State vectors describe pure quantum states. Density matrices extend the same mathematics to uncertainty about preparation, subsystems of entangled states and the probabilistic structure needed for realistic quantum systems.

The first four Quantum Mathematics guides built pure-state quantum mechanics using complex vectors, operators, tensor products and unitary evolution. That framework is elegant, but it is not enough for every situation. A laboratory may prepare different pure states with known probabilities. A subsystem of an entangled pair may not possess its own pure state vector. Noise may convert a pure state into a statistical mixture. In all of these cases, the density operator becomes the natural mathematical object.

Pure state vector → outer product → density matrix. Statistical ensemble → weighted sum of density matrices. Composite state → partial trace → reduced density matrix.

1. From a ket to a density matrix

For a normalised pure state |ψ⟩, define the density operator

ρ = |ψ⟩⟨ψ|.

If |ψ⟩=[α,β]ᵀ, then ⟨ψ|=[α*,β*], so

ρ = [[|α|², αβ*],[βα*, |β|²]].

The diagonal entries contain the computational-basis probabilities. The off-diagonal entries contain coherence information, including relative phase. This is why a density matrix remembers more than a list of measurement probabilities in one basis.

Worked example: the plus state

For |+⟩=(|0⟩+|1⟩)/√2,

ρ₊=(1/2)[[1,1],[1,1]].

Both diagonal entries are 1/2, but the off-diagonal entries are also 1/2. Those off-diagonal terms distinguish the coherent superposition from a classical fifty-fifty mixture of |0⟩ and |1⟩.

2. Statistical mixtures

Suppose a preparation device produces pure state |ψᵢ⟩ with classical probability pᵢ, where pᵢ≥0 and Σᵢpᵢ=1. The density matrix of the ensemble is

ρ = Σᵢ pᵢ |ψᵢ⟩⟨ψᵢ|.

This is a mixed state unless the resulting operator happens to have rank one. The mixture does not say the system is literally in several pure states at once. It says our operational preparation is described by a classical distribution over pure-state preparations, or equivalently by the resulting density operator.

Worked example: an equal classical mixture

Prepare |0⟩ half the time and |1⟩ half the time. Then

ρ = (1/2)|0⟩⟨0| + (1/2)|1⟩⟨1| = (1/2)[[1,0],[0,1]] = I/2.

The diagonal probabilities match those of |+⟩ in the computational basis, but the off-diagonal coherence has vanished. The states therefore behave differently under a change of basis.

3. Superposition is not mixture

Compare

ρ_superposition=(1/2)[[1,1],[1,1]]

with

ρ_mixture=(1/2)[[1,0],[0,1]].

Both produce 0 and 1 with equal probability in the computational basis. But an X-basis measurement distinguishes them. The plus state is an X eigenstate and gives the + outcome with certainty. The maximally mixed state gives the two X outcomes with equal probability.

Equal probabilities in one basis do not imply equal quantum states.

4. The three defining properties of a density matrix

In finite-dimensional quantum mechanics, a valid density matrix satisfies three core conditions:

Hermitian: ρ†=ρ.
Positive semidefinite: ⟨v|ρ|v⟩≥0 for every vector |v⟩.
Unit trace: Tr(ρ)=1.

Hermiticity ensures real expectation values. Positivity ensures probabilities cannot become negative. Unit trace ensures total probability is one.

5. The trace

The trace of a square matrix is the sum of its diagonal entries:

Tr(A)=ΣᵢAᵢᵢ.

The trace has several useful properties. It is linear, so Tr(aA+bB)=aTr(A)+bTr(B). It is cyclic for matrix products of compatible sizes: Tr(ABC)=Tr(BCA)=Tr(CAB). It is basis-independent, even though diagonal entries themselves depend on basis.

For a pure-state density matrix, Tr(|ψ⟩⟨ψ|)=⟨ψ|ψ⟩=1.

Worked example: trace check

Consider ρ=[[3/4,1/4],[1/4,1/4]]. The trace is 3/4+1/4=1. That is necessary, but not sufficient, for validity; positivity must also be checked.

6. Expectation values become trace formulas

For a pure state, Guide 2 used

⟨A⟩=⟨ψ|A|ψ⟩.

With density matrices, the same quantity becomes

⟨A⟩=Tr(ρA).

For ρ=|ψ⟩⟨ψ|, cyclicity of trace gives Tr(|ψ⟩⟨ψ|A)=⟨ψ|A|ψ⟩. The density-matrix formula therefore extends rather than replaces the pure-state formula.

Worked example: Z expectation from a mixed state

Take ρ=diag(3/4,1/4) and Z=diag(1,-1). Then

Tr(ρZ)=3/4-1/4=1/2.

This agrees with the classical weighted mean of the two Z outcomes.

7. Measurement probabilities using projectors

If Pₖ is the projector associated with a projective measurement outcome, then

p(k)=Tr(ρPₖ).

