1.3 Pure and Mixed States

There are situations where the state of a quantum mechanical system cannot be described with the help of a state vector. Here, we look at such situations and provide a mathematical tool for describing them.

1.3.1 Density matrix

Let us start with the state of a combined two-component physical system given by (1.2.5). Let ()_i=1,...,N and ()_j=1,...,M denote, respectively, the standard orthonormal bases of the Hilbert spaces of systems A and B:

where (α_i)_i=1,...,N and (β_j)_j=1,...,M are some probability amplitudes. The states that allow the state vector representation (1.3.1) are called pure states. In this case, the state of the combined system is

However, in general, the state of the combined system would look like

where γ_ij are probability amplitudes that may not necessarily be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M. If γ_ij cannot be factorised as α_iβ_j, then the component systems A and B are entangled and their states cannot be represented by the state vectors (1.3.1). Such states of systems A and B are called mixed states.

The more general setup is that of an ensemble of states of the form {p_k,}_k=1,…,N, where each  is a quantum state whose wavefunction is known with certainty (although this does not necessarily provide full knowledge of the measurement statistics), and each p_k is the associated probability (not amplitude) in [0,1]. In order to define properly pure and mixed states, introduce the density operator as follows:

Definition 7. A density operator ρ is a positive semidefinite Hermitian operator with unit trace and takes the form

where ∑ _k=1^Np_k = 1 and equals 1 if k = l and zero otherwise.

Mathematically, such a density operator ρ corresponds to a density matrix (ρ_kl)_k,l=1,…,N such that

1.3.2 Pure state

A pure state is one that can be represented by a state vector

where (α_i)_i=1,...,N are probability amplitudes in ℂ such that ∑ _i=1^N|α_i|² = 1. In the ensemble setup above, this means that there exists k^∗∈{1,…,N} such that p_k^∗ = 1 and hence = and therefore ρ = ⟨ψ|. The density matrix also allows us to compute expectations of the form (1.2.4):

Lemma 3. Let ρ be the density matrix associated to the pure state (1.3.2) and let 𝒜 be an observable (Hermitian operator), then

Proof. The lemma follows from the immediate computation

⟨ψ|𝒜	= ⟨ψ|𝒜∑ i=1Nα i
	= ∑ i=1Nα i ⟨ψ|𝒜
	= ∑ i=1N⟨ψ|𝒜
	= ∑ i=1N ⟨i|ρ𝒜 = Tr(ρ𝒜).

□

With the state given by (1.3.2), we obtain

At the same time we have

Comparison of (1.3.2) and (1.3.2) yields the following expression for the density matrix of a pure state:

Example: An example of a pure state is the Hadamard state

with corresponding density matrix

1.3.3 Mixed state

A mixed state is one that cannot be represented by a single pure state vector, and is therefore represented as a statistical distribution of pure states in the form of an ensemble of quantum states {p_k,}_k=1,…,N, where ∑ _k=1^Np_k = 1 and p_k ∈ [0,1] for each k. The density of a mixed state therefore reads

Similarly to Lemma 3, we can write expectations of observables with respect to mixed states using the density matrix:

Lemma 4. Let ρ be the density matrix associated to the mixed state (1.3.3) and let 𝒜 be an observable (Hermitian operator), then

Proof. The lemma follows from the immediate computation

Tr(ρ𝒜)	= ∑ i=1N ⟨i|ρ𝒜
	= ∑ i=1N ⟨i|𝒜
	= ∑ k=1Np k
	= ∑ k=1Np k ⟨ψk|𝒜.

□

Let us see now how the density matrix formalism can help us describe the state of a combined system. Consider an entangled state of two systems, A and B, given by (1.3.1), and a Hermitian operator 𝒜 that only acts within the Hilbert space of system A. What would be the expectation value of 𝒜 in this state? Starting with (1.2.4), we obtain

Since only terms with l = j survive in (1.3.3) due to the orthogonality of the basis states, we have

Thus, the density matrix that describes the mixed state of system A is

Note that in the case where the probability amplitudes γ_ij can be factorised as the product of probability amplitudes (α_i)_i=1,...,N and (β_j)_j=1,...,M, we obtain

which describes a pure state.

A simple criterion to distinguish a pure state from a mixed state is the following:

Lemma 5. Let ρ be a density matrix. The inequality Tr(ρ²) ≤ 1 always holds and Tr(ρ²) = 1 if and only if ρ corresponds to a pure state.

Proof. Consider an ensemble of pure states {p_i,}_i=1,…,N, with density matrix given by (1.3.3). Therefore

Tr(ρ2)	= Tr
	= Tr
	= Tr = ∑ i=1Np i2Tr⟨ψi| = ∑ i=1Np i2 = ∑ i=1Np i2,

which is smaller than 1 since the p_i are probabilities in [0,1] summing up to 1. Assume now that Tr(ρ²) equals one, then so does ∑ _i=1^Np_i². If p_i ∈ (0,1) for all i = 1,…,N, then

which is a contradiction, and therefore there exists i^∗∈{1,…,N} such that p_i^∗ = 1, so that ρ = ⟨ψ_i^∗| is a pure state. Conversely, if ρ = ⟨ψ_i| for some i ∈{1,…,N} represents a pure state, then

□

Example: An example of a mixed state is a statistical ensemble of states and . If a physical system is prepared to be either in state or state with equal probability, it can be described by the mixed state

Note that this is different from the density matrix of the pure state

which reads

Unlike pure quantum states, mixed quantum states cannot be described by a single state vector. However, the pure states and the mixed states can be described by the density matrix.