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Density Operators

Abstract

The details of how both quantum and classical uncertainty can be captured in a single model.

Keywords:StatesDensity OperatorsDensity Matrices

Definition

A density operator, ρ \rho is a positive semi-definite operator of trace one,

ρ≥0,  tr[ρ]=1.\rho \geq 0, ~~ \textrm{tr} \big[ \rho \big] = 1.
(1)

Density operators are vectors in a Hilbert space and are used to model ensembles of quantum states. They allow uncertainty in measurement outcomes arising from both quantum and classical origin’s to be modelled.

Physical Interpretation

Consider a source emitting a state from the ensemble (pi,∣ψi⟩)( p_i, \ket{\psi_i} ), meaning the state ∣ψi⟩ \ket{\psi_i} is emitted by the source with probability pi p_i , ∑ipi=1\sum_i p_i = 1. This ensemble is described by the density operator

ρ=∑ipi∣ψi⟩⟨ψi∣,\rho = \sum_i p_i \ket{\psi_i}\bra{\psi_i},
(2)

where ∣ψi⟩⟨ψi∣\ket{\psi_i}\bra{\psi_i} is the projector onto the state ∣ψi⟩ \ket{\psi_i} .

When performing a measurement on a state output from the source, there will be uncertainty in the measurement outcomes arising from both the classical uncertainty from it being unknown which state one has and from the quantum nature of the state one does have.

Properties

  1. ρ†=ρ\rho^{\dagger} = \rho, density operators are Hermitian operators.
  2. Given an observable O=∑λk∣λk⟩⟨λk∣O = \sum \lambda_k \ket{\lambda_k}\bra{\lambda_k}. The probability of measuring OO and getting the outcome λk\lambda_k is given by
prob(λk)=tr[ρ∣λk⟩⟨λk∣]=⟨λk∣ρ∣λk⟩.\textrm{prob}(\lambda_k) = \textrm{tr} \big[ \rho \ket{\lambda_k} \bra{\lambda_k} \big] = \bra{\lambda_k} \rho \ket{\lambda_k}.
(3)
  1. The positivity of the density operator ensures that all probabilities are positive as,
ρ≥0⟺⟨u∣ρ∣u⟩≥0 ∀ ∣u⟩.\rho \geq 0 \Longleftrightarrow \bra{u} \rho \ket{u} \geq 0~\forall~\ket{u}.
(4)
  1. The unit trace of the density operator ensures normalisation,
tr[ρ]=tr[∑ipi∣ψi⟩⟨ψi∣]=∑ipi⟨ψi∣ψi⟩=∑1pi=1\begin{align*} \textrm{tr} \big[ \rho \big] &= \textrm{tr} \bigg[ \sum_i p_i \ket{\psi_i} \bra{\psi_i} \bigg] \\ &= \sum_i p_i \braket{\psi_i | \psi_i} \\ &= \sum_1 p_i = 1 \end{align*}
(5)
  1. Different ensembles can lead to the same density operators. Given the density operator is all one needs to calculate possible measurement outcomes, it is impossible to distinguish two sources (two ensembles) with the same density operator.
  2. All density operators have at least one source that could generate it. Specifically, a density operator ρ \rho can be written in diagonal form as
ρ=∑nλn∣λn⟩⟨λn∣,\rho = \sum_n \lambda_n \ket{\lambda_n}\bra{\lambda_n},
(6)

where λn \lambda_n are the eigenvalues of ρ \rho with associated eigenvectors ∣λn⟩\ket{\lambda_n}. Given that λ≥0 \lambda \geq 0 (due to the positive of ρ \rho ) and ∑nλn=1 \sum_n \lambda_n = 1 (due to the unit trace condition) this density operator can be realised by a source outputting the ensemble (λn,∣λn⟩)( \lambda_n, \ket{\lambda_n}).

Density Matrix

A density operator written with respect to a particular basis gives the density matrix.

Let the set of vectors {∣i⟩}i=0d−1 \{ \ket{i} \}_{i=0}^{d-1} be a basis. The density matrix of a density operator ρ \rho with respect to this basis is given by

ρ=(⟨0∣ρ∣0⟩,⟨0∣ρ∣1⟩, … ,⟨0∣ρ∣d−1⟩⟨1∣ρ∣0⟩,⟨1∣ρ∣1⟩, … ,⟨1∣ρ∣d−1⟩⟨2∣ρ∣0⟩,⟨2∣ρ∣1⟩, … ,⟨2∣ρ∣d−1⟩⋮⟨d−1∣ρ∣0⟩,⟨d−1∣ρ∣1⟩,…,⟨d−1∣ρ∣d−1⟩)\rho = \begin{pmatrix} \bra{0} \rho \ket{0}, \bra{0} \rho \ket{1}, ~ \ldots ~, \bra{0} \rho \ket{d-1} \\ \bra{1} \rho \ket{0}, \bra{1} \rho \ket{1}, ~ \ldots ~, \bra{1} \rho \ket{d-1} \\ \bra{2} \rho \ket{0}, \bra{2} \rho \ket{1}, ~ \ldots ~, \bra{2} \rho \ket{d-1} \\ \vdots \\ \bra{d-1} \rho \ket{0}, \bra{d-1} \rho \ket{1}, \ldots, \bra{d-1} \rho \ket{d-1} \\ \end{pmatrix}
(7)

