。KCBS 不等式，Yu-Oh 不等式，或者一些 Bell 不等式。语境性的一个更一般的度量是上下文分数。

Given a set of measurement statistics e, consisting of a probability distribution over joint outcomes for each measurement context, we may consider factoring e into a noncontextual part eNC and some remainder e',

Given a set of measurement statistics e, consisting of a probability distribution over joint outcomes for each measurement context, we may consider factoring e into a noncontextual part eNC and some remainder e',

给定一组测量统计数据 e，包括每个测量上下文联合结果的概率分布，我们可以考虑将 e 分解为一个非上下文部分 eNC 和一些余数 e’,

[math]\displaystyle{ e = \lambda e^{NC} + (1-\lambda)e' \, . }[/math]


e = \lambda e^{NC} + (1-\lambda)e' \, .


e = \lambda e^{NC} + (1-\lambda)e' \, .

The maximum value of λ over all such decompositions is the noncontextual fraction of e denoted NCF(e), while the remainder CF(e)=(1-NCF(e)) is the contextual fraction of e. The idea is that we look for a noncontextual explanation for the highest possible fraction of the data, and what is left over is the irreducibly contextual part. Indeed for any such decomposition that maximises λ the leftover e' is known to be strongly contextual. This measure of contextuality takes values in the interval [0,1], where 0 corresponds to noncontextuality and 1 corresponds to strong contextuality. The contextual fraction may be computed using linear programming.

The maximum value of λ over all such decompositions is the noncontextual fraction of e denoted NCF(e), while the remainder CF(e)=(1-NCF(e)) is the contextual fraction of e. The idea is that we look for a noncontextual explanation for the highest possible fraction of the data, and what is left over is the irreducibly contextual part. Indeed for any such decomposition that maximises λ the leftover e' is known to be strongly contextual. This measure of contextuality takes values in the interval [0,1], where 0 corresponds to noncontextuality and 1 corresponds to strong contextuality. The contextual fraction may be computed using linear programming.

在所有这些分解上 λ 的最大值是 e 表示的 NCF (e)的非上下文分数，而余数 CF (e) = (1-NCF (e))是 e 的上下文分数。这个想法是，我们为数据的最大可能部分寻找一个非上下文的解释，剩下的部分是不可简化的上下文部分。事实上，对于任何这样的分解，最大化 λ 剩余的 e’已知是强上下文关系。上下文性的度量以[0,1]区间为单位，其中0代表非上下文性，1代表强烈的上下文性。上下文分数可以使用线性规划计算。

It has also been proved that CF(e) is an upper bound on the extent to which e violates any normalised noncontextuality inequality.[11] Here normalisation means that violations are expressed as fractions of the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for which this violation is attained. In this sense the contextual fraction is a more neutral measure of contextuality, since it optimises over all possible noncontextual inequalities rather than checking the statistics against one inequality in particular.

It has also been proved that CF(e) is an upper bound on the extent to which e violates any normalised noncontextuality inequality. Here normalisation means that violations are expressed as fractions of the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for which this violation is attained. In this sense the contextual fraction is a more neutral measure of contextuality, since it optimises over all possible noncontextual inequalities rather than checking the statistics against one inequality in particular.

证明了 CF (e)是 e 违反非上下文不等式规范化程度的一个上界。在这里，标准化意味着违反被表示为不平等的代数最大违反的分数。此外，对偶线性规划的 λ 最大化计算一个非上下文不等式，这种违背是达到的。在这个意义上，语境分数是语境性的一个更中性的衡量标准，因为它优化了所有可能的非语境不平等，而不是检查统计数据，特别是针对一个不平等。

Measures of (non)contextuality within the Contextuality-by-Default (CbD) framework

