We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert-Schmidt scalar product, and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed...

Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finitedimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave “like” the category of finite-dimensional Hilbert spaces in various ways...

A basic overview of quantum theory fundamentals (pure states, density matrix, observables) is presented, then density matrices in finite dimensional Hilbert spaces are examined in more detail. Then the problem of optimal quantum state determination is dealt with. This leads to definition of mutually unbiased bases. An explicit form of such bases for prime dimensions is given. An alternative pro...

Each matrix representation P: G —> GLn(K) of a finite Group G over a field K induces an action of G on the module A over the polynomial algebra A = K [ x 1 , . . . , xn]. The graded A-submodule M(P) of A generated by the orbit of (x1, ..., xn) is studied. A decomposition of MO) into generic modules is given. Relations between the numerical invariants of P and those of M(P), the latter being eff...

The goal of the present paper is a short introduction to a general module frame theory in C*-algebras and Hilbert C*modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We...

Unstable particles, together with their stable decay products, constitute probability collectives which are defined as Hilbert spaces with dimension higher than one, nondecomposable in a particle basis. Their structure is considered in the framework of Birkhoff-von Neumann’s Hilbert subspace lattices. Bases with particle states are related to bases with a diagonal scalar product by a Hilbert-be...

‎Frames generalize orthonormal bases and allow representation of all the elements of the space‎. ‎Frames play significant role in signal and image processing‎, ‎which leads to many applications in informatics‎, ‎engineering‎, ‎medicine‎, ‎and probability‎. ‎In this paper‎, ‎we introduce the concepts of operator frame for the space $End_{mathcal{A}}^{ast}(mathcal{H})$ of all adjointable operator...

Hilbert schemes are a basic topic in Algebraic Geometry [1, 2]. Methods related to Gröbner bases are fundamental here, as Hartshorne’s proof of the connectedness of Hilbert schemes via generic initial ideals demonstrates [3]. For many purposes, the explicit construction of Hilbert schemes is important. Classical approaches lead to prohibitively large equations. Newer ideas like Gröbner strata [...

A complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p, where p is a prime number. They mesh naturally with finite affine planes of order N , that exist when N = p. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is dir...