generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

the principle of strictly increasing mixing character in incompatible general measurement it is known that the density operators rho of a quantal system are grouped into classes of equally mixed ones the class to which rho belongs is its so called mixing character the set of classes is known to be a lattice the ruch and mead 1976 principle of increasing mixing character in complete measurement is extended to general observables it is shown that the principle of strictly increasing mixing character holds in general incompatible measurement and some consequences are discussed a sufficient condition for strictly mixing homomorphic functionals i e for those which preserve the strictly larger relation in the lattice is obtained density operator dependent strengthening of the natural pre order in the set of general observables is investigated

majorization entropic uncertainty relations entropic uncertainty relations in a finite dimensional hilbert space are investigated making use of the majorization technique we derive explicit lower bounds for the sum of r enyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix numerical simulations show that for a generic unitary matrix of size n 5 our bound is stronger than the well known result of maassen and uffink mu with a probability larger than 98 we also show that the bounds investigated are invariant under the dephasing and permutation operations finally we derive a classical analogue of the mu uncertainty relation which is formulated for stochastic transition matrices

optimal tight frames and quantum measurement tight frames and rank one quantum measurements are shown to be intimately related in fact the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank one generalized quantum measurements on that space using this relationship frame theoretical analogs of various quantum mechanical concepts and results are developed the analog of a least squares quantum measurement is a tight frame that is closest in a least squares sense to a given set of vectors the least squares tight frame is found for both the case in which the scaling of the frame is specified constrained least squares frame clsf and the case in which the scaling is chosen to minimize the least squares error unconstrained least squares frame ulsf the well known canonical frame is shown to be proportional to the ulsf and to coincide with the clsf with a certain scaling

a dynamical system approach to the self similar large time behavior in scalar convection diffusion equations abstract we present a dynamical system method that provides both the existence of self similar solutions and the self similar large time behavior for convection diffusion equations in r n this method avoids a direct study of the elliptic problem related to the self similar profiles we concentrate our attention on the model example u t u a u 1 n u g x t in r n 0 u 0 u 0 l 1 r n l r n where a r n and g x t t 1 n 2 1h x formula with h l2 r n exp x 2 4 and l r n such that h y dy 0 and h y y dy being small enough in the natural similarity variables the self similar profiles become stationary solutions of a new convection diffusion equation by using lyapunov type arguments that rely in an essential manner on the l1 contraction property of the system we prove that those stationary solutions exists and that any trajectory converges to one of them the limit of any trajectory is completely determined by its mass which is conserved along the time in order to ensure the relative compactness of trajectories we work in the functional framework of the weighted sobolev spaces introduced by escobedo and kavian

shape as a cantor completion process let x and y be metric compacta y embedded in the hilbert cube q for two maps f g x q the authors define f f g inf e 0 f is homotopic to g in the e neighborhood of y and a sequence of maps fk x q k n is said to be a cauchy sequence provided for every e 0 there is a k0 n such that f fk fk e whenever k k k 0 such sequences coincide with the approximative maps of k borsuk theory of shape pwn warsaw 1975 and represent shape morphisms from x to y the function f is not a pseudometric but defining d lim k f fk gk where the shape morphisms sh x y are represented by cauchy sequences fk gk the authors prove that sh x y d becomes a complete zero dimensional ultrametric space homeomorphic to a closed subset of the irrationals among other things the authors prove that if two compacta x and y are of the same shape then for every compactum z the spaces sh x z and sh y z are uniformly homeomorphic in the last section the authors show for example that for x compact and y fanr the space sh x y is countable give several characterizations of various kinds of movability and translate their results to z sets in q and sequences of proper maps between their complements

