quantum correlations and distinguishability of quantum states a survey of various concepts in quantum information is given with a main emphasis on the distinguishability of quantum states and quantum correlations covered topics include generalized and least square measurements state discrimination quantum relative entropies the bures distance on the set of quantum states the quantum fisher information the quantum chernoff bound bipartite entanglement the quantum discord and geometrical measures of quantum correlations the article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them and for mathematicians looking for an up to date introductory course on these subjects which are mainly developed in the physics literature

no hypersignaling principle a paramount topic in quantum foundations rooted in the study of the einstein podolsky rosen epr paradox and bell inequalities is that of characterizing quantum theory in terms of the spacelike correlations it allows here we show that to focus only on spacelike correlations is not enough we explicitly construct a toy model theory that while not contradicting classical and quantum theories at the level of spacelike correlations still displays an anomalous behavior in its timelike correlations we call this anomaly quantified in terms of a specific communication game the hypersignaling phenomena we hence conclude that the principle of quantumness if it exists cannot be found in spacelike correlations alone nontrivial constraints need to be imposed also on timelike correlations in order to exclude hypersignaling theories

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

decomposition of any quantum measurement into extremals we design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements we show that if one allows for a classical post processing step only extremal rank 1 positive operator valued measures are needed for a measurement with n elements on a d dimensional space our algorithm will decompose it into at most n 1 d 1 extremals whereas the best previously known upper bound scaled as d2 since the decomposition is not unique we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property

classical randomness in quantum measurements similarly to quantum states also quantum measurements can be mixed corresponding to a random choice within an ensemble of measuring apparatuses such mixing is equivalent to a sort of hidden variable which produces a noise of purely classical nature it is then natural to ask which apparatuses are indecomposable i e do not correspond to any random choice of apparatuses this problem is interesting not only for foundations but also for applications since most optimization strategies give optimal apparatuses that are indecomposable mathematically the problem is posed describing each measuring apparatus by a positive operator valued measure povm which gives the statistics of the outcomes for any input state the povms form a convex set and in this language the indecomposable apparatuses are represented by extremal points the analogous of pure states in the convex set of states differently from the case of states however indecomposable povms are not necessarily rank one e g von neumann measurements in this paper we give a complete classification of indecomposable apparatuses for discrete spectrum by providing different necessary and sufficient conditions for extremality of povms along with a simple general algorithm for the decomposition of a povm into extremals as an interesting application informationally complete measurements are analysed in this respect the convex set of povms is fully characterized by determining its border in terms of simple algebraic properties of the corresponding povms

extreme covariant quantum observables in the case of an abelian symmetry group and a transitive value space we represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact abelian symmetry group g the value space of such observables is a transitive g space we characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables the results are applied to position position difference and time observables

realizing a 2 d positive operator valued measure by local operations and classical communication we study the realization of a 2 d positive operator valued measure povm by local operations and classical communication locc we derive that any povm on a 2 d hilbert space in which each party s space is finite dimensional can be realized by one way locc this implies that any optimal discrimination of quantum states spanning such a 2 d hilbert space is possible by one way locc regardless of the optimality criterion used and how entangled the states are

on the convex structure of process positive operator valued measures measurements on quantum channels are described by so called process positive operator valued measures or process povms we study implementing schemes of extremal process povms as it turns out the corresponding measurement must satisfy certain extremality property which is stronger than the usual extremality given by the convex structure this property motivates the introduction and investigation of the a convex structure of povms which generalizes both the usual convex and c convex structures we show that extremal points and faces of the set of process povms are closely related to a extremal points and a faces of povms for a certain subalgebra a we also give a characterization of a extremal and a pure povms

effective methods for constructing extreme quantum observables we study extreme points of the set of finite outcome positive operator valued measures povms on finite dimensional hilbert spaces and particularly the possible ranks of the effects of an extreme povm we give results discussing ways of deducing new rank combinations of extreme povms from rank combinations of known extreme povms and using these results show ways to characterize rank combinations of extreme povms in low dimensions we show that when a rank combination together with a given dimension of the hilbert space solve a particular packing problem there is an extreme povm on the hilbert space with the given ranks this geometric method is particularly effective for constructing extreme povms with desired rank combinations

