5  Partial observing and measuring systems

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.

5.1 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 state in \(A\).

Definition 5.1 (Partially observing a probabilistic system) Given a distribution \(M\in \mathbb{R}^N\) 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 \[ \Pr[\text{outcome} = k] = \sum_{i\in A_k} M_i = \| v^{(k)}\|_1 \] \(\| \|_1\) denotes the 1-norm here. The distribution after the observation of the outcome \(k\) is given by the normalized conditional distribution. This is computed by

1. Computing the non-normalized conditional distribution \(v^{(k)}\) denoted by \(v^{(k)} := (v_1,\dots,v_N)\) with \[ v_i := \begin{cases} M_i & \text{if } i \in A_k\\ 0 & \text{else}\end{cases} \]
2. Computing the normalized conditional distribution by calculating \[ \frac{v^{(k)}}{\Pr[\text{outcome} = k]} \]

Note that similar to the full observation of a probabilistic system a partial observation has no impact on the system. We only get some new knowledge but do not influence the actual system by our observation.

5.2 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 5.2 (Partially measuring a quantum system) Given a quantum state \(\psi \in \mathbb{C}^N\) 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 the alternative \(k\) is given by \[ \Pr[\text{outcome} = k] = \sum_{i\in A_k} |\psi_i|^2 = \| \phi^{(k)}\|^2 \] \(\| \|\) denotes the Euclidean norm here. The post-measurement-state (p.m.s.) of the outcome \(k\) is computed as follows:

1. Computing the non-normalized post-measurement-state \(\phi^{(k)}\) denoted by \(\phi^{(k)} := (\phi_1,\dots,\phi_N)\) with \[ \phi_i := \begin{cases} \psi_i & \text{if } i \in A_k\\ 0 & \text{else}\end{cases} \]
2. Computing the normalized post-measurement-state by calculating \[ \frac{\phi^{(k)}}{\sqrt{\Pr[\text{outcome} = k]}} = \frac{\phi^{(k)}}{\|\phi^{(k)}\|} \]

As with the complete measurement for quantum systems, the measurement can change the system. Note that there exists other types of definitions for a measurement e.g. projective measurement, generalized measurements, POVMs, … The variant above can best be described as a “projective measurement in the computational basis”.