general entropy like uncertainty relations in finite dimensions we revisit entropic formulations of the uncertainty principle up for an arbitrary pair of positive operator valued measures povm a and b acting on finite dimensional hilbert space salicr generalized entropies including r nyi and tsallis ones among others are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables we obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals which is valid for both pure and mixed states the bound depends on the overlap triplet with ca respectively cb being the overlap between the elements of the povm a respectively b and the overlap between the pair of povm our approach is inspired by that of de vicente and s nchez ruiz 2008 phys rev a 77 042110 and consists in a minimization of the entropy sum subject to the landau pollak inequality that links the maximum probabilities of both observables we solve the constrained optimization problem in a geometrical way and furthermore when dealing with r nyi or tsallis entropic formulations of the up we overcome the h lder conjugacy constraint imposed on the entropic indices by the riesz thorin theorem in the case of nondegenerate observables we show that for given the bound obtained is optimal and that for r nyi entropies our bound improves deutsch one but maassen uffink bound prevails when finally we illustrate by comparing our bound with known previous results in particular cases of r nyi and tsallis entropies

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

new extremal problems for the riemannian recognition program via alexandrov geometry in its most general form the recognition problem in riemannian geometry asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold optimally one wants to recognize a manifold having made as few measurements as possible many results in riemannian geometry including pinching theorems can be viewed this way here we are only interested in measurements that assign real numbers to each complete riemannian manifold typical examples of such invariants are diameter volume curvature bounds etc when viewing one or several invariants i ii ii of this type as a map on a suitable class l of riemannian manifolds the following problems pose themselves

majorization properties of generalized thermal distributions abstract we examine the majorization properties of general thermal like mixed states depending on a set of parameters sufficient conditions which ensure the increase in mixedness and hence of any associated entropic form when these parameters are varied are identified we then discuss those exhibiting a power law distribution showing that they can be characterized by two distinct mixing parameters one associated with temperature and the other with the non extensivity index q illustrative numerical results are also provided

an alternative to the luders generalization of the von neumann projection and its interpretation an alternative to the luders generalization of the von neumann projection is formulated allowing a generalization to measurements represented by positive operator valued measures this generalization is based on the notions of non orthogonal projections and bi orthonormal systems in hilbert schmidt space and is closely related to phase space representations of quantum mechanics a number of applications will be discussed it is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state to be obtained by the measurement

mean value and uncertainty of optical phase a simple mechanical analogy there exist several definitions of the mean value and the phase uncertainty of an optical field we show that one of these definitions is especially simple and intuitive because of its mechanical analogy on this basis we also derive the summation rule and the tchebyshev inequality for random angular and phase variables and a number phase uncertainty relation

volume comparison and its applications in riemann finsler geometry abstract we introduce a new geometric quantity the mean covariation for finsler metrics and establish a volume comparison theorem as an application we obtain some precompactness and finiteness theorems for finsler manifolds

entropic measures of joint uncertainty effects of lack of majorization we compute renyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two dimensional quantum systems the relative amount of uncertainty between two states depends on the uncertainty measure used these results are not reproduced by a more standard duality relation we show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics

generalized entropy as a measure of quantum uncertainty abstract a generalized entropy is used in order to advance a different form of expressing the uncertainty principle of quantum mechanics we consider the generalized entropic formulation for different pairs of incompatible observables in particular we study the number phase entropic uncertainty measure for the case of coherent states within the pegg barnett theory we also tackle the situation of operators with continuous spectra where a correlation functional is calculated in terms of generalized joint and marginal entropies for harmonic oscillator wavefunctions

fisher information renyi entropy power and quantum phase transition in the dicke model abstract fisher information renyi entropy power and fisher renyi information product are presented for the dicke model there is a quantum phase transition in this quantum optical model it is pointed out that there is an abrupt change in the fisher information renyi entropy power the fisher shannon and renyi lengths at the transition point it is found that these quantities diverge as the characteristic length c 1 4 around the critical value of the coupling strength c for any value of the parameter

