We present a derivation and numerous applications of a compact explicit formula for the average fidelity of a quantum operation on a finite dimensional quantum system. The formula can be applied to averages over particularly relevant subspaces; it is easily generalized to multi-component systems, and as a special result, we show that when the same completely positive trace-preserving map is applied to a large number of qubits with one-bit fidelity F close to unity, the average fidelity of the operation on the full K-bit register scales as $F^{3K/2}$.Comment: 5 pages, no figures. The text has been modified to acknowledge that our Eq.(1) has appeared already in quant-ph/0503243 and quant-ph/051221

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σ n ⊆ R n+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in R n+1 , and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE.We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution Σ n is either a hyperplane R n , the round cylinder R × S n−1 of radius 2(n − 1), the round sphere S n of radius √ 2n, or is diffeomorphic to an S 1 × S n−1 (i.e. a "doughnut" as in the paper by Sigurd B. Angenent, 1992, which when n = 2 is a torus). In particular, for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.

Abstract. We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has 4g + 4 symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry.Each is at infinity asymptotic to the cone in R 3 over a 2π/(g + 1)-periodic graph on an equator of the unit sphere S 2 ⊆ R 3 , with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions.The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

When applied to different input states, an imperfect quantum operation yields output states with varying fidelities, defined as the absolute square of their overlap with the desired states. We present an expression for the distribution of fidelities for a class of operations applied to a general qubit state, and we present general expressions for the variance and input-space averaged fidelities of arbitrary linear maps on finite dimensional Hilbert spaces.Comment: 5 pages, 1 figur

While it is well known from examples that no interesting "halfspace theorem" holds for properly immersed n-dimensional selftranslating mean curvature flow solitons in Euclidean space R n+1 , we show that they must all obey a general "bi-halfspace theorem" (aka "wedge theorem"): Two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori-Yau maximum principle.As an application we classify the convex hulls of all properly immersed (possibly with compact boundary) n-dimensional mean curvature flow self-translating solitons Σ n in R n+1 , up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman-Meeks in 1989, for minimal submanifolds: All of R n , halfspaces, slabs, hyperplanes and convex compacts in R n .