No-Cloning Theorem,Kochen-Specker Theorem,and Quantum Measurement Theories

The usual no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we investigate a relation between the no-cloning theorem and the projective measurement theory that the results of measurements are either + 1 or − 1. We introduce the Kochen-Specker (KS) theorem with the projective measurement theory. We result in the fact that the two quantum states under consideration cannot be orthogonal if we avoid the KS contradiction. Thus the no-cloning theorem implies that the two quantum states underconsideration are identical in that case. It turns out that the KS theorem with the projective measurement theory says a new version of the no-cloning theorem. Next, we investigate a relation between the no-cloning theorem and the measurement theory based on the truth values that the results of measurements are either + 1 or 0. We return to the usual no-cloning theorem that the two quantum states are identical or orthogonal in the case.

     

QED in Krein Space Quantization

In this paper we consider the QED in Krein space quantization. We show that the theory is automatically regularized. The three primitive divergences integrals in usual QED are considered in Krein QED. The photon self energy, electron self energy and vertex function are calculated in this formalism. We show that these quantities are finite. The infrared and ultraviolet divergencies do not appear. We discuss that Krein space quantization is similar to Pauli-Villars regularization, so we have called it the “Krein regularization”.     

Kochen-Specker Theorem as a Precondition for Quantum Computing

We study the relation between the Kochen-Specker theorem (the KS theorem) and quantum computing. The KS theorem rules out a realistic theory of the KS type. We consider the realistic theory of the KS type that the results of measurements are either +1 or ?1. We discuss an inconsistency between the realistic theory of the KS type and the controllability of quantum computing. We have to give up the controllability if we accept the realistic theory of the KS type. We discuss an inconsistency between the realistic theory of the KS type and the observability of quantum computing. We discuss the inconsistency by using the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be for an easy detector to a Pauli observable. We cannot accept the realistic theory of the KS type to simulate the double-slit experiment in a significant specific case. The realistic theory of the KS type can not depicture quantum detector. In short, we have to give up both the observability and the controllability if we accept the realistic theory of the KS type. Therefore, the KS theorem is a precondition for quantum computing, i.e., the realistic theory of the KS type should be ruled out.     

Power Spectrum in Krein Space Quantization

The power spectrum of scalar field and space-time metric perturbations produced in the process of inflation of universe, have been presented in this paper by an alternative approach to field quantization namely, Krein space quantization (Gazeau et al. in Class. Quantum Gravity 17:1415, 2000; Takook in Int. J. Mod. Phys. E 11:509, 2002; Rouhani and Takook in ). Auxiliary negative norm states, the modes of which do not interact with the physical world, have been utilized in this method. Presence of negative norm states play the role of an automatic renormalization device for the theory.     

Ward Identity in Krein Space Quantization

In the previous paper, Krein space quantization has been studied for QED (Forghan et al. in Ann. Phys. 327:2388, 2012). In this paper, the relation between the vertex function and the electron self energy has been studied, showing that the Ward identity is correct for Krein space quantization.     

A Krein Space Approach to PT-symmetry

In this note we apply Krein space methods to PT-symmetric problems to obtain conditions for the spectrum to be real and estimates of the number of non-real spectral points. An erratum to this article is available at.     

Trans-Planckian Scale and Krein Space Quantization

In this work, Krein space quantization method is applied to eliminate the Ultraviolet divergence of Green functions. This paper shows that the power spectrum of scalar field fluctuations can be calculated in the limit of short distance physics, trans-Planckian physics, without using the usual re-normalization process.     

Coupling Constant of QED in Krein Space Quantization

In this paper, considering the photon self energy in Krein space quantization including quantum metric fluctuation, the running coupling constant of QED is calculated and compared with the conventional result in QED theory.     

Using Inequalities as Tests for the Kochen-Specker Theorem for Multiparticle States

The Kochen-Specker theorem is investigated for n spin-1/2 systems by using an inequality proposed in Nagata (J. Math. Phys. 46, 102101, 2005) on the basis on binary logic. A measurement theory based on the truth values (binary logic), i.e., the truth T (1) for true and the falsity F (0) for false is used. The values of measurement outcome are either + 1 or 0 (in \(\hbar /2\) unit). The quantum predictions by n-multipartite states violate the inequality by an amount that grows exponentially with n. The measurement theory based on the binary logic provides an exponentially stronger refutation of the existence of hidden-variable when the number of parties of the state increases more. It turns out that the Kochen-Specker theorem becomes a quite strong theorem when the dimension of the multipartite state highly increases, regardless of entanglement properties.

