New variants of the Bell-Kochen-Specker theorem
Adan Cabello, Jose M. Estebaranz, Guillermo Garcia Alcaine
本文讨论了Bell-Kochen-Specker定理的两个新证明：一个是基于Clifton提出的14个四维命题的状态无关证明，另一个是关于两个自旋为1/2的粒子的单重态的5个命题的状态特定证明。
We discuss two new demonstrations of the Bell-Kochen-Specker theorem: a state-independent proof using 14 four-dimensional propositions, based on a suggestion made by Clifton, and a state-specific proof involving 5 propositions on the singlet state of two spin-1/2 particles.
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The Kochen-Specker theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We show that within every Kochen-Specker graph, there exist interesting subgraphs which we term $01$-gadgets, that capture the essential contradiction necessary to prove the Kochen-Specker theorem, i.e,. every Kochen-Specker graph contains a $01$-gadget and from every $01$-gadget one can construct a proof of the Kochen-Specker theorem. Moreover, we show that the $01$-gadgets form a fundamental primitive that can be used to formulate state-independent and state-depend…
Two types of inequalities, Kochen-Specker inequalities and noncontextuality inequalities, are both used to demonstrate the incompatibility between the noncontextual hidden variable model and quantum mechanics. It has been thought that noncontextuality inequalities are much more potent than Kochen-Specker inequalities, since the latter are constrained by the Kochen-Specker rules, which are regarded as an extra constraint imposed on the noncontextual hidden variable model. However, we find that a noncontextuality inequality exists in a ray set if and only if a Kochen-Specker inequality exists in…
The generalized Kochen-Specker theorem
由于Cabello和Nakamura在二维中推广的Kochen-Specker定理的证明被扩展到所有更高的维度。一组18个国家在四个层面上是用来给密切相关的证明广义Kochen-Specker，Kochen-Specker和贝尔定理，揭示了这三个定理之间的关系。
Complementarity between Position and Momentum as a Consequence of Kochen-Specker Arguments
我们给出了两个简单的Kochen-Specker参数之间的互补的位置和动量分量的无自旋粒子，参数的结构是相同的佩雷斯和Mermin自旋1/2粒子。
Minimal Bell-Kochen-Specker proofs with POVMs on qubits
关于使用POVM的Bell-Kochen-Specker证明可能有许多不同的定义。在这里，我们提出并讨论了这些定义中的三个量子位的最小证明，并表明它们确实是最小的。
A Kochen-Specker Theorem for Imprecisely Specified Measurement
最近的一个说法，有限的精确度在设计的实际实验"无效"的影响的Kochen-Specker定理，是不支持的，因为连续性的概率测量结果下轻微的变化，在实验配置。
We put forward three simple algorithms to generate Kochen-Specker sets used for parity proof of Kochen-Specker theorem in three-qubit system. These algorithms enables us to generate 320, 640 and 64 Kochen-Specket sets with 36, 38 and 40 rays, respectively. No any computer calculation is required, every step in the algorithms is determined by the method of picking the rays that repeat 4 times in the KS sets.
A priori Knowledge and the Kochen-Specker Theorem
我们引入并形式化了关于量子系统的“先验知识”的概念，并展示了这种形式的知识的一些性质。最后，我们表明，Kochen-Specker定理直接从这项研究。这个版本是一个草稿版本，特别是参考书目是极其稀缺的。欢迎评论。
Kochen-Specker theorem for a single qubit using positive operator-valued measures
给出了单个二能级系统的Kochen-Specker定理的一个证明。它采用了五个八元正算子值的措施和一个简单的代数推理的基础上的几何形状的十二面体。
The Kochen-Specker theorem states that noncontextual hidden variable theories are incompatible with quantum mechanics. We provide a state independent proof of the Kochen-Specker theorem using the smallest number of projectors, i.e., thirty rank-2 projectors, associated with the Mermin pentagram for a three-qubit system.
We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthogonality conditions and thus admit no free parameters.
Kochen-Specker Obstruction for Position and Momentum Using a Single Degree of Freedom
它表明，给定一个合理的连续性假设拥有的值，它是可能的，以构建一个Kochen-Specker障碍的任何坐标和共轭动量，表明这两个量中的最多一个可以有一个noncontextual值。
The Kochen-Specker Theorem Revisited in Quantum Measure Theory
Kochen-Specker定理被广泛地解释为暗示与哥本哈根量子力学的预测一致的非上下文隐变量理论是不可能的。研究了Sorkin提出的一种新的与观察者无关的量子力学解释定理的重要性。
We present two geometric proofs of the Kochen-Specker theorem. A quite similar argument has been used by Cooke, Keane, and Moran, as well as by Kalmbach in her book to derive the Gleason theorem.