Several measures of the degree of contextuality in contextual systems were proposed within the CbD framework,[22] but only one of them, denoted CNT2, has been shown to naturally extend into a measure of noncontextuality in noncontextual systems, NCNT2. This is important, because at least in the non-physical applications of CbD contextuality and noncontextuality are of equal interest. Both CNT2 and NCNT2 are defined as the [math]\displaystyle{ L_1 }[/math]-distance between a probability vector [math]\displaystyle{ \mathbf{p} }[/math] representing a system and the surface of the noncontextuality polytope [math]\displaystyle{ \mathbb{P} }[/math] representing all possible noncontextual systems with the same single-variable marginals. For cyclic systems  of dichotomous random variables, it is shown[36] that if the system is contextual (i.e., [math]\displaystyle{ D\left(\mathcal{C}_{n}\right)\gt \Delta\left(\mathcal{C}_{n}\right) }[/math]),

Several measures of the degree of contextuality in contextual systems were proposed within the CbD framework, but only one of them, denoted CNT2, has been shown to naturally extend into a measure of noncontextuality in noncontextual systems, NCNT2. This is important, because at least in the non-physical applications of CbD contextuality and noncontextuality are of equal interest. Both CNT2 and NCNT2 are defined as the L_1-distance between a probability vector \mathbf{p} representing a system and the surface of the noncontextuality polytope \mathbb{P} representing all possible noncontextual systems with the same single-variable marginals. For cyclic systems  of dichotomous random variables, it is shown that if the system is contextual (i.e., D\left(\mathcal{C}_{n}\right)>\Delta\left(\mathcal{C}_{n}\right)),

= = = 在 Contextuality-by-Default (CbD)框架中的非上下文度量 = = = 在 CbD 框架中提出了上下文系统中上下文度量的几个度量，但是只有其中一个度量，即 CNT2，被证明自然地扩展为非上下文系统中的非上下文度量，NCNT2。这一点很重要，因为至少在 CbD 语境和非语境的非物理应用中是同样重要的。Cnt2和 ncnt2都被定义为表示一个系统的概率向量 mathbf { p }与表示具有相同单变量边界的所有可能非上下文系统的非上下文多边形 mathbb { p }的表面之间的 l1距离。对于二值随机变量的循环系统，证明了如果系统是上下文的(即，d left (mathcal { c } _ { n } right) > Delta left (mathcal { c } _ { n } right) ,

[math]\displaystyle{ \mathrm{CNT}_{2}=D\left(\mathcal{C}_{n}\right)-\Delta\left(\mathcal{C}_{n}\right), }[/math]

\mathrm{CNT}_{2}=D\left(\mathcal{C}_{n}\right)-\Delta\left(\mathcal{C}_{n}\right),

2} = d left (mathcal { c } _ { n } right)-Delta left (mathcal { c } _ { n } right) ,

and if it is noncontextual ( [math]\displaystyle{ D\left(\mathcal{C}_{n}\right)\leq\Delta\left(\mathcal{C}_{n}\right) }[/math]),

and if it is noncontextual ( D\left(\mathcal{C}_{n}\right)\leq\Delta\left(\mathcal{C}_{n}\right)),

如果它是非上下文的(d left (mathcal { c } _ { n } right) leq Delta left (mathcal { c } _ { n } right)) ,

[math]\displaystyle{ \mathrm{NCNT}_{2}=\min\left(\Delta\left(\mathcal{C}_{n}\right)-D\left(\mathcal{C}_{n}\right),m\left(\mathcal{C}_{n}\right)\right), }[/math]

\mathrm{NCNT}_{2}=\min\left(\Delta\left(\mathcal{C}_{n}\right)-D\left(\mathcal{C}_{n}\right),m\left(\mathcal{C}_{n}\right)\right),

