Tensor Product Of Spaces Research Articles

Stability results of octahedrality in tensor product spaces and finite representability inℓ1

Stability results of octahedrality in tensor product spaces and finite representability inℓ1

Ultimate Data Hiding in Quantum Mechanics and Beyond

The phenomenon of data hiding, i.e. the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under LOCC, is a distinctive signature of nonclassicality. The relevant figure of merit is the maximal ratio (called data hiding ratio) between the distinguishability norms associated with the two sets of measurements we are comparing, typically all measurements vs LOCC protocols. For a bipartite $n\times n$ quantum system, it is known that the data hiding ratio scales as $n$, i.e. the square root of the real dimension of the local state space of density matrices. We show that for bipartite $n_A\times n_B$ systems the maximum data hiding ratio against LOCC protocols is $\Theta\left(\min\{n_A,n_B\}\right)$. This scaling is better than the previously obtained upper bounds $O\left(\sqrt{n_A n_B}\right)$ and $\min\{n_A^2, n_B^2\}$, and moreover our intuitive argument yields constants close to optimal. In this paper, we investigate data hiding in the more general context of general probabilistic theories (GPTs), an axiomatic framework for physical theories encompassing only the most basic requirements about the predictive power of the theory. The main result of the paper is the determination of the maximal data hiding ratio obtainable in an arbitrary GPT, which is shown to scale linearly in the minimum of the local dimensions. We exhibit an explicit model achieving this bound up to additive constants, finding that the quantum mechanical data hiding ratio is only of the order of the square root of the maximal one. Our proof rests crucially on an unexpected link between data hiding and the theory of projective and injective tensor products of Banach spaces. Finally, we develop a body of techniques to compute data hiding ratios for a variety of restricted classes of GPTs that support further symmetries.

Eventually entanglement breaking maps

We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say that the map has a finite index of separability. In particular, we show that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have a finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. Our work is motivated by Christandl’s PPT-squared conjecture. This conjecture states that every PPT channel, when composed with itself, becomes entanglement breaking.

Tensor Characterizations of Summing Polynomials

Operators T that belong to some summing operator ideal can be characterized by means of the continuity of an associated tensor operator $${\overline{T}}$$ that is defined between tensor products of sequences spaces. In this paper, we provide a unifying treatment of these tensor product characterizations of summing operators. We work in the more general frame provided by homogeneous polynomials, where an associated “tensor” polynomial—which plays the role of $${\overline{T}}$$ —, needs to be determined first. Examples of applications are shown.

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118–A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy–Buniakowskii–Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted L^2 sense, then this CBS constant only depends on a pair of finite element spaces H_{1}, H_{2} associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of H_{1}, we investigate non-standard choices of H_{2} and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When H_1 and H_2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.

Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain D such that D̅ is the union of cells {D̅i}i∈I and we introduce a two-scale representation by identifying any function v(x) defined on D with a bi-variate function v(i,y), where i ∈ I relates to the index of the cell containing the point x and y ∈ Y relates to a local coordinate in a reference cell Y. We introduce a weak formulation of the problem in a broken Sobolev space V(D) using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying V(D) with a tensor product space ℝI⊗ V(Y) of functions defined over the product set I × Y. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.

Breaking the curse for uniform approximation in Hilbert spaces via Monte Carlo methods

We study the L∞-approximation of d-variate functions from Hilbert spaces via linear functionals as information. It is a common phenomenon in tractability studies that unweighted problems (with each dimension being equally important) suffer from the curse of dimensionality in the deterministic setting, that is, the number n(ϵ,d) of information needed in order to solve a problem within a given accuracy ϵ>0 grows exponentially in d. We show that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness r>1∕2, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability, namely n(ϵ,d)⪯ϵ−2d(1+logd). Similar benefits from Monte Carlo methods in terms of tractability have only been known for integration problems so far.

A fast sparse grid based space–time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space–time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin’s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the $$\mathcal {H}$$ -matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces.

Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals

A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. The method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

A simplified formalism of the algebra of partially transposed permutation operators with applications

Herein we continue the study of the representation theory of the algebra of permutation operators acting on the -fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced irreducible representations, which allows us to significantly simplify previously proved theorems and, most importantly, derive new results for irreducible representations of the mentioned algebra. In our analysis we are able to reduce the complexity of the central expressions by getting rid of sums over all permutations from the symmetric group, obtaining equations which are much more handy in practical applications. We also find relatively simple matrix representations for the generators of the underlying algebra. The obtained simplifications and developments are applied to derive the characteristics of a deterministic port-based teleportation scheme written purely in terms of irreducible representations of the studied algebra. We solve an eigenproblem for the generators of the algebra, which is the first step towards a hybrid port-based teleportation scheme and gives us new proofs of the asymptotic behaviour of teleportation fidelity. We also show a connection between the density operator characterising port-based teleportation and a particular matrix composed of an irreducible representation of the symmetric group, which encodes properties of the investigated algebra.

