In the previous chapter, we looked into observing a probabilistic and measuring a quantum system. In this approach, we alway looked at the full system. This means that we either have no measurement at all or we know the exact possibility, in which our system is.
For larger systems, this can become quite complicated, as we might not need the full measurement, but only some partial information. For example if we consider a dice throw, we might not need the final number of the dice, but we are only interested if it is an even or an odd number. To archive this, we can do a partial observation on a probabilistic system.
Partially observing a probabilistic system
To perform a partial observation on a probabilistic system, we first decide on which alternatives we want to distinguish. Each alternative is described by a set \(A\) of deterministic possibilities. By performing the partial observation, we will get for each alternative \(A\) the probability that the system is in a deterministic state in \(A\).
Definition 5.1 (Partially observing a probabilistic system) Given a probabilistic system with deterministic possibilities \(X = (x_1, \dots, x_n)\), a distribution \(\mu \in \mathbb{R}^N\) and a family of alternatives \(A_1,\dots,A_m\) with \(A_i \cap A_j = \emptyset\) and \(\bigcup_i A_i = X\), the probability of observing the alternative \(k\) is given by \[
\Pr[\text{outcome} = k] = \sum_{x_i \in A_k} \mu_{(x_i)} .
\] The distribution \(v\) after the observation of the outcome \(k\) is given by the (normalized) conditional distribution:
\[
v = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_N \end{pmatrix}
\text{ with }
v_i := \begin{cases} \frac{\mu_i}{\Pr[\text{outcome} = k]} & \text{if } x_i \in A_k \\ 0 & \text{if } x_i \notin A_k\end{cases}.
\]
Note: In this definition we were careful to distinguish between the names \(x_i\) of the deterministic possibilities and their number \(i\) (e.g. when writing \(\mu_i\)). We will often be less precise and simply pretend the deterministic possibilities are the number \(1,\dots,N\). That is, we would write the definition as follows and pretend it means the above:
Definition 5.2 (Partially observing a probabilistic system) Given a distribution \(\mu \in \mathbb{R}^N\) and a family of alternatives \(A_1,\dots,A_m \subseteq \{1,\dots,N\}\) with \(A_i \cap A_j = \emptyset\) and \(\bigcup_i A_i = \{1,\dots,N\}\), the probability of observing the alternative \(k\) is given by \[
\Pr[\text{outcome} = k] = \sum_{i \in A_k} \mu_i .
\] The distribution \(v\) after the observation of the outcome \(k\) is given by the conditional distribution:
\[
v = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_N \end{pmatrix}
\text{ with }
v_i := \begin{cases} \frac{\mu_i}{\Pr[\text{outcome} = k]} & \text{if } i \in A_k \\ 0 & \text{if } i \notin A_k\end{cases}.
\]
Note that similar to the full observation of a probabilistic system a partial observation does not actually change the system. We only get some new knowledge. In particular, a third person can never notice wether we observed the system or not.
A fair dice was rolled and it is only known that it is not a 5. Thus the distribution \(\mu\) is given by
\[
\mu = \begin{pmatrix} \frac{1}{5} \\[3pt] \frac{1}{5} \\[3pt] \frac{1}{5} \\[3pt] \frac{1}{5} \\[3pt] 0 \\[3pt] \frac{1}{5} \end{pmatrix}.
\]
Now we want to observe whether the number is low (\(\leq 3\)) or high \((\geq 4)\). This means we have two alternatives: \(A_{\text{low}} = \{1,2,3\}\) and \(A_{\text{high}} = \{4,5,6\}\). We therefore obtain the following probabilities for these two alternatives \[\Pr[\text{outcome = low}] = \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3}{5},\] \[\Pr[\text{outcome = hight}] = \frac{1}{5} + 0 + \frac{1}{5} = \frac{2}{5}.\]
The conditional distribution after the outcome low is \[
\begin{pmatrix} \frac{1}{5}/\frac{3}{5} \\[3pt] \frac{1}{5}/\frac{3}{5} \\[3pt] \frac{1}{5}/\frac{3}{5} \\[3pt] 0 \\[3pt] 0 \\[3pt] 0 \end{pmatrix}
=
\begin{pmatrix} \frac{1}{3} \\[3pt] \frac{1}{3} \\[3pt] \frac{1}{3} \\[3pt] 0 \\[3pt] 0 \\[3pt] 0 \end{pmatrix}.
