gaussian quantum marginal problem the quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system this setting also referred to as the question of the compatibility of local spectra has several applications in quantum information theory here we introduce the analogue of this statement for gaussian states for any number of modes and solve it in generality for pure and mixed states both concerning necessary and sufficient conditions formally our result can be viewed as an analogue of the sing thompson theorem respectively horn s lemma characterizing the relationship between main diagonal elements and singular values of a complex matrix we find necessary and sufficient conditions for vectors d1 dn and c1 cn to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix respectively more physically speaking this result determines what local temperatures or entropies are consistent with a pure or mixed gaussian state of several modes we find that this result implies a solution to the problem of sharing of entanglement in pure gaussian states and allows for estimating the global entropy of non gaussian states based on local measurements implications to the actual preparation of multi mode continuous variable entangled states are discussed we compare the findings with the marginal problem for qubits the solution of which for pure states has a strikingly similar and in fact simple form

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

stabilizer information inequalities from phase space distributions the shannon entropy of a collection of random variables is subject to a number of constraints the best known examples being monotonicity and strong subadditivity it remains an open question to decide which of these laws of information theory are also respected by the von neumann entropy of many body quantum states in this article we consider a toy version of this difficult problem by analyzing the von neumann entropy of stabilizer states we find that the von neumann entropy of stabilizer states satisfies all balanced information inequalities that hold in the classical case our argument is built on the fact that stabilizer states have a classical model provided by the discrete wigner function the phase space entropy of the wigner function corresponds directly to the von neumann entropy of the state which allows us to reduce to the classical case our result has a natural counterpart for multi mode gaussian states which sheds some light on the general properties of the construction we also discuss the relation of our results to recent work by linden ruskai and winter the quantum entropy cone of stabiliser states e print arxiv 1302 5453

even partitions in plethysms abstract we prove that for all natural numbers k n d with k d and every partition of size kn with at most k parts there exists an irreducible gl d c representation of highest weight 2 in the plethysm sym k sym 2 n c d this gives an affirmative answer to a conjecture by weintraub steven h weintraub some observations on plethysms j algebra 129 1 1990 103 114 our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory

comment on some results of erdahl and the convex structure of reduced density matrices in j math phys 13 1608 1621 1972 erdahl10 1063 1 1665885 considered the convex structure of the set of n representable 2 body reduced density matrices in the case of fermions some of these results have a straightforward extension to the m body setting and to the more general quantum marginal problem we describe these extensions but cannot resolve a problem in the proof of erdahl s claim that every extreme point is exposed in finite dimensions nevertheless we can show that when 2m n every extreme point of the set of n representable m body reduced density matrices has a unique pre image in both the symmetric and anti symmetric setting moreover this extends to the quantum marginal setting for a pair of complementary m body and n m body reduced density matrices

a linear time algorithm for obtaining the convex hull of a simple polygon in this paper a linear time algorithm is described for finding the convex hull of a simple i e non self intersecting polygon

on state versus channel quantum extension problems exact results for u otimes u otimes u symmetry we develop a framework which unifies seemingly different extension or joinability problems for bipartite quantum states and channels this includes known extension problems such as optimal quantum cloning and quantum marginal problems as special instances central to our generalization is a variant of the jamiolkowski isomorphism between bipartite states and linear transformations which we term the homocorrelation map in contrast to the better known choi isomorphism which emphasizes the preservation of the positivity constraint use of the jamiolkowski isomorphism allows one to characterize the preservation of the statistical correlations of bipartite states and quantum channels the resulting homocorrelation map thus acquires a natural operational interpretation we define and analyze state joining channel joining and local positive joining problems in three party settings with collective symmetry obtaining exact analytical characterizations in low dimensions we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree whereas quantum channels are limited in the degree to which their measurement outcomes may disagree loosely speaking quantum mechanics enforces an upper bound on the strength of positive correlation across two subsystems at a single time as well as on the strength of negative correlation between the state of a single system across two instants of time we argue that these general statistical bounds inform the quantum joinability limitations and show that they are in fact sufficient for the three party invariant setting

reducible correlations in dicke states we apply a simple observation to show that the generalized dicke states can be determined from their reduced subsystems in this framework it is sufficient to calculate the expression for only the diagonal elements of the reduced density matrices in terms of the state coefficients we prove that the correlation in generalized dicke states gd l n can be reduced to 2l partite level application to the quantum marginal problem is also discussed

