Local Realism and Conditional Probability

Emilio Santos has argued (Santos, Studies in History and Philosophy of Physics http: //arxiv-org/abs/quant-ph/0410193) that to date, no experiment has provided a loophole-free refutation of Bell’s inequalities. He believes that this provides strong evidence for the principle of local realism, and argues that we should reject this principle only if we have extremely strong evidence. However, recent work by Malley and Fine (Non-commuting observables and local realism, http: //arxiv-org/abs/quant-ph/0505016) appears to suggest that experiments refuting Bell’s inequalities could at most confirm that quantum mechanical quantities do not commute. They also suggest that experiments performed on a single system could refute local realism. In this paper, we develop a connection between the work of Malley and Fine and an argument by Bub from some years ago [Bub, The Interpretation of Quantum Mechanics, Chapter VI(Reidel, Dodrecht,1974)]. We also argue that the appearance of conflict between Santos on the one hand and Malley and Fine on the other is a result of differences in the way they understand local realism.     

Extending States on Finite Concrete Logics

In this note we collect several observations on state extensions. They may be instrumental to anyone who pursues the theory of quantum logics. In particular, we find out when extensions (resp. signed extensions) exist in the “concrete” concrete logic of all even-element subsets of an even-element set (Th. 2.3 and Th. 2.9). We also mildly add to the study of difference-closed logics as initiated in Ovchinnikov (1999) by finding an extension theorem for subadditive states. Our results suplement the research carried on in De Simone (2000), Gudder (1979), Gudder and Marchand (1980), Navara and Pták (1983), Navara (1989), Ovchinnikov (1999), Pták (2000), Pták and Pulmannová (1991), Prather (1980), Sherstnev (1968), Sultanbekov (1992), and Svozil (1998). 2000 AMS Classification: 06C15, 81P10. The authors want to express their gratitude to the referee for valuable suggestions which corrected the original version. Originally published in International Journal of Theoretical Physics, Vol. 44, No. 7, 2005, Due to a publishing error, authorship of the article was credited incorrectly. The corrected article is reprinted in its entirety here. The online version of the original article can be found at     

Absolute Continuity of States on Concrete Logics

By concrete logic we mean a quantum logic which is set-representable, and byVitali—Hahn—Saks logic (VHS-logic) we mean a concrete logic for which theVitali—Hahn—Saks theorem holds true. In this note we investigate the size of theclass of VHS-logics, showing among others that each concrete logic can beconcretely enlarged to a VHS-logic as well as to a non-VHS-logic.     

A Conditional Mutual Information Estimator for Mixed Data and an Associated Conditional Independence Test

In this study, we focus on mixed data which are either observations of univariate random variables which can be quantitative or qualitative, or observations of multivariate random variables such that each variable can include both quantitative and qualitative components. We first propose a novel method, called CMIh, to estimate conditional mutual information taking advantages of the previously proposed approaches for qualitative and quantitative data. We then introduce a new local permutation test, called LocAT for local adaptive test, which is well adapted to mixed data. Our experiments illustrate the good behaviour of CMIh and LocAT, and show their respective abilities to accurately estimate conditional mutual information and to detect conditional (in)dependence for mixed data.     

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.     

Laziness of Symmetric and Separable States

An N-partite state is considered lazy, if the entropy rate of one subsystem with respect to time is zero under any coupling to the other subsystems. In this paper, we show that all biaxial or purely second rank tensor polarized systems are lazy. Such a system can be produced in the laboratory by the interaction of a spin-1 nuclei with non-zero quadrupole moment like H ², N ¹⁴ with an external quadrupole field found in suitable crystal lattice. We then investigate the ’laziness’(property of the system to be lazy) of N-qubit mixed symmetric separable states and enumerate the conditions for them to be lazy. Further, we study the laziness of direct product states on application of a global and local noisy channels.     

Spin States and Probability Distribution Functions

Formulation of the conventional quantum mechanics in which a state is described by probability instead of wave function and density matrix is presented. We consider the possibility of constructing the invertable map of spinors onto positive probability distributions. For any value of spin, the basis of the irreducible representation of a rotation group is realized by a family of probability distributions of the spin projection parametrized by points on a sphere. Quantum states of a symmetric top described by the probability distributions are discussed.     

Determinacy of States and Independence of Operator Algebras

The aim of this paper is to summarize, deepen,and comment upon recent results concerning states onoperator algebras and their extensions. The first partis focused on the relationship between pure states and singly generated subalgebras. Among otherswe show that every pure state on a separablealgebra A is uniquely determined by some element of Awhich exposes . The main part of this paper is the second section, dealing with characterizationof various types of independence conditions arising inthe axiomatics of quantum field theory. These twotopics, seemingly different, are connected by a common extension technique based on determinacy ofpure states.     

