Noncommutative Grothendieck Research Articles

On the power of quantum entanglement in multipartite quantum XOR games

AbstractWe show that, given , there exist ‐player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one‐way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.

Expanding the reach of quantum optimization with fermionic embeddings

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

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.

Localizations and sheaves of glider representations

The notion of a glider representation of a chain of normal subgroups of a group is defined by a new structure, i.e. a fragment for a suitable filtration on the group ring. This is a special case of general glider representations defined for a positively filtered ring R with filtration FR and subring S=F0R. Nice examples appear for chains of groups, chains of Lie algebras, rings of differential operators on some variety or V-gliders for W for algebraic varieties V and W. This paper aims to develop a scheme theory for glider representations via the localizations of filtered modules. With an eye to non-commutative geometry we allow schemes over non-commutative rings with particular attention to so called almost commutative rings. We consider particular cases of Proj R (e.g. for some P.I. ring R) in terms of prime ideals, R-tors in terms of torsion theories and W_(R) in terms of a non-commutative Grothendieck topology based on words of Ore set localizations.

Tight Hardness of the Non-Commutative Grothendieck Problem

$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight $\textsf{NP}$-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009).

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.

Efficient rounding for the noncommutative Grothendieck inequality

$ \newcommand{\cclass}[1]{{\textsf{#1}}} $The classical Grothendieck inequality has applications to the design of approximation algorithms for $\cclass{NP}$-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a polynomial-time constant-factor approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principal component analysis and the orthogonal Procrustes problem.

Dilations of 𝑉-bounded stochastic processes indexed by a locally compact group

It is proved that a stochastic process (i.e., a Hilbert space valued function) indexed by a locally compact group is V V -bounded (i.e., weakly harmonizable in an appropriate sense) if, and only if, it can be expressed as an orthogonal projection of a process whose covariance function R R satisfies R ( s , t ) = ρ ( t − 1 s ) + ρ ( s t − 1 ) R(s,t) = \rho ({t^{ - 1}}s) + \rho (s{t^{ - 1}}) for some continuous positive-definite function ρ \rho . The result generalizes a well-known theorem due to H. Niemi, and depends on the noncommutative Grothendieck type inequality of G. Pisier.