nonvanishing of kronecker coefficients for rectangular shapes we prove that for any partition 1 d2 of size d there exists k 1 such that the tensor square of the irreducible representation of the symmetric group sk d with respect to the rectangular partition k k contains the irreducible representation corresponding to the stretched partition k 1 k d 2 we also prove a related approximate version of this statement in which the stretching factor k is eectively bounded in terms of d we further discuss the consequences for geometric complexity theory which provided the motivation for this work

on the use of group theoretical and graphical techniques toward the solution of the general n body problem group theoretic and graphical techniques are used to derive the n body wave function for a system of identical bosons with general interactions through first order in a perturbation approach this method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions the symmetric structure at lowest order has a point group isomorphic with the sn group the symmetric group of n particles and the resulting perturbation expansion of the hamiltonian is order by order invariant under the permutations of the sn group this invariance under sn imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series we show here that these blocks can be decomposed into a basis of binary tensors invariant under sn this basis is small 25 terms at first order in the wave function independent of n and is derived using graphical techniques this checks the n6 scaling of these terms at first order by effectively separating the n s

narain gupta s three normal subgroup problem and group homology abstract this paper is about application of various homological methods to classical problems in the theory of group rings it is shown that the third homology of groups plays a key role in narain gupta s three normal subgroup problem for a free group f and its normal subgroups r s t and the corresponding ideals in the integral group ring z f r r 1 z f s s 1 z f t t 1 z f a complete description of the normal subgroup f 1 rst is given provided r t and the third fourth and fifth homology groups of r r s are torsion groups

irreducible multiparty correlations in quantum states without maximal rank the correlations of an n partite quantum state are classified into a series of irreducible k party ones 2 ensuremath le k ensuremath le n with the irreducible k party correlation being the correlation in the states of k parties but nonexistent in the states of k ensuremath 1 parties a measure of the degree of irreducible k party correlation is defined based on the principle of maximal entropy adopting a continuity approach we overcome the difficulties in calculating the degrees of irreducible multiparty correlations for the multipartite states without maximal rank in particular we obtain the degrees of irreducible multiparty correlations in the n qubit stabilizer states and the n qubit generalized greenberger horne zeilinger states which reveals the distribution of multiparty correlations

the pauli principle revisited by the pauli exclusion principle no quantum state can be occupied by more than one electron one can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1 shortly after its discovery the pauli principle was replaced by anti symmetry of the multi electron wave function in this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix that goes far beyond the original pauli principle our approach uses berenstein and sjamaar s theorem on the restriction of an adjoint orbit onto a subgroup and allows us to treat any type of permutational symmetry

one qubit reduced states of a pure many qubit state polygon inequalities we show that a necessary and sufficient condition for a set of n one qubit mixed states to be the reduced states of a pure n qubit state is that their smaller eigenvalues should satisfy polygon inequalities each of them must be no greater than the sum of the others

kronecker coefficients for some near rectangular partitions abstract we give formulae for computing kronecker coefficients occurring in the expansion of s s where both and are nearly rectangular and have smallest parts equal to either 1 or 2 in particular we study s n n 1 1 s n n s n 1 n 1 1 s n n 1 s n 1 n 1 2 s n n s n 1 n 1 1 1 s n n and s n n 1 s n n 1 our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group mainly employing the pieri rule and a useful identity of littlewood as a consequence of these formulae we also derive an expression enumerating certain standard young tableaux of bounded height in terms of the motzkin and catalan numbers

categorical quantum circuits in this paper we extend previous work carried out on the application of the mathematics of category theory to quantum information science specifically we present a realization of a dagger compact category that can model finite dimensional quantum systems and explicitly allows for the interaction of systems of arbitrary possibly unequal dimensions hence our framework can handle generic tensor network states including matrix product states our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science the circuit diagrams themselves now become morphisms in a category making quantum circuits a special case of a much more general mathematical framework we introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information

riemann roch for tensor powers mapping a locally free module on a scheme to its l th tensor power gives rise to a natural map from the grothendieck group of all locally free modules to the grothendieck group of all locally free representations of the l th symmetric group in this paper we prove some formulas of riemann roch type for the behavior of this tensor power operation with respect to the push forward homomorphism associated with a projective morphism between schemes we furthermore establish analogous formulas for higher k groups

