arXiv subjects

Tim Netzer

Publications and source records attributed to Tim Netzer.

At least 37 records · Page 2Linked to original sources

arXiv2021

Halos and undecidability of tensor stable positive maps

A map $\mathcal{P}$ is tensor stable positive (tsp) if $\mathcal{P}^{\otimes n}$ is positive for all $n$, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

quant-ph↗

arXiv2021

Abstract Operator Systems over the Cone of Positive Semidefinite Matrices

There are several important abstract operator systems with the convex cone of positive semidefinite matrices at the first level. Well-known are the operator systems of separable matrices, of positive semidefinite matrices, and of block positive matrices. In terms of maps, these are the operator systems of entanglement breaking, completely positive, and positive linear maps, respectively. But there exist other interesting and less well-studied such operator systems, for example those of completely copositive maps, doubly completely positive maps, and decomposable maps, which all play an important role in quantum information theory. We investigate which of these systems is finitely generated, and which admits a finite-dimensional realization in the sense of the Choi-Effros Theorem. We answer this question for all of the described systems completely. Our main contribution is that decomposable maps form a system which does not admit a finite-dimensional realization, though being finitely generated, whereas the system of doubly completely positive maps is not finitely generated, though having a finite-dimensional realization. This also implies that there cannot exist a finitary Choi-type characterization of doubly completely positive maps.

math.OA↗

arXiv2021

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

math-ph↗

arXiv2021

Approximate Pythagoras Numbers on $*$-algebras over $\mathbb{C}$

The Pythagoras number of a sum of squares is the shortest length among its sums of squares representations. In many algebras, for example real polynomial algebras in two or more variables, there exists no upper bound on the Pythagoras number for all sums of squares. In this paper, we study how Pythagoras numbers in $*$-algebras over $\mathbb{C}$ behave with respect to small perturbations of elements. More precisely, the approximate Pythagoras number of an element is the smallest Pythagoras number among all elements in its $\varepsilon$-ball. We show that these approximate Pythagoras numbers are often significantly smaller than their exact versions, and allow for (almost) dimension-independent upper bounds. Our results use low-rank approximations for Gram matrices of sums of squares and estimates for the operator norm of the Gram map.

math.NT↗

arXiv2021

Quantum Information Theory and Free Semialgebraic Geometry: One Wonderland Through Two Looking Glasses

We illustrate how quantum information theory and free (i.e. noncommutative) semialgebraic geometry often study similar objects from different perspectives. We give examples in the context of positivity and separability, quantum magic squares, quantum correlations in non-local games, and positivity in tensor networks, and we show the benefits of combining the two perspectives. This paper is an invitation to consider the intersection of the two fields, and should be accessible for researchers from either field.

quant-ph↗

arXiv2020

Approximate Completely Positive Semidefinite Factorizations and their Ranks

In this paper we show the existence of approximate completely positive semidefinite (cpsd) factorizations with a cpsd-rank bounded above (almost) independently from the cpsd-rank of the initial matrix. This is particularly relevant since the cpsd-rank of a matrix cannot, in general, be upper bounded by a function only depending on its size. For this purpose, we make use of the Approximate Caratheodory Theorem in order to construct an approximate matrix with a low-rank Gram representation. We then employ the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.

math.AG↗

arXiv2020

Cats climb entails mammals move: preserving hyponymy in compositional distributional semantics

To give vector-based representations of meaning more structure, one approach is to use positive semidefinite (psd) matrices. These allow us to model similarity of words as well as the hyponymy or is-a relationship. Psd matrices can be learnt relatively easily in a given vector space $M\otimes M^*$, but to compose words to form phrases and sentences, we need representations in larger spaces. In this paper, we introduce a generic way of composing the psd matrices corresponding to words. We propose that psd matrices for verbs, adjectives, and other functional words be lifted to completely positive (CP) maps that match their grammatical type. This lifting is carried out by our composition rule called Compression, Compr. In contrast to previous composition rules like Fuzz and Phaser (a.k.a. KMult and BMult), Compr preserves hyponymy. Mathematically, Compr is itself a CP map, and is therefore linear and generally non-commutative. We give a number of proposals for the structure of Compr, based on spiders, cups and caps, and generate a range of composition rules. We test these rules on a small sentence entailment dataset, and see some improvements over the performance of Fuzz and Phaser.

cs.CL↗

arXiv2020

Approximate tensor decompositions: disappearance of many separations

It is well-known that tensor decompositions show separations, that is, that constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we show that many of these separations disappear in the approximate case. Specifically, for every approximation error $\varepsilon$ and norm, we define the approximate rank as the minimum rank of an element in the $\varepsilon$-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten $p$-norms. For nonnegative tensors, we show that the separations between rank, positive semidefinite rank, and nonnegative rank disappear for all $\ell_p$-norms with $p>1$. For the trace norm ($p = 1$), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of Carath\'eodory's Theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.

