on bipartite operators defined by sets of completely different permutations we introduce a class of bipartite operators acting on h h h being an n dimensional hilbert space defined by a set of n completely different permutations cdps bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems it turns out that any set of cdps gives rise to a certain direct sum decomposition of the total hilbert space which enables one simple construction of the corresponding bipartite operator interestingly if set of cdps defines an abelian group then the corresponding bipartite operator displays an additional property the partially transposed operator again corresponds to in general different set of cdps therefore our technique may be used to construct new classes of the so called ppt positive under partial transposition states which are of great importance for quantum information using well known relation between bipartite operators and linear maps one can use also construct linear maps related to cdps we introduce a class of bipartite operators acting on h h h being an n dimensional hilbert space defined by a set of n completely different permutations cdps bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems it turns out that any set of cdps gives rise to a certain direct sum decomposition of the total hilbert space which enables one simple construction of the corresponding bipartite operator interestingly if set of cdps defines an abelian group then the corresponding bipartite operator displays an additional property the partially transposed operator again corresponds to in general different set of cdps therefore our technique may be used to construct new classes of the so called ppt positive under partial transposition states which are of great importance for quantum information using well known relation between bipartite operators and linear maps one can use also construct linear maps relate

genuine multipartite entanglement criteria based on positive maps positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement in the multipartite case however the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition we thus generalise this bipartite concept to the multipartite setting by introducing non positive maps that are positive on the subset of biseparable states but can map to a non positive element if applied to a genuine multipartite entangled state we explicitly construct examples of multipartite non positive maps obtained from positive maps via a lifting procedure that in this fashion can reveal genuine multipartite entanglement in a robust way

complete positivity of the map from a basis to its dual basis the dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well the purpose of these notes is to give a condition that describes when the linear map taking a basis of mn to its dual basis is a complete order isomorphism we exhibit natural orthonormal bases for mn such that this map is an order isomorphism but not a complete order isomorphism included among such bases is the pauli basis our results generalize the choi matrix by giving conditions under which the role of the standard basis eij can be replaced by other bases

positive finite rank elementary operators and characterizing entanglement of states in this paper some indecomposable positive finite rank elementary operators of order n n are constructed this allows us to give a simple necessary and sufficient criterion for the separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order 2 2 and obtain some new mixed entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion

optimal entanglement witnesses from generalized reduction and robertson maps we provide a generalization of the reduction and robertson positive maps in matrix algebras they give rise to a new class of optimal entanglement witnesses their structural physical approximation is analyzed as a byproduct we provide new examples of ppt positive partial transpose entangled states

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

construction of 3 3 entangled edge states with positive partial transposes we construct a class of 3 3 entangled edge states with positive partial transposes using indecomposable positive linear maps this class contains several new types of entangled edge states with respect to the range dimensions of themselves and their partial transposes

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

extended reduction criterion and lattice states we study a particular class of states of a bipartite system consisting of two four level parties by means of an adapted extended reduction criterion we associate their entanglement properties to the geometric patterns of a planar lattice consisting of 16 points

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

a series of absolutely indecomposable positive maps in matrix algebras abstract we provide examples of positive maps in m n c n 4 which cannot be decomposed into a sum of a 2 positive map and a 2 copositive map

the structural physical approximation conjecture it was conjectured that the structural physical approximation spa of an optimal entanglement witness is separable or equivalently that the spa of an optimal positive map is entanglement breaking this conjecture was disproved first for indecomposable maps and more recently for decomposable maps the arguments in both cases are sketched along with important related results this review includes background material on topics including entanglement witnesses optimality duality of cones decomposability and the statement and motivation for the spa conjecture so that it should be accessible for a broad audience

totally acyclic complexes over noetherian schemes abstract we define a notion of total acyclicity for complexes of flat quasi coherent sheaves on a semi separated noetherian scheme and study these complexes using the pure derived category of flat quasi coherent sheaves we prove that a scheme is gorenstein if and only if every acyclic complex of flat quasi coherent sheaves is totally acyclic our formalism also removes the need for a dualising complex in several known results for rings including jorgensen s proof of the existence of gorenstein projective precovers