     

Proof of Kochen-Specker 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.     

A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.     

Parity Proofs of the Kochen-Specker Theorem Based on the 24 Rays of Peres

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.     

Generalized Kochen-Specker theorem

A generalized Kochen-Specker theorem is proved. It is shown that there exist sets of n projection operators, representing n yes-no questions about a quantum system, such that none of the 2 possible answers is compatible with sum rules imposed by quantum mechanics. Namely, if a subset of commuting projection operators sums up to a matrix having only even or only odd eigenvalues, the number of yes answers ought to he even or odd, respectively. This requirement may lead to contradictions. An example is provided, involving nine projection operators in a 4-dimensional space.Dedicated to Professor Max Jammer on the occasion of his 80th birthday.I am grateful to N. D. Mermin for patiently explaining to me that ref. 11 was a Kochen-Specker argument, not one about locality, as I had wrongly thought. This work was supported by the Gerard Swope Fund, and the Fund for Encouragement of Research.     

Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations

Any attempt to construct a realistinterpretation of quantum theory founders on theKochen–Specker theorem, which asserts theimpossibility of assigning values to quantum quantitiesin a way that preserves functional relations between them. We constructa new type of valuation which is defined on alloperators, and which respects an appropriate version ofthe functional composition principle. The truth-values assigned to propositions are (i) contextual and(ii) multivalued, where the space of contexts and themultivalued logic for each context come naturally fromthe topos theory of presheaves. The first step in our theory is to demonstrate that theKochen–Specker theorem is equivalent to thestatement that a certain presheaf defined on thecategory of self-adjoint operators has no globalelements. We then show how the use of ideas drawn from the theory ofpresheaves leads to the definition of a generalizedvaluation in quantum theory whose values are sieves ofoperators. In particular, we show how each quantum state leads to such a generalized valuation. Akey ingredient throughout is the idea that, in asituation where no normal truth-value can be given to aproposition asserting that the value of a physical quantity A lies in a subset , it is nevertheless possible toascribe a partial truth-value which is determined by theset of all coarse-grained propositions that assert thatsome function f(A) lies in f(), and that are true in a normalsense. The set of all such coarse-grainings forms asieve on the category of self-adjoint operators, and ishence fundamentally related to the theory ofpresheaves.     

New Kochen-Specker sets in four dimensions

We show that all possible 388 4-dim Kochen-Specker (KS) (vector) sets (of yes-no questions) with 18 through 23 vectors and 844 sets with 24 vectors all with component values from {−1,0,1} can be obtained by stripping vectors off a single system provided by Peres 20 years ago. In addition to them, we have found a number of other KS sets with 22 through 24 vectors. We present the algorithms we used and features we found, such as, for instance, that Peres' 24-24 KS set has altogether six critical KS subsets.     

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space, we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.     

Nonlocality and the Kochen-Specker paradox

A new proof of the impossibility of reconciling realism and locality in quantum mechanics is given. Unlike proofs based on Bell's inequality, the present work makes minimal and transparent use of probability theory and proceeds by demonstrating a Kochen-Specker type of paradox based on the value assignments to the spin components of two spatially separated spin-1 systems in the singlet state of their total spin. An essential part of the argument is to distinguish carefully two commonly confused types of contextuality; we call them ontological and environmental contextuality. These in turn are associated with two quite distinct senses of nonlocality. We indicate the relevance of our treatment to other related discussions in recent literature on the philosophy of quantum mechanics.     

Kochen-Specker theorem for finite precision spin-one measurements

Unsharp spin-one observables arise from the fact that a residual uncertainty about the actual orientation of the measurement device remains. If the uncertainty is below a certain level, and if the distribution of measurement errors is covariant under rotations, a Kochen-Specker theorem for the unsharp spin observables follows: There are finite sets of directions such that not all the unsharp spin observables in these directions can consistently be assigned approximate truth values in a noncontextual way.     

A Kochen-Specker inequality from a SIC

Yu and Oh (eprint) [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 (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality.