Table of contents

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W, the three statements are equivalent: (i) the set of product vectors obeying ⟨e, f|W|e, f⟩ = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = Q^Γ, with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in span after partial conjugation the whole space. On the other hand, already in the case of the Hilbert space, there exist DEWs for which (iii) does not imply (i). Consequently, either (i) does not imply (ii) or (ii) does not imply (iii), and the above transparent characterization, obeyed by qubit–qunit DEWs, does not hold in general.

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagomé, diced and decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.

A new approach to the proof of the Arrhenius formula of kinetic theory is proposed. We prove this formula by starting from the equation of diffusion in a potential. We put this diffusion equation in the form of evolutionary equation generated by some Schrödinger operator. We show that the Arrhenius formula for the rate of the over-barrier transitions follows from the formula for the rate of quantum tunnel transitions for the considered Schrödinger operator. The relation of the proposed approach and the Witten method of the proof of the Morse inequalities is discussed. In our approach, the Witten spectral asymptotics takes the form of the low-temperature limit and the Arrhenius formula is a correction to the Witten asymptotics.

When the independent variable is close to a critical point, it is shown that PVI can be asymptotically reduced to PIII. In this way, it is possible to compute the leading term of the critical behaviors of PVI transcendents starting from the behaviors of PIII transcendents.

From the three-dimensional space fractional Schrödinger equation, a generalized Lippmann–Schwinger equation for the fractional quantum mechanics is obtained for both scattering and bound states. We apply the generalized integral equation to study the fractional quantum scattering problem and give the approximate scattering wavefunction of first order and higher orders.

We prove a density result which allows us to justify the application of the method of fundamental solutions to solve the buckling eigenvalue problem of a plate. We address an example of an analytic convex domain for which the first eigenfunction does change the sign and present a large-scale numerical study with polygons providing numerical evidence to some new conjectures.

We obtain a general expression for a Wigner transform (Wigner function) on symmetric spaces of non-compact type and study the Weyl calculus of pseudodifferential operators on them.

The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t₀) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V₀, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V₀, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V₀ → 0, t − t₀ ∼ ℏ/2V₀ → ∞ (2V₀(t − t₀)/ℏ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V₀ including the Zeno case (V₀ → ∞) and the normalized TOA rate can be introduced for all values of V₀ as a probability distribution. The latter is illustrated for the Gaussian wave packet.

We study faithful teleportation systematically with arbitrary entangled states as resources. The necessary conditions of mixed states to complete perfect teleportation are proved. Based on these results, the necessary and sufficient conditions of faithful teleportation of an unknown state |ϕ⟩ in with an entangled resource ρ in and are derived. It is shown that for ρ in , ρ must be a maximally entangled state, while for ρ in , ρ must be a pure maximally entangled state. Moreover, we show that the sender's measurements must be all projectors of maximally entangled pure states. The relations between the entanglement of the formation of the resource states and faithful teleportation are also discussed.

Our preferences depend on the circumstances in which we reveal them. We will introduce a dependence which allows us to illustrate the relation between the possibility of winning of a particular candidate in an election and the type of preference. It is generally observed that if voters start to clearly prefer one of the candidates, the significance of intransitive preferences in the quantum model decreases. This dynamic change cannot be observed in the case of the classical model.

Generalized coherent states which are associated with the multiplicative group of non-zero complex numbers are introduced. They have the property of temporal stability, and they form a total set in some subspace of the full Hilbert space. They are highly non-classical states, in the sense that in general, their Bargmann functions have zeros which are related to regions where their Wigner functions take negative values. States with Bargmann functions sin (πAz) and exp (Bz)/[Γ(Az)]^k are examples of the general formalism, and they are discussed in detail.

In this paper, some indecomposable positive finite rank elementary operators of order (n, n) are constructed. This allows us to give a simple necessary and sufficient criterion for the separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order (2, 2) and obtain some new mixed entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion.

A problem of finding stationary states of open quantum systems is addressed. We focus our attention on a generic type of open system: a qubit coupled to its environment. We apply the theory of block operator matrices and find stationary states of two-level open quantum systems under certain conditions applied on both the qubit and the surrounding.

A method to construct a non-Dirac–Hermitian supersymmetric quantum system that is isospectral with a Dirac–Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving both bosonic and fermionic degrees of freedom in terms of non-Dirac–Hermitian operators which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. A pseudo-Hermitian realization of the Clifford algebra for a pre-determined positive-definite metric is used to construct supersymmetric systems with one or many degrees of freedom. It is shown that exactly solvable non-Dirac–Hermitian supersymmetric quantum systems can be constructed corresponding to each exactly solvable Dirac–Hermitian system. Examples of non-Dirac–Hermitian (i) non-relativistic Pauli Hamiltonian, (ii) super-conformal quantum system, and (iii) supersymmetric Calogero-type models admitting entirely real spectra are presented.