Quantum entanglement of unitary operators on bi-partite systems

Quantum entanglement plays a key role in quantum information theory. In recent years, there has been a lot of efforts to characterize the entanglement of quantum states both qualitatively and quantitatively. Entangled states can be generated from disentangled states by the action of a non-local Hamitonians. That means these Hamiltonians have the ability to entangle quantum states. It is therefore natural to investigate the entangling abilities of non-local Hamiltonians and the corresponding unitary evolution operators. The first steps along this direction have been performed [1,2] recently. In Ref. [1] it has been analyzed the entangling capabilities of unitary operators on a d1 × d2 systems and introduced an entangling power measure given by the mean linear entropy produced by acting with the unitary operator on a given distribution of product states. Dür et al. [2] investigated the entanglement capability of an arbitrary two-qubit non-local Hamiltonian and designed an optimal strategy for entanglement production. Cirac et al. [3] studied which physical operations acting on two spatially separated systems are capable of producing entanglement and shows how one can implement certain nonlocal operations if one shares a small amount of entanglement and is allowed to perform local operations and classical communications. The notion of entanglement of quantum evolutions e.g., unitary operators, has been introduced in Ref. [4] and there quantified by linear entropy [4]. As discussed in that paper, this notion arises in a very natural way once one recalls that unitary operators of a multipartite system belong to a multipartite state space as well, the socalled Hilbert-Schmidit space. It follow that one can lift all the notions developed for entanglement of quantum states to that of quantum evolutions. In this report we shall make a further step by studying the entanglement of a class of useful unitary operators e.g., quantum gates, on general i.e., d1 × d2 bipartite quantum systems.