We consider the quantum complexity of computing Schatten p-norms and related quantities, and find that the problem of estimating these quantities is closely related to the one clean qubit model of computation. We show that the problem of approximating Tr(|A|) for a log-local n-qubit Hamiltonian A and p = poly(n), up to a suitable level of accuracy, is contained in DQC1; and that approximating t...

This paper studies the minimax detection of a small submatrix of elevated mean in a p× p matrix contaminated by additive Gaussian noise where p tends to infinity. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent in the sense of Le Cam. Under...

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomor-phic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative L p (N;)-spaces deened by a semi-nite, normal, faithful trace on a von Neumann algebra N. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.

let $g$ be a locally compact group, $h$ be a compact subgroup of $g$ and $varpi$ be a representation of the homogeneous space $g/h$ on a hilbert space $mathcal h$. for $psi in l^p(g/h), 1leq p leqinfty$, and an admissible wavelet $zeta$ for $varpi$, we define the localization operator $l_{psi,zeta} $ on $mathcal h$ and we show that it is a bounded operator. moreover, we prove that the localizat...

We study when multiplication by a weight can turn non-compact composition operator on \(H^2\) into compact operator, and it be in Schatten classes. The \(q\)-summing case \(H^p\) is considered. also this one.

We show convergence of the weak dual greedy algorithm in wide class of Banach spaces, extending our previous result where it was shown to converge in subspaces of quotients of Lp (for 1 < p < ∞). In particular, we show it converges in the Schatten ideals Sp when 1 < p < ∞ and in any Banach lattice which is p-convex and q-concave with constants one, where 1 < p < q < ∞. We also discuss convergen...

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full group, Schatten perturbations identity and groups finite von Neumann algebras. The structures are defined by translation different norms on tangent space at identity. We first prove a result for metric derived from operator norm group. also strong results squared ...

Additive commutators of operators belonging to two{sided ideals of B(H) are studied. J){commutators is characterized and found to equal IJ ; B(H)]. It is shown that three commutators are enough in I; B(H)], and that four are enough in I; J ]. Finally, some results about single commutators are proved, for example that every element of the Schatten class C p is a single (C p ; B(H)){commutator if...

We prove that non-commutative martingale transforms are of weak type (1, 1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1 is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x = (xn) ∞ n=1 in L 1(M), adapted to (Mn)n=1, and any sequence of signs (εn) ∞ n=1, ∥∥∥∥ε1x1 + N ∑ n=2 εn(xn...

This paper is concerned with linear stochastic control systems in state space. The integral of the squared norm of the system output over a bounded time interval is interpreted as energy. The cumulants of the output energy in the infinite-horizon limit are related to Schatten norms of the system in the Hardy space of transfer functions and the risk-sensitive performance index. We employ a novel...