## 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 *any* vector in *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: *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