Projective Measurements

Alexei Gilchrist

Observables and Projective measurements. The Born rule. Expectation and variance of observables and uncertainty principle.

In classical physics the measurements are often implicit. The assumption is that any variables in the system are measureable, at least in principle. In contrast, in quantum mechanics the process of measurement is prescribed by it’s own set of postulates.

In the following, the simplest formulation of quantum measurements is presented. This can also be considered the core formulation as more general measurement theory is applying this core in combination with adding and removing systems.

1 Born rule

The Born Rule states that if a quantum system is in state \(|\psi\rangle\), the probability of measuring it to be in some other state \(|\phi\rangle\) is

\[p = |\langle \phi| \psi\rangle|^2.\]

As a quick sanity-check of the rule, imagine the quantum state is \(|\psi\rangle\) and we ask what is the probability that the system is in state \(|\psi\rangle\)? The answer should be 1, and this is indeed the case as \(|\langle \psi| \psi\rangle|^2 = 1\). Note the requirement of representing states by normalised vectors for this to work. Similarly, if we ask what is the probability that the system is in a state \(|\psi_\perp\rangle\) orthogonal to \(|\psi\rangle\), the answer should be 0 as it is: \(|\langle \psi_\perp| \psi\rangle|^2 = 0\).

Now imagine we ask what is the probability that it’s in state \(e^{i\theta}|\phi\rangle\)? Now we get

\[p = |\langle \phi|e^{-i\theta}|\psi\rangle|^2 = |e^{-i\theta}\langle \phi| \psi\rangle|^2 = |\langle \phi| \psi\rangle|^2.\]
That is, a global phase has no affect on probabilities of measurement and is effectively invisible.

The form of Born’s rule is suggestive, since we can write

\[p = |\langle \phi| \psi\rangle|^2 = \langle \psi| \phi\rangle\langle \phi| \psi\rangle = \langle \psi|\mathbb{P}_\phi|\psi\rangle,\]
where \(\mathbb{P}_\phi=|\phi\rangle\langle \phi|\) is a rank-1 projector into the space spanned by \(|\phi\rangle\). Are higher rank projectors also associated with probabilities of outcomes?

2 Observables

We’ve seen that Hermitian operators have real eigenvalues and their eigenvectors form a complete orthonormal basis. Both of these features are key for modelling observable quantities in quantum mechanics.

First, let’s review some of the properties of Hermitian, or self-adjoint operators. A Hermitian operator \(A\) has a specral decomposition, which we can write grouping by distinct eigenvallues:

\[A = \sum_j a_j \mathbb{P}_j,\]
where \(\mathbb{P}_j\) is a projector onto the eigenspace of eigenvalue \(a_j\). A reminder: the projectors will have the form \(\mathbb{P}_j = \sum_k |\lambda_k\rangle\langle \lambda_k|\). The number of terms in this expansion is called the rank of the projector and the rank will be greater than one if the eigenvalue is degenerate. Also remember that projectors for distinct eigenvalues are orthogonal, so all the \(\mathbb{P}_j\) in the expansion above are orthogonal to each other.

The operator \(A\) represents an observable quantity. If the system is in state \(\rho\), measuring \(A\) will have the following aspects:

  1. Measurement results: A measurement result will yield one of the eigenvalues \(a_j\).
  2. Probability: Result \(a_j\) is obtained with probability
    \[p(a_j) = \mathrm{Tr}(\mathbb{P}_j\;\rho\;\mathbb{P}_j) = \mathrm{Tr}(\mathbb{P}_j\;\rho).\]
  3. Post-measurement state: The process of measurement may change the system’s state. If result \(a_j\) is obtained then the state after the measurement is
    \[\rho_j = \frac{\mathbb{P}_j\;\rho\;\mathbb{P}_j}{p(a_j)}.\]

These rules describe an ideal measureent process known as projective measurements, or von Neumann measurements.

First let’s look at some simplifying cases

  • If the projector is rank-1, i.e. \(\mathbb{P}_j = |a_j \rangle\langle a_j|\), then probability simplifies:
    \[p(a_j) = \mathrm{Tr}(|a_j \rangle\langle a_j|\rho) = \langle a_j|\rho |a_j\rangle,\]
    and the post-measurement state is the eigen-state \(|a_j\rangle\):
    \[\rho_j = \frac{|a_j \rangle\langle a_j|\rho|a_j \rangle\langle a_j|}{\langle a_j|\rho |a_j\rangle} = |a_j \rangle\langle a_j|.\]
  • If the system is in a pure state e.g. \(\rho = |\psi\rangle\langle \psi|\) then
    \[\begin{gather*}p(a_j) = \mathrm{Tr}(\mathbb{P}_j|\psi\rangle\langle \psi|) = \langle \psi|\mathbb{P}_j |\psi\rangle \\ \rho_j = \frac{\mathbb{P}_j|\psi\rangle\langle \psi|\mathbb{P}_j}{ \langle \psi|\mathbb{P}_j |\psi\rangle }.\end{gather*}\]
    In this case we can express the effect of the measurement using only the state vectors:
    \[\begin{gather*}p(a_j) = \lVert \mathbb{P}_j |\psi\rangle\rVert^2 \\ |\psi_j\rangle = \frac{\mathbb{P}_j|\psi\rangle}{\lVert \mathbb{P}_j |\psi\rangle\rVert^2 }.\end{gather*}\]
  • If both the projector is rank-1 and the state is pure, then we recover the Born rule:
    \[p(a_j) = |\langle a_j| \psi\rangle|^2,\]

