An efficient numerical method for calculating the entanglement of formation of arbitrary mixed quantum states of any dimension

Abstract

We present a conjugate gradient method for calculating the entanglement of formation of arbitrary mixed quantum states of any dimension and with any bipartite division of the Hilbert space. The development of the gradient used by the algorithm, its implications for the number of states required in the optimal decomposition, and the way that conjugate gradient minimization has been adapted for this particular problem are outlined. We have found that the algorithm exhibits linear convergence for general mixed states, and that it correctly reproduces the known results for pairs of qubits and for isotropic states. The results of an example application of the code are discussed: calculating the entanglement of formation of a Psi+ Bell state of two qutrits when one of those qutrits is subject to various decoherence channels. The results for qutrits are contrasted with those for qubits: for the types of decoherence considered here, qutrit entanglement appears to be more robust than qubit entanglement.