on a generalized entropic uncertainty relation in the case of the qubit we revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two dimensional hilbert space r nyi entropy is used as an uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables we derive a general expression for the tight lower bound of the sum of r nyi entropies for any couple of positive entropic indices thus we have overcome the h lder conjugacy constraint imposed on the entropic indices by riesz thorin theorem in addition we present an analytical expression for the tight bound inside the square 0 1 2 2 in the plane and a semi analytical expression on the line it is seen that previous results are included as particular cases moreover we present a semi analytical suboptimal bound for any couple of indices in all cases we provide the minimizing states

simplified dynamic model for the motility of irregular echinoids inspired by the locomotion mechanism of sea urchins we study the locomotion of an irregular echinoid by means of a simplified dynamical model we prove that if two conjectures are assumed the geometrical arrangement of the five ambulacral petals of irregular echinoids should form a eutactic star in order to optimize motility we firstly propose an adequate measure of eutacticity that allows us to to verify the statistical tendency to such a property for a representative collection of fossil specimens next regarding dynamics the biological advantage of eutactic stars is addressed as a minimal path problem finally we study the stability of some eutactic stars under small perturbations

a radius sphere theorem the purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated rauch berger klingenberg sphere theorem and the diameter sphere theorem in riemannian geometry there has lately been considerable interest in studying spaces which are more singular than riemannian manifolds a natural reason for doing this is because gromov hausdorff limits of riemannian manifolds are almost never riemannian manifolds but usually only inner metric spaces with various nice properties the kind of spaces we wish to study here are the so called alexandrov spaces alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the distance comparison sense this definition might seem a little ambiguous since there are many ways in which one can define finite dimensionality and lower curvature bounds the foundational work by plaut in pi however shows that these different possibilities for definitions are equivalent the structure of alexandrov spaces was studied in bgp pi and p in particular if x is an alexandrov space and p e x then the space of directions ep at p is an alexandrov space of one less dimension and with curvature 1 furthermore a neighborhood of p in x is homeomorphic to the linear cone over one of the important implications of this is that the local structure of n dimensional alexandrov spaces is determined by the structure of n l dimensional alexandrov spaces with curvature 1 sphere theorems in this context seem to be particularly interesting for if one can give geometric characterizations of

on information gain by quantum measurements of continuous observables a generalization of shannon s amount of information into quantum measurements of continuous observables is introduced a necessary and sufficient condition for measuring processes to have a non negative amount of information is obtained this resolves groenewold s conjecture completely including the case of measurements of continuous observables as an application the approximate position measuring process considered by von neumann and later by davies is shown to have a non negative amount of information

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

uncertainty in quantum measurements the object of this letter is to show that except in the case of canonically conjugate observables the generalized heisenberg inequality does not properly express the quantum uncertainty principle it is in general too weak an inequality is obtained which does express the principle

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

superspace geometry the exact uncertainty relationship between complementary aspects after reviewing the standard uncertainty relations due to heisenberg 1927 robertson 1929 and schrodinger 1930 as well as the relations of deutsch and of maassen and uffink including the so called entropic relations the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry an approach not hitherto employed two incompatible properties a sigma alpha aalpha mod alpha alpha mod and n sigma nnn mod n n mod belong to a pair of complementary aspects defined by two orthonormal bases mod alpha and mod n in the hilbert space h if the state is mod psi then p alpha mod alpha mod psi mod 2 is the probability of obtaining the value aalpha in a measurement of a and p n mod n mod psi mod 2 is the probability of obtaining the value nn in a measurement of n the two aspects are characterised relative to mod psi by the numbers so called purities pi sigma alpha p alpha 2 and pi sigma n p n 2 both or 1 he gives a complete characterisation of the uncertainty relationship between a and n more precisely between their aspects in terms of the range of joint values of pi pi for arbitrary initial states pure as well as mixed a theorem of lenard is given an alternative proof employing only elementary superspace geometry the results depend on two angles phi m minimal angle and phi m maximal angle between the two aspects exact expressions for phi m and phi m are obtained in terms of the overlap matrix lambda lambda alpha n mod alpha mod n mod 2 as a corollary he finds the uncertainty relation for a pure state mod psi pi pi or 1 1 g 1 1 g cos phi m where g dim h and a sharper one for mixed states pi pi 2 is obtainable if and only if the intersection of the aspects holds a pure state if phi m pi 2 maximal incompatibility then pi pi or 1 1 g is a special case of a stronger relation sigma mu g o pi mu 2 which one obtains for g 1 maximally incompatible aspects by means of a thereom of ivanovic

