Quantum decision theory as quantum theory of measurement

We present a general theory of quantum information processing devices, that can be applied to human decision makers, to atomic multimode registers, or to molecular high-spin registers. Our quantum decision theory is a generalization of the quantum theory of measurement, endowed with an action ring, a prospect lattice and a probability operator measure. The algebra of probability operators plays the role of the algebra of local observables. Because of the composite nature of prospects and of the entangling properties of the probability operators, quantum interference terms appear, which make actions noncommutative and the prospect probabilities nonadditive. The theory provides the basis for explaining a variety of paradoxes typical of the application of classical utility theory to real human decision making. The principal advantage of our approach is that it is formulated as a self-consistent mathematical theory, which allows us to explain not just one effect but actually all known paradoxes in human decision making. Being general, the approach can serve as a tool for characterizing quantum information processing by means of atomic, molecular, and condensed-matter systems.     

Total correlations and mutual information

In quantum information theory it is generally accepted that quantum mutual information is an information-theoretic measure of total correlations of a bipartite quantum state. We argue that there exist quantum states for which quantum mutual information cannot be considered as a measure of total correlations. Moreover, for these states we propose a different way of quantifying total correlations.     

Wave equations with energy-dependent potentials

We study wave equations with energy-dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy-dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory. This work was supported by the agreement between IN2P3 and ASCR (collaboration no. 97-13) and by the Grant Agency of ASCR (J.M., grant No.A1048305).     

The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem

A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial deterministic decoherence-like process, a subsequent indeterministic collapse-like process, and a final deterministic purification-like process. We also show that quantum probabilities can be generally interpreted as the probabilities of a first-order non-classical theory, describing situations of maximal lack of knowledge regarding the process of actualization of potential interactions, during a measurement.     

Electron scattering by a ring-shaped barrier in magnetic fields

We investigate the transport phenomena through a region containing a ring-shaped barrier in a quasi-one-dimensional quantum wire in magnetic fields. The calculated magnetoconductance curve shows a periodic dip structure, which is superimposed upon by another quasi-periodic dip structure. The current distributions for resonant states and the magnetoconductance features are well explained on the basis of the magnetic field dependence of the eigenvalue in the two-dimensional system.     

Numerical simulation of one- and two-dimensional ESEEM experiments

Numerical simulation has become an indispensable tool for the interpretation of pulse EPR experiments. In this work it is shown how automatic orientation selection, grouping of operator factors, and direct selection and elimination of coherences can be used to improve the efficiency of time-domain simulations of one- and two-dimensional electron spin echo envelope modulation (ESEEM) spectra. The program allows for the computation of magnetic interactions of any symmetry and can be used to simulate spin systems with an arbitrary number of nuclei with any spin quantum number. Experimental restrictions due to finite microwave pulse lengths are addressed and the enhancement of forbidden coherences by microwave pulse matching is illustrated. A comparison of simulated and experimental HYSCORE (hyperfine sublevel correlation) spectra of ordered and disordered systems with varying complexity shows good qualitative agreement.     

Sphaleron glueballs in NBI theory with symmetrized trace

We derive a closed expression for the SU(2) Born–Infeld action with the symmetrized trace for static spherically symmetric purely magnetic configurations. The lagrangian is obtained in terms of elementary functions. Using it, we investigate glueball solutions to the flat space NBI theory and their self-gravitating counterparts. Such solutions, found previously in the NBI model with the “square root–ordinary trace” lagrangian, are shown to persist in the theory with the symmetrized trace lagrangian as well. Although the symmetrized trace NBI equations differ substantially from those of the theory with the ordinary trace, a qualitative picture of glueballs remains essentially the same. Gravity further reduces the difference between solutions in these two models, and, for sufficiently large values of the effective gravitational coupling, solutions tends to the same limiting form. The black holes in the NBI theory with the symmetrized trace are also discussed.     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.     

Computing spin networks

We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU (2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3nj coefficients connecting inequivalent binary coupling schemes of n + 1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6j symbols) which satisfy algebraic identities that make the structure of the model similar to “state sum models” employed in discretizing topological quantum field theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j) is discussed.     

Simultaneous selective detection of multiple quantum spectra

A three-dimensional multiple-quantum NMR experiment that produces individual spectra of all quantum orders is described. The separation of different quantum orders is accomplished via Fourier transformation with respect to the phase of the first two pulses of a generic three-pulse multiple-quantum sequence. This dramatically reduces the time required to obtain several selectively detected spectra and enhances the sensitivity and digital resolution from that obtained using the original two-dimensional technique. The experiment is demonstrated on the protons of para-chlorotoluene dissolved in the nematic liquid crystal Merck ZLI-1132.     

Globally conformal invariant gauge field theory with rational correlation functions

Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields V_κ(x₁,x₂) of dimension (κ,κ). For a globally conformal invariant (GCI) theory we write down the OPE of V_κ into a series of twist (dimension minus rank) 2κ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field.

We argue that the theory of a GCI hermitian scalar field of dimension 4 in D=4 Minkowski space such that the 3-point functions of a pair of 's and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density .     

