A simple realization of Weyl-Heisenberg covariant measurements

Abstract

Informationally complete (IC) measurements are fundamental tools in quantum information processing, yet their physical implementation remains challenging. By the Naimark extension theorem, an IC measurement may be realized by a von Neumann measurement on an extended system after a suitable interaction. In this work, we elaborate on a simple algorithm for realizing Naimark extensions for rank-one Weyl-Heisenberg covariant informationally complete measurements in arbitrary finite dimensions. Exploiting Weyl-Heisenberg covariance, we show that the problem reduces to determining a $d \times d$ unitary from which the full $d^2 \times d^2$ unitary interaction can be constructed. The latter unitary enjoys a block-circulant structure which allows e.g., for an elegant optical implementation. We illustrate the procedure with explicit calculations for qubit, qutrit, and ququart SIC-POVMs. Finally, we show that from another point of view, this method amounts to preparing an ancilla system according to a so-called fiducial state, followed by a generalized Bell-basis measurement on the system and ancilla. These results provide a straightforward framework for implementing informationally complete measurements in the laboratory suitable for both qubit and qudit based systems.

Figures (2)

• Figure 1: Schematic diagram illustrating the implementation of a rank-one Weyl--Heisenberg--covariant POVM in arbitrary dimension $d$. The input state $|\psi\rangle$ is injected through the modes $(0,\, d,\, \ldots,\, d(d-1))$. The Fourier transform $F_d$ and its inverse $F_d^\dagger$ act on the modes indicated by the color coding in the diagram. The unitary $U_0$ operates on modes $(0,1,\ldots,d-1)$, $U_1$ on $(d,d+1,\ldots,2d-1)$, and, more generally, $U_{k}$ acts on $(kd, kd+1,\ldots,(k+1)d-1)$ for $k=0,\ldots,d-1$. The detection events at the numbered output ports correspond to the outcomes of the measurement.
• Figure 2: Schematic diagram illustrating the optical implementation of a rank-one Weyl--Heisenberg covariant measurement in $d=2$. The input state $|\psi\rangle$ is injected through inputs $(0,2)$. The Fourier matrix $F_2$ and its adjoint $F_2^\dagger$ act on the inputs as indicated by the color coding in the diagram, while the unitary matrices $U_0$ and $U_1$ operate on inputs $(0,1)$ and $(2,3)$, respectively. Each detection event at a numbered output port corresponds to a distinct measurement outcome.