数学{ NCNT } _ {2} = min left (Delta left (mathcal { c } _ { n } right)-d left (mathcal { c } _ { n } right) ，m left (mathcal { c } _ { n } right) ,

where [math]\displaystyle{ m\left(\mathcal{C}_{n}\right) }[/math] is the [math]\displaystyle{ L_1 }[/math]-distance from the vector [math]\displaystyle{ \mathbf{p}\in\mathbb{P} }[/math] to the surface of the box circumscribing the noncontextuality polytope. More generally, NCNT2 and CNT2 are computed by means of linear programming.[22] The same is true for other CbD-based measures of contextuality. One of them, denoted CNT3, uses the notion of a quasi-coupling, that differs from a coupling in that the probabilities in the joint distribution of its values are replaced with arbitrary reals (allowed to be negative but summing to 1). The class of quasi-couplings [math]\displaystyle{ S }[/math] maximizing the probabilities [math]\displaystyle{ \Pr\left[S_{q}^{c}=S_{q}^{c'}\right] }[/math] is always nonempty, and the minimal total variation of the signed measure in this class is a natural measure of contextuality.[37]

where m\left(\mathcal{C}_{n}\right) is the L_1-distance from the vector \mathbf{p}\in\mathbb{P} to the surface of the box circumscribing the noncontextuality polytope. More generally, NCNT2 and CNT2 are computed by means of linear programming. The same is true for other CbD-based measures of contextuality. One of them, denoted CNT3, uses the notion of a quasi-coupling, that differs from a coupling in that the probabilities in the joint distribution of its values are replaced with arbitrary reals (allowed to be negative but summing to 1). The class of quasi-couplings S maximizing the probabilities \Pr\left[S_{q}^{c}=S_{q}^{c'}\right] is always nonempty, and the minimal total variation of the signed measure in this class is a natural measure of contextuality.

其中 m 左(mathcal { c }{ n } right)是从 mathbb { p }中的向量 mathbf { p }到包围非上下文多面体的盒面的 l _ 1距离。更一般地说，ncnt2和 cnt2是通过线性规划计算得到的。其他基于 cbd 的语境性度量也是如此。其中一个代号为 CNT3，使用了准耦合的概念，与耦合的不同之处在于，其值的联合分布的概率被任意的雷亚尔替代(允许为负数，但求和为1)。拟耦合类 s 最大概率 Pr 左[ s _ { q } ^ { c } = s _ { q } ^ { c’}右]总是非空的，这类有符号测度的最小总变差是上下文性的自然测度。

Contextuality as a resource for quantum computing

Recently, quantum contextuality has been investigated as a source of quantum advantage and computational speedups in quantum computing.

Recently, quantum contextuality has been investigated as a source of quantum advantage and computational speedups in quantum computing.

= = 语境性作为量子计算的资源 = = 最近，量子语境性作为量子计算的优势和计算速度的来源被研究。

Magic state distillation

Magic state distillation is a scheme for quantum computing in which quantum circuits constructed only of Clifford operators, which by themselves are fault-tolerant but efficiently classically simulable, are injected with certain "magic" states that promote the computational power to universal fault-tolerant quantum computing.[38] In 2014, Mark Howard, et al. showed that contextuality characterises magic states for qudits of odd prime dimension and for qubits with real wavefunctions.[39] Extensions to the qubit case have been investigated by Juani Bermejo-Vega et al.[35] This line of research builds on earlier work by Ernesto Galvão,[34] which showed that Wigner function negativity is necessary for a state to be "magic"; it later emerged that Wigner negativity and contextuality are in a sense equivalent notions of nonclassicality.[40]

Magic state distillation is a scheme for quantum computing in which quantum circuits constructed only of Clifford operators, which by themselves are fault-tolerant but efficiently classically simulable, are injected with certain "magic" states that promote the computational power to universal fault-tolerant quantum computing. In 2014, Mark Howard, et al. showed that contextuality characterises magic states for qudits of odd prime dimension and for qubits with real wavefunctions. Extensions to the qubit case have been investigated by Juani Bermejo-Vega et al. This line of research builds on earlier work by Ernesto Galvão, which showed that Wigner function negativity is necessary for a state to be "magic"; it later emerged that Wigner negativity and contextuality are in a sense equivalent notions of nonclassicality.

