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.

Classification of four qubit states and their stabilisers under SLOCC operations

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group SL(2,C)^4 on the Hilbert space H_4 = (C^2)^{\otimes 4}. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of SL(2,C)^4-orbits on H_4. It follows that an element of H_4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in SL(2,C)^4. We also present a complete and irredundant classification of elements and stabilisers up to the action of the semidirect product Sym_4\ltimes\SL(2,C)^4 where Sym_4 permutes the four tensor factors of H_4.

Stochastic emulation of quantum algorithms

Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of $\nb$ qubits by $2(\nb+1)$ classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. We implement several well-known quantum algorithms, analyse the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of the emulation.

On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

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.

Completing bases in four dimensions

Criteria for the completion of an incomplete basis of, or context in, a four-dimensional Hilbert space by (in)decomposable vectors are given. This, in particular, has consequences for the task of ``completing'' one or more bases or contexts of a (hyper)graph: find a complete faithful orthogonal representation (aka coordinatization) of a hypergraph when only a coordinatization of the intertwining observables is known. In general indecomposability and thus physical entanglement and the encoding of relational properties by quantum states ``prevails'' and occurs more often than separability associated with well defined individual, separable states.

Unraveling the origin of higher success probabilities in quantum annealing versus semi-classical annealing

Quantum annealing aims at finding optimal solutions to complex optimization problems using a suitable quantum many body Hamiltonian encoding the solution in its ground state. To find the solution one typically evolves the ground state of a soluble, simple initial Hamiltonian adiabatically to the ground state of the designated final Hamiltonian. Here we explore whether and when a full quantum representation of the dynamics leads to higher probability to end up in the desired ground when compared to a classical mean field approximation. As simple but nontrivial example we target the ground state of interacting bosons trapped in a tight binding lattice with small local defect by turning on long range interactions. Already two atoms in four sites interacting via two cavity modes prove complex enough to exhibit significant differences between the full quantum model and a mean field approximation for the cavity fields mediating the interactions. We find a large parameter region of highly successful quantum annealing, where the semi-classical approach largely fails. Here we see strong evidence for the importance of entanglement to end close to the optimal solution. The quantum model also reduces the minimal time for a high target occupation probability. Surprisingly, in contrast to naive expectations that enlarging the Hilbert space is beneficial, different numerical cut-offs of the Hilbert space reveal an improved performance for lower cut-offs, i.e. an nonphysical reduced Hilbert space, for short simulation times. Hence a less faithful representation of the full quantum dynamics sometimes creates a higher numerical success probability in even shorter time. However, a sufficiently high cut-off proves relevant to obtain near perfect fidelity for long simulations times in a single run. Overall our results exhibit a clear improvement to find the optimal solution based on a quantum model versus simulations based on a classical field approximation.