One of the recent no-go theorems on Ψ-epistemic interpretations of quantum proves that there are no 'maximally epistemic' interpretations of quantum theory. The proof utilises similar arrangements to Clifton's quantum contextuality proof and has parallels to Harrigan and Rudolph's quantum deficiency no-go theorem, itself based on the Kochen-Specker quantum contextuality proof. This paper shows how the Kochen-Specker theorem can also be turned into a no 'maximally epistemic' theorem, but of a more limited kind.
Quantum contextuality is one of the fundamental notions in quantum mechanics. Proofs of the Kochen-Specker theorem and noncontextuality inequalities are two means for revealing the contextuality phenomenon in quantum mechanics. It has been found that some proofs of the Kochen-Specker theorem, such as those based on rays, can be converted to a state-independent noncontextuality inequality, but it remains open whether it is true in general, i.e., whether any proof of the Kochen-Specker theorem can always be converted to a noncontextuality inequality. In this paper, we address this issue. We prov…
Proposed experimental tests of the Bell-Kochen-Specker theorem
对于一个两粒子两态系统，存在着一系列相容的命题，对于这些命题，量子力学和非上下文隐变量理论对每一个单独的系统都做出了相互矛盾的预测，而不管它的量子态是什么。这使得贝尔-科亨-斯派克定理可以用一个简单的全或无状态无关的实验验证。
Kochen-Specker theorem for 8-dimensional space
在实8维希尔伯特空间中，用36个向量产生了一个Kochen-Specker矛盾。这些向量可以组合成30个不同的投影算子（14个秩为2，16个秩为1）。这种矛盾的状态特定变体只需要13个向量，对于8维来说是非常低的数字。
A Kochen-Specker inequality
通过概率的方法，语境性的概念得到了扩展，使其可以用于非理想的情况。提出了一个不等式，至少在原则上使测试，以丢弃非上下文的隐变量模型在低错误率，在Kochen-Specker定理的精神。假设误差是独立的，一个显式的误差界为1.42%，低于此Kochen-Specker矛盾发生。
In the paper it is argued that the Kochen-Specker theorem necessitates a conclusion that for a quantum system it is possible to find a set of projection operators which is not truth-value bivalent; that is, a bivalent truth-value assignment function imposed on such a set cannot be total. This means that at least one proposition associated with the said set must be neither true nor false.
A number of new proofs of the Kochen-Specker theorem are given based on the observables of the three-qubit Pauli group. Each proof is presented in the form of a diagram from which it is obvious by inspection. Each of our observable-based proofs leads to a system of projectors and bases that generally yields a large number of "parity proofs" of the Kochen-Specker theorem. Some examples of such proofs are given and some of their applications are discussed.
Finite-precision measurement does not nullify the Kochen-Specker theorem
事实证明，迈耶[Phys. Rev. Lett.]提出的任何类型的隐变量理论都是有效的。{ \bf 83}，3751（1999）]，Kent [{ \em ibid.} { \bf 83}，3755（1999）]和Clifton和Kent [Proc. R. Soc. London，Ser.A { \bf 456}，2101（2000）]导致实验上可检验的预言，这些预言与量子力学的预言相矛盾。因此，有人认为，一旦考虑到实际测量的有限精度，稠密Kochen-Specker可着色集的存在就不能被解释为Kochen-Specker定理的物理影响无效。
A Simple Proof of the Kochen-Specker Theorem on the Problem of Hidden Variables
在这次演讲中，我提出了一个简单的推导老结果的Kochen和Specker，这显然是无关的著名工作贝尔隐变量，但大概是同样重要的。Kochen和Specker在1967年证明了量子力学不能嵌入经典的随机理论，只要量子理论的概率分布被重现，并且满足一个额外的非常理想的性质。这以一种惊人的方式表明，在实现量子力学的“完整”版本的爱因斯坦方案时，有哪些困难。
A simple three rules supplemented by five steps scheme is proposed to produce Kochen-Specker (KS) sets with 30 rank-2 projectors that occur twice each. The KS sets provide state-independent proof of KS theorem based on a system of three qubits. A small adjustment of the scheme enables us to manually generate a large number of KS sets with a mixture of rank-1 and rank-2 projectors.
A diagrammatic representation is given of the 24 rays of Peres that makes it easy to pick out all the 512 parity proofs of the Kochen-Specker theorem contained in them. The origin of this representation in the four-dimensional geometry of the rays is pointed out.
