4.3 The projection rule; 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.

  • The orthonormality condition: \langle e_k|e_l\rangle = \delta_{kl} i.e. the basis consists of unit vectors that are pairwise orthogonal.
  • The completeness condition: \sum_k|e_k\rangle\langle e_k| = \mathbf{1} i.e. any vector in \mathcal{H} can be expressed as the sum of the orthogonal projections on the |e_k\rangle.

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} 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 on them.

  • The orthogonality conditions for projectors: P_k P_l = P_k\delta_{kl}
  • The projector decomposition of the identity: \sum_k P_k = \mathbf{1}

For any decomposition of the identity into orthogonal projectors P_k, there exists a measurement that takes a quantum system in state |\psi\rangle, outputs label 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}}.