Description:
Description: We derive an asymptotic bound for the error of state estimation when we are allowed to use the quantum correlation in the measuring apparatus. It is also proven that this bound can be achieved in any statistical model in the qubit system. Moreover, we show that this bound cannot be attained by any quantum measurement with no quantum correlation in the measuring apparatus except for several specific statistical models. That is, in such a statistical model, th...
Description: We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein–Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional dif...
Description: The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z = 0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that i...
Description: Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a compl...
Description: We study the structure of the spectrum of a two-level quantum system weakly coupled to a boson field (spin-boson model). Our analysis allows to avoid the cutoff in the number of bosons, if their spectrum is bounded below by a positive constant. We show that, for small coupling constant, the lower part of the spectrum of the spin-boson Hamiltonian contains (one or two) isolated eigenvalues and (one or two) manifolds of atom +1-boson states indexed by the boso...
Description: Consider a three dimensional system which looks like a cross connected pipe system, i.e., a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We...
Description: A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output p-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to ...
Description: A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in H that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interfe...
Description: We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.
Description: We consider the three-dimensional semirelativistic Hartree model for fast quantum mechanical particles moving in a self-consistent field. Under appropriate assumptions on the initial density matrix as a (fully) mixed quantum state we prove by using Wigner transformation techniques that its classical limit yields the well known relativistic Vlasov–Poisson system. The result holds for the case of attractive and repulsive mean-field interactions, with an additi...
Description: Let n ≥ 2 be an integer. To each irreducible representation σ of O(1), an O(1)-Kepler problem in dimension n is constructed and analyzed. This system is super integrable and when n = 2 it is equivalent to a generalized MICZ (McIntosh-Cisneros-Zwanziger)-Kepler problem in dimension 2. The dynamical symmetry group of this system is (2n,) with the Hilbert space of bound states H(σ) being the unitary highest weight representation of (2n,) with highest weight wh...
Description: The quantum analogs of the derivatives with respect to coordinates qk and momenta pk are commutators with operators Pk and Qk. We consider quantum analogs of fractional Riemann–Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann–Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for analytic functions. To define a quantum analog of the fractional Li...
Description: We use the stochastic quantization method to study systems with complex valued path integral weights. We assume a Langevin equation with a memory kernel and Einstein’s relations with colored noise. The equilibrium solution of this non-Markovian Langevin equation is analyzed. We show that for a large class of elliptic non-Hermitian operators acting on scalar functions on Euclidean space, which define different models in quantum field theory, converge to an eq...
Description: We discuss (extended) super-Schrödinger algebras obtained as subalgebras of the superconformal algebra psu(2,2∣4). The Schrödinger algebra with two spatial dimensions can be embedded into so(4,2). In the superconformal case the embedded algebra may be enhanced to the so-called super-Schrödinger algebra. In fact, we find an extended super-Schrödinger subalgebra of psu(2,2∣4). It contains 24 supercharges (i.e., 3/4 of the original supersymmetries) and the gene...
Description: We revisit Weyl geometry in the context of recent higher-dimensional theories of space-time. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian space-time may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical con...
Description: We consider the system = a(z−a1x3−a2x2−bx), = −z, = −b1x+y+b2z, where a and b are parameters and b1 = 7/10, b2 = 6/25, a1 = 44/3, and a2 = 41/2. We analyze the existence of local and global analytic first integrals.
Description: We study a differential-difference equation of the form tx(n+1) = f(t(n),t(n+1),tx(n)) with unknown t = t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF = 0 and DI = I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp...
Description: The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining approximate solutions. This article shows that the RG method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold.
Description: The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented.