Finite Heisenberg group and its application in quantum computing Introduction Wigner Function Entanglement properties in two qubit systems We first introduce The Heisenberg Group : 1-Continuous Heisenberg Group: 2-Discrete Heisenberg Group: For a particle in a 1 dimension the Wigner function, has the following properties: (q,p are position and momentum) 1- is real valued. 2- The inner product between states , is 3- The integral of WF over the infinite strip of phase space between any two parallel lines( , ) equal to the probability that the operator will be found to take a value between c , d . is the expectation value of an operator , known as ‘’phase point operator”: , The operator , that parametrically depends on , is: In the case of 2-qubit the WF becomes Given a 2-qubit density matrix , such a state is said to be separable if there exists a decomposition , . A non separable state is said to be entangled. The WF associated to a separable state ( ) is The transposition action on a single-qubit WF and on its characteristic function gives In the two qubit case , , It is known that the operator relevant to a separable-state density operator possesses non-negative eigenvalues. If has all nonnegative eigenvalues, then , for all density matrices , thus giving which is a necessary condition for separability. Finite Heisenberg group and its application in quantum computing and Information M. A. Jaafarizadeh L. Shaabani Laaya_shaabani@yahoo.com 3-Finite Heisenberg Group: is the cyclic group of order Abstract The Heisenberg group has been instrumental in the development of a wide range of mathematical topics. It is well known that the Heisenberg group appears in various areas such as Quantum Theory, Signal Theory, Theory of theta functions, Quantum Tomography, Quantum Graph, Discrete Wigner Function, and Number Theory. For each , is a nilpotent group of class 2 and of order ,with the ordinary product of matrices: The center is given by The generators of are: Given a state , if both and have non-negative elements, than is separable. Viceversa, if a state is entangled, then or has negative values. This is a necessary condition for entanglement and sufficient condition for separability. To illustrate this result, we considered the and Bell decomposable states. we showed that the entanglement condition which we get, is corresponding with the condition that we earn by the positive partial transposition. The entanglement properties within the Wigner Function Based on the Partial Transposition criterion, we establish the relation between the separability of a density matrix and the non-negativity of the WF ’s relevant both to such a density matrix and to the partially transposed thereof. The Finite Heisenberg group is the semi- direct product of the Abelian normal subgroup With the subgroup is isomorphic to a semi-direct product of the form The appropriate homomorphism is given by The set forms an orthogonal basis for the set of hermitian operators acting on a single qubit. Thus any density matrix for a single qubit can be written as While the characteristic function for a single qubit is where In the case of 2 qubit Function is connected with the discrete WF by a discrete Fourier transform. In the single qubit case, , and in 2 qubit case