Narrow Results

We consider finite-dimensional Hilbert spaces with with n ≥ 2 and unitary operators. In particular, we consider the case n = 2^m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU^* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.

In this paper, two novel schemes for bidirectional controlled remote state preparation (BCRSP) and asymmetric bidirectional controlled remote state preparation (ABCRSP) in four-dimensional systems are proposed. The first scheme uses a six-particle entangled state as the quantum channel. Under Charlie’s supervision, Alice and Bob can simultaneously create a single-particle four-dimensional equatorial state at each other’s location. Moreover, in the second scheme, using a seven-particle entangled state as the quantum channel, Alice remotely prepares a two-particle four-dimensional equatorial state for Bob, while Bob remotely prepares another single-particle four-dimensional equatorial state for Alice under the supervision of controller Charlie. Both schemes employ quantum operations such as control-not, projective measurement and unitary transformation in four-dimensional systems. It is worth noting that the proposed schemes have a total success probability of 100% in ideal environment. In addition, we also investigate the proposed two schemes in the noisy environment. The fidelities of output states are calculated and the effect factors are discussed. Finally, we give the advantages of these schemes compared to other schemes.

By using unitary transformation and representation theory of quantum mechanics, we obtain two forms of density matrix respectively in (x₁, p₂) and (x₁, x₂) representations for an electron, with an anisotropic effective mass, confined in quantum dots under a uniform magnetic field (UMF). We find that both forms of density matrix can play an implemental role in the thermodynamical calculation. But, in order to calculate thermal average of angular momentum in z direction for this system, the density matrix in (x₁, p₂) representations is a more convenient form. When the confinement to electron in quantum dots disappears, we encounter a divergence problem of thermal average for some physical quantities. However, in the calculation of the thermal average of Hamiltonian, these divergent quantities cancelled each other out. And we eventually obtain a reasonable result.

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.

An exact solution for single electron states on mesoscopic rings with the Rashba coupling and in the presence of external magnetic and electric fields is derived by means of a unitary transformation. The transformation maps the model to a bare ring, which gives the possibility of a very simple formulation of single or many electron problems. As an example some exact results for spin and energy levels are presented.

The influences of temperature and cyclotron frequency of a magnetic field on the ground state energy and mean number of phonons (MNP) of strong-coupling magnetopolarons in an asymmetric Gaussian potential quantum well(AGPQW) are researched by employing the linear-combination operator method and the unitary transformation. It was demonstrated through the numerical calculations that the ground state energy and the MNP increase with higher magnetic field cyclotron frequencies and temperature. In addition, increasing of the barrier of asymmetric Gaussian potential (AGP) causes the ground state energy to decrease while increasing the MNP of magnetopolarons.

Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.

Starting from Kirchhoff's equation for each mesoscopic coupled circuit, we investigated the quantum fluctuations of a dissipative mesoscopic capacitance–resistance–inductance coupled circuit using canonical transformation and unitary transformation. It is shown that the quantum fluctuations of charges and currents are not only related to the parameters of components in the self-circuit and coupled part, but also influenced by the parameters of other circuit. The uncertainty relations of the charges and currents are analyzed briefly.

Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.

The expectation value of the vacuum energy density tied to the cosmological constant is derived from the non-relativistic quantum description of the reheating process using the unitary transformation method. Our research is performed for two special cases, i.e. under the assumption of two different duration terms for the radiation-dominated era. We suppose that the radiation-dominated era lasted for 3,000 years in the first case, and for a very short time so that it can be ignored in the other case. For the former case, the resulting present cosmological constant is much higher than the previously known one. However, for the latter case, it is in good agreement with that obtained from recent cosmic microwave background radiation measurements. Hence, we cannot exclude the possibility that the change of the universe from the radiation-dominated era to the matter-dominated era might have progressed very swiftly after the beginning of reheating. The analysis shows that the vacuum energy density has been dissipated by the nonconservative force acting on the coherent oscillations of the scalar field according to the expansion of the universe.

In this paper, we present a scheme which can realize quantum cloning of an unknown N-particle entangled state and its orthogonal complement state with assistance offered by a state preparer. The first stage of the scheme requires usual teleportation via N non-maximally entangled particle pairs as quantum channel. In the second stage of the scheme, with the assistance (through a N-particle projective measurement) of the preparer, the perfect copies and complement copies of an unknown N-particle entangled state can be obtained.

In this paper, we propose a scheme for probabilistic remote preparation of a two-particle nonmaximally entangled state. In this scheme, two pairs of the two-particle partially entangled state are used as quantum channel. Alice performs the single-particle measurement three times and sends the information to the receiver. In accordance with the information, the receiver can construct the original state. By this method, the total probability of successful preparation is enhanced.

This paper is a summary of research on the relationship between quantum topology and quantum information theory.

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85-th birthday.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work. In particular, we give quantum algorithms for computing colored Jones polynomials and the Witten-Reshetikhin-Turaev invariant of three-manifolds.

The recent debate on hyper-computation has raised new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics. We propose here the idea of "geometry of effective physical process" as the essentially physical notion of computation. A key element to understand the shape of the geometry in Quantum mechanics is the Fisher metric whose meaning is connected with the stabilization of the Classical mechanics by Quantum mechanics. Different researchers define the stabilization as the manifestation of dissipative processes into the vacuum. We argue that no dissipation takes place, but a more general change occurs in Quantum mechanics. In Quantum mechanics we cannot use the traditional Euclidean geometry but we need to introduce more sophisticated non Euclidean geometry which includes a new kind of information diffused in the whole universe and that we can represent as Fisher information or active information. We remark that from the Fisher information we can obtain the Bohm and Hiley quantum potential and the classical Schrödinger equation. The bridge between Quantum mechanics and Classical mechanics is given by the Fisher metric in statistical geometry whose value is obtained through Quantum mechanics. We can see that the quantum phenomena do not affect a limited region of the space but are reflected in a change of the geometry of the whole universe. In conclusion any local physical change or physical process is reflected in the whole universe by the change of its geometry, This is the deepest meaning of the entanglement in Quantum mechanics and quantum computing. We stress the connection between metric and information as a measure of change. We analyze how the standard form (quantum gates) and the non-standard form of the quantum computing can be seen as a particular case of the metric of the parameter space in the distribution of the probabilities. Because computation is not restricted to calculus but produces the environment changing via physical processes, super-Turing potentialities derive from an incomputable information source embedded into the geometry of the universe in accordance with Bell's constraints. On condition that we consider the formal concept of "universality" as a particular case of the universal geometry of the probabilistic space with its transformations we open the possibility that quantum oracles can be reachable. In this way computation is led back to the hidden universal geometry of the physical world. In the general relativity we define the geometry of the space time. In our approach quantum phenomena define the geometry of the parameters of the probability distribution that include also the space time parameters. To study this new approach to the computation we use the new theory of Morphogenic systems.

Unitary integral transforms play an important role in mathematical physics. A primary example is the Fourier transform whose kernel is of the form k(x,y) = k(xy), i.e., of the product type. Here we consider the determination of spectrum for such unitary operators as the issue is important in the solvability of the corresponding inhomogeneous Fredholm integral equation of the second kind. A Main Theorem is proven that characterizes the spectral set. Properties of eigenfunctions and eigenspace dimensions are further derived as consequences of the Main Theorem. Concrete examples are also offered as applications.