Sensitivity enhancement in structural measurements by solid state NMR through pulsed spin locking

Free induction decay (FID) signals in solid state NMR measurements performed with magic angle spinning can often be extended in time by factors on the order of 10 by a simple pulsed spin locking technique. The sensitivity of a structural measurement in which the structural information is contained in the dependence of the integrated FID amplitude on a preceding evolution period can therefore be enhanced substantially by pulsed spin locking in the signal detection period. We demonstrate sensitivity enhancements in a variety of solid state NMR techniques that are applicable to selectively isotopically labeled samples, including 13C-15N rotational echo double resonance (REDOR), 13C-13C dipolar recoupling measurements using the constant-time finite-pulse radio-frequency-driven recoupling (fpRFDR-CT) and constant-time double-quantum-filtered dipolar recoupling (CTDQFD) techniques, and torsion angle measurements using the double quantum chemical shift anisotropy (DQCSA) technique. Further, we demonstrate that the structural information in the solid state NMR data is not distorted by pulsed spin locking in the detection period.     

Insufficiency of the quantum state for deducing observational probabilities

It is usually assumed that the quantum state is sufficient for deducing all probabilities for a system. This may be true when there is a single observer, but it is not true in a universe large enough that there are many copies of an observer. Then the probability of an observation cannot be deduced simply from the quantum state (say as the expectation value of the projection operator for the observation, as in traditional quantum theory). One needs additional rules to get the probabilities. What these rules are is not logically deducible from the quantum state, so the quantum state itself is insufficient for deducing observational probabilities. This is the measure problem of cosmology.     

Quantum dots in aperiodic order

We study numerically with a Green-function technique one-dimensional arrays of quantum dots with two different models. The arrays are ordered according to the Fibonacci, the Thue–Morse, and the Rudin–Shapiro sequences. As a comparison, results from a periodically ordered chain and also from a random chain are included. The focus is on how the conductance (calculated within the Landauer–Büttiker formalism) depends on the Fermi level. In the first model, we find that in some cases rather small systems (≈60 dots) behave in the same manner as very large systems (>16,000 dots) and this makes it possible in these cases to interpret our results for the small systems in terms of the spectral properties of the infinite systems. In particular, we find that it is possible to see some consequences of the singular continuous spectra that some of the systems possess, at least for temperatures up to 100 mK. In the second model, we study the phenomenon ohmic addition, i.e. when the resistances of the constrictions add up to the total resistance. It results that of the systems studied, it is only the Rudin–Shapiro system that has this behaviour for large structures, while the resistances of the Fibonacci and the Thue–Morse systems might reach a limiting value (as a periodic system does).     

Solitonic mechanism of structural transformations in a nondegenerate chain of particles with anharmonic and competing interactions

A new type of dynamics of an infinite atomic chain of particles with anharmonic and competing interactions is investigated in the general case when its homogeneous equilibrium states have different energies. Cooperative transformations realized by topological and nontopological solitons are revealed. The soliton velocity spectrum is calculated in the framework of an approximate continuous second-order theory. Solitons with vanishing velocity are shown to be in good asymptotical correspondence with the exact static solutions of the Reichert-Schilling model.     

量子理论若干基本问题研究的新进展

本文结合最近的典型量子物理实验 ,如用冷原子Bragg散射实现的“which way”实验 ,量子退相干过程的微腔QED检验和C60 分子的量子干涉等 ,比较系统地介绍了量子理论基本问题若干研究的新进展 ,特别强调了处于其核心的量子测量问题及其相关的基本概念和基本思想 ,如EPR佯谬和Bell不等式 ,量子退相干和量子纠缠。从理论和实验结合的角度 ,本文阐述了被测系统和测量仪器的相互作用怎样导致量子测量的一般动力学过程。由此还讨论了外部环境和内部运动怎样诱导量子退相干和量子耗散 ,对“薛定谔猫佯谬”和“宏观物体空间局域化描述”给出了可能的物理解释。最后 ,通过具体例子 ,本文简单地讨论了量子物理基本问题的研究结果对量子信息的应用     

Another derivation of Weinberg's formula

To investigate how quantum effects might modify special relativity, we will study a Lorentz transformation between classical and quantum reference frames and express it in terms of the four-dimensional (4D) momentum of the quantum reference frame. The transition from the classical expression of the Lorentz transformation to a quantum-mechanical one requires us to symmetrize the expression and replace all its dynamical variables with the corresponding operators, from which we can obtain the same conclusion as that from quantum field theory (given by Weinberg's formula): owing to the Heisenberg's uncertainty relation, a particle (as a quantum reference frame) can propagate over a spacelike interval.     

Calculating the C Operator in PT-symmetric Quantum Mechanics

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.     

量子理论若干基本问题研究的新进展

本结合最后的典型量子物理实验,如用冷原子Bragg散射实现的“which-way”实验,量子退相干过程的微腔QED检验和C60分子的量子干涉等,比较系统地介绍了量子理论基本问题若干研究的新进展,特别强调了处于其核心的量子测量问题及其相关的基本概念和基本思想,如EPR佯谬和Bell不等式,量子退相干和量子纠缠,从理论和实验结合的角度本阐述了被测系统和测量仪器的相互作用怎样导致量子测量的一般动力学过程,由此还讨论了外部环境和内部运动怎样诱导量子退相干和量子耗散,对“薛定谔猫佯谬”和“宏观物体空间局域化描述”给出了可能的物理解释,最后,通过具体例子,本简单地讨论了量子物理基本问题的研究结果对量子信息的应用。