幻态蒸馏是量子计算的一种方案，量子线路只由克利福德算子构成，这些算子本身是容错的，但在经典模拟中是高效的，它们被注入了某些“幻态”，这些“幻态”将计算能力提升到普遍的容错量子计算。2014年，马克 · 霍华德等人。结果表明，上下文性是奇素数量子和实波函数量子比特的魔幻状态的特征。量子比特案件的扩展已经由 Juani Bermejo-Vega 等人进行了调查。这一系列的研究建立在 Ernesto Galvão 的早期工作的基础上，该工作表明 Wigner 函数消极性对于一个状态是“魔法”是必要的; 后来发现 Wigner 消极性和语境性在某种意义上等价于非经典性的概念。

Measurement-based quantum computing

Measurement-based quantum computation (MBQC) is a model for quantum computing in which a classical control computer interacts with a quantum system by specifying measurements to be performed and receiving measurement outcomes in return. The measurement statistics for the quantum system may or may not exhibit contextuality. A variety of results have shown that the presence of contextuality enhances the computational power of an MBQC.

Measurement-based quantum computation (MBQC) is a model for quantum computing in which a classical control computer interacts with a quantum system by specifying measurements to be performed and receiving measurement outcomes in return. The measurement statistics for the quantum system may or may not exhibit contextuality. A variety of results have shown that the presence of contextuality enhances the computational power of an MBQC.

= = = = 基于测量的量子计算机 = = = 基于测量的量子计算机(MBQC)是一种量子计算模型，在这种模型中，经典的控制计算机通过指定要执行的测量并接收测量结果来与量子系统相互作用。量子系统的测量统计数据可能会也可能不会表现出语境性。大量的实验结果表明，语境的存在增强了 MBQC 的计算能力。

In particular, researchers have considered an artificial situation in which the power of the classical control computer is restricted to only being able to compute linear Boolean functions, i.e. to solve problems in the Parity L complexity class ⊕L. For interactions with multi-qubit quantum systems a natural assumption is that each step of the interaction consists of a binary choice of measurement which in turn returns a binary outcome. An MBQC of this restricted kind is known as an l2-MBQC.[41]

In particular, researchers have considered an artificial situation in which the power of the classical control computer is restricted to only being able to compute linear Boolean functions, i.e. to solve problems in the Parity L complexity class ⊕L. For interactions with multi-qubit quantum systems a natural assumption is that each step of the interaction consists of a binary choice of measurement which in turn returns a binary outcome. An MBQC of this restricted kind is known as an l2-MBQC.

特别是，研究人员考虑了一种人为的情况，其中经典的控制计算机的能力被限制为只能计算线性布尔函数，即。解决奇偶校验 l 复杂类中的问题。对于与多量子比特量子系统的相互作用，一个自然的假设是，相互作用的每一个步骤都由一个二进制度量选择组成，这个二进制度量选择反过来又返回一个二进制结果。这种受限制的 MBQC 称为 l2-MBQC。

Anders and Browne

In 2009, Janet Anders and Dan Browne showed that two specific examples of nonlocality and contextuality were sufficient to compute a non-linear function. This in turn could be used to boost computational power to that of a universal classical computer, i.e. to solve problems in the complexity class P.[42] This is sometimes referred to as measurement-based classical computation.[43] The specific examples made use of the Greenberger–Horne–Zeilinger nonlocality proof and the supra-quantum Popescu–Rohrlich box.

In 2009, Janet Anders and Dan Browne showed that two specific examples of nonlocality and contextuality were sufficient to compute a non-linear function. This in turn could be used to boost computational power to that of a universal classical computer, i.e. to solve problems in the complexity class P. This is sometimes referred to as measurement-based classical computation. The specific examples made use of the Greenberger–Horne–Zeilinger nonlocality proof and the supra-quantum Popescu–Rohrlich box.