Bell-Kochen-Specker theorem: A proof with 18 vectors
本文仅用18个四维向量对Bell-Kochen-Specker定理给出了一个“状态无关”的证明，这是这类证明的一个记录.这组向量包含子集，使我们能够开发一个具有10个向量（也是一个记录）的“状态特定”证明和一个具有7个向量的“概率”证明，这反映了哈代非局部性定理的代数结构。
Yu and Oh [1] have given a state independent proof of the Kochen-Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute a SIC and four Mutually Unbiased Bases.
Contextual Value-definiteness and the Kochen-Specker Paradox
价值确定性和因果关系的现实主义租户之间的兼容性受到质疑的几个现实主义不可能性的证明，其中他们的形式要素显示冲突。我们回顾这是如何在Kochen-Specker和冯诺依曼证明，并指出他们的关键假设之间的连接：通过在后一个证明中的可加性，noncontextuality在前者的现实因果关系的约束。我们的结论是，价值确定性和语境性确实不是相互排斥的。
Recursive proof of the Bell-Kochen-Specker theorem in any dimension $n>3$
我们提出了一种方法，从$n$维度中的类似集合（$3\leq d<n\leq 2d$）中获得证明$n$维度中的Bell-Kochen-Specker定理的向量集合。作为该方法的一个应用，我们发现已知的最小证明在维度五（29向量），六（31）和七（34），和不同的集匹配当前记录（36）在维度八。
An obstruction based approach to the Kochen-Specker theorem
文献[1]证明了Kochen-Specker定理可以用Hilbert空间上投影算子的布尔子代数的偏序集上的某个变集的整体元素的不存在性来表示。在本文中，我们将展示如何建设的障碍，这种全球性的元素出现，以及如何这提供了一种新的方式来看待证明定理。
Realization of All-or-nothing-type Kochen-Specker Experiment with Single Photons
利用I型相位匹配BBO晶体中的自发参量下转换过程作为单光子源，实现了Simon \QTR{it}{et al}提出的全有或全无型Kochen-Specker实验. [Phys. Rev. Lett. \QTR{bf}{85}，1783（2000）]来验证非上下文隐变量或量子力学是否正确。所得结果与量子力学基本一致。
Kochen-Specker Theorem for Finite Precision Spin One Measurements
非锐自旋1观测值的产生是因为测量设备的实际方向仍然存在残余不确定性。如果不确定性低于一定水平，并且测量误差的分布在旋转下是协变的，则非锐化自旋观测量的Kochen-Specker定理如下：存在有限的方向集，使得不是所有在这些方向上的非锐化自旋观测量都可以以非上下文的方式被一致地分配近似真值。
It is pointed out that the 60 complex rays in four dimensions associated with a system of two qubits yield over 10^9 critical parity proofs of the Kochen-Specker theorem. The geometrical properties of the rays are described, an overview of the parity proofs contained in them is given, and examples of some of the proofs are exhibited.
The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefi…
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Kochen和Specker定理可以看作是Gleason定理和逻辑紧性的一个推论。类似的紧凑性参数导致更强的结果有限集的射线在希尔伯特空间，我们也证明了直接建设。最后，我们证明了格里森定理本身有一个建设性的证明，基于一个通用的，有限的，有效生成的一组射线，每个量子态可以近似。
On the connection between the theorems of Gleason and of Kochen and Specker
本文通过函数 m 在单位球面上取值的线性方程，给出 Gleason 定理与 Kochen-Specker 定理简化版本的初等证明，揭示二者分别依赖整个球面与有限点集的联系。
The Kochen-Specker theorem theoretically shows evidence of the incompatibility of noncontextual hidden variable theories with quantum mechanics. Quantum contextuality is a more general concept than quantum non-locality which is quite well tested in experiments by using Bell inequalities. Within neutron interferometry we performed an experimental test of the Kochen-Specker theorem with an inequality, which identifies quantum contextuality, by using spin-path entanglement in a single neutron system. Here entanglement is achieved not between different particles, but between degrees of freedom, i.…
A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category
我们扩展了在以前的论文中给出的拓扑理论的处理，在量子理论中给量赋值，以及相关的问题，如Kochen-Specker定理。这个扩展有两个主要部分：使用冯·诺依曼代数作为基范畴（第2节）;以及我们的广义赋值与（i）实数区间量的赋值和（ii）粗粒度预层的子对象的概念的关系（第3节）。
Experimental Test of the Kochen-Specker Theorem for Single Qubits using Positive Operator-Valued Measures
我们提出了一个实验方案的实施任意广义测量，代表正算子值的措施，对单光子的偏振，使用线性光学器件。此外，我们实验测试的Kochen-Specker定理的单量子比特使用积极的运营商值的措施。我们的实验结果第一次反驳了非上下文隐藏变量理论，即使是单量子比特。
In a recent work, it was shown by one of us (EGC) that Bell-Kochen-Specker inequality violations in phenomena satisfying the no-disturbance condition (a generalisation of the no-signalling condition) cannot in general be explained with a faithful classical causal model -- that is, a classical causal model that satisfies the assumption of no fine-tuning. The proof of that claim however was restricted to Bell scenarios involving 2 parties or Kochen-Specker-contextuality scenarios involving 2 measurements per context. Here we show that the result holds in the general case of arbitrary numbers of …
