A Novel LSB Based Quantum Watermarking

Quantum watermarking is a technique which embeds the invisible quantum signal such as the owners identification into quantum multimedia data (such as audio, video and image) for copyright protection. In this paper, using a quantum representation of digital images a new quantum watermarking protocol including quantum image scrambling based on Least Significant Bit (LSB) is proposed. In this protocol, by using m-bit embedding key K₁ and m-bit extracting key K₂ a m-pixel gray scale image is watermarked in a m-pixel carrier image by the original owner of the carrier image. For validation of the presented scheme the peak-signal-to-noise ratio (PSNR) and histogram graphs of the images are analyzed.     

Quantum Image Encryption Based on Iterative Framework of Frequency-Spatial Domain Transforms

A novel quantum image encryption and decryption algorithm based on iteration framework of frequency-spatial domain transforms is proposed. In this paper, the images are represented in the flexible representation for quantum images (FRQI). Previous quantum image encryption algorithms are realized by spatial domain transform to scramble the position information of original images and frequency domain transform to encode the color information of images. But there are some problems such as the periodicity of spatial domain transform, which will make it easy to recover the original images. Hence, we present the iterative framework of frequency-spatial domain transforms. Based on the iterative framework, the novel encryption algorithm uses Fibonacci transform and geometric transform for many times to scramble the position information of the original images and double random-phase encoding to encode the color information of the images. The encryption keys include the iterative time t of the Fibonacci transform, the iterative time l of the geometric transform, the geometric transform matrix G i which is n × n matrix, the classical binary sequences K (\(k_{0}k_{1}{\ldots } k_{2^{2n}-1}\)) and \(D(d_{0}d_{1}{\ldots } d_{2^{2n}-1}\)). Here the key space of Fibonacci transform and geometric transform are both estimated to be 2²⁶. The key space of binary sequences is (2 ^n×n ) × (2 ^n×n ). Then the key space of the entire algorithm is about \(2^{2{n^{2}}+52}\). Since all quantum operations are invertible, the quantum image decryption algorithm is the inverse of the encryption algorithm. The results of numerical simulation and analysis indicate that the proposed algorithm has high security and high sensitivity.     

Action Quantization,Energy Quantization,and Time Parametrization

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \(\psi \) that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem generates a microstate-dependent time parametrization \(t-\tau =\partial _E W\) even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.     

Quantum chaos in nuclear physics

A definition of classical and quantum chaos on the basis of the Liouville–Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated.     

Optimal Synthesis of the Joint Unitary Evolutions

Joint unitary operations play a central role in quantum communication and computation. We give a quantum circuit for implementing a type of unconstructed useful joint unitary evolutions in terms of controlled-NOT (CNOT) gates and single-qubit rotations. Our synthesis is optimal and possible in experiment. Two CNOT gates and seven R _x , R _y or R _z rotations are required for our synthesis, and the arbitrary parameter contained in the evolutions can be controlled by local Hamiltonian or external fields.     

Renormalons and analytic properties of the β function

The presence or absence of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann-Low function β(g) and some other functions. A constructive criterion for the absence of singularities consists in the proper behavior of the β function and its Borel image at infinity, β(g) ∝ g^α and B(z) ∝ z^α with α ≤ 1. This criterion is probably fulfilled for the ?⁴ theory, quantum electrodynamics, and quantum chromodynamics, but is violated in the O(n)-symmetric sigma model with n → ∞.     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

Control of Quantum Echo and Bell State Swapping of Two Atomic Qubits in the Two-Mode Vacuum Field Environment

We investigate quantum echo control and Bell state swapping for two atomic qubits (TAQs) coupling to two-mode vacuum cavity field (TMVCF) environment via two-photon resonance. We discuss the effect of initial entanglement factor ?? and relative coupling strength R=g₁/g₂ on quantum state fidelity of TAQs, and analyze the relation between three kinds of quantum entanglement(C(ρ_a),C(ρ_f),S(ρ_a)) and quantum state fidelity, then reveal physical essence of quantum echo of TAQs. It is shown that in the identical coupling case R=1, periodic quantum echo of TAQs with π cycle is always produced, and the value of fidelity can be controlled by choosing appropriate ?? and atom-filed interaction time. In the non-identical coupling case R≠1, quantum echoes with periods of π, 2π and 4π can be formed respectively by adjusting R. The characteristics of quantum echo results from the non-Markovianity of TMVCF environment, and then we propose Bell state swapping scheme between TAQs and two-mode cavity field.     

Complex Wavelet Transform of the Two-mode Quantum States

By employing the bipartite entangled state representation and the technique of integration within an ordered product of operators, the classical complex wavelet transform of a complex signal function can be recast to a matrix element of the squeezing-displacing operator U ₂(μ, σ) between the mother wavelet vector 〈ψ| and the two-mode quantum state vector |f〉 to be transformed. 〈ψ|U ₂(μ, σ)|f〉 can be considered as the spectrum for analyzing the two-mode quantum state |f〉. In this way, for some typical two-mode quantum states, such as two-mode coherent state and two-mode Fock state, we derive the complex wavelet transform spectrum and carry out the numerical calculation. This kind of wavelet-transform spectrum can be used to recognize quantum states.     