the capacity of the quantum depolarizing channel the information carrying capacity of the d dimensional depolarizing channel is computed it is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space and using measurements which are products of projections onto this same orthonormal basis in other words neither entangled signal states nor entangled measurements give any advantage for information capacity the result follows from an additivity theorem for the product channel spl delta spl ominus spl psi where spl delta is the depolarizing channel and spl psi is a completely arbitrary channel we establish the amosov holevo werner see probl inform transm vol 36 p 305 313 2000 p norm conjecture for this product channel for all p spl ges 1 and deduce from this the additivity of the minimal entropy and of the holevo quantity spl chi sup

eigenvalue distributions of reduced density matrices given a random quantum state of multiple distinguishable or indistinguishable particles we provide an effective method rooted in symplectic geometry to compute the joint probability distribution of the eigenvalues of its one body reduced density matrices as a corollary by taking the distribution s support which is a convex moment polytope we recover a complete solution to the one body quantum marginal problem we obtain the probability distribution by reducing to the corresponding distribution of diagonal entries i e to the quantitative version of a classical marginal problem which is then determined algorithmically this reduction applies more generally to symplectic geometry relating invariant measures for the coadjoint action of a compact lie group to their projections onto a cartan subalgebra and can also be quantized to provide an efficient algorithm for computing bounded height kronecker and plethysm coefficients

the uniqueness of plethystic factorisation we prove that a plethysm product of two schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products

rank reduction for the local consistency problem we address the problem of how simple a solution can be for a given quantum local consistency instance more specifically we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible we prove that any compatible local density operators can be satisfied by a low rank global density operator then we study both fermionic and bosonic versions of the n representability problem as applications after applying the channel state duality we prove that any compatible local channels can be obtained through a global quantum channel with small kraus rank

a fermionic de finetti theorem quantum versions of de finetti s theorem are powerful tools yielding conceptually important insights into the security of key distribution protocols or tomography schemes and allowing one to bound the error made by mean field approaches such theorems link the symmetry of a quantum state under the exchange of subsystems to negligible quantum correlations and are well understood and established in the context of distinguishable particles in this work we derive a de finetti theorem for finite sized majorana fermionic systems it is shown much reflecting the spirit of other quantum de finetti theorems that a state which is invariant under certain permutations of modes loses most of its anti symmetric character and is locally well described by a mode separable state we discuss the structure of the resulting mode separable states and establish in specific instances a quantitative link to the quality of the hartree fock approximation of quantum systems we hint at a link to generalized pauli principles fo

limitations of semidefinite programs for separable states and entangled games semidefinite programs sdps are a framework for exact or approximate optimization with widespread application in quantum information theory we introduce a new method for using reductions to construct integrality gaps for sdps meaning instances where the sdp value is far from the true optimum these are based on new limitations on the sum of squares sos hierarchy in approximating two particularly important sets in quantum information theory where previously no omega 1 round integrality gaps were known integrality gaps for the 2 rightarrow 4 norm had previously been sought due to its connection to small set expansion sse and unique games ug in both cases no go theorems were previously known based on computational assumptions such as the exponential time hypothesis eth which asserts that 3 sat requires exponential time to solve our unconditional results achieve the same parameters as all of these previous results for separable states or as some of the previous results for quantum correlations in some cases we can make use of the framework of lee raghavendra steurer lrs to establish integrality gaps for any sdp extended formulation not only the sos hierarchy our hardness result on separable states also yields a dimension lower bound of approximate disentanglers answering a question of watrous and aaronson et al these results can be viewed as limitations on the monogamy principle the ppt test the ability of tsirelson type bounds to restrict quantum correlations as well as the sdp hierarchies of doherty parrilo spedalieri navascues pironio acin and berta fawzi scholz indeed a wide range of past work in quantum information can be described as using an sdp on one of the above two problems and our results put broad limits on these lines of argument