If noncontextuality is defined as the robustness of a system's response to a measurement against other simultaneous measurements, then the Kochen-Specker arguments do not provide an algebraic proof for quantum contextuality. Namely, for the argument to be effective, (i) each operator must be uniquely associated with a measurement and (ii) commuting operators must represent simultaneous measurements. However, in all Kochen-Specker arguments discussed in the literature either (i) or (ii) is not met. Arguments meeting (i) contain at least one subset of mutually commuting operators which do not re…
Quantum cryptographic protocols based on complementarity are nonsecure against attacks in which complementarity is imitated with classical resources. The Kochen-Specker (KS) theorem provides protection against these attacks, without requiring entanglement or spatially separated composite systems. We analyze the maximum tolerated noise to guarantee the security of a KS-protected cryptographic scheme against these attacks, and describe a photonic realization of this scheme using hybrid ququarts defined by the polarization and orbital angular momentum of single photons.
MGP versus Kochen-Specker condition in hidden variables theories
量子力学的隐变量理论通常假设满足KS条件。贝尔-科亨-斯佩克定理表明，这些理论必然是上下文相关的。但是KS条件可以从操作的角度进行批评，这意味着应该采用一个较弱的条件（MGP）来代替它。这导致人们引入一类隐参数理论，在这种理论中，由于Bell-Kochen-Specker定理的证明失败，原则上可以避免上下文。作者最近为量子力学的客观解释提供了一个简单的模型，它可以被看作是一个非上下文的隐参数理论，这表明这种理论实际上是存在的。
Contextuality provides one of the fundamental characterizations of quantum phenomena, and can be used as a resource in lots of quantum information processing. In this paper, we summarize and derive some equivalent noncontextual inequalities from different noncontextual models of the proofs for Kochen-Specker theorem based on Greenberger-Horne-Zeilinger states. These noncontextual inequalities are equivalent up to some correlation items which hold both for noncontextual hidden variable theories and quantum mechanics. Therefore, using single-photon hyperentangled Greenberger-Horne-Zeilinger stat…
A Kochen-Specker Theorem for Unsharp Spin 1 Observables
不尖锐的自旋可观的事实，即剩余的不确定性的测量设备的实际对准仍然存在。如果不确定性低于一定水平，并且如果分布失调在旋转下是协变的，则不清晰自旋观测量的Kochen-Specker定理如下：存在有限的方向集，使得不是所有在这些方向上的不清晰自旋观测量都可以以非上下文的方式被一致地分配近似真值。
Ontological Models for Quantum Mechanics and the Kochen-Specker theorem
某些具体的“本体论模型”的量子力学（模型中的测量结果是确定性的和量子态是等效的经典概率分布在一些空间的“隐藏变量”）进行了检查。这些模型是Kochen和Specker的这种模型对于单个二维系统的推广-特别是详细考虑了三维量子系统的模型。不幸的是，这些模型似乎不能完全再现量子力学统计。然而，它们确实接近于这样做，而且由于它们仅仅涉及复射影空间上的概率分布，它们确实再现了量子力学中几乎所有其他的东西。Kochen-Specker定理根据这些模型进行了研究，并讨论了所表现的上下文性的相当温和的性质。
Generalized Kochen-Specker Theorem
证明了一个推广的Kochen-Specker定理。它表明，存在的$n$投影算子，表示$n$是-否的问题，量子系统，使没有$2^n$可能的答案是兼容的总和规则强加的量子力学。也就是说，如果一个可交换投影算子的子集和成一个只有偶数或奇数特征值的矩阵，那么“是”的答案的数量应该分别是偶数或奇数。这一要求可能导致矛盾。提供了一个例子，涉及9个投影算子在一个4维空间。
We show that a reduced form of the structural requirements for deterministic hidden variables used in Bell-Kochen-Specker theorems is already sufficient for the no-go results. Those requirements are captured by the following principle: an observable takes a spectral value x if and only if the spectral projector associated with x takes the value 1. We show that the only if part of this condition suffices. The proof identifies an important structural feature behind the no-go results; namely, if at least one projector is assigned the value 1 in any resolution of the identity, then at most one is.
