Joint measurements of spin, operational locality, and uncertainty

Joint, or simultaneous, measurements of noncommuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin-$1∕2$ particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.

Figure 1

For a joint measurement of spin along both $\mathbf{a}$ and ${\mathbf{a}}^{\prime }$ to be possible, the sum of the diagonals in a parallellogram with $\alpha \mathbf{a}$ and ${\alpha }^{\prime }{\mathbf{a}}^{\prime }$ as its sides must be less than 2. Unless $\mathbf{a}$ and ${\mathbf{a}}^{\prime }$ are parallel, this forces both $\mid \alpha \mid$ and $\mid {\alpha }^{\prime }\mid$ to be strictly less than 1.

Figure 2

We are proposing to make a joint measurement of spin along directions $\mathbf{a}$ and ${\mathbf{a}}^{\prime }$ by making a measurement along either direction $\mathbf{c}$ or direction ${\mathbf{c}}^{\prime }$. Direction $\mathbf{c}$ will lie somewhere between $\mathbf{a}$ and ${\mathbf{a}}^{\prime }$. Direction ${\mathbf{c}}^{\prime }$ will lie somewhere between $\mathbf{a}$ and $-{\mathbf{a}}^{\prime }$. In this picture, orthogonal spin states $\mid {+}_a⟩$ and $\mid {-}_a⟩$ correspond to opposite directions $\mathbf{a}$ and $-\mathbf{a}$, and similarly for the other directions. If we want to think in terms of photon polarization, the orthogonal polarization states $\mid H⟩$ and $\mid V⟩$ will also be represented by opposite vectors in the picture. The vectors $\mathbf{a},{\mathbf{a}}^{\prime }$, etc., are the Bloch vectors for the corresponding polarization states.

Figure 3

One way to realize a measurement of polarization either along one direction or along another, would be to use a nonpolarizing beam splitter (BS) with a suitable splitting ratio, and to measure polarization along the first direction in one arm, and along the second direction in the other arm. The wave plates WP1 and WP2 rotate the polarization in the beams in a suitable way so that the polarization measurements may be done using polarizing beam splitters PBS, which separate horizontal and vertical polarization. The detectors are not shown in the picture. With this setup, there are four possible measurement outcomes, even for a single photon.