In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are [Formula: see text] matrices, then [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are non-negative continuous functions such that [Formula: see text] and [Formula: see...

In this appendix, we prove Theorem 4 and Lemmas 7 – 12 in order. A.1. Proof of Theorem 4. We first need a lemma for perturbation bound of square root matrices. Lemma 16. Let A, B be positive semi-definite matrices, and then for any unitarily invariant norm ï¿¿·ï¿¿, ï¿¿A 1/2 − B 1/2 ï¿¿ ≤ 1 σ min (A 1/2) + σ min (B 1/2) ï¿¿A − Bï¿¿. Proof. The proof essentially follows the idea of [27]. Let D = ...

We give a lower bound for the error of any unitarily invariant algorithm learning half-spaces against the uniform or related distributions on the unit sphere. The bound is uniform in the choice of the target half-space and has an exponentially decaying deviation probability in the sample. The technique of proof is related to a proof of the Johnson Lindenstrauss Lemma. We argue that, unlike prev...

We first show that every γ-contractive commuting multioperator is unitarily equivalent to the restriction of S⊕W to an invariant subspace, where S is a backwards multi-shift and W a γ-isometry. We then describe γ-isometries in terms of (γ, 1)-isometries, and establish that under an additional assumption on T , W above can be chosen to be a commuting multioperator of isometries. Our methods prov...

We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This result is then used to analyze the average error correcting performance of codes that are randomly drawn from unitarily invariant code ensembles. Our results co...

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds ...

Let M be the flat Minkowski space. The solutions of the ware equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H,5 , such that the representation U,$ will act u...