Entanglement on mixed stabiliser states--I: Normal Forms and Reduction Procedures
The stabiliser formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of N-qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabiliser states correspond to projectors on subspaces associated with stabiliser codes. Here, we derive efficient reduction procedures to obtain various useful normal forms for stabiliser states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilisers. We obtain two single-party normal forms. The simplest, the row-reduced echelon form, is useful to calculate partial traces of stabiliser states. The second is the fully reduced form and allows for the efficient calculation of the overlap between two stabiliser states, as well as of the Uhlmann fidelity between them, and their Bures distance. We also find a reduction procedure of bipartite stabiliser states, where the operations involved are restricted to be local ones. Using this two-party normal form, we prove that every bipartite mixed stabiliser state is locally equivalent to a direct product of a number of maximally entangled states and a separable state. Using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabiliser states.