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

unconditional security from noisy quantum storage we consider the implementation of two party cryptographic primitives based on the sole assumption that no large scale reliable quantum storage is available to the cheating party we construct novel protocols for oblivious transfer and bit commitment and prove that realistic noise levels provide security even against the most general attack such unconditional results were previously only known in the so called bounded storage model which is a special case of our setting our protocols can be implemented with present day hardware used for quantum key distribution in particular no quantum storage is required for the honest parties

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

quantifying measurement incompatibility of mutually unbiased bases quantum measurements based on mutually unbiased bases are commonly used in quantum information processing as they are generally viewed as being maximally incompatible and complementary here we quantify precisely the degree of incompatibility of mutually unbiased bases mub using the notion of noise robustness specifically for sets of k mub in dimension d we provide upper and lower bounds on this quantity notably we get a tight bound in several cases in particular for complete sets of k d 1 mub using the standard construction for d being a prime power on the way we also derive a general upper bound on the noise robustness for an arbitrary set of quantum measurements moreover we prove the existence of sets of k mub that are operationally inequivalent as they feature different noise robustness and we provide a lower bound on the number of such inequivalent sets up to dimension 32 finally we discuss applications of our results for einstein podolsky rosen steering

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

decomposition rules for quantum renyi mutual information with an application to information exclusion relations we prove decomposition rules for quantum renyi mutual information generalizing the relation i a b h a h a b to inequalities between renyi mutual information and renyi entropy of different orders the proof uses beigi s generalization of reisz thorin interpolation to operator norms j math phys 54 12 122202 2013 and a variation of the argument developed by dupuis j math phys 56 022203 2015 which was used to show chain rules for conditional renyi entropies the resulting decomposition rule is then applied to establish an information exclusion relation for renyi mutual information generalizing the original relation by hall phys rev lett 74 3307 1995

preparation uncertainty implies measurement uncertainty in a class of generalized probabilistic theories in quantum theory it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them they are called preparation uncertainty and measurement uncertainty respectively and research has unveiled that they are not independent from but related with each other in a quantitative way this study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories gpts in particular a certain class of gpts is considered which can be characterized by transitivity and self duality and regarded as extensions of quantum theory it is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories if preparation uncertainty exists then measurement uncertainty also exists and they are described by similar inequalities our results manifest that their correspondences are not specific to quantum theory but more universal ones

complementary reductions for two qubits reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system in connection with optimal state determination for two qubits the question was raised about the maximum number of pairwise complementary reductions the main result of the paper tells that the maximum number is 4 that is if a1 a2 ak are pairwise complementary or quasiorthogonal subalgebras of the algebra m4 c of all 4 4 matrices and they are isomorphic to m2 c then k 4 the proof is based on a cartan decomposition of su 4 in the way to the main result contributions are made to the understanding of the structure of complementary reductions

equivariant analytic torsion for compact lie group actions abstract we define the equivariant analytic torsion for a compact lie group action and study its dependence on the geometric data

uncertainty inequalities among frequency moments and radial expectation values applications to atomic systems rigorous and universal bounds on frequency moments of one particle densities in terms of radial expectation values in the conjugate space are obtained the results valid for any d dimensional quantum mechanical system are derived by using renyi like position momentum inequalities in an information theoretical framework especially interesting are the upper bounds on the dirac exchange and thomas fermi kinetic energies as well as the disequilibrium or self similarity of both position and momentum distributions a variety of bounds for these functionals in a given space are known but most usually in terms of quantities defined within the same space very few results including a density functional on one space and expectation values on the conjugate one are found in the literature a pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work a numerical study of the aforementioned relationships is carried out for atomic systems in their ground state some results ar

equivalence theorem of uncertainty relations we present an equivalence theorem to unify the two classes of uncertainty relations i e the variance based ones and the entropic forms showing that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators this means that an uncertainty relation in the language of entropy may be mapped onto a variance based one and vice versa employing the equivalence theorem alternative formulations of entropic uncertainty relations are obtained for the qubit system that are stronger than the existing ones in the literature and variance based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations

gelfand yaglom perez theorem for generalized relative entropy functionals the measure theoretic definition of kullback leibler relative entropy or simply kl entropy plays a basic role in defining various classical information measures on general spaces entropy mutual information and conditional forms of entropy can be expressed in terms of kl entropy and hence properties of their measure theoretic analogs will follow from those of measure theoretic kl entropy these measure theoretic definitions are key to extending the ergodic theorems of information theory to non discrete cases a fundamental theorem in this respect is the gelfand yaglom perez gyp theorem m s pinsker information and information stability of random variables and process 1960 holden day san francisco ca english ed 1964 translated and edited by amiel feinstein theorem 2 4 2 which states that measure theoretic relative entropy equals the supremum of relative entropies over all measurable partitions this paper states and proves the gyp theorem for renyi relative entropy of order greater than one consequently the result can be easily extended to tsallis relative entropy

higher entropic uncertainty relations for anti commuting observables uncertainty relations provide one of the most powerful formulations of the quantum mechanical principle of complementarity yet very little is known about such uncertainty relations for more than two measurements here we show that sufficient unbiasedness for a set of binary observables in the sense of mutual anticommutation is good enough to obtain maximally strong uncertainty relations in terms of the shannon entropy we also prove nearly optimal relations for the collision entropy this is the first systematic and explicit approach to finding an arbitrary number of measurements for which we obtain maximally strong uncertainty relations our results have immediate applications to quantum cryptography