a bound on the mutual information and properties of entropy reduction for quantum channels with inefficient measurements the holevo bound is a bound on the mutual information for a given quantum encoding in 1996 schumacher westmoreland and wootters phys rev lett 76 3452 1996 derived a bound that reduces to the holevo bound for complete measurements but that is tighter for incomplete measurements the most general quantum operations may be both incomplete and inefficient here we show that the bound derived by sww can be further extended to obtain one that is yet again tighter for inefficient measurements this allows us in addition to obtain a generalization of a bound derived by hall and to show that the average reduction in the von neumann entropy during a quantum operation is concave in the initial state for all quantum operations this is a quantum version of the concavity of the mutual information we also show that both this average entropy reduction and the mutual information for pure state ensembles are schur concave for unitarily covariant measurements that is for these measurements information

generalized coherent states and extremal positive operator valued measures we present a correspondence between positive operator valued measures povms and sets of generalized coherent states povms describe quantum observables and similar to quantum states quantum observables can also be mixed we show how the formalism of generalized coherent states leads to a useful characterization of extremal povms we prove that covariant phase space observables related to squeezed states are extremal while those related to number states are not extremal this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

clean positive operator valued measures in quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure povm a quantum channel transforms povms into povms generally irreversibly thus losing some of the information retrieved from the measurement this poses the problem of which povms are undisturbed i e they are not irreversibly connected to another povm we will call such povms clean in a sense the clean povms would be perfect since they would not have any additional extrinsical noise quite unexpectedly it turns out that such a cleanness property is largely unrelated to the convex structure of povms and there are clean povms that are not extremal and vice versa in this article we solve the cleannes classification problem for number n of outcomes n d d dimension of the hilbert space and we provide a set of either necessary or sufficient conditions for n d along with an iff condition for the case of informationally complete povms for n d2

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

extreme commutative quantum observables are sharp it is well known that in the description of quantum observables positive operator valued measures povms generalize projection valued measures pvms and they also turn out be more optimal in many tasks we show that a commutative povm is an extreme point in the convex set of all povms if and only if it is a pvm this results implies that non commutativity is a necessary ingredient to overcome the limitations of pvms

convex optimization based parameter estimation and experiment design for pauli channels in this technical note we study the problem of parameter estimation for quantum channels which are completely positive maps acting on quantum states we focus on an important subclass of channels called pauli channels which are characterized by a certain set of vectors directions and a number of scalar parameters for the case where the directions are known a special parametrization turns the parameter estimation problem into a convex optimization problem for the case of unknown directions we give a simple algorithm to estimate the directions for qubit pauli channels these results assume that the identification experiment configuration is given namely a set of quantum states are given on which the channel acts and a set of positive measurement operator s are fixed from which information is gathered in the second part of the technical note we consider the problem of determining the experiment design namely determining states and measurements for optimal parameter identification of the channels we formalize this problem as a maximization problem for fisher information and assuming known channel directions prove that this problem is convex we also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal for qubit pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions we illustrate the results of the technical note with numerical examples and simulations

simulating positive operator valued measures with projective measurements standard projective measurements pms represent a subset of all possible measurements in quantum physics defined by positive operator valued measures we study what quantum measurements are projective simulable that is can be simulated by using projective measurements and classical randomness we first prove that every measurement on a given quantum system can be realized by classical randomization of projective measurements on the system plus an ancilla of the same dimension then given a general measurement in dimension two or three we show that deciding whether it is pm simulable can be solved by means of semidefinite programming we also establish conditions for the simulation of measurements using projective ones valid for any dimension as an application of our formalism we improve the range of visibilities for which two qubit werner states do not violate any bell inequality for all measurements from an implementation point of view our work provides bounds on the amount of white noise a measurement tolerates before losing any advantage over projective ones

