Grothendieck Inequality Research Articles

Jordan norms for multilinear maps on $C^{\ast}$-algebras and Grothendieck's inequalities

There exists a generalization of the concept completely bounded norm, for multilinear maps on $C^{\ast }$-algebras. We will use the word Jordan norm, for this norm and denote it by $\lVert \cdot \rVert _J$. The Jordan norm $\lVert\Phi\rVert_J$ of a multilinear map is obtained via factorizations of $\Phi$ in the form $$\Phi (a_1, \dots , a_n) = T_0 \sigma _1(a_1)T_1 \cdots T_{(n-1)}\sigma _n(a_n)T_n ,$$ where the maps $\sigma _i$ are Jordan homomorphisms. We show that any bounded bilinear form on a pair of $C^{\ast }$-algebras is Jordan bounded and satisfies $\lVert B\rVert _J \leq 2\lVert B\rVert $.

Some points of view on Grothendieck's inequalities

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constantkGC.

Unique matrix factorizations associated to bilinear forms and Schur multipliers

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m×n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.

Large deviations for mean field model in Erdős–Rényi graph

In this paper, we study a particle systems (or interacting diffusions) on an Erdős–Rényi graph with the parameter pN∈(0,1] that behaves like a mean-field system up to large deviations. Our aim is to establish the large deviations for the empirical measure process of particle systems under the condition NpN4→∞ as N→∞, where N is the number of particles. The result is obtained by proving the exponential equivalence between our systems and general interacting systems without random graphs. The multilinear extensions of Grothendieck inequality play a crucial role in our proof.

Ultra-quantum coherent states in a single finite quantum system

A set of n coherent states is introduced in a quantum system with d-dimensional Hilbert space H(d). It is shown that they resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits. A n-tuple representation of arbitrary states in H(d), analogous to the Bargmann representation, is defined. There are two other important properties of these coherent states which make them ‘ultra-quantum’. The first property is related to the Grothendieck formalism which studies the ‘edge’ of the Hilbert space and quantum formalisms. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form that uses complex numbers in the unit disc, and a ‘quantum’ quadratic form that uses vectors in the unit ball of the Hilbert space. It shows that if , the corresponding might take values greater than 1, up to the complex Grothendieck constant . related to these coherent states is shown to take values in the ‘Grothendieck region’ , which is classically forbidden in the sense that does not take values in it. The second property complements this, showing that these coherent states violate logical Bell-like inequalities (which for a single quantum system are quantum versions of the Frechet probabilistic inequalities). In this sense also, our coherent states are deep into the quantum region.

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex $m \times n$ matrices. Based on the theory of operator spaces and completely bounded maps we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of bilinear forms and of Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant

We address the problem of testing the quantumness of two-dimensional systems in the prepare-and-measure (PM) scenario, using a large number of preparations and a large number of measurement settings, with binary outcome measurements. In this scenario, we introduce constants, which we relate to the Grothendieck constant of order 3. We associate them with the white noise resistance of the prepared qubits and to the critical detection efficiency of the measurements performed. Large-scale numerical tools are used to bound the constants. This allows us to obtain new bounds on the minimum detection efficiency that a setup with 70 preparations and 70 measurement settings can tolerate.

Grothendieck bound in a single quantum system

Grothendieck’s bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a ‘classical’ quadratic form that uses complex numbers in the unit disc, and takes values less than 1. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the ‘quantum’ quadratic form might take values greater than 1, up to the complex Grothendieck constant . The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the ‘Grothendieck region’ , which is a classically forbidden region in the sense that cannot take values in it. Necessary (but not sufficient) conditions for taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with six and 12-dimensional Hilbert space, are shown to lead to in the Grothendieck region . They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.

On optimality of constants in the Little Grothendieck Theorem

We explore the optimality of the constants making valid the recently established little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB$^*$-algebra $B$ into a

Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

In the paper, the authors 1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $; 2. find closed-form formulas for the sequence $ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $ where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials; 3. derive series representations of generalized logsine functions. The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.

Mapping ideals of quantum group multipliers

We study the dual relationship between quantum group convolution maps L1(G)→L∞(G) and completely bounded multipliers of Gˆ. For a large class of locally compact quantum groups G, we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with Mcb(L1(Gˆ)), yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter coinciding with ℓ1(bGˆ), where bG is the quantum Bohr compactification of G. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with C(bG). Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of inner co-amenable quantum group, that we introduce and for which we exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras VN(G) for quasi-SIN locally compact groups G.

Quasirandom quantum channels

Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or `quantum', case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.