This formula works for pure and mixed states. For a computational-basis measurement of a qubit, P₀=|0⟩⟨0| and P₁=|1⟩⟨1|. The probabilities are simply the corresponding diagonal entries of ρ in that basis.

8. State update after an ideal projective outcome

In the standard ideal projective model, if outcome k occurs with projector Pₖ, the post-measurement state is

ρ' = PₖρPₖ / Tr(ρPₖ)

provided the denominator is non-zero. The numerator projects onto the relevant subspace; the denominator renormalises the result.

This is a special measurement model. Guide 7 will generalise measurement beyond orthogonal projectors.

9. Purity

A useful scalar diagnostic is

γ=Tr(ρ²).

For a finite-dimensional density matrix, 1/d ≤ Tr(ρ²) ≤ 1, where d is the Hilbert-space dimension. A pure state has Tr(ρ²)=1. Mixed states have smaller purity. The maximally mixed state I/d has minimum purity 1/d.

Worked example: qubit purity

For ρ=I/2,

ρ²=I/4, so Tr(ρ²)=2×(1/4)=1/2.

For ρ₊=|+⟩⟨+|, projectors satisfy ρ₊²=ρ₊, so the purity is one.

10. Eigenvalues of a density matrix

Because ρ is Hermitian and positive semidefinite, its eigenvalues are real and non-negative. Because the trace is one, the eigenvalues sum to one.

Thus the eigenvalues of ρ behave like a probability distribution. The spectral decomposition is

ρ=Σᵢ λᵢ |i⟩⟨i|.

A pure state has spectrum {1,0,0,...}. A maximally mixed d-dimensional state has spectrum {1/d,...,1/d}.

11. Different ensembles can give the same density matrix

The same mixed density operator can often be decomposed into pure-state ensembles in many different ways. For example, the maximally mixed qubit state can be written as

I/2=(1/2)|0⟩⟨0|+(1/2)|1⟩⟨1|

and also as

I/2=(1/2)|+⟩⟨+|+(1/2)|−⟩⟨−|.

No measurement on the qubit alone can distinguish which ensemble decomposition was used if both preparations produce exactly the same density matrix. Operational predictions depend on ρ, not on one preferred decomposition.

12. The Bloch-vector form of a qubit state

Every qubit density matrix can be written

ρ=(1/2)(I+rₓX+rᵧY+r_zZ)

or compactly ρ=(I+r·σ)/2, where r=(rₓ,rᵧ,r_z) is the Bloch vector and σ=(X,Y,Z).

Physical qubit states satisfy |r|≤1. Pure states lie on the surface of the Bloch sphere with |r|=1. Mixed states lie inside the sphere. The maximally mixed state sits at the centre with r=0.

Purity in Bloch form

For a qubit,

Tr(ρ²)=(1+|r|²)/2.

This makes the geometry immediate: longer Bloch vectors correspond to greater purity.

13. Composite density matrices

If subsystem A has density matrix ρ_A and subsystem B has density matrix ρ_B and the joint state is uncorrelated, then the composite state is

ρ_AB=ρ_A⊗ρ_B.

But not every joint state factors this way. Correlated classical mixtures and entangled quantum states require more general density matrices on the tensor-product space.

14. Why subsystems need the partial trace

Suppose we know the joint state ρ_AB but want predictions for measurements performed only on subsystem A. We need an object ρ_A such that

Tr(ρ_A A)=Tr(ρ_AB(A⊗I_B))

for every observable A acting on subsystem A. The unique standard construction is the partial trace:

ρ_A=Tr_B(ρ_AB).

The partial trace removes subsystem B while preserving all measurement predictions accessible locally to A.

15. Computing a partial trace in a basis

If {|j⟩_B} is an orthonormal basis for B, then

Tr_B(ρ_AB)=Σ_j (I_A⊗⟨j|)ρ_AB(I_A⊗|j⟩).

This formula looks technical, but the job is simple: sum over the basis index belonging to the subsystem being discarded.

Worked example: product state

If ρ_AB=|0⟩⟨0|⊗|+⟩⟨+|, then tracing out B returns ρ_A=|0⟩⟨0|. Tracing out A returns ρ_B=|+⟩⟨+|. Product states therefore recover their individual pure states.

16. Worked example: partial trace of a Bell state

Take

|Φ+⟩=(|00⟩+|11⟩)/√2.

The joint density matrix is

ρ_AB=(1/2)(|00⟩⟨00|+|00⟩⟨11|+|11⟩⟨00|+|11⟩⟨11|).

Tracing out B removes the cross terms because ⟨0|1⟩=0. The surviving terms give

ρ_A=(1/2)(|0⟩⟨0|+|1⟩⟨1|)=I/2.

The same holds for B. The joint Bell state is pure, but each subsystem alone is maximally mixed.

A whole can be pure while each part is mixed.

