Postulate 2: Measurement & Born's Rule

We are now in a position to state (a simplified version) of the second postulate of quantum mechanics, that is, how measurement is performed and what happens afterwards.

Some treatments of quantum mechanics will introduce this as the third postulate, but we prefer this ordering.

We will break the postulate into two parts before giving the formal statement.

Born's Rule

The first part of the second postulate of quantum mechanics is frequently referred to as Born’s Rule.

Born's Rule: For a quantum state $|\psi\rangle$ and a measurement with mutually exclusive events $i$, (described by elements of an orthonormal basis $\{| \phi_i \rangle\}$, where $i = 1, 2, 3, \ldots, n$), the probability of observing the outcome $i$, or equivalently, the probability of measuring $\ket{\psi}$ to be in the state $\ket{\phi_{i}}$ is given by:

$$P(i) = |\text{proj}_{|\phi_i\rangle} |\psi\rangle| = |\langle \phi_i | \psi \rangle|^2.$$

Where $|\text{proj}_{|\phi_i\rangle} |\psi\rangle|$ refers to the scalar projection of $|\psi\rangle$ onto $|\phi_i\rangle$.

We may denote this probability in a few different ways, depending on the clarity of the calculation and/or the context.

For instance: $P(i), P(i)_{\ket{\psi}}, P(\ket{\phi_{i}}),$ or $P(\ket{\phi_{i}})_{\ket{\psi}}$, would all be equivalent ways of denoting the probability of getting outcome $i$ when measuring the quantum state $\ket{\psi}$ in the measurement basis $\{\ket{\phi_{i}}\}_{i\in\{1,...,n\}}$. 

Physical interpretation of Born's Rule

One may very well ask: what does it mean to "observe outcome $i$" or "measuring $\ket{\psi}$ to be in the state $\ket{\phi_{i}}$"?

Recall that every quantum measurement corresponds to some experimental set up in a lab. So "outcome $i$" simply corresponds to one of the exclusive states that that measurement apparatus is configured to measure.

For example, it may be that $0$ is the outcome that corresponds to measuring the electron in an atom in its ground state, and $1$ corresponds to the outcome of measuring the electron in its excited state, and $2$ corresponds to the outcome of measuring the electron in its next excited state, etc.

Or it may be that $0$ is the outcome that corresponds to measuring a photon to be horizontally polarized, and $1$ corresponds to the outcome of measuring a photon to be vertically polarized.

For us, and for many working theoretical physicists, the specific laboratory implementation is not relevant, and often times even the specific physical system being considered is not relevant.

The rules of quantum theory apply just the same.

It's important to emphasize that Born's rule tells us the probability of measuring the quantum state $\ket{\psi}$ to be in the state $\ket{\phi_{i}}$, as well as "getting outcome $i$". They are equivalent statements, and we will say more about this in the second part of the second postulate.

Application of Born's rule to computational basis measurements

Let's apply Born's rule to a scenario with which we are already familiar. Suppose we have a qubit:

$$...$$