We study a variant of quantum hypothesis testing wherein an additional ‘inconclusive’ measurement outcome is added, allowing one to abstain from attempting discriminate the hypotheses. The error probabilities are then conditioned on successful attempt, with inconclusive trials disregarded. completely characterise this task in both single-shot and asymptotic regimes, providing exact formulas for...

First note that for any ρ ∈ (0, 1) and x ∈ R the function f (x) = x − 1 2 log ρ 2m 2 2x + 2πe(1 − ρ 2m) (1) is an monotonically increasing function with respect to x, because f ′ (x) = 2πe(1 − ρ 2m) ρ 2m 2 2x + 2πe(1 − ρ 2m) > 0. (2) By applying Shannon's EPI we have. h(s b |f a) ≥ 1 2 log ρ 2m 2 2h(sa|fa) + 2πe(1 − ρ 2m) (3) and thus, h(s a |f a) − h(s b |f a) ≤ h(s a |f a) − 1 2 log ρ 2m 2 2h...

We introduce a new class of nonlinear equations admitting a representation in terms of Darboux-covariant compatibility conditions. Their special cases are, in particular, (i) the “general” von Neumann equation iρ̇ = [H, f(ρ)], with [f(ρ), ρ] = 0, (ii) its generalization involving certain functions f(ρ) which are non-Abelian in the sense that [f(ρ), ρ] 6= 0, and (iii) the Nahm equations.

Let G be a complex reductive algebraic group and let Γ be a finitely generated group. In this paper we study properties of irreducible and completely reducible representations ρ : Γ → G in the context of the geometric invariant theory of the G-action on Hom(Γ, G) by conjugation. In particular, we prove that ρ is poly-stable (i.e. its orbit is closed) if and only if ρ is completely reducible. We...

be a continuous irreducible representation unramified away from finitely many primes. In [FM95], Fontaine and Mazur conjecture that if ρ is semi-stable at p, then either ρ is the Tate twist of an even representation with finite image or ρ is modular. In [Kis09], Kisin establishes this conjecture in almost all cases under the additional hypotheses that ρ|Dp has distinct Hodge–Tate weights and ρ ...

Preface In this thesis we give a new approach to the classical problems of finite and infinite packings and lattice packings of convex bodies. This approach is based on the introduction of parameterized densities δ ρ (K, C), ρ ∈ R >0. The parameterized density of a finite packing set C of a convex body K with respect to the parameter ρ is defined by δ ρ (K, C) = #(C)V (K) V (conv(C) + ρK) , whe...

In this Note, we prove three theorems. Throughout, F will denote a number field with absolute Galois group GF = Gal(F̄ /F ), and the adele ring AF = F∞ × AF,f . When ρ is an irreducible continuous 2– dimensional C representation of GF , one says that it is icosahedral, resp. octahedral, resp. tetrahedral, resp. dihedral when the projective image of ρ(GF ) is A5, resp. S4, resp. A4, resp. D2m for...

The main purpose of this note is to prove a convolution formula for conjugacy classes in symmetric groups suggested in [7] (formula (2.2), see also [8]). Given a partition ρ ⊢ r of a positive integer r, where r ≤ n, denote by ρ = ρ ∪ (1 n−r) the partition of n obtained from ρ by adding an appropriate number of unities. Let C ρ;n be the sum of permutations w ∈ S n of cycle type ρ. We also define...

A quantum system consisting of two subsystems is separable if its density matrix can be written as ρ = ∑ A wA ρ ′ A ⊗ ρ ′′ A, where ρ ′ A and ρ ′′ A are density matrices for the two subsytems. In this Letter, it is shown that a necessary condition for separability is that a matrix, obtained by partial transposition of ρ, has only non-negative eigenvalues. This criterion is stronger than the Bel...