Reexamination of the Kochen-Specker theorem: Relaxation of the completeness assumption
本文通过放宽完备性假设，证明可在任意 Kochen-Specker 集合上构造联合准概率分布，并指出可观测完备性可能只是次级性质而非基本特征。
A feasible "Kochen-Specker" experiment with single particles
我们提出了一个简单的实验方案，可以用来证明非上下文隐变量和量子力学之间的全有或全无类型的矛盾。该方案的灵感来自Cabello和Garcia-Alfred最近的想法，表明即使对于单个粒子，路径和自旋信息也不能以非上下文的方式预先确定。
I review the two theorems referred to in the title, and then suggest that it would be interesting to know how much of Hilbert space one can use without forcing the proof of these theorems. It would also be interesting to know what parts of Hilbert space that are essential for the proofs. I go on to discuss cubes, graphs, and pentagrams.
Coloring the rational quantum sphere and the Kochen-Specker theorem
本文扩展了Godsil和Zaks关于有理单位球构造性着色的研究，并探讨了实球面一致着色需额外颜色对Kochen-Specker定理的影响。
We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics and, thus, bypasses the constraints imposed by the Kochen-Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute fr…
In this paper we attempt to physically interpret the Modal Kochen- Specker (MKS) theorem. In order to do so, we analyze the features of the possible properties of quantum systems arising from the elements in an orthomodular lattice and distinguish the use of "possibility" in the classical and quantum formalisms. Taking into account the modal and many worlds non-collapse interpretation of the projection postulate, we discuss how the MKS theorem rules the constraints to actualization, and thus, the relation between actual and possible realms.
Comment on a suggested Kochen-Specker test
有一个建议是，对量子力学和非关联隐变量理论之间的差异进行观测测试，需要测量两个可交换的可观测量的乘积，而不需要分别测量任何一个可观测量。有人提出了这样做的建议，但这是有问题的。
Kochen-Specker (KS) sets are key tools for proving some fundamental results in quantum theory and also have potential applications in quantum information processing. However, so far, their intrinsic complexity has prevented experimentalists from using them for any application. The KS set requiring the smallest number of contexts has been recently found. Relying on this simple KS set, here we report an input state-independent experimental technique to certify whether a set of measurements is actually accessing a preestablished quantum six-dimensional space encoded in the transverse momentum of …
We discuss two approaches to producing generalized parity proofs of the Kochen-Specker theorem. Such proofs use contexts of observables whose product is $I$ or $-I$; we call them constraints. In the first approach, one starts with a fixed set of constraints and methods of linear algebra are used to produce subsets that are generalized parity proofs. Coding theory methods are used for enumeration of the proofs by size. In the second approach, one starts with the combinatorial structure of the set of constraints and one looks for ways to suitably populate this structure with observables. As well…
Kochen-Specker non-contextuality through the lens of quantization
本文探讨了量子理论通过量化经典理论获得时，科赫恩-斯佩克尔非上下文性条件的合理性，通过变形量子化和Weyl量化等方法分析了代数关系的变化。
In this paper we attempt to discuss what has Kochen-Specker (KS) theorem to say about physical invariance and quantum individuality. In particular, we will discuss the impossibility of making reference to objective physical properties within the orthodox formalism of quantum mechanics. Through an analysis of the meaning of physical invariance and quantum contextuality we will derive a Corollary to KS theorem that proves that a vector in Hilbert space cannot be interpreted coherently as an object possessing (objective) physical properties. As a consequence, the notion of quantum object can be o…
Some remarks on the theorems of Gleason and Kochen-Specker
在量子比特情形下，证明了两类限制性信息完备POVM的Gleason型定理。一个特殊的（不完全）Kochen-Specker着色，建议Appleby在三维，推广到任意尺寸。我们调查其有效性作为一个功能的尺寸，使用两种不同的措施。特别是，我们将得出一个限制的分数的领域，可以令人满意地着色使用广义Appleby建设的维数接近无穷大。第二个，也是物理上更相关的有效性测量，是查看适当着色的可能ON碱基的分数。利用这个测度，我们将推导出三维和四维实空间中的“上界的下界”。
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