17. Entanglement from reduced-state mixedness

For a bipartite pure state, there is a powerful test: the state is separable exactly when either reduced density matrix is pure. If the reduced state is mixed, the global pure state is entangled.

This follows from the Schmidt decomposition introduced in Guide 3. More than one non-zero Schmidt coefficient gives a reduced density matrix with more than one non-zero eigenvalue.

Be careful: for globally mixed states, local mixedness alone does not prove entanglement. Classical correlations can also make reduced states mixed.

18. Unitary evolution in density-matrix form

If a state evolves by |ψ'⟩=U|ψ⟩, then its density matrix evolves as

ρ' = UρU†.

This works for pure and mixed states. Unitary evolution preserves trace, positivity and the eigenvalues of ρ, so it also preserves purity and von Neumann entropy.

19. The von Neumann equation

For a closed system obeying Schrödinger evolution, the density matrix satisfies

iℏ dρ/dt = [H,ρ]=Hρ-ρH.

This is the density-operator analogue of the Schrödinger equation. If [H,ρ]=0, the density matrix is stationary under that time-independent Hamiltonian.

20. Purification

Any finite-dimensional mixed state can be represented as the reduced state of a larger pure state. Suppose

ρ_A=Σᵢ λᵢ|i⟩⟨i|.

Introduce an auxiliary system B with orthonormal states |i⟩_B and define

|Ψ⟩_AB=Σᵢ √λᵢ |i⟩_A|i⟩_B.

Then Tr_B(|Ψ⟩⟨Ψ|)=ρ_A. This construction is called purification. It shows that mixedness can be understood mathematically as arising from ignored correlations with a larger system, although different physical preparations can lead to the same reduced state.

21. Common misconception: a mixed state is an imperfect pure state

Not necessarily. Mixed states are legitimate quantum states with their own density operators. They may arise from classical uncertainty, entanglement with an unobserved environment, deliberate randomisation or open-system dynamics. Calling them merely “imperfect” hides the mathematical role they play.

22. Common misconception: diagonal density matrices are always classical

Diagonal form is basis-dependent. Every Hermitian density matrix can be diagonalised in its eigenbasis. What matters is which basis is operationally significant and what transformations or measurements are being considered. A density matrix that is diagonal in one basis can contain off-diagonal elements in another.

23. Common misconception: partial trace means averaging amplitudes

No. The partial trace acts on the density operator, not directly by averaging state-vector amplitudes. It removes degrees of freedom while preserving all local expectation values. Trying to “average away” one component at the ket level generally gives the wrong object.

24. Worked synthesis problem

Consider the two-qubit pure state

|ψ⟩=√(3/4)|00⟩+(1/2)|11⟩.

Step 1: Check normalisation. 3/4+1/4=1.

Step 2: Form the joint density matrix. ρ_AB=|ψ⟩⟨ψ|, including diagonal terms (3/4)|00⟩⟨00| and (1/4)|11⟩⟨11| plus two coherence terms.

Step 3: Trace out B. The cross terms vanish under the partial trace, leaving

ρ_A=(3/4)|0⟩⟨0|+(1/4)|1⟩⟨1|.

Step 4: Compute purity. Tr(ρ_A²)=9/16+1/16=10/16=5/8<1.

Step 5: Interpret. The global state is pure but entangled; subsystem A is mixed. Its Z expectation is 3/4-1/4=1/2.

25. Practice set

Construct the density matrix of |−⟩=(|0⟩-|1⟩)/√2.
Construct the equal mixture of |0⟩ and |1⟩.
Show that the two matrices in Questions 1 and 2 are different.
Verify that Tr(ρ)=1 for ρ=diag(0.8,0.2).
Compute Tr(ρ²) for that state.
Find the Z expectation value for ρ=diag(0.8,0.2).
Find the reduced state of the first qubit for |00⟩.
Find the reduced state of the first qubit for |Φ+⟩.
State the density-matrix form of unitary evolution.
Explain why a subsystem of a pure entangled state can be mixed.

Answers

(1/2)[[1,-1],[-1,1]].
I/2.
The superposition has off-diagonal entries -1/2; the mixture has zero off-diagonal entries.
0.8+0.2=1.
0.8²+0.2²=0.68.
0.8-0.2=0.6.
|0⟩⟨0|.
I/2.
ρ' = UρU†.
The local system is correlated with the other subsystem; tracing out that partner removes joint coherence from the local description.

26. Where this sits in the Quantum Mathematics estate

This guide is the bridge from pure-state quantum mechanics to realistic quantum information. Once density operators are available, we can describe noise, irreversible-looking local behaviour, generalised measurements and entropy without abandoning the linear-algebraic framework built in Batch 01.

Educational note: this guide focuses on finite-dimensional density operators. Infinite-dimensional systems require additional functional-analytic care, including trace-class operators and domain questions.

Small Group Tutorials

Quantum Mathematics Learning Guide 5: Density Matrices, Mixed States, Trace and Partial Trace