the complexity of the local hamiltonian problem the k locham problem is a natural complete problem for the complexity class qma the quantum analogue of np it is similar in spirit to sc max k sat which is np complete for k geq 2 it was known that the problem is qma complete for any k geq 3 on the other hand 1 locham is in p and hence not believed to be qma complete the complexity of the 2 locham problem has long been outstanding here we settle the question and show that it is qma complete we provide two independent proofs our first proof uses only elementary linear algebra our second proof uses a powerful technique for analyzing the sum of two hamiltonians this technique is based on perturbation theory and we believe that it might prove useful elsewhere using our techniques we also show that adiabatic computation with 2 local interactions on qubits is equivalent to standard quantum computation

nonlocality is transitive we show a transitivity property of nonlocal correlations there exist tripartite nonsignaling correlations of which the bipartite marginals between a and b as well as b and c are nonlocal and any tripartite nonsignaling system between a b and c consistent with them must be such that the bipartite marginal between a and c is also nonlocal this property represents a step towards ruling out certain alternative models for the explanation of quantum correlations such as hidden communication at finite speed whereas it is not possible to rule out this model experimentally it is the goal of our approach to demonstrate this explanation to be logically inconsistent either the communication cannot remain hidden or its speed has to be infinite the existence of a three party system that is pairwise nonlocal is of independent interest in the light of the monogamy property of nonlocality

nonzero kronecker coefficients and what they tell us about spectra a triple of spectra ra rb rab is said to be admissible if there is a density operator ab with

optimal cloning and singlet monogamy the inability to produce two perfect copies of an unknown state is inherently linked with the inability to produce maximal entanglement between multiple spins despite this there is no quantitative link between how much entanglement can be generated between spins and how well an unknown state can be cloned this situation is remedied by giving a set of sufficient conditions such that a completely positive map can be optimally implemented as a teleportation operation into a standard reference state the case of arbitrary 1 ensuremath rightarrow n asymmetric cloning of d dimensional spins can then be solved exactly yielding the concept of singlet monogamy the utility of this relation is demonstrated by calculating properties of heisenberg systems and contrasting them with the results from standard monogamy arguments

the spectra of quantum states and the kronecker coefficients of the symmetric group determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics in this paper we consider the spectra of a bipartite quantum state and its two marginal states to each spectrum we can associate a representation of the symmetric group defined by a young diagram whose normalised row lengths approximate the spectrum we show that for allowed spectra the representation of the composite system is contained in the tensor product of the representations of the two subsystems this gives a new physical meaning to representations of the symmetric group it also introduces a new way of using the machinery of group theory in quantum informational problems which we illustrate by two simple examples

lower bound on entanglement in subspaces defined by young diagrams eigenvalues of 1 particle reduced density matrices of n fermion states are upper bounded by 1 n resulting in a lower bound on entanglement entropy we generalize these bounds to all other subspaces defined by young diagrams in the schur weyl decomposition of ncd eigenvalues of 1 particle reduced density matrices of n fermion states are upper bounded by 1 n resulting in a lower bound on entanglement entropy we generalize these bounds to all other subspaces defined by young diagrams in the schur weyl decomposition of ncd

connecting n representability to weyl s problem the one particle density matrix for n 3 and r 6 an analytic proof of the necessity of the borland dennis conditions for 3 representability of a one particle density matrix with rank 6 is given this may shed some light on klyachko s recent use of schubert calculus to find general conditions for n representability

description of object shapes by apparent boundary and convex hull the notions of the apparent boundary and the strict apparent boundary of an object are introduced which together with the convex hull provide an automatic approach for object shape analysis various properties of the apparent boundary and the strict apparent boundary are established which show that they are good approximations of the real object boundary applications of the apparent boundary and the convex hull to the automatic shape characterization of planktonic foraminifera are presented

sufficient conditions for pure state n representability a simple scheme is presented allowing the construction of most of the known and a wide variety of new sufficient conditions for pure state n representability of reduced one electron density operators

symmetry properties of product states for the system of n n level atoms suppose that the state of a system of n n level atoms is given by a tensor product of n identical density matrices the exact formulas are presented that describe the probability that such a system may be found in a pure state with a given symmetry with respect to permutations of atoms the asymptotic form of these probabilities valid for large n is also derived

an algorithm to explore entanglement in small systems a quantum state s entanglement across a bipartite cut can be quantified with entanglement entropy or more generally schmidt norms using only schmidt decompositions we present a simple iterative algorithm to maximize schmidt norms depending on the choice of norm the optimizing states maximize or minimize entanglement possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace recognizing that convergence but not success is certain we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels

