Implementing general measurements on linear optical and solid-state qubits

We show a systematic construction for implementing general measurements on a single qubit, including both strong (or projection) and weak measurements. We mainly focus on linear optical qubits. The present approach is composed of simple and feasible elements, i.e., beam splitters, wave plates, and polarizing beam splitters. We show how the parameters characterizing the measurement operators are controlled by the linear optical elements. We also propose a method for the implementation of general measurements in solid-state qubits.

Figure 1

Schematic diagram of a proposal for implementing symmetric, arbitrary-strength two-outcome measurements on a single-photon's polarization. This setup is essentially equal to the one in Ref. [22]. Three kinds of field operators (i.e., ${\widehat{a}}_{\mathrm{in},n}$, ${\widehat{a}}_{\mathrm{int},n}$, and ${\widehat{a}}_{\mathrm{out},n}$) are needed for representing the path degrees of freedom. A single photon with arbitrary polarization (described by $|\psi \rangle$) enters a polarizing beam splitter (PBS). Next, the polarization is transformed by a wave plate (WP) in each arm. Then, the two arms are recombined in a beam splitter (BS). After performing unitary gates ${\widehat{V}}_1$ and ${\widehat{V}}_2$ for compensation purposes, the resultant polarization states are expressed in terms of the measurement operators ${\widehat{M}}_1$ and ${\widehat{M}}_2$ given in Eqs. (1) and (2).

Figure 2

Measurement direction of the measurement operators (1) and (2) for the measurement strength $\varepsilon =0.9$ (red solid), $0.6$ (blue dashed), and $0.3$ (magenta dotted). The horizontal axis represents the reflection coefficient $r$ of the beam splitter, while the vertical axis is the polar angle $\theta$ of the measurement direction. All curves merge at $(r,\theta )=(1/\sqrt{2},\pi /2)$. This case corresponds to the use of a 50:50 beam splitter in Fig. 1.

Figure 3

Schematic diagrams of the construction for general measurements with (a) two outcomes and (b) $N$ outcomes ($N=4$). In (a), the interferometer for a symmetric arbitrary-strength two-outcome measurement followed by a beam splitter leads to general two-outcome measurements. The final box leading to ${\widehat{M}}_2^{\prime }|\psi \rangle$ after the last beam splitter represents a unitary gate for compensation purposes. Depending on system's parameters, this unitary gate becomes either a phase-shift gate or the identity operator (i.e., does nothing). In (b), the repeated application of general two-outcome measurements (GTOMs) allows the construction of a POVM with multiple outcomes.