= = = Anders and Browne = = = = 2009年，Janet Anders 和 Dan Browne 证明了非局部性和上下文性的两个具体例子足以计算一个非线性函数。这反过来又可以用来提高计算能力的普遍经典计算机，即。来解决复杂类 p 中的问题。这有时被称为基于测量的经典计算。具体的例子使用了 Greenberger-Horne-Zeilinger 非定域证明和超量子 Popescu-Rohrlich 盒。

Raussendorf

In 2013, Robert Raussendorf showed more generally that access to strongly contextual measurement statistics is necessary and sufficient for an l2-MBQC to compute a non-linear function. He also showed that to compute non-linear Boolean functions with sufficiently high probability requires contextuality.[41]

In 2013, Robert Raussendorf showed more generally that access to strongly contextual measurement statistics is necessary and sufficient for an l2-MBQC to compute a non-linear function. He also showed that to compute non-linear Boolean functions with sufficiently high probability requires contextuality.

= = = Raussendorf = = = = 2013年，Robert Raussendorf 更广泛地表明，获得强背景测量统计数据对于一个 l2-MBQC 计算一个非线性函数是必要的和充分的。他还指出，计算具有足够高概率的非线性布尔函数需要上下文。

Abramsky, Barbosa and Mansfield

Abramsky, Barbosa and Mansfield

= 艾布拉姆斯基，巴博萨和曼斯菲尔德 =

A further generalisation and refinement of these results due to Samson Abramsky, Rui Soares Barbosa and Shane Mansfield appeared in 2017, proving a precise quantifiable relationship between the probability of successfully computing any given non-linear function and the degree of contextuality present in the l2-MBQC as measured by the contextual fraction.[11] Specifically, [math]\displaystyle{ (1-p_s) \geq \left( 1-CF(e) \right) . \nu(f) }[/math]where [math]\displaystyle{ p_s, CF(e), \nu(f) \in [0,1] }[/math] are the probability of success, the contextual fraction of the measurement statistics e, and a measure of the non-linearity of the function to be computed [math]\displaystyle{ f }[/math], respectively.

A further generalisation and refinement of these results due to Samson Abramsky, Rui Soares Barbosa and Shane Mansfield appeared in 2017, proving a precise quantifiable relationship between the probability of successfully computing any given non-linear function and the degree of contextuality present in the l2-MBQC as measured by the contextual fraction. Specifically, (1-p_s) \geq \left( 1-CF(e) \right) . \nu(f) where p_s, CF(e), \nu(f) \in [0,1] are the probability of success, the contextual fraction of the measurement statistics e, and a measure of the non-linearity of the function to be computed f , respectively.

2017年，Samson Abramsky，Rui Soares Barbosa 和 Shane Mansfield 对这些结果进行了进一步的推广和改进，证明了成功计算任何给定非线性函数的概率与 l2-MBQC 中上下文分数测量的上下文度之间的精确量化关系。具体地说，(1-p _ s) geq 左(1-CF (e)右) 。Nu (f)其中 p _ s，CF (e) ，nu (f)在[0,1]中分别是成功的概率，测量统计量 e 的上下文分数，以及计算 f 的函数的非线性度量。

Further examples

Further examples

= = 更多的例子 = =

  • The above inequality was also shown to relate quantum advantage in non-local games to the degree of contextuality required by the strategy and an appropriate measure of the difficulty of the game.[11]
  • Similarly the inequality arises in a transformation-based model of quantum computation analogous to l2-MBQC where it relates the degree of sequential contextuality present in the dynamics of the quantum system to the probability of success and the degree of non-linearity of the target function.[33]
  • Preparation contextuality has been shown to enable quantum advantages in cryptographic random-access codes[44] and in state-discrimination tasks.[45]
  • In classical simulations of quantum systems, contextuality has been shown to incur memory costs.[46]
  • The above inequality was also shown to relate quantum advantage in non-local games to the degree of contextuality required by the strategy and an appropriate measure of the difficulty of the game.
  • Similarly the inequality arises in a transformation-based model of quantum computation analogous to l2-MBQC where it relates the degree of sequential contextuality present in the dynamics of the quantum system to the probability of success and the degree of non-linearity of the target function.
  • Preparation contextuality has been shown to enable quantum advantages in cryptographic random-access codes and in state-discrimination tasks.
  • In classical simulations of quantum systems, contextuality has been shown to incur memory costs.