That last point explains what it means to “measure the system to be in state \(|\phi\rangle\)”. The observable we are measuring in such cases is of the form

\[k |\phi\rangle\langle \phi| + 0 (I-|\phi\rangle\langle \phi| )\]
for some eigenvalue \(k\ne 0\). This is an observable yielding result \(k\) for “the state is \(|\phi\rangle\)”, and result 0 for “it’s not \(|\phi\rangle\)”.

One of the key properties of projective measurements is the Principle of Repeatability: if we measure a system using an ideal measurement apparatus and obtain outcome \(a\), then immediately re-measure the system with the same apparatus, we should again obtain outcome \(a\). After all, nothing has changed in between the two measurements. This principle is held up by the orthogonality of projectors for different measurement results

\[\begin{gather*}\lVert \mathbb{P}_j\mathbb{P}_k|\psi\rangle\rVert = \delta_{jk}\\ \mathbb{P}_k\mathbb{P}_k|\psi\rangle = \mathbb{P}_k|\psi\rangle.\end{gather*}\]

3 Expectation of observables

Starting with some classical definitions, say we have a set of \(n\) values \(\{x_1,x_2,\ldots,x_n\}\), and value \(x_j\) occurs with probability \(p_j\). Then the average, or mean of the set is

\[\langle x \rangle = \sum_j p_j x_j.\]
and the average of some function \(f(x)\) is
\[\langle f(x) \rangle = \sum_j p_j f(x_j).\]
Note we are using the notation \(\langle x\rangle\) to denote the average rather than the more common \(\bar{x}\) as there is an elegant visual connection with averages in quantum mechanics. If the state of a system is the pure state \(|\psi\rangle\), we can use the above to calculate the average value of an observable, the measurement results will be the eigenvalues \(a_j\) and the probability will be \(\langle \psi| \mathbb{P}_j |\psi\rangle\) so
\[\langle A \rangle = \sum_j a_j \langle \psi| \mathbb{P}_j |\psi\rangle = \langle \psi| \sum_j a_j\mathbb{P}_j |\psi\rangle = \langle \psi|A |\psi\rangle.\]
and you see how the angle brackets evokes the bra and ket.

More generally, if the system is in state \(\rho\) then the average of some observable \(A\) is

\[\langle A \rangle = \sum_j a_j \mathrm{Tr}(\mathbb{P}_j \rho) = \mathrm{Tr}\left(\sum_j a_j\mathbb{P}_j \rho\right) = \mathrm{Tr}(A \rho).\]

Again, starting with the classical definition, the variance of the set \(\{x_j\}\), which is a measure of the width of the distribution of values, is given by

\[\mathrm{Var}(x) = \langle\left(x-\langle x\rangle\right)^2\rangle = \langle x^2\rangle - \langle x\rangle^2\]

If \(A\) is Hermitian then \(A^2\) is also Hermitian (i.e. \(A^2 = \sum_{jk} a_j a_k \mathbb{P}_j\mathbb{P}_k = \sum_j a_j \mathbb{P}_j\)) so the variance of an observable is

\[\mathrm{Var}(A) = \langle \psi| A^2 |\psi\rangle - \langle \psi| A |\psi\rangle^2\]
for a pure state \(|\psi\rangle\), and
\[\mathrm{Var}(A) = \mathrm{Tr}(A^2\rho) - \mathrm{Tr}(A\rho)^2\]
for the general state \(\rho\).

4 Uncertainty principle

Say we have two observables \(A\) and \(B\). In general they won’t commute so \([A,B]\ne 0\). If the state of the quantum system is \(|\psi\rangle\) then there is an uncertainty principle between the two observables know as Robertson’s inequality:

\[\Delta A\Delta B \ge \frac{1}{2}|\langle \psi|[A,B]|\psi\rangle|,\]
where \(\Delta A = \sqrt{\mathrm{Var(A)}}\) is the standard deviation of the results upon measuring \(A\).

© Copyright 2022 Alexei Gilchrist