distinguishability of quantum states by separable operations in this paper we study the distinguishability of multipartite quantum states by separable operations we first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations an analytical version of this condition is derived for the case of d 1 pure states where d is the total dimension of the state space under consideration a number of interesting consequences of this result are then carefully investigated remarkably we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication before our work only a class of 3 otimes 3 nonlocal separable operations was known bennett phys rev a 59 1070 1999 we also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states i e has an orthogonal schmidt number not less than three thus generalize the recent work about indistinguishable bipartite subspaces watrous phys rev lett 95 080505 2005 notably we obtain an explicit construction of indistinguishable subspaces of dimension 7 or 6 by considering a composite quantum system consisting of two qutrits resp three qubits which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8

computing the homology of koszul complexes let r be a commutative ring and i an ideal in r which is locally generated by a regular sequence of length d then each projective r i module v has an r projective resolution p of length d in this paper we compute the homology of the n th koszul complex associated with the homomorphism p 1 p 0 for all n if d 1 this computation yields a new proof of the classical adams riemann roch formula for regular closed immersions which does not use the deformation to the normal cone any longer furthermore if d 2 we compute the homology of the complex n sym 2 k p where k and n denote the functors occurring in the dold kan correspondence

the subgroup determined by a certain ideal in a free group ring abstract for normal subgroups r and s of a free group f an identification of the subgroup f 1 r f s is derived and it is shown that the quotient f 1 r f s r s r s r s r s r s r s is in general non trivial

quantum computational complexity of the n representability problem qma complete we study the computational complexity of the n representability problem in quantum chemistry we show that this problem is quantum merlin arthur complete which is the quantum generalization of nondeterministic polynomial time complete our proof uses a simple mapping from spin systems to fermionic systems as well as a convex optimization technique that reduces the problem of finding ground states to n representability

testing product states quantum merlin arthur games and tensor optimization we give a test that can distinguish efficiently between product states of n quantum systems and states that are far from product if applied to a state whose maximum overlap with a product state is 1 e the test passes with probability 1 e regardless of n or the local dimensions of the individual systems the test uses two copies of we prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarizing channel a key application of the test is to quantum merlin arthur games with multiple merlins where we obtain several structural results that had been previously conjectured including the fact that efficient soundness amplification is possible and that two merlins can simulate many merlins qma k qma 2 for k 2 building on a previous result of aaronson et al this implies that there is an efficient quantum algorithm to verify 3 sat with constant soundness given two unentangled proofs of o n qubits we also show how qma 2 with log sized proofs is equivalent to a large number of problems some related to quantum information such as testing separability of mixed states as well as problems without any apparent connection to quantum mechanics such as computing injective tensor norms of 3 index tensors as a consequence we obtain many hardness of approximation results as well as potential algorithmic applications of methods for approximating qma 2 acceptance probabilities finally our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product which is a generalization of classical linearity testing

entropic inequalities and marginal problems a marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables here we point out that the existence of such a joint distribution imposes nontrivial conditions already on the level of shannon entropies of the given marginals these entropic inequalities are necessary but not sufficient criteria for the existence of a joint distribution for every marginal problem a list of such shannon type entropic inequalities can be calculated by fourier motzkin elimination and we offer a software interface to a fourier motzkin solver for doing so for the case that the hypergraph of given marginals is a cycle graph we provide a complete analytic solution to the problem of classifying all relevant entropic inequalities and use this result to bound the decay of correlations in stochastic processes furthermore we show that shannon type inequalities for differential entropies are not relevant for continuous variable marginal problems non shannon type inequalities are both in the discrete and in the continuous case in contrast to other approaches our general framework easily adapts to situations where one has additional conditional independence requirements on the joint distribution as in the case of graphical models we end with a list of open problems a complementary article discusses applications to quantum nonlocality and contextuality

classification of n qubit states with minimum orbit dimension the group of local unitary transformations acts on the space of n qubit pure states decomposing it into orbits in a previous paper we proved that a product of singlet states together with an unentangled qubit for a system with an odd number of qubits achieves the smallest possible orbit dimension equal to 3n 2 for n even and 3n 1 2 for n odd where n is the number of qubits in this paper we show that any state with minimum orbit dimension must be of this form and furthermore such states are classified up to local unitary equivalence by the sets of pairs of qubits entangled in singlets

