4.2 Back to qubits; complete measurements

A projector is any Hermitian operator P=P^\dagger which is idempotent (P^2=P). The rank of P is evaluated using \operatorname{tr}(P). In the Dirac notation, |e\rangle\langle e| is a rank one projector on the subspace spanned by the unit vector |e\rangle, and it acts on any vector |v\rangle as (|e\rangle\langle e|)|v\rangle = |e\rangle\langle e|v\rangle.

The most common measurement in quantum information science is the standard measurement on a qubit, also referred to as the measurement in the standard (or computational) basis \{|0\rangle,|1\rangle\}. When we draw circuit diagrams it is tacitly assumed that such a measurement is performed on each qubit at the end of quantum evolution.

The standard (computational) basis defines the standard measurements.

Figure 4.1: The standard (computational) basis defines the standard measurements.

However, if we want to emphasise the role of the measurement, then we can include it explicitly in the diagram as a special quantum gate, e.g. as

or, in an alternative notation, as

As we can see, if the qubit is prepared in state |\psi\rangle = \alpha_0|0\rangle + \alpha_1|1\rangle and subsequently measured in the standard basis state, then the outcome is |k\rangle (for k=0,1) with probability63 \begin{aligned} |\alpha_k|^2 &= |\langle k|\psi\rangle|^2 \\&= \underbrace{\langle\psi|k\rangle}_{\alpha_k^\star} \underbrace{\langle k|\psi\rangle}_{\alpha_k} \\&= \langle\psi| \underbrace{|k\rangle\langle k|}_{\text{projector}} |\psi\rangle \\&= \langle\psi|P_k|\psi\rangle \end{aligned} where P_k=|k\rangle\langle k| is the projector on |k\rangle. If the outcome of the measurement is k, then the output state of the measurement gate is |k\rangle. The original state |\psi\rangle is irretrievably lost. This sudden change of the state, from the pre-measurement state |\psi\rangle to the post-measurement state, either |0\rangle or |1\rangle, is often called a collapse or a reduction of the state.

So it looks like there are two distinct ways for a quantum state to change: on the one hand we have unitary evolutions, and on the other hand we have an abrupt change during the measurement process. Surely, the measurement process is not governed by any different laws of physics?

No, it is not!

A measurement is a physical process and can be explained without any “collapse”, but it is usually a complicated process in which one complex system (a measuring apparatus or an observer) interacts and gets correlated with a physical system being measured. We will discuss this more later on, but for now let us accept a “collapse” as a convenient mathematical shortcut, and describe it in terms of projectors rather than unitary operators.


  1. This slick argument is a good example of how nice the bra-ket notation can be.↩︎