Density Matrices by Example: From Superposition to Partial Trace
The density matrix is one of the most powerful tools in quantum mechanics. It generalises the familiar state vector formalism and allows us to handle not only pure superpositions but also statistical mixtures and subsystems. In this tutorial we will build up the concept step by step, beginning with a simple electron in a hydrogen atom and ending with an entangled two-spin system where the reduced density matrix becomes mixed.
1. A superposed hydrogen electron
Imagine an electron in a hydrogen atom that can only occupy the first three energy eigenstates. Its quantum state is written as
|ψ⟩ = C1|E1⟩ + C2|E2⟩ + C3|E3⟩, with |C1|² + |C2|² + |C3|² = 1.
Here the coefficients C1, C2 and C3 are complex amplitudes, and |E1⟩, |E2⟩ and |E3⟩ are the energy eigenvectors forming the basis.
If you wish, you may picture |E1⟩ as the 1s orbital, |E2⟩ as 2s, and |E3⟩ as 3s, but the mathematics is basis-independent.
2. Measurement as projection
If we measure the energy of this electron, the appropriate operators are projection operators. The projector onto state |Ei⟩ is
Pi = |Ei⟩⟨Ei|.
The probability of finding the electron in state |Ei⟩ is then given by
p(Ei) = ⟨ψ|Pi|ψ⟩ = |Ci|².
Thus the squared amplitudes give us the probabilities, as expected.
3. Constructing the density matrix
We now form the density operator, which is defined for a pure state as
ρ = |ψ⟩⟨ψ|.
In the ordered basis {|E1⟩, |E2⟩, |E3⟩}, this has the explicit matrix form
ρ =
[ |C1|² C1C2* C1C3* ][ C2C1* |C2|² C2C3* ][ C3C1* C3C2* |C3|² ]
We immediately see some useful properties: ρ is Hermitian, its trace is 1, and since this is a pure state, we also have ρ² = ρ.
4. Probabilities and coherences
The diagonal entries of ρ are just the measurement probabilities in this basis:
p(E1) = ρ11 = |C1|², p(E2) = ρ22 = |C2|², p(E3) = ρ33 = |C3|².
The off-diagonal entries, however, are not so trivial. These so-called coherences encode information about the relative phases between the states. They become important when we measure in a basis other than the energy eigenbasis.
5. An interference example
Suppose we want to know the probability that the electron is in the superposition state
|φ⟩ = (|E1⟩ + |E2⟩)/√2.
The relevant probability is
p(φ) = ⟨φ|ρ|φ⟩ = ½(|C1|² + |C2|² + 2 Re[C1C2*]).
Notice the extra term 2 Re[C1C2*]. This arises precisely from the off-diagonal entries of ρ. The probability now depends not only on the magnitudes of C1 and C2 but also on their relative phase. This is interference in action, expressed in the language of density matrices.
6. Expectation values
To calculate an expectation value of the energy H we use the simple rule
⟨H⟩ = Tr(ρH).
So in the restricted 3-dimensional subspace we were using, Ĥ is represented as a diagonal matrix:
E1 0 0 0 E2 0 0 0 E3
That is, H = diag(E1, E2, E3).
For the Hamiltonian itself, which is diagonal in the energy eigenbasis, only the diagonal terms of ρ contribute. Thus
⟨H⟩ = |C1|² E1 + |C2|² E2 + |C3|² E3.
Here the off-diagonal terms vanish, reflecting the fact that the Hamiltonian is measured in its own eigenbasis.
7. Subsystems and the partial trace
The true power of density matrices emerges when we study composite systems. Consider two spin-½ particles, A and B, in the entangled singlet state
|Ψ⁻⟩ = (|↑⟩A|↓⟩B − |↓⟩A|↑⟩B)/√2.
The joint density matrix is ρAB = |Ψ⁻⟩⟨Ψ⁻| - a large and complex expression. To find the state of subsystem A alone, we take the partial trace over subsystem B:
ρA = TrB(ρAB).
Carrying out the trace explicitly yields
ρA = [ 1/2 0 ][ 0 1/2 ]
This is the maximally mixed state. Although the overall two-particle state is pure, each particle by itself looks completely random. Entanglement in the global state has destroyed purity in the subsystem.
8. Conclusion
We have seen how to construct a density matrix from a pure state, how to read probabilities and expectation values from it, how the off-diagonal terms encode coherence, and finally how partial tracing over one part of an entangled system produces a mixed state.
These are the core ideas that make density matrices indispensable in modern quantum mechanics. From here one can go on to study thermal ensembles, time evolution, and open quantum systems, all of which rely on the same underlying formalism.