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

approximate transformations of bipartite pure state entanglement from the majorization lattice abstract we study the problem of deterministic transformations of an initial pure entangled quantum state into a target pure entangled quantum state by using local operations and classical communication locc a celebrated result of nielsen 1999 gives the necessary and sufficient condition that makes this entanglement transformation process possible indeed this process can be achieved if and only if the majorization relation holds where and are probability vectors obtained by taking the squares of the schmidt coefficients of the initial and target states respectively in general this condition is not fulfilled however one can look for an approximate entanglement transformation vidal et al 2000 have proposed a deterministic transformation using locc in order to obtain a target state opt most approximate to in terms of maximal fidelity between them here we show a strategy to deal with approximate entanglement transformations based on the properties of the majorization lattice more precisely we propose as approximate target state one whose schmidt coefficients are given by the supremum between and our proposal is inspired on the observation that fidelity does not respect the majorization relation in general remarkably enough we find that for some particular interesting cases like two qubit pure states or the entanglement concentration protocol both proposals are coincident

blow up and propagation of disturbances for fast diffusion equations abstract this paper is concerned with the cauchy problem for the fast diffusion equation u t u m u p 1 in r n n 1 where m 0 1 p 1 1 and 0 the initial condition u 0 is assumed to be continuous nonnegative and bounded using a technique of subsolutions we set up sufficient conditions on the initial value u 0 so that u t x blows up in finite time and we show how to get estimates on the profile of u t x for small enough values of t 0

extremum entrop y based heisenberg like uncertainty relations in this work we use the extremization method of various information theoretic measures fisher information shannon entropy tsallis entropy for d dimensional quantum systems which complementarily describe the spreading of the quantum states of natural systems under some given constraints usually one or two radial expectation values this variational method allows us to determine an extremum entropy distribution which is the least biased one to characterize the state among all those compatible with the known data then we use it together with the spin dependent uncertainty like relations of daubechies thakkar type as a tool to obtain relationships between the position and momentum radial expectation values of the type for d dimensional systems of n fermions with spin s the resulting uncertainty like products which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system are shown to often improve the best corresponding relationships existing in the literature

universal uncertainty relations uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined the modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring noncommuting observables however there is no fundamental reason for using entropies as quantifiers any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation starting from a very reasonable assumption of invariance under mere relabeling of the measurement outcomes we show that schur concave functions are the most general uncertainty quantifiers we then discover a fine grained uncertainty relation that is given in terms of the majorization order between two probability vectors significantly extending a majorization based uncertainty relation first introduced in m h partovi phys rev a 84 052117 2011 such a vector type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers our relation is therefore universal and captures the essence of uncertainty in quantum theory

concepts of conditional expectations in quantum theory a concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the cycon hellwig conditional expectations and a posteriori states which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables those related to subalgebras and conditional probability distributions these three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes for the above purpose we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures eventually it is proved that in the von neumann algebraic formulation the cycon hellwig conditional expectations always exist as completely positive map valued measures

