Uniform probability distribution over all density matrices

Let $$\mathscr {H}$$ H be a finite-dimensional complex Hilbert space and $$\mathscr {D}$$ D the set of density matrices on $$\mathscr {H}$$ H , i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on $$\mathscr {D}$$ D that can be regarded as the uniform distribution over $$\mathscr {D}$$ D . We propose a measure on $$\mathscr {D}$$ D , argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.

On Numerical Radius Bounds Involving Generalized Aluthge Transform

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator S defined on a complex Hilbert space with polar decomposition S = U ∣ S ∣ , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

Quantum theory based on real numbers can be experimentally falsified

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

Linear maps on ℬ(ℋ) preserving some operator properties

In this paper, for a complex Hilbert space ℋ with dim ℋ ≥ 2, we study the linear maps on ℬ(ℋ), the bounded linear operators on ℋ, that preserves projections and idempotents. As a result, we characterize the linear maps on ℬ(ℋ) that preserves involutions in both directions.

The homeomorphism of Minkowski space and the separable complex Hilbert space: the physical, mathematical and philosophical interpretations

A homeomorphism is built between the separable complex Hilbert space and Minkowski space by meditation of quantum information. That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering. Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy.

Regularization of the Factorization Method applied to diffuse optical tomography

In this paper, we develop a new regularized version of the Factorization Method for positive operators mapping a complex Hilbert Space into it’s dual space. The Factorization Method uses Picard’s Criteria to define an indicator function to image an unknown region. In most applications the data operator is compact which gives that the singular values can tend to zero rapidly which can cause numerical instabilities. The regularization of the Factorization Method presented here seeks to avoid the numerical instabilities in applying Picard’s Criteria. This method allows one to image the interior structure of an object with little a priori information in a computationally simple and analytically rigorous way. Here we will focus on an application of this method to diffuse optical tomography where will prove that this method can be used to recover an unknown subregion from the Dirichlet-to-Neumann mapping. Numerical examples will be presented in two dimensions.

Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the followingmonotonic integral transformwhere the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)2 ≤ Δ for some constants α, β, δ, Δ, thenandwhere is the second derivative of as a real function.Applications for power function and logarithm are also provided.

On Relation between the Joint Essential Spectrum and the Joint Essential Numerical Range of Aluthge Transform

Associated with every commuting m-tuples of operators on a complex Hilbert space X is its Aluthge transform. In this paper we show that every commuting m-tuples of operators on a complex Hilbert space X and its Aluthge transform have the same joint essential spectrum. Further, it is shown that the joint essential spectrum of Aluthge transform is contained in the joint essential numerical range of Aluthge transform.

On class (BD) Operators

In this paper, we introduce the class of (BD) operators acting on a complex Hilbert space H. An operator if T ∈ B (H) is said to belong to class (BD) if T * 2 (TD) 2 commutes with (T *TD) 2 equivalently [T * 2 (TD) 2, (T *TD) 2] = 0. We investigate the properties of this class and we also analyze the relation of this class to D-operator and then generalize it to class (nBD) and analyze its relation to the class of n-power D-operator through complex symmetric operators.