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders-von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are： the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic （events） belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.     

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

In the quantum mechanical Hilbert space formalism, the probabilisticinterpretation is a later ad-hoc add-on, more or less enforced by theexperimental evidence, but not motivated by the mathematical model itself. Amodel involving a clear probabilistic interpretation from the very beginningis provided by the quantum logics with unique conditional probabilities. Itincludes the projection lattices in von Neumann algebras and hereprobability conditionalization becomes identical with the state transitionof the Lüders - von Neumann measurement process. This motivates thedefinition of a hierarchy of five compatibility and comeasurability levelsin the abstract setting of the quantum logics with unique conditionalprobabilities. Their meanings are: the absence of quantum interference orinfluence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.     

Conditional Information and Hidden Correlations in Single-qudit States

We discuss the notions of mutual information and conditional information for noncomposite systems, classical and quantum; both the mutual information and the conditional information are associated with the presence of hidden correlations in the state of a single qudit. We consider analogs of the entanglement phenomena in the systems without subsystems related to strong hidden quantum correlations.     

Geometry and Symmetric Coherent States of Three Qubits Systems

In this paper, we first generalize the previous results that relate 1- and 2-qubit geometries to complex and quaternionic Möbius transformations respectively, to the case of 3-qubit states under octonionic Möbius transformations. This completes the correspondence between the qubit geometries and the four normed division algebras. Thereby, new systems of symmetric coherent states with 2 and 3 qubits can be constructed by mapping the spin coherent states to their antipodal symmetric ponits on the generalized Bloch spheres via Möbius transformations in corresponding dimensions. Finally, potential applications of the normed division algebras in physics are discussed.     

Symmetric Telecloning and Entanglement Distribution of Spin Quantum States

We propose a physical realization of symmetric telecloning machine for spin quantum states. The concept of area average fidelity is introduced to describe the telecloning quality. It is indicated that for certain input states this quantity may come to an enough high level to satisfy the need of quantum information processing. We also study the properties of entanglement distribution via the spin chain for arbitrary two-qubit entangled pure states as inputs and find that the decay ratio of entanglement for the output states is only determined by the parameters of spin chain and waiting time, independent of the initial input states.     

Probability Representation of Quantum Observables and Quantum States

We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.     

利用分束器通过条件测量制备非经典光场态和量子纠缠态

把相干态和单粒子数态作为分束器的输入态，通过条件测量可以剪去输入相干态中的任意粒子数态，并研究了这些被剪切了的光场态的性质。结果显示剪去真空态和单粒子数态的输出态具有较强的压缩和亚泊松分布等非经典效应。把剪切了的输出态和真空态输入分束器，得到的输出态具有量子纠缠性质，从而制备出量子纠缠态，同时也验证了被剪切的输出态的非经典性。     

Symmetric extensions of quantum States and local hidden variable theories

While all bipartite pure entangled states violate some Bell inequality, the relationship between entanglement and nonlocality for mixed quantum states is not well understood. We introduce a simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states. The method is based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a local hidden variable model with a certain number of settings for the observers Alice and Bob.     

Thermal States in Conformal QFT. II

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net of von Neumann algebras on . In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir₁ with the central charge c = 1, whilst for the Virasoro net Vir _c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets and is the fixed point of w.r.t. a compact gauge group, then any locally normal, primary KMS state on extends to a locally normal, primary state on , KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.     

The Exit Distribution for Smart Kinetic Walk with Symmetric and Asymmetric Transition Probability

It has been proved that the distribution of the point where the smart kinetic walk (SKW) exits a domain converges in distribution to harmonic measure on the hexagonal lattice. For other lattices, it is believed that this result still holds, and there is good numerical evidence to support this conjecture. Here we examine the effect of the symmetry and asymmetry of the transition probability on each step of the SKW on the square lattice and test if the exit distribution converges in distribution to harmonic measure as well. From our simulations, the limiting exit distribution of the SKW with a non-uniform but symmetric transition probability as the lattice spacing goes to zero is the harmonic measure. This result does not hold for asymmetric transition probability. We are also interested in the difference between the SKW with symmetric transition probability exit distribution and harmonic measure. Our simulations provide strong support for a explicit conjecture about this first order difference. The explicit formula for the conjecture will be given below.