Uncertainty Relations for the Entanglement Between a Qubit and a Qutrit

Abstract

We consider a qubit-qutrit system interacting each other through an isotropic Heisenberg Hamiltonian in a uniform magnetic field. An expression for the entanglement between the two parties as a function of both the anisotropy factor and the expansion coefficients is found. It is also calculated the uncertainty in the concurrence C (measured as the standard deviation) as a function of the anisotropy factor. We make a strictly mathematical assumption that there exists a canonical conjugate variable (called π) to the concurrence C. Furthermore, we assume that such two variables satisfy (Robertson, Phys. Rev. 34, 163 1929) uncertainty relation of the form (A)2(B)2 > | 12 ψ|[A, B]|ψ|2. From such an inequality we impose bounds for the uncertainty of π for several values of the anisotropy factor. The above may help to open a novel vision about the properties of the concurrence between a qubit and a qutrit.