\] That is, we know we have a uniformly random number from \(,1,2,3\), but don’t know which. And after the outcome high the conditional distribution is \[
\begin{pmatrix} 0 \\[3pt] 0 \\[3pt] 0 \\[3pt] \frac{1}{5}/\frac{2}{5} \\[3pt] 0/\frac{2}{5} \\[3pt] \frac{1}{5}/\frac{2}{5} \end{pmatrix}
=
\begin{pmatrix} 0 \\[3pt] 0 \\[3pt] 0 \\[3pt] \frac{1}{2} \\[3pt] 0 \\[3pt] \frac{1}{2} \end{pmatrix}.
\]
Partially measuring a quantum system
Similar to the partial observation of a probabilistic system, we can perform a partial measurement on a quantum system.
Definition 5.3 (Partially measuring a quantum system) Given a quantum system with classical possibilities \(X = (x_1, \dots, x_n)\), a quantum state \(\mu \in \mathbb{C}^N\) and a family of alternatives \(A_1,\dots,A_m\) with \(A_i \cap A_j = \emptyset\) and \(\bigcup_i A_i = X\), the probability of observing the alternative \(k\) is given by \[
\Pr[\text{outcome} = k] = \sum_{x_i \in A_k} |\psi_{(x_i)}|^2.
\] The post-measurement-state of the outcome \(k\) is computed as follows:
- Computing the non-normalized post-measurement-state \(\phi^{(k)}\) denoted by \(\phi^{(k)} := (\phi_1,\dots,\phi_N)\) with \[
\phi_i := \begin{cases} \psi_i & \text{if } x_i \in A_k \\ 0 & \text{if } x_i \in A_k \end{cases}.
\]
- Computing the normalized post-measurement-state by calculating: \[
\text{post-measurement-state }
\coloneqq \frac{\phi^{(k)}}{\|\phi^{(k)}\|}
= \frac{\phi^{(k)}}{\sqrt{\Pr[\text{outcome} = k]}}.
\]
As in Definition 5.1, we were precise about the difference between the classical possibility \(x_i\) and their numbers but will not always be so precise in the future.
As with the complete measurement for quantum systems, the measurement can change the system. Note that there exist other types of definitions for a measurement e.g. projective measurements, generalized measurements, POVMs, \(\dots\) The variant above can best be described as a “projective measurement in the computational basis”.
There is a slight difference between this definition and Definition 4.2, namely if you compute the post-measurement-state, you may get a different result. The two post-measurement-states can differ by a factor \(c \in \mathbb{C}\) with \(|c| = 1\), called a “global phase”. Such a global phase makes no observable physical difference, so this “contradiction” is not a problem.
A photon is in superposition between the 4 paths left, right, top and bottom:
\[
\psi= \begin{pmatrix} \frac{1}{10} \\[3pt] -\frac{3}{10} \\[3pt] \frac{9}{10}i \\[3pt] \frac{3}{10} \end{pmatrix}.
\]
There are two alternatives: \(A_{\text{horizontal}}\), so that the photon is in the left or right path, and \(A_{\text{vertical}}\), so that the photon is in the top or bottom path. We therefore obtain the following probabilities for these two alternatives \[
\Pr[A_{\text{horizontal}}] = \left|\frac{1}{10}\right|^2 + \left|-\frac{3}{10}\right|^2 = \frac{1}{100} + \frac{9}{100} = \frac{1}{10},
\] \[
\Pr[A_{\text{vertical}}] = \left|\frac{9}{10}i\right|^2 + \left|\frac{3}{10}\right|^2 = \frac{81}{100} + \frac{9}{100} = \frac{9}{10}.
\]
The normalized post-measurement-state for the alternative \(A_{\text{horizontal}}\) is \[
\begin{pmatrix} \frac{1}{10} \\[3pt] -\frac{3}{10} \\[3pt] 0 \\[3pt] 0 \end{pmatrix}/\sqrt{\frac{1}{10}}
= \begin{pmatrix} \frac{1}{\sqrt{10}} \\[3pt] -\frac{3}{\sqrt{10}} \\[3pt] 0 \\[3pt] 0 \end{pmatrix}
\]
For the alternative \(A_{\text{vertical}}\) the normalized post-measurement-state is \[
\begin{pmatrix} 0 \\[3pt] 0 \\[3pt] \frac{9}{10}i \\[3pt] \frac{3}{10} \end{pmatrix}/\sqrt{\frac{9}{10}}
= \begin{pmatrix} 0 \\[3pt] 0 \\[3pt] \frac{3}{\sqrt{10}}i \\[3pt] \frac{1}{\sqrt{10}} \end{pmatrix}.
\]