Total variance and invariant information in complementary measurements

We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner–Zeilinger invariant information.By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner–Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner–Zeilinger invariant information.     

Uncertainty Relations for Coherence

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases(MUBs), and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement(reference basis) such that the coherence of these states can be simultaneously small.     

Uncertainty relations for quantum coherence with respect to mutually unbiased bases

The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.     

Average versus maximal coherence

Coherence is a fundamental feature of quantum mechanics and plays a crucial role in the quantum realm. The issue of quantification of coherence has received considerable interests in recent years, and a variety of coherence quantifiers have been introduced. With these measures, one can investigate coherence in a more quantitative way. Based on the Wigner-Yanase skew information, we evaluate average coherence of a state with respect to arbitrary mutually unbiased bases as well as with respect to all orthonormal bases, and demonstrate their equivalence. We further evaluate the maximum coherence, and reveal a remarkable fact that in high dimensions, the coherence of a generic state is nearly maximal with respect to almost all reference bases. A similar concentration phenomenon also manifests itself when the coherence is quantified by the relative entropy of coherence. This observation concerning the almost equivalence between the average coherence and the maximal coherence for generic states may have theoretical implications for quantum information and quantum thermodynamics.     

Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography

A unique feature of polarization-sensitive Mueller optical coherence tomography is that, by measuring Jones or Mueller matrices, it can reveal the complete polarization properties of biological samples, even in the presence of diattenuation. We map local polarization properties for the first time to our knowledge by using polar decomposition in combination with least-squares fitting to differentiate measured integrated Jones matrices with respect to depth. We also introduce the new concept of dual attenuation coefficients to characterize diattenuation per unit infinitesimal length in tissues. We experimentally verify the algorithm using measurements of a section of porcine tendon and the septum of a rat heart.     

Convection-compensating diffusion experiments with phase-sensitive double-quantum filtering

We present a design scheme for phase-sensitive, convection-compensating diffusion experiments with gradient-selected homonuclear double-quantum filtering. The scheme consists of three blocks: a 1/2J evolution period during which antiphase single-quantum coherences are created; a period of double-quantum evolution; and another 1/2J period, during which antiphase single-quantum coherences are converted back into an in-phase state. A single coherence transfer pathway is selected using an asymmetric set of gradient pulses, and both diffusion sensitization and convection compensation are built into the gradient coherence transfer pathway selection. Double-quantum filtering can be used either for solvent suppression or spectral editing, and we demonstrate examples of both applications. The new experiment performs well in the absence of a field-frequency lock and does not require magnitude Fourier transformation. The proposed scheme may offer advantages in diffusion measurements of spectrally crowded systems, particularly small molecules solubilized in colloidal solutions or bound to macromolecules.     

Significance and measurability of the phase of a spatially coherent optical field

In usual measurements of the phase of an optical field it is generally assumed that the field is monochromatic. In reality this assumption is never justified. The distinction between monochromaticity and complete spatial coherence is first discussed, and it is then shown that with every spatially coherent field (e.g., a laser mode) one can associate a monochromatic wave that, in a well-defined sense, represents the average behavior of the field. Its phase can be measured by standard interferometric techniques and also by techniques developed in recent years for the measurement of the spectral degree of coherence of fields of arbitrary states of coherence.     

Hierarchical Characterization of Complex Networks

While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be obtained by considering further neighborhoods. The current work considers the concept of virtual hierarchies established around each node and the respectively defined hierarchical node degree and clustering coefficient (introduced in cond-mat/0408076), complemented by new hierarchical measurements, in order to obtain a powerful set of topological features of complex networks. The interpretation of such measurements is discussed, including an analytical study of the hierarchical node degree for random networks, and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random, scale-free and regular network models as well as real data (airports, proteins and word associations). The enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network, a possibility which is also illustrated in the current article.     

An analysis of the validity of local causality at the statistical level in Einstein-Podolsky-Rosen-Type situations

We investigate the question of local causality at the statistical level in Einstein-Podolsky-Rosen (EPR) type situations, taking into account the most general class of measurements envisaged in quantum theory. The condition for local causality at the statistical level used in this paper pertains to the invariance of statistics of measurements on one sub-system with respect to the choice and type of measurements on its correlated partner in the EPR-type examples. Our analysis is based on a criterion for measurements performed on one of the EPR sub-systems, which is more general than the criterion used in the earlier treatments. We discuss both non-absorptive measurements (where the system is available for further observation after the measurement is performed) as well as absorptive measurements (where the system is absorbed in the process of a particular outcome being realized). We show that in the case of arbitrary non-absorptive measurements characterized by operationvalued measures, the requirement of local causality at the statistical level is satisfied and in the process we identify the key inputs in such a proof. We also obtain the specific conditions under which an absorptive measurement satisfies local causality at the statistical level.     

Map for Simultaneous Measurements for a Quantum Logic

In this paper we will study a function of simultaneous measurements for quantum events (s-map) which will be compared with the conditional states on an orthomodular lattice as a basic structure for quantum logic. We will show the connection between s-map and a conditional state. On the basis of the Rényi approach to the conditioning, conditional states, and the independence of events with respect to a state are discussed. Observe that their relation of independence of events is not more symmetric contrary to the standard probabilistic case. Some illustrative examples are included.     