constructing positive maps from block matrices positive maps are useful for detecting entanglement in quantum information theory any entangled state can be detected by some positive map in this paper the relation between positive block matrices and completely positive trace preserving maps is characterized consequently a new method for constructing decomposable maps from positive block matrices is derived in addition a method for constructing positive but not completely positive maps from hermitian block matrices is also obtained

generalising wigner s theorem we analyse linear maps of operator algebras mapping the set of rank k projectors onto the set of rank l projectors surjectively a complete characterisation of such maps for prime is provided a particular case corresponding to is well known as wigner s theorem hence our result may be considered as a generalisation of this celebrated wigner s result

brown representability for the dual let t be the homotopy category of all spectra brown proved that a homological functor h t o p ab is representable if it takes coproducts to products that is the functors h may be characterised as the homological functors taking coproducts to products in this article we will prove the dual a covariant functor h t ab which takes products to products is representable it is of the form h

on completely copositive and decomposable linear transformations abstract characterization theorems and other results for the cone of completely copositive linear transformations are given further results including the cone of decomposable linear transformations and its dual follow

limitations on separable measurements by convex optimization we prove limitations on locc and separable measurements in bipartite state discrimination problems using techniques from convex optimization specific results that we prove include an exact formula for the optimal probability of correctly discriminating any set of either three or four bell states via locc or separable measurements when the parties are given an ancillary partially entangled pair of qubits an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements and an optimal bound on the success probability for any locc or separable measurement for the recently proposed state discrimination problem of yu duan and ying

complex wkb evolution of markovian open systems we derive a consistent complex wkb expression for the characteristic function that is the fourier transform of the wigner function of a markovian open system defined by a generic lindblad equation the hamiltonian can be a general function of positions and momenta so the only restriction is that linear coupling to the environment is assumed the propagation is shown to correspond to a classical evolution of the liouville type in a complex double phase space the imaginary part of the double hamiltonian being responsible for decoherence whereas dissipation is included in the real part the theory exact in the quadratic case is designed to describe the decoherent and dissipative evolution of localized states such as the interference terms of a schrodinger cat state as well as the evolution of extended states which were the exclusive subjects of our previous real wkb approximation the present rederivation allows us to interpret the real formalism as a first order classical perturbation of the complex theory and to discuss its validity this is also clarified by a simple cubic hamiltonian that illustrates various levels of approximation derived from the complex wkb theory

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

pure state correlations chords in phase space the intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations in the case of pure states these are invariant with respect to fourier transformation the overlaps themselves are here studied in terms of the wigner function and its fourier transform i e the characteristic function or chord function unlike the wigner function the chord function need not be real but eventual symmetry with respect to reflections about a phase space point may relate these representations semiclassical approximations for the classical like region of small chords and for large chords are derived these lead to an interpretation of the fourier invariance in terms of conjugate chords the interrelation of large and small sub planck phase space structures previously noted in the literature is thus reinterpreted

positive maps from irreducibly covariant operators in this paper we discuss positive maps induced by irreducibly covariant linear operators for finite groups the application of group theory methods allows deriving some new results of a different kind in particular a family of necessary conditions for positivity for such objects is derived stemming either from the definition of a positive map or the novel method inspired by the inverse reduction map in the low dimensional cases for the permutation group s 3 and the quaternion group q the necessary and sufficient conditions are given together with the discussion on their decomposability in higher dimensions we present positive maps induced by a three dimensional irreducible representation of the permutation group s 4 and d dimensional representation of the monomial unitary group mu d in the latter case we deliver if and only if condition for the positivity and compare the results with the method inspired by the inverse reduction we show that the generalised choi map can be obtained by considered covariant maps induced by the monomial unitary group as an additional result a novel interpretation of the fujiwara algolet conditions for positivity and complete positivity is presented finally in the end a new form of an irreducible representation of the symmetric group s n is constructed allowing us to simplify the form of certain choi jamiolkowski images derived for irreducible representations of the symmetric group