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

asymptotic entropic uncertainty relations we analyze entropic uncertainty relations for two orthogonal measurements on a n dimensional hilbert space performed in two generic bases it is assumed that the unitary matrix u relating both bases is distributed according to the haar measure on the unitary group we provide lower bounds on the average shannon entropy of probability distributions related to both measurements the bounds are stronger than those obtained with use of the entropic uncertainty relation by maassen and uffink and they are optimal up to additive constants we also analyze the case of a large number of measurements and obtain strong entropic uncertainty relations which hold with high probability with respect to the random choice of bases the lower bounds we obtain are optimal up to additive constants and allow us to prove a conjecture by wehner and winter on the asymptotic behavior of constants in entropic uncertainty relations as the dimension tends to infinity as a tool we develop estimates on the maximum operator norm of a submatrix of a fixed size of a random unitary matrix distributed according to the haar measure which are of independent interest

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

unified entropy trade off relations for a single quantum channel many important properties of quantum channels are quantified by means of entropic functionals characteristics of such a kind are closely related to different representations of a quantum channel in the jamiolkowski choi representation the given quantum channel is described by the so called dynamical matrix entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence within the so called natural representation the quantum channel is formally posed by another matrix obtained as the reshuffling of the dynamical matrix the corresponding entropies characterize an amount with which the receiver a priori knows the channel output as was previously shown the map and receiver entropies are mutually complementary characteristics indeed there exists a non trivial lower bound on their sum first we extend the concept of receiver entropy to the family of unified entropies developing the previous results we further derive non trivial lower bounds on the sum of the map and receiver q s entropies the derivation is essentially based on some inequalities with the schatten norms and anti norms

randomizing quantum states constructions and applications the construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver we show that if only near perfect security is required the size of the key can be reduced by a factor of two more specifically we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing as compared to the d2 operators required to randomize perfectly aside from the private quantum channel variations of this construction can be applied to many other tasks in quantum information processing we show for instance that it can be used to construct locc data hiding schemes for bits and qubits that are much more efficient than any others known allowing roughly log d qubits to be hidden in 2 log d qubits the method can also be used to exhibit the existence of quantum states with locked classical correlations an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key our construction also provides the basic building block for a method of remotely preparing arbitrary d dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement

an entropy based uncertainty principle for a locally compact abelian group abstract we classify all functions on a locally compact abelian group giving equality in an entropy inequality generalizing the heisenberg uncertainty principle in particular for functions on a real line we proof a conjecture of hirschman published in 1957

a transform of complementary aspects with applications to entropic uncertainty relations even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood here we construct special sets of up to 2n 1 mutually unbiased bases mubs in dimension d 2 n which have particularly beautiful symmetry properties derived from the clifford algebra more precisely we show that there exists a unitary transformation that cyclically permutes such bases this unitary can be understood as a generalization of the fourier transform which exchanges two mubs to multiple complementary aspects we proceed to prove a lower bound for min entropic entropic uncertainty relations for any set of mubs and show that symmetry plays a central role in obtaining tight bounds for example we obtain for the first time a tight bound for four mubs in dimension d 4 which is attained by an eigenstate of our complementarity transform finally we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete wigner functions are directly related to min entropic uncertainty relations for mubs

entropic heisenberg limits and uncertainty relations from the holevo information bound strong and general entropic and geometric heisenberg limits are obtained for estimates of multiparameter unitary displacements in quantum metrology such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions a key ingredient is the holevo bound on the shannon mutual information of a quantum communication channel this leads to a bayesian bound on performance in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group a geometric measure of performance related to entropy is proposed for general parameter estimation it is also shown how strong entropic uncertainty relations for mutually unbiased observables such as number and phase position and momentum energy and time and orthogonal spin 1 2 directions can be obtained from elementary applications of holevo s bound a geometric interpretation of results is emphasised in terms of the volumes of quantum and classical statistical ensembles

