On Schatten p-norms of commutators

The Quantum Complexity of Computing Schatten $p$-norms

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...

Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are contactions in a C*-algebra A, with x normal and with spectrum within the domain of f . In particular we discuss ‖[x, y]‖ and ‖[x, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖ we can rigorous calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf (δ) for many f . ...

Notes on super-operator norms induced by schatten norms

Let Φ be a super-operator, i.e., a linear mapping of the form Φ : L(F) → L(G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by ‖Φ‖q→p = sup{‖Φ(X)‖p/‖X‖q : X 6= 0}, induced by Schatten norms for 1 ≤ p, q ≤ ∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is prov...

Script of a Talk on the Ky Fan and Schatten P Norms

Commutators on L p , 1 ≤ p < ∞ †

The operators on Lp = Lp[0, 1], 1 ≤ p < ∞, which are not commutators are those of the form λI + S where λ , 0 and S belongs to the largest ideal in L(Lp). The proof involves new structural results for operators on Lp which are of independent interest.