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

disproving the conjecture on the structural physical approximation to optimal decomposable entanglement witnesses we disprove the conjecture that structural physical approximations to optimal entanglement witnesses are separable states the conjecture holds true for extremal decomposable entanglement witnesses

features of moyal trajectories we study the moyal evolution of the canonical position and momentum variables we compare it with the classical evolution and show that contrary to what is commonly found in the literature the two dynamics do not coincide we prove that this divergence is quite general by studying hamiltonians of the form p2 2m v q several alternative formulations of moyal dynamics are then suggested we introduce the concept of star function and use it to reformulate the moyal equations in terms of a system of ordinary differential equations on the noncommutative moyal plane we then use this formulation to study the semiclassical expansion of moyal trajectories which is cast in terms of a order by order in recursive hierarchy of i first order partial differential equations as well as ii systems of first order ordinary differential equations the latter formulation is derived independently for analytic hamiltonians as well as for the more general case of locally integrable ones we present various examples i

choi matrices norms and entanglement associated with positive maps on matrix algebras abstract we study positive maps of b k into b h for finite dimensional hilbert spaces k and h our main emphasis is on how choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps special attention will be given to entanglement properties and k positive maps

rank two bipartite bound entangled states do not exist we explore the relation between the rank of a bipartite density matrix and the existence of bound entanglement we show a relation between the rank marginal ranks and distillability of a mixed state and use this to prove that any rank n bound entangled state must have support on no more than an n n hilbert space a direct consequence of this result is that there are no bipartite bound entangled states of rank two we also show that a separability condition in terms of a quantum entropy inequality is associated with the above results we explore the idea of how many pure states are needed in a mixture to cancel the distillable entanglement of a schmidt rank n pure state and provide a lower bound of n 1 we also prove that a mixture of a non zero amount of any pure entangled state with a pure product state is distillable

separable states and the structural physical approximation of a positive map abstract we introduce a necessary condition for a state to be separable and apply this condition to the structural physical approximation of an optimal positive map and give a proof of the fact that the spa need not be separable

generation of mapping cones from small sets we answer in the affirmative a recently posed question that asked if there exists an untypical convex mapping cone i e one that does not arise from the transpose map and the cones of k positive and k superpositive maps we explicitly construct such a cone based on atomic positive maps our general technique is to consider the smallest convex mapping cone generated by a single map and we derive several results on such mapping cones we use this technique to also present several other examples of untypical mapping cones including a family of cones generated by spin factors we also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality

a class of atomic positive linear maps in matrix algebras we find a large class of atomic positive linear maps between n dimensional matrix algebras and study their properties in connection with complete positivity 2 positivity and schwarz inequality

parity measurements decoherence and spiky wigner functions notwithstanding radical conceptual differences between classical and quantum mechanics it is usually assumed that physical measurements concern observables common to both theories not so with the eigenvalues 1 of the parity operator the effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks the result is a set of definite parity though otherwise mixed here we derive the general form of a parity collapsed state whether pure or mixed the signature of positive or negative parity is a corresponding spike in the wigner function which is sharpened by decoherence we conjecture that states with pure parity always have negative values in their wigner functions

symplectic evolution of wigner functions in markovian open systems the wigner function is known to evolve classically under the exclusive action of a quadratic hamiltonian if the system also interacts with the environment through lindblad operators that are complex linear functions of position and momentum then the general evolution is the convolution of a non hamiltonian classical propagation of the wigner function with a phase space gaussian that broadens in time we analyze the consequences of this in the three generic cases of elliptic hyperbolic and parabolic hamiltonians the wigner function always becomes positive in a definite time which does not depend on the initial pure state we observe the influence of classical dynamics and dissipation upon this threshold we also derive an exact formula for the evolving linear entropy as the average of a narrowing gaussian taken over a probability distribution that depends only on the initial state this leads to a long time asymptotic formula for the growth of linear entropy we finally discuss the possibility of recovering the initial state

