Narrow Results

Several mathematical frameworks of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find that they are related to the Riemann zeta function and to incomplete Gauss sums, respectively.

We develop a comprehensive approach of stabilizer codes and provide a scheme generating equientangled basis interpolating between the product basis and maximally entangled basis. The key ingredient is the theory of phase states for finite-dimensional systems (qudits). In this respect, we derive entangled phase states for a multiqudit system whose dynamics is governed by a two-qudit interaction Hamiltonian. We construct the stabilizer codes for this family of entangled phase states. The stabilizer phase states are defined as the common eigenvectors of the stabilizer group generators which are explicitly specified. Furthermore, we construct equally entangled bases from bipartite as well as multipartite entangled qudit phase states.

The description of qudits in a formalism based on a generalized variant of Weyl–Heisenberg algebras is discussed. The unitary phase operators for a multi-qudit system and the corresponding phase states (the eigenstates of the phase operator) are constructed. We discuss the dynamics of multi-qudit phase states governed by a generalized Hamiltonian involving one- and two-body interactions which offer a remarkable connection between phase states, generalized graph states and the mutually unbiased bases. The entangled phase states are shown to possess the following properties simultaneously, namely the mutually unbiasedness of phase states resulting from the one-body generalized oscillator Hamiltonian and the entanglement properties of generalized graph states resulting from the two-body interaction Hamiltonian.

We derive absolutely maximally entangled (AME) states from phase states for a multi-qudit system whose dynamics is governed by a two-qudit interaction Hamiltonian of Heisenberg type. AME states are characterized by being maximally entangled for all bipartitions of the multi-qudit system and present absolute multipartite entanglement. The key ingredient of this approach is the theory of phase states for finite-dimensional systems (qudits). We define further the unitary phase operators of $N$-qudit systems and we give next the corresponding separable phase states. Using a qudit–qudit Hamiltonian acting as entangling operator on separable phase states, we generate entangled phase states. Finally, from the labeled entangled phase states, we derive the absolutely maximally entangled states.

The main aim of this work is to build unitary phase operators and the corresponding temporally stable phase states for the $su(n+1)$ Lie algebra. We first introduce an irreducible finite-dimensional Hilbertian representation of the $su(n+1)$ Lie algebra which is suitable for our purpose. The phase operators obtained from the $su(n+1)$ generators are defined and the phase states are derived as eigenstates associated to these unitary phase operators. The special cases of $su(3)$ and $su(2)$ Lie algebras are also explicitly investigated.

This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.