measurement theory in local quantum physics in this paper we aim to establish foundations of measurement theory in local quantum physics for this purpose we discuss a representation theory of completely positive cp instruments on arbitrary von neumann algebras we introduce a condition called the normal extension property nep and establish a one to one correspondence between cp instruments with the nep and statistical equivalence classes of measuring processes we show that every cp instrument on an atomic von neumann algebra has the nep extending the well known result for type i factors moreover we show that every cp instrument on an injective von neumann algebra is approximated by cp instruments with the nep the concept of posterior states is also discussed to show that the nep is equivalent to the existence of a strongly measurable family of posterior states for every normal state two examples of cp instruments without the nep are obtained from this result it is thus concluded that in local quantum physics not every cp instrument represents a measuring process but in most of physically relevant cases every cp instrument can be realized by a measuring process within arbitrary error limits as every approximately finite dimensional von neumann algebra on a separable hilbert space is injective to conclude the paper the concept of local measurement in algebraic quantum field theory is examined in our framework in the setting of the doplicher haag roberts and doplicher roberts theory describing local excitations we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a cp instrument with the nep provided that the split property holds for the net of local algebras

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

renyi formulation of the entropic uncertainty principle for povms uncertainty relations for a pair of arbitrary measurements are posed in the form of inequalities using the renyi entropies the formulation deals with povm measurements on finite dimensional hilbert space both the entropic uncertainty relations with state dependent and state independent bounds are presented the results are illustrated on the example of measurements for state discrimination

sur une question d erdos et schinzel ii let f n denote a polynomial with integer coefficients and define h f x y z to be the number of integers not exceeding x for which f n has at least one divisor d such that y

realization schemes for quantum instruments in finite dimensions we present a general dilation scheme for quantum instruments with continuous outcome space in finite dimensions in terms of a measurement on a finite dimensional ancilla described by a positive operator valued measure povm the general result is then applied to a large class of instruments generated by operator frames which contains group covariant instruments as a particular case and allows one to construct dilation schemes based on a measurement on the ancilla followed by a conditional feed forward operation on the output in the case of tight operator frames our construction generalizes quantum teleportation and telecloning producing a whole family of generalized teleportation schemes in which the instrument is realized via a joint povm at the sender combined with a conditional feed forward operation at the receiver

some extensions of the uncertainty principle abstract we study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities extending results recently derived by bialynicki birula i bialynicki birula formulation of the uncertainty relations in terms of the renyi entropies physical review a 74 5 2006 052101 and zozor et al s zozor c vignat on classes of non gaussian asymptotic minimizers in entropic uncertainty principles physica a 375 2 2007 499 517 those inequalities can be considered as generalizations of the heisenberg uncertainty principle since they measure the mutual uncertainty of a wave function and its fourier transform through their associated renyi entropies with conjugated indices we consider here the general case where the entropic indices are not conjugated in both cases where the state space is discrete and continuous we discuss the existence of an uncertainty inequality depending on the location of the entropic indices and in the plane our results explain and extend a recent study by luis a luis quantum properties of exponential states physical review a 75 2007 052115 where states with quantum fluctuations below the gaussian case are discussed at the single point 2 2

reliable entanglement detection under coarse grained measurements we derive reliable entanglement witnesses for coarse grained measurements on continuous variable systems these witnesses never return a false positive for identification of entanglement under any degree of coarse graining we show that even in the case of gaussian states entanglement witnesses based on the shannon entropy can outperform those based on variances we apply our results to experimental identification of spatial entanglement of photon pairs

fringe visibility and which way information an inequality an inequality is derived according to which the fringe visibility in a two way interferometer sets an absolute upper bound on the amount of which way information that is potentially stored in a which way detector in some sense this inequality can be regarded as quantifying the notion of wave particle duality the derivation of the inequality does not make use of heisenberg s uncertainty relation in any form

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

canonical approximate quantum measurements in order to generalize the wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable which generalizes von neumann s model of measuring interaction and davies s covariant instrument for approximate position measurement calculations of two types of root mean square errors show that this model clears certain requirements for good approximate measurements a modified wigner formula for successive canonical approximate measurements of arbitrary observables is established it is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von neumann luders collapse postulate by a minor modification in the process of measurement