1. 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)
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
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