Measurement with Density Matrices

Overview

Born’s rule was originally introduced for pure states: probabilities come from squared magnitudes of amplitudes.

Density matrices let us express the same rule in a way that works for:

• pure states,
• mixed states,
• and subsystems of larger systems.

This page introduces the density-matrix form of Born’s rule and explains (conceptually) expectation values, while keeping computations simple.

Intuition

When you measure, you are asking a question with multiple possible outcomes. For each outcome, the theory assigns a number between 0 and 1: its probability.

For a pure state, the probability is “overlap squared.” For a mixed state, you should average those overlaps over the classical randomness.

The density matrix $\rho$ is designed so that you can compute the averaged probabilities directly, without tracking the underlying ensemble.

Formal Description

Projectors for measurement outcomes

For a measurement in an orthonormal basis $\{|b_k\rangle\}$, each outcome corresponds to a projector:

$$\Pi_k = |b_k\rangle\langle b_k|.$$

• $|b_k\rangle$ is the basis state for outcome $k$.
• $\langle b_k|$ is its bra.
• $\Pi_k$ is the operator that “selects the component along $|b_k\rangle$.”

Born rule with density matrices

If the system is described by a density matrix $\rho$, then the probability of outcome $k$ is

$$P(k) = \mathrm{Tr}(\rho\,\Pi_k).$$

Here:

• $\rho\,\Pi_k$ is a matrix product.
• $\mathrm{Tr}(\cdot)$ means “sum of diagonal entries.”

You can view this formula as the matrix version of “overlap squared,” generalized to mixtures.

Expectation values (conceptual)

Sometimes you want a single number summarizing a measurement, not the full probability table. In many cases that number is an expectation value.

If an observable (a measurement quantity) is represented by an operator $A$, then its expected value in state $\rho$ is

$$\mathbb{E}[A] = \mathrm{Tr}(\rho A).$$

You do not need to treat $A$ abstractly yet; the key idea is: density matrices make “averaging” linear and straightforward.

Worked Example

Consider the mixed state

$$\rho = \tfrac{1}{2}|0\rangle\langle 0| + \tfrac{1}{2}|1\rangle\langle 1|.$$

Measure in the computational basis.

The projectors are:

$$\Pi_0 = |0\rangle\langle 0|,\quad \Pi_1 = |1\rangle\langle 1|.$$

Compute $P(0)$:

$$P(0)=\mathrm{Tr}(\rho\Pi_0) = \mathrm{Tr}\left(\left(\tfrac{1}{2}|0\rangle\langle 0| + \tfrac{1}{2}|1\rangle\langle 1|\right)|0\rangle\langle 0|\right).$$

Distribute the product:

$$= \tfrac{1}{2}\mathrm{Tr}(|0\rangle\langle 0|0\rangle\langle 0|) + \tfrac{1}{2}\mathrm{Tr}(|1\rangle\langle 1|0\rangle\langle 0|).$$

Use orthonormality: $\langle 1|0\rangle=0$ and $\langle 0|0\rangle=1$. Intuitively, the second term vanishes.

So the result is:

$$P(0)=\tfrac{1}{2}.$$

Similarly, $P(1)=1/2$.

This matches the ensemble intuition: it’s a 50/50 classical mixture.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Density matrices are especially useful for subsystems and correlations. Next we’ll revisit correlations with sharper language: what can be explained by classical shared randomness, and what requires genuinely quantum entanglement.