Measurement scheme for purity based on two two-body gates

A scheme for measuring the purity of a quantum system with a finite number of levels is presented. The method makes use of two $\sqrt{\text{swap}}$ gates and hinges only on measurements performed on a reference system, prepared in a certain pure state and coupled with the target system. Neither tomographic methods, with the complete reconstruction of the state, nor interferometric setups are needed.

Figure 1

Schematic diagram of the procedure. Evolution due to Eqs. (3) and (4): the output of the first gate, whose inputs are $\rho$ and $\omega$, is fed to the second gate, together with the other $\rho$. A projective measurement $\mathcal{O}$ is performed on the ancilla but not on the target states (that are traced away).