Grothendieck constant is norm of Strassen matrix multiplication tensor

We show that two important quantities from two disparate areas of complexity theory --- Strassen's exponent of matrix multiplication $\omega$ and Grothendieck's constant $K_G$ --- are intimately related. They are different measures of size for the same underlying object --- the matrix multiplication tensor, i.e., the $3$-tensor or bilinear operator $\mu_{l,m,n} : \mathbb{F}^{l \times m} \times \mathbb{F}^{m \times n} \to \mathbb{F}^{l \times n}$, $(A,B) \mapsto AB$ defined by matrix-matrix product over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$. It is well-known that Strassen's exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of $\mu_{l,m,n}$. We will show that Grothendieck's constant is the least upper bound on a tensor norm of $\mu_{l,m,n}$, taken over all $l, m, n \in \mathbb{N}$. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequality \[ \lVert\mu_{l,m,n}\rVert_{1,2,\infty} =\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\lVert X\rVert_{1,2}\lVert Y\rVert_{2,\infty}\lVert M\rVert_{\infty,1}}\le K_G. \] We prove that Grothendieck's inequality is unique: If we generalize the $(1,2,\infty)$-norm to arbitrary $p,q, r \in [1, \infty]$, \[ \lVert\mu_{l,m,n}\rVert_{p,q,r}=\max_{X,Y,M\neq0}\frac{|\operatorname{tr}(XMY)|}{\|X\|_{p,q}\|Y\|_{q,r}\|M\|_{r,p}}, \] then $(p,q,r )=(1,2,\infty)$ is, up to cyclic permutations, the only choice for which $\lVert\mu_{l,m,n}\rVert_{p,q,r}$ is uniformly bounded by a constant independent of $l,m,n$.

An elementary and unified proof of Grothendieck’s inequality

We present an elementary, self-contained proof of Grothendieck’s inequality that unifies the real and complex cases and yields both the Krivine and Haagerup bounds, the current best-known explicit bounds for the real and complex Grothendieck constants respectively. This article is intended to be pedagogical, combining and streamlining known ideas of Lindenstrauss–Pełczynski, Krivine, and Haagerup into a proof that needs only univariate calculus, basic complex variables, and a modicum of linear algebra as prerequisites.

Powers of quasi-Banach lattices and orthogonally additive polynomials

The paper aims to examine convexity and concavity conditions for homogeneous orthogonally additive polynomials in quasi-Banach lattices. We characterize (p,q)-convex quasi-Banach lattice by the property that every positive orthogonally additive polynomial defined on such lattice is (p,q)-convex as well as obtain some versions of Krivine factorization result and Grothendieck's inequality for orthogonally additive polynomials in quasi-Banach lattices.

Qutrit witness from the Grothendieck constant of order four

In this paper, we prove that $K_G(3)<K_G(4)$, where $K_G(d)$ denotes the Grothendieck constant of order $d$. To this end, we use a branch-and-bound algorithm commonly used in the solution of NP-hard problems. It has recently been proven that $K_G(3)\le 1.4644$. Here we prove that $K_G(4)\ge 1.4841$, which has implications for device-independent witnessing dimensions greater than two. Furthermore, the algorithm with some modifications may find applications in various black-box quantum information tasks with large number of inputs and outputs.

Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constantKG(3)

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner stateρ=v|ψ−⟩⟨ψ−|+(1−v)14via a local hidden variable (LHV) model, where|ψ−⟩denotes the singlet state. We show analytically that these correlations are local forv=999×689×10−6cos2⁡(π/50)≃0.6829. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constantKG(3)≤1/v≃1.4644. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state forv≃0.4553. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.

Quasirandom Cayley graphs

We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, R\"odl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.

Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups.

The little Grothendieck problem consists of maximizing Σ ij Cijxixj for a positive semidef-inite matrix C, over binary variables xi ∈ {±1}. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ ℝ dn × dn a positive semidefinite matrix, the objective is to maximize [Formula: see text] restricting Oi to take values in the group of orthogonal matrices [Formula: see text], where Cij denotes the (ij)-th d × d block of C. We propose an approximation algorithm, which we refer to as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices [Formula: see text] and show a constant approximation ratio. Our method is based on semidefinite programming. For a given d ≥ 1, we show a constant approximation ratio of αℝ(d)2, where αℝ(d) is the expected average singular value of a d × d matrix with random Gaussian [Formula: see text] i.i.d. entries. For d = 1 we recover the known αℝ(1)2 = 2/π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group [Formula: see text]. Orthogonal-Cut also serves as an approximation algorithm for several applications, including the Procrustes problem where it improves over the best previously known approximation ratio of [Formula: see text]. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d = 1, the sharp, known ratios.

Grothendieck’s inequality in the noncommutative Schwartz space

In the spirit of Grothendieck's famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Fr\'echet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.