Proving Kochen-Specker Theorem Using Projection Measurement and Positive Operator-Valued Measure

One of the main theorems on the impossibility of hidden variables in quantum mechanics is Kochen-Specker theorem (KS). This theorem says that any hidden variable theory that satisfies quantum mechanics must be contextual. More specifically, it asserts that, in Hilbert space of dimension ≥ 3, it is impossible to associate definite numerical values, 1 or 0, with every projection operator Pm, in such a way that, if a set of commuting Pm satisfies 1=ΣmP, the corresponding values will also satisfy . Since the first proof of Kochen and Specker using 117 vectors in R3, there were many attempts to reduce the number of vector either via conceiving ingenious models or extending the system being considered to higher dimension. By considering eight dimensional three qubits system, we found a state dependent proof that requires only five vectors. The state that we assign value of 1 is the ray that arises from intersection of two planes. The recent advancements show that the KS theorem proof can be extended to two dimensional quantum system through generalized measurement represented by positive operator-valued measured (POVM). In POVMs the number of available outcomes of a measurement may be higher than the dimensionality of the Hilbert space and N-outcome generalized measurement is represented by N-element POVM which consists of N positive semidefinite operators {}dE that sum to identity. Each pair of elements is not mutually orthogonal if the number of outcome of measurements is bigger than the dimensionality. In terms of POVM, Kochen-Specker theorem asserts that and could not be satisfied for . We developed a general model that enables us to generate different sizes of the POVM for the proof of the Kochen-Specker theorem. We show that the current simplest Nakamura model is in fact a special case of our model. W also provide another model which is as simple as the Nakamura’s but consists of different sets of POVM

Cosmic Crystallography: CCP-Index of Thurston Manifold

The universe is assumed to have negative spatial curvature with 3-dimensional hyperbolic Thurston manifold as the fundamental domain. The universal covering space of the universe is tessellated by fundamental domain through holonomy group. Collecting correlated pair method (CCP-method) is implemented to this model to compute CCP-index which indicates the multi-connectedness of the universe

Proof of Kochen¨CSpecker Theorem: conversion of product Rule to sum rule

Valuation functions of observables in quantum mechanics are often expected to obey two constraints called the sum rule and product rule. However, the Kochen–Specker (KS) theorem shows that for a Hilbert space of quantum mechanics of dimension d ≥ 3, these constraints contradict individually with the assumption of value definiteness. The two rules are not irrelated and Peres [Found. Phys. 26 (1996) 807] has conceived a method of converting the product rule into a sum rule for the case of two qubits. Here we apply this method to a proof provided by Mermin based on the product rule for a three-qubit system involving nine operators. We provide the conversion of this proof to one based on sum rule involving ten operators

Kochen-Specker Theorem for a Single Qubit

The Kochen-Specker theorem (KS) states that Ilonconrextual hidden variable theories are incompatible with quantum mechanics. Since the first proof of the theorem, von Neumann projection measurement has been used and the quantum system considered has dimensionality of at least three. However, recently generalized measurements represented by positive-operator-va!ued measures (POVM) have been applied to extend the proof to two dimensional quantum systems. This note gives numerical calculations on the first KS proof for a single qubit

Kochen-Specker theorem for a three-qubit system: a state-dependent proof with seventeen rays

We consider Kochen–Specker theorem for three-qubit system with eight-dimensional state space. Reexamining the proof given by Kernaghan and Peres, we make some clarifications on the orthogonality of rays and rank-two projectors found by them. Basing on their five groups of orthogonal octad, we then show a proof that requires only seventeen rays

No-Go theorems and quantization

In this review, we would like to highlight the three known no-go theorems in quantum physics in relation to the process of quantization that maps classical observables to quantum ones. The quantization approach considered is a mixture of Isham’s group-theoretic quantization and geometric quantization with special emphasis on underlying compact phase space geometry of spheres. The first is Groenewold-van Hove theorem that states the obstruction of quantizing the full algebra of observables and in the sphere case, only limited to the spin observables plus the constant functions. The other two are theorems of Bell and Kochen-Specker stating that the only hidden variable theories allowed by quantum physics are nonlocal and contextual ones. We give simple examples of these no-go theorems and indicate some interesting problems arising from them for the field of quantizatio

Classical tautology versus quantum mechanical contradiction

This is a pedagogical note introducing the basic idea of contextuality through a pairof spin- 1/2 particles system, quantum mechanically interpreting Schütte’s tautology, and using the later to generate a set of uncolourable rays in three-dimensional Hilbert space in order to prove Kochen-Specker Theorem

Banking in Vietnam

Vietnam, an isolated land for the past decades, is currently undergoing a series of reforms to open its doors and bring in a new era of changes and challenges. Just like China, another growing “tiger” in the region, Vietnam is now the new area of focus for investors in the Asia-Pacific region. Its economy has been growing substantially for the past years. New trade links, increased foreign investments and more foreign diplomatic ties are now taking place within the country. In the coming years, Vietnam will continue to grow with good prospects for investments, but this is, however, restricted by the existence of the United States trade embargo. Once the embargo is completely lifted, the development of Vietnam towards a conducive business environment for foreign investors will then accelerate.BUSINES

Complementary Sequential Circulating Tumor Cell (CTC) and Cell-Free Tumor DNA (ctDNA) Profiling Reveals Metastatic Heterogeneity and Genomic Changes in Lung Cancer and Breast Cancer

Introduction Circulating tumor cells (CTCs) and cell-free tumor DNA (ctDNA) are tumor components present in circulation. Due to the limited access to both CTC enrichment platforms and ctDNA sequencing in most laboratories, they are rarely analyzed together. Methods Concurrent isolation of ctDNA and single CTCs were isolated from lung cancer and breast cancer patients using the combination of size-based and CD45-negative selection method via DropCell platform. We performed targeted amplicon sequencing to evaluate the genomic heterogeneity of CTCs and ctDNA in lung cancer and breast cancer patients. Results Higher degrees of genomic heterogeneity were observed in CTCs as compared to ctDNA. Several shared alterations present in CTCs and ctDNA were undetected in the primary tumor, highlighting the intra-tumoral heterogeneity of tumor components that were shed into systemic circulation. Accordingly, CTCs and ctDNA displayed higher degree of concordance with the metastatic tumor than the primary tumor. The alterations detected in circulation correlated with worse survival outcome for both lung and breast cancer patients emphasizing the impact of the metastatic phenotype. Notably, evolving genetic signatures were detected in the CTCs and ctDNA samples during the course of treatment and disease progression. Conclusions A standardized sample processing and data analysis workflow for concurrent analysis of CTCs and ctDNA successfully dissected the heterogeneity of metastatic tumor in circulation as well as the progressive genomic changes that may potentially guide the selection of appropriate therapy against evolving tumor clonality