Statistics of dynamic speckles in application to distance measurements

We present an analysis of statistical properties of dynamic speckles to estimate the limiting accuracy of measurements achievable in a distance sensor using spatially filtered dynamic speckles. The main reason for inaccurate measurements using dynamic speckles is their stochastic nature. It is shown that the average lifetime of dynamic speckles is the key factor defining the measurement accuracy. Main conclusions of the theoretical analysis were confirmed in an experiment carried out with a fast moving rough surface. Special attention is paid to a recently proposed range sensor using dynamic speckles generated by a fast-deflecting laser beam. It is shown that this sensor possesses the best combination of accuracy and response time.     

Continuous Quantum Nondemolition Measurements of a Particle in Electromagnetic and Gravitational Fields

The detection of a particle in electromagnetic plus gravitational fields is investigated. We obtain a set of quantum nondemolition variables. The continuous measurements of these nondemolition parameters are analyzed in the framework of restricted path integral formalism. We manipulate the corresponding propagators, and deduce the probabilities associated with the possible measurement outputs.     

Continuous Quantum Nondemolition Measurements of a Particle in Electromagnetic and Gravitational Fields

The detection of a particle in electromagnetic plus gravitational fields is investigated. We obtain a set of quantum nondemolition variables. The continuous measurements of these nondemolition parameters are analyzed in the framework of restricted path integral formalism. We manipulate the corresponding propagators, and deduce the probabilities associated with the possible measurement outputs.     

Quantum theory of continuous measurements and its applications in quantum optics

We present an overview of the quantum theory of continuous measurements and discuss some of its important applications in quantum optics. Quantum theory of continuous measurements is the appropriate generalization of the conventional formulation of quantum theory, which is adequate to deal with counting experiments where a detector monitors a system continuously over an interval of time and records the times of occurrence of a given type of event, such as the emission or arrival of a particle. We first discuss the classical theory of counting processes and indicate how one arrives at the celebrated photon counting formula of Mandel for classical optical fields. We then discuss the inadequacies of the so called quantum Mandel formula. We explain how the unphysical results that arise from the quantum Mandel formula are due to the fact that the formula is obtained on the basis of an erroneous identification of the coincidence probability densities associated with a continuous measurement situation. We then summarize the basic framework of the quantum theory of continuous measurements as developed by Davies. We explain how a complete characterization of the counting process can be achieved by specifying merely the measurement transformation associated with the change in the state of the system when a single event is observed in an infinitesimal interval of time. In order to illustrate the applications of the quantum theory of continuoius measurements in quantum optics, we first derive the photon counting probabilities of a single-mode free field and also of a single-mode field in interaction with an external source. We then discuss the general quantum counting formula of Chmara for a multi-mode electromagnetic field coupled to an external source. We explain how the Chmara counting formula is indeed the appropriate quantum generalization of the classical Mandel formula. To illustrate the fact that the quantum theory of continuous measurements has other diverse applications in quantum optics, besides the theory of photodetection, we summarize the theory of ‘quantum jumps’ developed by Zoller, Marte and Walls and Barchielli, where the continuous measurements framework is employed to evaluate the statistics of photon emission events in the resonance fluorescence of an atomic system.     

Two-dimensional depth-resolved Mueller matrix characterization of biological tissue by optical coherence tomography

We built a polarization-sensitive optical coherence tomographic system and measured the two-dimensional depth-resolved full 4 x 4 Mueller matrix of biological tissue for what is believed to be the first time. The Mueller matrix measurements, which we made by varying the polarization states of the light source and the detector, yielded a complete characterization of the polarization property of the tissue sample. The initial experimental results indicated that this new approach reveals some tissue structures that are not perceptible in standard optical coherence tomography.     

Super-symmetric informationally complete measurements

Symmetric informationally complete measurements (SICs in short) are highly symmetric structures in the Hilbert space. They possess many nice properties which render them an ideal candidate for fiducial measurements. The symmetry of SICs is intimately connected with the geometry of the quantum state space and also has profound implications for foundational studies. Here we explore those SICs that are most symmetric according to a natural criterion and show that all of them are covariant with respect to the Heisenberg–Weyl groups, which are characterized by the discrete analog of the canonical commutation relation. Moreover, their symmetry groups are subgroups of the Clifford groups. In particular, we prove that the SIC in dimension 2, the Hesse SIC in dimension 3, and the set of Hoggar lines in dimension 8 are the only three SICs up to unitary equivalence whose symmetry groups act transitively on pairs of SIC projectors. Our work not only provides valuable insight about SICs, Heisenberg–Weyl groups, and Clifford groups, but also offers a new approach and perspective for studying many other discrete symmetric structures behind finite state quantum mechanics, such as mutually unbiased bases and discrete Wigner functions.     

Template matching of mixed quantum states

We consider the problem of optimal classification of an unknown input mixed quantum state with respect to a set of predefined patterns C_i, each represented by a known mixed quantum template . The performance of the matching strategy is addressed within a Bayesian formulation where the cost function, as suggested by the theory of monotone distances between quantum states, is chosen to be the fidelity or the relative entropy between the input and the templates. We investigate various examples of quantum template matching for the case of a finite number of copies of a two-level input state and for a generic, group covariant, set of two-level template states.     

Classical Invasive Description of Informationally-Complete Quantum Processes

In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, conditions that characterize uniquely classical processes that are probed by a reasonable class of such invasive measurements are derived. We then analyze under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally-complete measurements. It is expected that this investigation will help build a bridge between two fundamental traits of non-classicality, namely, coherence and contextuality.     

Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d?3?c(d) α(d) for some ${c(d)\in[1,2]}$ and α(d) being the number of ones appearing in the binary expansion of (d?1).