on positive linear maps between matrix algebras abstract there is a concrete example of a positive linear map from m 2 to m 4 which is not decomposable modification of this map gives an explicit counterexample to a conjecture of woronowicz on the strong kadison inequality

on structural physical approximations and entanglement breaking maps very recently a conjecture saying that the so called structural physical approximations spas to optimal positive maps optimal entanglement witnesses give entanglement breaking eb maps separable states has been posed korbicz et al 2008 phys rev a 78 062105 the main purpose of this contribution is to explore this subject first we extend the set of entanglement witnesses supporting the conjecture then we ask whether spas constructed from other than the depolarizing channel maps also lead to eb maps and show that in general this is not the case on the other hand we prove an interesting fact that for any positive map there exists an eb channel such that the spa of constructed with the aid of is again an eb channel finally we ask similar questions in the case of continuous variable systems we provide a simple way of constructing spa and prove that in the case of the transposition map it gives the eb channel

the popescu gabriel theorem for triangulated categories abstract the popescu gabriel theorem states that each grothendieck abelian category is a localization of a module category in this paper we prove an analogue where grothendieck abelian categories are replaced by triangulated categories which are well generated in the sense of neeman and algebraic in the sense of keller the role of module categories is played by derived categories of small differential graded categories an analogous result for topological triangulated categories has recently been obtained by a heider

entanglement witnesses arising from choi type positive linear maps we construct optimal pptes witnesses to detect 3 3 ppt entangled edge states of type 6 8 constructed recently by kye and osaka 2012 j math phys 53 052201 to do this we consider positive linear maps which are variants of the choi type map involving complex numbers and examine several notions related to the optimality of those entanglement witnesses throughout the discussion we suggest a method to check the optimality of entanglement witnesses without the spanning property

a class of exposed indecomposable positive maps exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a class of indecomposable positive maps in the algebra of 2n 2n complex matrices with n 2 it is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states

exposed positive maps a sufficient condition exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a sufficient condition for a positive map to be exposed this is an analogue of a spanning property which guarantees that a positive map is optimal we analyze a class of decomposable maps for which this condition is also necessary

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

indecomposable positive maps in low dimensional matrix algebras abstract we classify a series of positive maps in low dimensional matrix algebras with respect to the degree of indecomposability and provide stronger indecomposable maps in m n we also present a basic result about the atomic property

a note on the optimality of decomposable entanglement witnesses and completely entangled subspaces entanglement witnesses ews constitute one of the most important entanglement detectors in quantum systems nevertheless their complete characterization in particular with respect to the notion of optimality is still missing even in the decomposable case here we show that for any qubit qunit decomposable ew dew w the three statements are equivalent i the set of product vectors obeying e f w e f 0 spans the corresponding hilbert space ii w is optimal and iii w q with q denoting a positive operator supported on a completely entangled subspace ces and standing for the partial transposition while implications i ii and ii iii are known here we prove that iii implies i this is a consequence of a more general fact saying that product vectors orthogonal to any ces in span after partial conjugation the whole space on the other hand already in the case of the hilbert space there exist dews for which iii does not imply i consequently either i does not imply ii or ii does not imply iii and the above transparent characterization obeyed by qubit qunit dews does not hold in general

the structural physical approximations and optimal entanglement witnesses we introduce the notions of positive and copositive types for entanglement witnesses depending on the distance to the positive part and copositive part an entanglement witness w is of positive type if and only if its partial transpose w is of copositive type we show that if the structural physical approximation spa of w is separable then w should be of copositive type and the spa of w is never separable unless w is of both positive and copositive types this shows that the spa conjecture is meaningful only for those of copositive type we provide examples to show that the spa conjecture fails even for the case of copositive types

a class of positive atomic maps we construct a new class of positive indecomposable maps in the algebra of d d complex matrices these maps are characterized by the weakest positivity property and for this reason they are called atomic this class provides a new rich family of atomic entanglement witnesses which define an important tool for investigating quantum entanglement it turns out that they are able to detect states with the weakest quantum entanglement