the parts determine the whole in a generic pure quantum state we show that almost every pure state of multiparty quantum systems each of whose local hilbert space has the same dimension is completely determined by the state s reduced density matrices of a fraction of the parties this fraction is less than about two thirds of the parties for states of large numbers of parties in other words once the reduced states of this fraction of the parties have been specified there is no further freedom in the state

generic pure quantum states as steady states of quasi local dissipative dynamics we investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality constrained purely dissipative markovian dynamics in the tripartite setting we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one solely depending on the subsystems dimension a complete analytical characterization is given when the central subsystem is a qubit in the n partite case we provide conditions on the subsystems size and the nature of the locality constraint under which random pure states cannot be quasi locally stabilized generically also allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic our results offer insights into the extent to which random pure states may arise as unique ground states of frustration free parent hamiltonians we further argue that to a high probability pure quantum states sampled from a t design enjoy the same stabilizability properties of haar random ones as long as suitable dimension constraints are obeyed and t is sufficiently large lastly we demonstrate a connection between the tasks of quasi local state stabilization and unique state reconstruction from local tomographic information and provide a constructive procedure for determining a generic n partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties improving over existing results

a linear time algorithm for computing the convex hull of an ordered crossing polygon in this paper we describe a linear time algorithm for finding the convex hull of an ordered crossing polygon and prove its correctness

hypothesis testing on invariant subspaces of the symmetric group part i quantum sanov s theorem and arbitrarily varying sources we report a proof of the quantum sanov theorem by the elementary application of basic facts about representations of the symmetric group together with a complete characterization of the optimal error exponent in a situation where the null hypothesis is given by an arbitrarily varying quantum source instead our approach differs from previous ones in two ways first it supports a reasoning inspired by the method of types second the measurement scheme we propose to distinguish the two alternatives not only does that job asymptotically perfectly but also yields additional information about the null hypothesis an example of that is given the measurement is composed of projections onto permutation invariant subspaces thus providing a direct link between one of the most basic tasks in quantum information on the one hand and fundamental objects in representation theory on the other we additionally connect to the representation theory by proving a relation between kostka numbers and quantum states and to state estimation via a generalization of a well known spectral estimation theorem to non i i d sequences

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

product rule for gauge invariant weyl symbols and its application to the semiclassical description of guiding centre motion we derive a product rule for gauge invariant weyl symbols which provides a generalization of the well known moyal formula to the case of non vanishing electromagnetic fields applying our result to the guiding centre problem we expand the guiding centre hamiltonian into an asymptotic power series with respect to both planck s constant and an adiabaticity parameter already present in the classical theory this expansion is used to determine the influence of quantum mechanical effects on guiding centre motion

canonical form of three fermion pure states with six single particle states we construct a canonical form for pure states in 3 c6 the three fermion system with six single particle states under local unitary lu transformations i e the unitary group u 6 we also construct a minimal set of generators of the algebra of polynomial u 6 invariants on 3 c6 it turns out that this algebra is isomorphic to the algebra of polynomial lu invariants of three qubits which are additionally invariant under qubit permutations as a consequence of this surprising fact we deduce that there is a one to one correspondence between the u 6 orbits of pure three fermion states in 3 c6 and the lu orbits of pure three qubit states when qubit permutations are allowed as an important byproduct we obtain a new canonical form for pure three qubit states under lu transformations u 2 u 2 u 2 no qubit permutations allowed

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

a complete basis for a perturbation expansion of the general n body problem we discuss a basis set developed to calculate perturbation coefficients in an expansion of the general n body problem this basis has two advantages first the basis is complete order by order for the perturbation series second the number of independent basis tensors spanning the space for a given order does not scale with n the number of particles despite the generality of the problem at first order the number of basis tensors is 25 for all n i e the problem scales as n0 although one would initially expect an n6 scaling at first order the perturbation series is expanded in inverse powers of the spatial dimension this results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group sn the resulting perturbation series is order by order invariant under the n operations of the sn point group which is responsible for the slower than exponential growth of the basis in this paper we demonstrate the completeness of the basis and perform the first test of this formalism through first order by comparing to an exactly solvable fully interacting problem of n particles with a two body harmonic interaction potential