Some results on the projective cone normed tensor product spaces over banach algebras

For two real Banach algebras $\mathbb{A}_1$ and $\mathbb{A}_2$, let $K_p$ be the projective cone in $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$. Using this we define a cone norm on the algebraic tensor product of two vector spaces over the Banach algebra $\mathbb{A}_1\otimes_\gamma \mathbb{A}_2$ and discuss some properties. We derive some fixed point theorems in this projective cone normed tensor product space over Banach algebra with a suitable example. For two self mappings $S$ and $T$ on a cone Banach space over Banach algebra, the stability of the iteration scheme $x_{2n+1}=Sx_{2n}$, $x_{2n+2}=Tx_{2n+1},\;n=0,1,2,...$ converging to the common fixed point of $S$ and $T$ is also discussed here.

Some new results on certain types of proximinality in Banach spaces

In this paper, we prove that any convex set in a normed space is $\varepsilon -$proximinal. Consequently, every subspace in a Banach space is $\varepsilon -$proximinal. Some other results of proximinality in tensor product spaces are given.

Concerning $q$-summable Szlenk index

For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^{*}$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms under $\ell_{r}$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_{p}$ and $c_{0}$ direct sums of operators.

Some solutions of fractional inverse problems

In this paper we find some types of solutions for certain degenerate and non degenerate fractional inverse problems. The main idea of the proofs is based on theory of tensor product of Banach spaces.

Formula omitted]-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs

In this article, we present a cost–benefit analysis of the approximation in tensor products of Hilbert spaces of Sobolev-analytic type. The Sobolev part is defined on a finite dimensional domain, whereas the analytical space is defined on an infinite dimensional domain. As main mathematical tool, we use the ε-dimension in Hilbert spaces which gives the lowest number of linear information that is needed to approximate an element from the unit ball W in a Hilbert space Y up to an accuracy ε>0 with respect to the norm of a Hilbert space X. From a practical point of view this means that we a priori fix an accuracy and ask for the amount of information to achieve this accuracy. Such an analysis usually requires sharp estimates on the cardinality of certain index sets which are in our case infinite-dimensional hyperbolic crosses. As main result, we obtain sharp bounds of the ε-dimension of the Sobolev-analytic-type function classes which depend only on the smoothness differences in the Sobolev spaces and the dimension of the finite dimensional domain where these spaces are defined. This implies in particular that, up to constants, the costs of the infinite dimensional (analytical) approximation problem is dominated by the finite-variate Sobolev approximation problem. We demonstrate this procedure with examples of functions spaces stemming from the regularity theory of parametric partial differential equations.

Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups

In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels. An application to the construction of $d$-positive maps on matrix algebras is also presented.

Kähler quantization and entanglement

For a very ample line bundle L on a compact connected complex manifold X, with a real structure, we discuss entanglement properties of certain sequences of vectors in tensor products of spaces of holomorphic sections of powers of L.

Can a quantum state over time resemble a quantum state at a single time?

The standard formalism of quantum theory treats space and time in fundamentally different ways. In particular, a composite system at a given time is represented by a joint state, but the formalism does not prescribe a joint state for a composite of systems at different times. If there were a way of defining such a joint state, this would potentially permit a more even-handed treatment of space and time, and would strengthen the existing analogy between quantum states and classical probability distributions. Under the assumption that the joint state over time is an operator on the tensor product of single-time Hilbert spaces, we analyse various proposals for such a joint state, including one due to Leifer and Spekkens, one due to Fitzsimons, Jones and Vedral, and another based on discrete Wigner functions. Finding various problems with each, we identify five criteria for a quantum joint state over time to satisfy if it is to play a role similar to the standard joint state for a composite system: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time steps is associative. We show that no construction satisfies all these requirements. If Hermiticity is dropped, then there is an essentially unique construction that satisfies the remaining four criteria.

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs Bn(H) in B(H)n1+⋯+nk, where n=(n1,…,nk)∈Nk and B(H) is the algebra of all bounded linear operators on a Hilbert space H. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on Bn(H), study their properties, and obtain hyperbolic versions of Schwarz–Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case n1=⋯=nk=1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk) of all free holomorphic automorphisms of Dk(H), and the δH-topology induced on Dk(H) is the usual operator norm topology. The restriction of δH to the scalar polydisk Dk is equivalent to the Kobayashi distance on Dk. Most of the results of this paper are presented in the more general setting of Harnack (resp. Poisson) parts of the closed polyball Bn(H)−.

Nonlinear Volterra‐type diffusion equations and theory of quantum fields

We study nonlinear Volterra‐type evolution integral equations of the form: in a C∗‐algebra or in a Hilbert algebra of Dixmier‐Segal type, acting on a Hilbert space tensor product , where denotes a Hilbert space and is the Boson‐Einstein (Fermion‐Dirac) Fock space, over a complex Hilbert space . Under suitable Carathéodory‐type conditions on the corresponding Nemytskii operator Φ of f and assuming that k is a quantum dynamical‐type semigroup, we obtain exactly one classical global solution in the space of bounded continuous (operator‐valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in (respectively in , applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that k is a quantum dynamical semigroup acting on , where Φ defines a positive (respectively completely positive) quantum stochastic process.