NONLOCAL CHARACTERIZATION OF TWO-QUBIT GATES
Entanglement is a nonlocal quantum correlation between different parts of a composite system having no classical analog. Such correlation has been recognized as useful resource in quantum teleportation, dense coding and in various information processing techniques. Hence, the investigations on characterizing entangled states of qubits enjoy with much attention. On the other hand, it is also equally important to study quantum operators (gates) as they are capable of generating entangled states. Finding the set of universal gates is one of the requirements for the realization of practical quantum computer. Since an arbitrary n-qubit gate can be decomposed in terms of two-qubit and single qubit gates, investigations on entangling (nonlocal) attributes of quantum operators is reduced to the level of two-qubit gates. Few characterizing tools are available to understand the nonlocal attributes of two-qubit gates.
One important tool is a pair of local invariants which is useful to check the equivalence of quantum circuit. A gate is called as perfect entangler if it produces maximal entangled state when acting on some input product state. Entangling power of a gate is the average entanglement produced when acting on all input product states. One can also characterize the two-qubit gates by extending the notion of entanglement in state space to that in operator space. In this approach, entanglement of an operator may be quantified using two measures, namely Schmidt strength and linear entropy. Nonlocal characteristics of an arbitrary two-qubit gate can be well appreciated through the canonical form whose geometry takes the shape of tetrahedron (Weyl Chamber). In this representation, a point in the Weyl chamber corresponds to a two-qubit gate.
Our investigations particularly concentrate on the families of gates which lie on the edges of Weyl chamber. It is shown that all the gates lying on the edges are one parametric and hence simple to study. It has been identified that SWAP^(\alpha) and SWAP^(-\alpha), with \alpha between 0 and 1, families of gates form two edges of the Weyl Chamber. These families are known to be fundamental for circuit construction. Since all the above characterizing tools are derived from the geometrical representation, it is possible to find useful relations between them. In particular, there exists a simple relation between entangling power and a local invariant. There also exists an expression for linear entropy in terms of local invariants. These relations demonstrate that local invariants play a pivotal role in the nonlocal characterization of two-qubit gates. We have also found that there is no one-to-one relation between Schmidt strength and linear entropy, except for few families of gates.