Inner Product

The Hilbert Schmit inner product, (⋅,⋅):H×H→C1(\cdot, \cdot): \mathcal{H} \times \mathcal{H} \rightarrow \mathbb{C}^{1}, is an inner product between two operators. When considered between two density operators ρ \rho and σ \sigma it is given by

(ρ,σ)=tr[ρ†σ].(\rho, \sigma) = \textrm{tr} \big[ \rho^{\dagger} \sigma \big].
(8)

Mixed and Pure States

A source that emits a single quantum state with probability one is describe by a pure state. All uncertainties in measurement outcomes on pure state are therefore only quantum mechanical in origin.

Put alternatively, a density operator is in general a probabilistic mixture of quantum states. Pure states are then those quantum states that cannot be written as a probabilistic mixture of other quantum states.

The convex hull of the set of pure states is the set of density operators.

A density operator ρ \rho is pure if and only if

  1. ρ \rho is a rank one projector onto a quantum state ∣ψ⟩\ket{\psi},
ρ=∣ψ⟩⟨ψ∣.\rho = \ket{\psi} \bra{\psi}.
(9)
  1. tr[ρ2]=1\textrm{tr} \big[ \rho^2 \big] = 1.
    • This leads to a purity measure, with γ=tr[ρ2] \gamma = \textrm{tr} \big[ \rho^2 \big], with 1/d≤γ≤1 1/d \leq \gamma \leq 1, where γ=1\gamma = 1 if and only if ρ \rho is pure and γ=1/d \gamma = 1/d if ρ \rho is maximally mixed.

The Reduced Density Operator

Consider a state that exists across multiple systems

∣Ψ⟩⟨Ψ∣∈HA⊗HB.\ket{\Psi}\bra{\Psi} \in \mathcal{H}_A \otimes \mathcal{H}_B.
(10)

The reduced density operator of AA is an operator on HA\mathcal{H}_A that contains all the information about measurement outcomes one could get if measurements were only performed on the part of the state in system AA.

The reduced density operator on system AA is found by tracing out system BB, whilst the reduced density operator on system BB is found by tracing out system AA,

ρA=trB[∣Ψ⟩⟨Ψ∣],ρB=trA[∣Ψ⟩⟨Ψ∣],\begin{align*} \rho_A = \textrm{tr}_{B} \big[ \ket{\Psi}\bra{\Psi} \big], \\ \rho_B = \textrm{tr}_{A} \big[ \ket{\Psi}\bra{\Psi} \big], \end{align*}
(11)

where trθ\textrm{tr}_{\theta} an a map known as the partial trace.

The notion of the reduced density operator can be generalised to any number of systems and also applies to mixed states across multiple systems.

Limitations

It is important to note that the reduced density operator does not tell one about the correlations between the system A and B. It only contains information about measurement outcomes from measurements made on the part of the state in system A. With just ρA \rho_A one cannot say with certainty what measurement outcomes they would get from measuring the part of the state in system BB (they would need ρB \rho_B for this) or what measurement outcomes they would get from measuring the global state ∣Ψ⟩⟨Ψ∣ \ket{\Psi}\bra{\Psi} . In addition, the reduced state ρA \rho_A does not lead to a unique global state, hence, from ρA \rho_A one cannot find the global state.

Properties of the Reduced Density Operator

  1. ρA†=ρA\rho_A^{\dagger} = \rho_A, the reduced density operator is Hermitian.
  2. ρA≥0\rho_A \geq 0, the reduced density operator is positive.
  3. tr[ρA]=1\textrm{tr} \big[ \rho_A \big] = 1, the reduced density operator has unit trace.
The Partial Trace

The partial trace is used to find the reduced density operator and is given by the following map

trB[∣i⟩⟨j∣A⊗∣l⟩⟨k∣B]=⟨k∣l⟩∣i⟩⟨j∣A.\textrm{tr}_{B} \big[ \ket{i}\bra{j}_A \otimes \ket{l}\bra{k}_B \big] = \braket{k|l}\ket{i}\bra{j}_A.
(12)

Here, one would say they are tracing out system BB. This means applying the trace operation to only the BB system.

The partial trace is a quantum channel, and hence it is a linear, completely positive, trace-preserving map.

The full trace can be decomposed into the partial trace over all the different sub-systems

tr[∣Ψ⟩⟨Ψ∣]=trB(trA[∣Ψ⟩⟨Ψ∣])=trA(trB[∣Ψ⟩⟨Ψ∣])\begin{align*} \textrm{tr} \big[ \ket{\Psi}\bra{\Psi} \big] &= \textrm{tr}_B \bigg( \textrm{tr}_A \big[ \ket{\Psi}\bra{\Psi} \big] \bigg) \\ &= \textrm{tr}_A \bigg( \textrm{tr}_B \big[ \ket{\Psi}\bra{\Psi} \big] \bigg) \end{align*}
(13)

The notion of the partial trace can be generalised to any number of systems.

Quantum Formalism
Observables
Quantum Formalism
Quantum Channels