Measuring the tangle of three-qubit states, by A. Pérez-Salinas, C. Bravo-Prieto, D. García-Martín and J. I. Latorre
In this paper, the authors propose a method for transforming an unknown three-qubit state into its canonical form up to phases. This transformation can be achieved variationally and may be used to estimate the tangle. Simulations on this method have been performed.
The tangle is a three-qubit entanglement invariant that quantifies genuine tripartite entanglement. The state with maximum tangle is the GHZ state (and its local transformations). The distribution of tangle for random three-qubit states is depicted below.
The canonical form up to phases can be achieved when the number of measurements of three possible elements in the computational basis are zero. In this form, the tangle is easily measured. The canonical form can be cast by applying local unitaries on each qubit.
The tangle is affected by errors in the circuit. To mitigate these errors, a post-selection scheme can be applied after measuring the output state. This post-selection consists of discarding those results that should not appear in the canonical form.