  • 上述不等式还表明，非局部博弈中的量子优势与策略所要求的语境程度和博弈难度的适当度量有关。
  • 类似地，这个不等式出现在类似于 l2-MBQC 的基于变换的量子计算模型中，它将量子系统动力学中的顺序上下文程度与成功的概率和目标函数的非线性程度联系起来。
  • 准备语境性已被证明在加密随机访问码和状态识别任务中具有量子优势。
  • 在量子系统的经典模拟中，情境性已被证明会产生记忆成本。

See also

  • Kochen–Specker theorem
  • Mermin–Peres square
  • KCBS pentagram
  • Quantum nonlocality
  • Quantum foundations

= = =

  • Kochen-Specker 定理
  • Mermin-Peres 方
  • KCBS 五角星
  • 量子非定域性
  • 量子基础

References

  1. ↑ S. Kochen and E.P. Specker, "The problem of hidden variables in quantum mechanics", Journal of Mathematics and Mechanics 17, 59–87 (1967)
  2. ↑ Gleason, A. M, "Measures on the closed subspaces of a Hilbert space", Journal of Mathematics and Mechanics 6, 885–893 (1957).
  3. ↑ 3.0 3.1 3.2 Abramsky, Samson; Brandenburger, Adam (2011-11-28). "The Sheaf-Theoretic Structure Of Non-Locality and Contextuality". New Journal of Physics. 13 (11): 113036. arXiv:1102.0264. Bibcode:2011NJPh...13k3036A. doi:10.1088/1367-2630/13/11/113036. ISSN 1367-2630. S2CID 17435105.
  4. ↑ Cabello, Adan; Severini, Simone; Winter, Andreas (2014-01-27). "Graph-Theoretic Approach to Quantum Correlations". Physical Review Letters. 112 (4): 040401. arXiv:1401.7081. Bibcode:2014PhRvL.112d0401C. doi:10.1103/PhysRevLett.112.040401. ISSN 0031-9007. PMID 24580419. S2CID 34998358.
  5. ↑ 5.0 5.1 Acín, Antonio; Fritz, Tobias; Leverrier, Anthony; Sainz, Ana Belén (2015-03-01). "A Combinatorial Approach to Nonlocality and Contextuality". Communications in Mathematical Physics. 334 (2): 533–628. arXiv:1212.4084. Bibcode:2015CMaPh.334..533A. doi:10.1007/s00220-014-2260-1. ISSN 1432-0916. S2CID 119292509.
  6. ↑ Abramsky, Samson; Mansfield, Shane; Barbosa, Rui Soares (2012-10-01). "The Cohomology of Non-Locality and Contextuality". Electronic Proceedings in Theoretical Computer Science. 95: 1–14. arXiv:1111.3620. doi:10.4204/EPTCS.95.1. ISSN 2075-2180. S2CID 9046880.
  7. ↑ 7.0 7.1 Dzhafarov, Ehtibar N.; Kujala, Janne V. (2016-09-07). "Probabilistic foundations of contextuality". Fortschritte der Physik. 65 (6–8): 1600040. arXiv:1604.08412. Bibcode:2016arXiv160408412D. doi:10.1002/prop.201600040. ISSN 0015-8208. S2CID 56245502.
  8. ↑ 8.0 8.1 Fine, Arthur (1982-02-01). "Hidden Variables, Joint Probability, and the Bell Inequalities". Physical Review Letters. 48 (5): 291–295. Bibcode:1982PhRvL..48..291F. doi:10.1103/PhysRevLett.48.291.
  9. ↑ Raussendorf, Robert (2013-08-19). "Contextuality in measurement-based quantum computation". Physical Review A. 88 (2): 022322. arXiv:0907.5449. Bibcode:2013PhRvA..88b2322R. doi:10.1103/PhysRevA.88.022322. ISSN 1050-2947. S2CID 118495073.
  10. ↑ Howard, Mark; Wallman, Joel; Veitch, Victor; Emerson, Joseph (June 2014). "Contextuality supplies the 'magic' for quantum computation". Nature. 510 (7505): 351–355. arXiv:1401.4174. Bibcode:2014Natur.510..351H. doi:10.1038/nature13460. ISSN 0028-0836. PMID 24919152. S2CID 4463585.