Quantum to classical transition in the ground state of a spin-S quantum antiferromagnet

We study a frustrated spin-S staggered-dimer Heisenberg model on square lattice by using the bond-operator representation for quantum spins, and investigate the emergence of classical magnetic order from the quantum mechanical (staggered-dimer singlet) ground state for increasing S. Using triplon analysis, we find the critical couplings for this quantum phase transition to scale as 1 /S(S + 1). We extend the triplon analysis to include the effect of quintet dimer-states, which proves to be essential for establishing the classical order (Néel or collinear in the present study) for large S, both in the purely Heisenberg case and also in the model with single-ion anisotropy.     

Higgs Bundles,Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.     

Generic Representation of Y(so(3)) Based on the Lie Algebraic Basis of so(3)

We focus on constructing a generic representation of Y(so(3)) based on the Lie algebraic basis of so(3) basis, and further developing transition of Yangian operator \(\hat {\mathbf {Y}}\). As an application of Y(so(3)), we calculate all the matrix elements of unit vector operators \(\hat {\mathbf {n}}\) in angular momentum basis. It is also discovered that the Yangian operator \(\hat {\mathbf {Y}}\) may work in quantum vector space. In addition, some shift operators \(\hat {O}^{(\pm )}_{\mu }\) are naturally built on the basis of the representation of Y(so(3)). As an another application of Y(so(3)), we can derive the CG cofficients of two coupled angular momenta from the down-shift operator \(\hat {O}^{(-)}_{-1}\) acting on a so(3)-coupled tensor basis. This not only explores that Yangian algebras can work in quantum tensor space, but also provides a novel approach to solve CG coefficients for two coupled angular momenta.     

New Method of Calculating a Multiplication by using the Generalized Bernstein-Vazirani Algorithm

We present a new method of more speedily calculating a multiplication by using the generalized Bernstein-Vazirani algorithm and many parallel quantum systems. Given the set of real values \(\{a_{1},a_{2},a_{3},\ldots ,a_{N}\}\) and a function \(g:\textbf {R}\rightarrow \{0,1\}\), we shall determine the following values \(\{g(a_{1}),g(a_{2}),g(a_{3}),\ldots , g(a_{N})\}\) simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of \(N\). Next, we consider it as a number in binary representation; M₁ = (g(a₁),g(a₂),g(a₃),…,g(a _N )). By using \(M\) parallel quantum systems, we have \(M\) numbers in binary representation, simultaneously. The speed of obtaining the \(M\) numbers is shown to outperform the classical case by a factor of \(M\). Finally, we calculate the product; \( M_{1}\times M_{2}\times \cdots \times M_{M}. \) The speed of obtaining the product is shown to outperform the classical case by a factor of N × M.     

Conformal Geometry of the Supercotangent and Spinor Bundles

We study the actions of local conformal vector fields \({X \in {\rm conf}(M,g)}\) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle \({\mathcal{M}}\) of (M, g). We first deal with the classical framework and determine the Hamiltonian lift of conf (M, g) to \({\mathcal{M}}\) . We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.The quantum and classical actions of conf (M, g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf (M, g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf (M, g)-modules, in particular the conformally odd powers of the Dirac operator.     

Bit-Oriented Quantum Public-Key Cryptosystem Based on Bell States

Quantum public key encryption system provides information confidentiality using quantum mechanics. This paper presents a quantum public key cryptosystem (QPKC) based on the Bell states. By Holevo’s theorem, the presented scheme provides the security of the secret key using one-wayness during the QPKC. While the QPKC scheme is information theoretic security under chosen plaintext attack (CPA). Finally some important features of presented QPKC scheme can be compared with other QPKC scheme.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

Quantenelektrodynamik bei nichtverschwindender Photonmasse

Quantum electrodynamics with non-vanishing photon mass is written down in interaction representation. To apply the Wick decomposition formalism of theS-matrix one can introduce an indefinite metricη, similar to that of Gupta-Bleuler's quantum electrodynamics with vanishing photon mass. It will be shown that the complementary photons can be eliminated from the formalism with the help of the subsidiary condition. By a succeeding unitary transformation allx-singularities (x=photon mass) can be removed. The limiting processx→0, which then becomes possible, leads to the well-known so-called ‘reduced’ theory of quantum electrodynamics. A physical interpretation of this limiting process will be tried using, as a simple example, the radiation of an electric dipole.     

A q-boson representation of Zamolodchikov-Faddeev algebra for stochastic R matrix of $$\varvec{U_ q(A^{(1)}_n)}$$U q(An(1))

We construct a q-boson representation of the Zamolodchikov-Faddeev algebra whose structure function is given by the stochastic R matrix of \(U_q(A^{(1)}_n)\) introduced recently. The representation involves quantum dilogarithm type infinite products in the \(n(n-1)/2\)-fold tensor product of q-bosons. It leads to a matrix product formula of the stationary probabilities in the \(U_q(A_n^{(1)})\)-zero range process on a one-dimensional periodic lattice.