## 4.3 The projection rule, and incomplete measurements

So far we have identified measurements with orthonormal bases, or, if you wish, with a set of orthonormal projectors on the basis vectors.

An orthonormal basis satisfies two conditions:

• Orthonormality: \langle e_k|e_l\rangle = \delta_{kl}
• Completeness: \sum_k|e_k\rangle\langle e_k| = \mathbf{1}

Given a quantum system in state |\psi\rangle such that |\psi\rangle = \sum_k \alpha_k|e_k\rangle, we can write \begin{aligned} |\psi\rangle &= \mathbf{1}|\psi\rangle \\&= \sum_k (|e_k\rangle\langle e_k|) |\psi\rangle \\&= \sum_k |e_k\rangle\langle e_k|\psi\rangle \\&= \sum_k |e_k\rangle\alpha_k \\&= \sum_k \alpha_k|e_k\rangle \end{aligned} which tells us that any vector in \mathcal{H} can be expressed as the sum of the orthogonal projections on the |e_k\rangle, whence the name of the “completeness” condition. This says that the measurement in the basis \{|e_i\rangle\} gives the outcome labelled by e_k with probability |\langle e_k|\psi\rangle|^2 = \langle\psi|e_k\rangle\langle e_k|\psi\rangle and leaves the system in state |e_k\rangle. This is a complete measurement, which represents the best we can do in terms of resolving state vectors in the basis states. But sometimes we do not want our measurement to distinguish all the elements of an orthonormal basis.

For example, a complete measurement in a four-dimensional Hilbert space will have four distinct outcomes: |e_1\rangle, |e_2\rangle, |e_3\rangle, and |e_4\rangle, but we may want to lump together some of the outcomes and distinguish, say, only between \{|e_1\rangle, |e_2\rangle\}, and \{|e_3\rangle,|e_4\rangle\}. In other words, we might be trying to distinguish one subspace from another, without separating vectors that lie in the same subspace. Such measurements (said to be incomplete) are indeed possible, and they can be less disruptive than the complete measurements.

Intuitively, an incomplete measurement has fewer outcomes and is hence less informative, but the state after such a measurement is usually less disturbed.

In general, instead of projecting on one dimensional subspaces spanned by vectors from an orthonormal basis, we can decompose our Hilbert space into mutually orthogonal subspaces of various dimensions and project onto them.

A full system of projectors satisfies two conditions: Conditions on projectors:

• Orthogonality: P_k P_l = P_k\delta_{kl}
• Completeness: \sum_k P_k = \mathbf{1}

For any decomposition of the identity into orthogonal projectors P_k (using the completeness condition), there exists a measurement that takes a quantum system in state |\psi\rangle, gives the output labelled k with probability \langle\psi|P_k|\psi\rangle, and leaves the system in the state P_k|\psi\rangle (multiplied by the normalisation factor, i.e. divided by the length of P_k|\psi\rangle): |\psi\rangle \mapsto \frac{P_k|\psi\rangle}{\sqrt{\langle\psi|P_k|\psi\rangle}}.