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_{i}β_{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}^{N}p_{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=1}^{N}α _{i} |
= ∑ _{i=1}^{N}α _{i} ⟨ψ|𝒜 | |
= ∑ _{i=1}^{N}⟨ψ|𝒜 | |
= ∑ _{i=1}^{N} ⟨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}^{N}p_{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=1}^{N} ⟨i|ρ𝒜 |
= ∑ _{i=1}^{N} ⟨i|𝒜 | |
= ∑ _{k=1}^{N}p _{k} | |
= ∑ _{k=1}^{N}p _{k} ⟨ψ_{k}|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(ρ^{2}) ≤ 1 always holds and Tr(ρ^{2}) = 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(ρ^{2}) | = Tr |
= Tr | |
= Tr = ∑ _{i=1}^{N}p _{i}^{2}Tr⟨ψ_{i}| = ∑ _{i=1}^{N}p _{i}^{2} = ∑ _{i=1}^{N}p _{i}^{2}, |
which is smaller than 1 since the p_{i} are probabilities in [0,1] summing up to 1. Assume now that Tr(ρ^{2}) equals one, then so does ∑ _{i=1}^{N}p_{i}^{2}. 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.