finite blocklength converse bounds for quantum channels we derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability for both entanglement assisted and unassisted codes in terms of a unifying framework of quantum hypothesis testing with restricted measurements our bounds do not depend on any special property of the channel such as memorylessness and generalize both a classical converse of polyanskiy poor and verdu as well as a quantum converse of renner and wang and have a number of desirable properties in particular our bound on entanglement assisted codes is a semidefinite program and for memoryless channels its large blocklength limit is the well known formula for entanglement assisted capacity due to bennett shor smolin and thapliyal

generalized probabilistic theories and conic extensions of polytopes generalized probabilistic theories gpt provide a general framework that includes classical and quantum theories it is described by a cone c and its dual c we show that whether some one way communication complexity problems can be solved within a gpt is equivalent to the recently introduced cone factorization of the corresponding communication matrix m we also prove an analogue of holevo s theorem when the cone c is contained in the classical capacity of the channel realized by sending gpt states and measuring them is bounded by polytopes and optimising functions over polytopes arise in many areas of discrete mathematics a conic extension of a polytope is the intersection of a cone c with an affine subspace whose projection onto the original space yields the desired polytope extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself the existence of a conic extension of a polytope is equivalent to that of a cone factorization of the slack matrix of the polytope on the same cone we show that all polytopes whose vertices can be recognized by a polynomial size circuit which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimization have small conic extension complexity when the cone is the completely positive cone using recent exponential lower bounds on the linear extension complexity of polytopes this provides an exponential gap between the communication complexity of gpt based on the completely positive cone and classical communication complexity and a conjectured exponential gap with quantum communication complexity our work thus relates the communication complexity of generalizations of quantum theory to questions of mainstream interest in the area of combinatorial optimization

global well posedness of the cauchy problem for the navier stokes equations of nonisentropic flow with discontinuous initial data abstract we prove the global existence uniqueness and continuous dependence on initial data for discontinuous solutions of the navier stokes equations for nonisentropic compressible flow in one space dimension a great deal of information is obtained concerning the qualitative behavior of the solution including an analysis of the convection and evolution of jump discontinuities the derivation of sharp rates of smoothing and the l asymptotic behavior

semigroups and linear partial differential equations with delay abstract we prove the equivalence of the well posedness of a partial differential equation with delay and an associated abstract cauchy problem this is used to derive sufficient conditions for well posedness exponential stability and norm continuity of the solutions applications to a reaction diffusion equation with delay are given

barycentric decomposition of quantum measurements in finite dimensions we analyze the convex structure of the set of positive operator valued measures povms representing quantum measurements on a given finite dimensional quantum system with outcomes in a given locally compact hausdorff space the extreme points of the convex set are operator valued measures concentrated on a finite set of k d2 points of the outcome space d being the dimension of the hilbert space we prove that for second countable outcome spaces any povm admits a choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure in the general case krein milman theorem is invoked to represent povms as barycenters of a certain set of povms concentrated on k d2 points of the outcome space

complete characterization of extreme quantum observables in infinite dimensions we give a complete characterization for extreme quantum observables i e for normalized positive operator valued measures povms which are extremals in the convex set of all povms the characterization is valid both in discrete and continuous cases and also in the case of an infinite dimensional hilbert space we show that sharp povms are pre processing clean i e they cannot be irreversibly connected to other povms via quantum channels

quantum steering with positive operator valued measures we address the problem of quantum nonlocality with positive operator valued measures povm in the context of einstein podolsky rosen quantum steering we show that given a candidate for local hidden state lhs ensemble the problem of determining the steerability of a bipartite quantum state of finite dimension with povms can be formulated as a nesting problem of two convex objects one consequence of this is the strengthening of the theorem that justifies choosing the lhs ensemble based on symmetry of the bipartite state as a more practical application we study the classic problem of the steerability of two qubit werner states with povms we show strong numerical evidence that these states are unsteerable with povms up to a mixing probability of within an accuracy of 10 3