Protocol for fermionic positive-operator-valued measures

In this paper we present a protocol for the implementation of a positive-operator-valued measure (POVM) on massive fermionic qubits. We present methods for implementing nondispersive qubit transport, spin rotations, and spin polarizing beam-splitter operations. Our scheme attains linear opticslike control of the spatial extent of the qubits by considering ground-state electrons trapped in the minima of surface acoustic waves in semiconductor heterostructures. Furthermore, we numerically simulate a high-fidelity POVM that carries out Procrustean entanglement distillation in the framework of our scheme, using experimentally realistic potentials. Our protocol can be applied not only to pure ensembles with particle pairs of known identical entanglement, but also to realistic ensembles of particle pairs with a distribution of entanglement entropies. This paper provides an experimentally realizable design for future quantum technologies.

Figure 1

AP POVM from Ref. [7]. Qubit rotations of the POVM are denoted by shaded rectangles, the phase shifts by open rectangles, and the spatial degrees of freedom by the states $|i\rangle , |s_{1,2}\rangle , |t_{1,2,3,4}\rangle$, or $|p_{1,2}\rangle$. Single and double diagonal lines indicate polarizing beam splitters and reflecting mirrors, respectively.

Figure 2

Mach-Zehnder interferometer for massive particles at four different time steps. The potential is infinite in the striped area and zero elsewhere. The beam splitters of the MZI are indicated with gray diagonal lines.

Figure 3

Simulation of a massive wave packet traveling through a POVM device. The electrostatic potential of the proposed semiconductor device is represented by gray contour lines. Dashed lines indicate the position of the spin beam splitters. After each beam splitter, a magnetic field, represented by the shaded areas, is applied for spin rotations according to Fig. 1. The arrow in the projected Bloch spheres indicates the wave function's spin orientation in their respective regions. In the simulation presented, the electron wave function is split with equal probability between the $|p_1\rangle$ and $|p_2\rangle$ outputs.

Figure 4

(a) Difference between the initial and the final ensemble mean entropy of entanglement (contour from color bar). The horizontal axis shows the POVM parameter, $\phi$, and the vertical axis shows the initial mean value of the entropy of entanglement. (b) Probability density as a function of ${\left|\alpha \right|}^2$, of two example input distributions, (1) and (2), and their corresponding non-normalized $|p_1\rangle$ output distributions, (1*) and (2*).