  11. ↑ 11.0 11.1 11.2 11.3 11.4 Abramsky, Samson; Barbosa, Rui Soares; Mansfield, Shane (2017-08-04). "Contextual Fraction as a Measure of Contextuality". Physical Review Letters. 119 (5): 050504. arXiv:1705.07918. Bibcode:2017PhRvL.119e0504A. doi:10.1103/PhysRevLett.119.050504. ISSN 0031-9007. PMID 28949723. S2CID 206295638.
  12. ↑ Bermejo-Vega, Juan; Delfosse, Nicolas; Browne, Dan E.; Okay, Cihan; Raussendorf, Robert (2017-09-21). "Contextuality as a Resource for Models of Quantum Computation with Qubits". Physical Review Letters. 119 (12): 120505. arXiv:1610.08529. Bibcode:2017PhRvL.119l0505B. doi:10.1103/PhysRevLett.119.120505. ISSN 0031-9007. PMID 29341645. S2CID 34682991.
  13. ↑ Crull, Elise; Bacciagaluppi, Guido (2016). Grete Hermann - Between Physics and Philosophy. Netherlands: Springer. pp. 154. ISBN 978-94-024-0968-0. 
  14. ↑ Carsten, Held (2000-09-11). "The Kochen–Specker Theorem". plato.stanford.edu. Retrieved 2018-11-17.
  15. ↑ 15.0 15.1 Abramsky, Samson; Soares Barbosa, Rui; Kishida, Kohei; Lal, Raymond; Mansfield, Shane (2015). "Contextuality, Cohomology and Paradox". Schloss Dagstuhl - Leibniz-Zentrum für Informatik GMBH, Wadern/Saarbruecken, Germany. Leibniz International Proceedings in Informatics (LIPIcs). 41: 211–228. arXiv:1502.03097. Bibcode:2015arXiv150203097A. doi:10.4230/lipics.csl.2015.211. ISBN 9783939897903. S2CID 2150387.
  16. ↑ Abramsky, Samson (2013), Tannen, Val; Wong, Limsoon; Libkin, Leonid; Fan, Wenfei (eds.), "Relational Databases and Bell's Theorem", In Search of Elegance in the Theory and Practice of Computation: Essays Dedicated to Peter Buneman, Lecture Notes in Computer Science, Springer Berlin Heidelberg, vol. 8000, pp. 13–35, doi:10.1007/978-3-642-41660-6_2, ISBN 9783642416606, S2CID 18824713
  17. ↑ Abramsky, Samson; Sadrzadeh, Mehrnoosh (2014), Casadio, Claudia; Coecke, Bob; Moortgat, Michael; Scott, Philip (eds.), "Semantic Unification", Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday, Lecture Notes in Computer Science, Springer Berlin Heidelberg, pp. 1–13, arXiv:1403.3351, doi:10.1007/978-3-642-54789-8_1, ISBN 9783642547898, S2CID 462058
  18. ↑ Abramsky, Samson; Dawar, Anuj; Wang, Pengming (2017). "The pebbling comonad in Finite Model Theory". 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). pp. 1–12. arXiv:1704.05124. doi:10.1109/LICS.2017.8005129. ISBN 9781509030187. 
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  46. ↑ Kleinmann, Matthias; Gühne, Otfried; Portillo, José R.; Larsson, Jan-\AAke; Cabello, Adán (November 2011). "Memory cost of quantum contextuality". New Journal of Physics. 13 (11): 113011. arXiv:1007.3650. Bibcode:2011NJPh...13k3011K. doi:10.1088/1367-2630/13/11/113011. ISSN 1367-2630. S2CID 13466604.

Category:Quantum mechanics

类别: 量子力学


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