math.OC↗

arXiv2020

Optimal bounds on the positivity of a matrix from a few moments

In many contexts one encounters Hermitian operators $M$ on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such $M$ is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten $p$-norm for all integer $p\in[1,\infty)$, assuming that we know the moments $\mathbf{tr}(M^k)$ up to a certain order $k=1,\ldots, m$. We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra.

math.AG↗

arXiv2020

Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices

We study six natural decompositions of mixed states in one spatial dimension: the Matrix Product Density Operator (MPDO) form, the local purification form, the separable decomposition (for separable states), and their three translational invariant (t.i.) analogues. For bipartite states diagonal in the computational basis, we show that these decompositions correspond to well-studied factorisations of an associated nonnegative matrix. Specifically, the first three decompositions correspond to the minimal factorisation, the nonnegative factorisation, and the positive semidefinite factorisation. We also show that a symmetric version of these decompositions corresponds to the symmetric factorisation, the completely positive factorisation, and the completely positive semidefinite transposed factorisation, respectively. We leverage this correspondence to characterise the six decompositions of mixed states.

quant-ph↗

arXiv2019

Quantum magic squares: dilations and their limitations

Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff--von Neumann Theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries, and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.

quant-ph↗

arXiv2019

Separability for mixed states with operator Schmidt rank two

The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them. In contrast, MPDOs of bond dimension three can contain an unbounded amount of classical correlations.

quant-ph↗

arXiv2019

Tensor decompositions on simplicial complexes with invariance

We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group action. We prove that this decomposition exists for all invariant tensors after possibly enriching the simplicial complex. As a special case we recover tensor networks with translational invariance and the symmetric tensor decomposition. We also define an invariant separable decomposition and purification form, and prove similar existence results. Associated to every decomposition there is a rank, and we prove several inequalities between them. For example, we show by how much the rank increases when imposing invariance in the decomposition, and that the tensor rank is the largest of all ranks. Finally, we apply our framework to nonnegative tensors, where we define a nonnegative and a positive semidefinite decomposition on arbitrary simplicial complexes with group action. We show a correspondence to the previous ranks, and as a very special case recover the nonnegative, the positive semidefinite, the completely positive and the completely positive semidefinite transposed decomposition.

math.CO↗

arXiv2019

A note on non-commutative polytopes and polyhedra

It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones. This was obtained as a byproduct in an earlier paper. In this note we give a direct and constructive proof of the statement. Our proof also yields a surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size $2$, independent of dimension and complexity of the initial convex cone.

math.AG↗

arXiv2019

Free Semialgebraic Geometry

This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and differences as well as some important results in the free setup.

math.AG↗

arXiv2017

On Projections of Free Semialgebraic Sets

An important result in real algebraic geometry is the projection theorem: every projection of a semialgebraic set is again semialgebraic. This theorem and some of its conclusions lie at the basis of many other results, for example the decidability of the theory of real closed fields, and almost all Positivstellensätze. Recently, non-commutative real algebraic geometry has evolved as an exciting new area of research, with many important applications. In this paper we examine to which extend a projection theorem is possible in the non-commutative (=free) setting. Although it is not yet clear what the correct notion of a free semialgebraic set is, we review and extend some results that count against a full free projection theorem. For example, it is undecidable whether a free statement holds for all matrices of at least one size. We then prove a weak version of the projection theorem: projections along linear and separated variables yields a semi-algebraically parametrised free semi-algebraic set.

math.FA↗

arXiv2017

Non-Nudgable Subgroups of Permutations

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is called non-nudgable, if the cardinality of this set always remains the same when replacing the initial permutation with its inverse. It is called nudgable otherwise. We show that all full permutation groups, standard dihedral groups, half of the alternating groups, and any abelian subgroup are non-nudgable. In the right probabilistic sense, it is thus quite likely that a randomly generated subgroup is non-nudgable. However, the other half of the alternating groups are nudgable. We also construct a smallest possible nudgable group, a 6-element subgroup of the permutation group on 4 elements.

math.CO↗

arXiv2017

Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

Containment problems for polytopes and spectrahedra appear in various applications, such as linear and semidefinite programming, combinatorics, convexity and stability analysis of differential equations. This paper explores the theoretical background of a method proposed by Ben-Tal and Nemirovksi. Their method provides a strengthening of the containment problem, that is algorithmically well tractable. To analyze this method, we study abstract operator systems, and investigate when they have a finite-dimensional concrete realization. Our results give some profound insight into their approach. They imply that when testing the inclusion of a fixed polyhedral cone in an arbitrary spectrahedron, the strengthening is tight if and only if the polyhedral cone is a simplex. This is true independent of the representation of the polytope. We also deduce error bounds in the other cases, simplifying and extending recent results by various authors.

math.FA↗