We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to arbitrary (in particular, possibly non-free) orientation-preserving any compact 3-manifold $N$ with boundary $\partial N = T$. This implies negative solution conjecture Dom\'inguez Segovia, as well Uribe's evenness for equivariant unitary bordism gr...

Two distinct projections of finite rank m are adjacent if their difference is an operator two or, equivalently, the intersection images (m−1)-dimensional. We extend this adjacency relation on other conjugacy classes finite-rank self-adjoint operators which leads to a natural generalization Grassmann graphs. Let C be class formed by with eigenspaces dimension greater than 1. Under assumption tha...

One of the central problems in representation theory is understanding irreducible unitary representations. The reason is that in many applications of linear algebra (like those of representation theory to harmonic analysis) the notion of length of vectors is fundamentally important. Unitary representations are exactly those preserving a good notion of length. The paper [4] provides an algorithm...

We provide a general and consistent formulation for linear subsystem quantum dynamical maps, developed from a minimal set of postulates, primary among which is a relaxation of the usual, restrictive assumption of uncorrelated initial system-bath states. We describe the space of possibilities admitted by this formulation, namely that, far from being limited to only completely positive (CP) maps,...

Let A ⊆ Mn(C) be a unital ∗-subalgebra of the algebra Mn(C) of all n × n complex matrices and let B be an hermitian matrix. Let Un(B) denote the unitary orbit of B in Mn(C) and let EA denote the trace preserving conditional expectation onto A. We give an spectral characterization of the set EA(Un(B)) = {EA(U B U) : U ∈ Mn(C), unitary matrix}. We obtain a similar result for the contractive orbit...

We derive a family of high-order, structure-preserving approximations of the Riemannian exponential map on several matrix manifolds, including the group of unitary matrices, the Grassmannian manifold, and the Stiefel manifold. Our derivation is inspired by the observation that if Ω is a skew-Hermitian matrix and t is a sufficiently small scalar, then there exists a polynomial of degree n in tΩ ...

Misra, Prigogine and Courbage (MPC) demonstrated the possibility of obtaining stochastic Markov processes from deterministic dynamics simply through a ”change of representation” which involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. From a mathematical point of view, MPC theory is a theory of positivity preser...

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. We describe how randomized benchmarking can be used to reconstruct the unital part of any trace-preserving quantum ...