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

entropy power uncertainty relations towards a tight inequality for all gaussian pure states we show that a proper expression of the uncertainty relation for a pair of canonically conjugate continuous variables relies on entropy power a standard notion in shannon information theory for real valued signals the resulting entropy power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to bialynicki birula and mycielski but can be further extended to rotated variables hence based on a reasonable assumption we give a partial proof of a tighter form of the entropy power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity interestingly it implies the generalized rotation invariant schrodinger robertson uncertainty relation exactly as the original entropy power uncertainty relation implies heisenberg relation it is saturated for all gaussian pure states in contrast with hitherto known entropic formulations of the uncertainty principle

visibility for anharmonic fringes we examine a new approach to the visibility of interference fringes defined as the distance l2 to the uniform distribution we show that this approach is superior to the standard approach when examining anharmonic fringes while they coincide for harmonic fringes we demonstrate the close relationship between this formalism and other well studied measures of quantum uncertainty and information we also show that this approach preserves the connection between visibility and correlation functions irrespective of the complexity of the interferometric pattern

y system and β deformed n 4 super yang mills we show how the perturbation theory results recently obtained by f fiamberti a santambrogio c sieg and d zanon for operator anomalous dimensions of deformed super yang mills theory can be reproduced from the ads5 cft4 y system proposed by ng v kazakov and p vieira to do this we obtain the general twisted asymptotic solution of this y system of functional equations we show that existence of an additional parameter in the deformed theory allows us to extract rich information about the perturbation theory integrals directly from the y system using this method we found a simple generating function for a broad class of such integrals

phase space localization of antisymmetric functions upper and lower bounds are written down for the minimum information entropy in phase space of an antisymmetric wave function in any number of dimensions similar bounds are given when the wave function is restricted to belong to any of the proper subspaces of the fourier transform operator

extended quantum conditional entropy and quantum uncertainty inequalities quantum states can be subjected to classical measurements whose incompatibility or uncertainty can be quantified by a comparison of certain entropies there is a long history of such entropy inequalities between position and momentum recently these inequalities have been generalized to the tensor product of several hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized our proofs utilize the technique of the original derivation of strong subadditivity of the von neumann entropy

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

entropic uncertainty relations and the stabilizer formalism entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy central questions include the derivation of lower bounds on the total uncertainty for given observables the characterization of observables that allow strong uncertainty relations and the construction of such relations for the case of several observables we demonstrate how the stabilizer formalism can be applied to these questions we show that the maassen uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases we compare the relative strengths of variance based and various entropic uncertainty relations for dichotomic anticommuting observables

uncertainty relations for joint localizability and joint measurability in finite dimensional systems two quantities quantifying uncertainty relations are examined busch and pearson j math phys 48 082103 2007 investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width in this paper we show a simple relationship between these quantities for finite dimensional systems our result indicates that if there is a bound on joint localizability it is possible to obtain a similar bound on joint measurability for finite dimensional systems uncertainty relations for a pair of general projection valued measures are obtained as by products

stronger uncertainty relations for all incompatible observables the heisenberg robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator notably it does not capture the concept of incompatible observables because it can be trivial i e the lower bound can be null even for two noncompatible observables here we give two stronger uncertainty relations relating to the sum of variances whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system

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

reformulation for arbitrary mixed states of jones bayes estimation of pure states abstract jones has cast the problem of estimating the pure state of a d dimensional quantum system into a bayesian framework the normalized uniform ray measure over such states is employed as the prior distribution the data consist of observed eigenvectors k k 1 n from an n trial analyzer that is a collection of n bases of the hilbert space c d the desired posterior inferred distribution is then simply proportional to the likelihood of k 1n k 2 here jones approach is extended to the more realistic experimental case of mixed input states as the unnormalized prior over the d d density matrices the recently developed reparameterization and unitarily invariant measure 2d 1 is utilized the likelihood is then taken to be k 1n k k reducing to that of jones when corresponds to a pure state the case of a pure state however the associated prior and posterior probabilities are then zero some analytical results for the case d 2 are presented

quantifying the incompatibility of quantum measurements relative to a basis motivated by quantum resource theories we introduce a notion of incompatibility for quantum measurements relative to a reference basis the notion arises by considering states diagonal in that basis and investigating whether probability distributions associated with different quantum measurements can be converted into one another by probabilistic postprocessing the induced preorder over quantum measurements is directly related to multivariate majorization and gives rise to families of monotones i e scalar quantifiers that preserve the ordering for the case of orthogonal measurement we establish a quantitative connection between incompatibility quantum coherence and entropic uncertainty relations we generalize the construction to include arbitrary positive operator valued measurements and report complete families of monotones

pitt s inequality and the uncertainty principle associated with the quaternion fourier transform abstract the quaternion fourier transform a generalized form of the classical fourier transform has been shown to be a powerful analyzing tool in image and signal processing this paper investigates pitt s inequality and uncertainty principle associated with the two sided quaternion fourier transform it is shown that by applying the symmetric form f f 1 i f 2 f 3 j i f 4 j of quaternion from hitzer and the novel module or l p norm of the quaternion fourier transform f then any nonzero quaternion signal and its quaternion fourier transform cannot both be highly concentrated two part results are provided one part is heisenberg weyl s uncertainty principle associated with the quaternion fourier transform it is formulated by using logarithmic estimates which may be obtained from a sharp of pitt s inequality the other part is the uncertainty principle of donoho and stark associated with the quaternion fourier transform