The Density Operator

The Density Operator

Suppose a quantum system is in state |ψ_ii with probability p_i, we call {p_i, |ψ_ii} an ensemble of quantum states. The density operator (or density matrix) for the system is defined as

ρ ≡ ^X

i

p_i|ψ_iihψ_i|.

The evolution of the density matrix (of a closed system) described by unitary operator U ^is

ρ = ^X

i

p_i|ψ_iihψ_i| −→^{U X}

i

p_iU|ψ_iihψ_i|U^{† =} UρU^†. After measurement described by M_m^{, outcome} m occurs with probability

p(m) = ^X

i

p_ip(m|i)

= ^X

i

p_ihψ_i|M^†_mM_m|ψ_ii

= ^X

i

p_itr ^M^†_mM_m|ψ_iihψ_i|^

= tr ^M^†_mM_mρ^.

1

while the new state (density operator) in this case is

ρ⁰ = ^X

i

p(i|m) M_m|ψ_iihψ_i|M^†_m tr ^M^†_mM_m|ψ_iihψ_i|^

= ^X

i

p(m|i)p_i p(m)

M_m|ψ_iihψ_i|M^†_m tr^M^†_mM_m|ψ_iihψ_i|^

= ^X

i

p_iM_m|ψ_iihψ_i|M^†_m tr^M^†_mM_mρ^

= M_mρM^†_m tr^M^† M_mρ^

Pure and Mixed States

States described by a state vector |ψi are called pure states. Pure state density matrices have the form ρ = |ψihψ|, density matrices not ex- pressable in this form is in a mixed state. For example, the density matrix ρ = |0ih0| + |1ih1|

for a qubit is in a mixed state.

The density operator represents a pure state if and only if tr ^ρ^2 = 1.

2

A density operator can also be formed from an ensemble of density operators {p_i, ρ_i}, each of which arises from some ensemble {p_ij, |ψ_iji}, so that each |ψ_iji has probability p_ip_ij,

ρ = ^X

ij

p_ip_ij|ψ_ijihψ_ij| = ^X

i

p_iρ_i.

We can say that the density matrix ρ is a mix- ture of density matrices ρ_i, each of which is a mixture of quantum states |ψ_iji.

General Properties of the Density Operator

An operator ρ is a density operator for some ensemble {p_i, |ψ_ii} if and only if

1. ρ is self-adjoint.

2. tr (ρ) = 1.

3. ρ is positive.

4

The Postulates Restated With Density Operators

Postulate 1 The state of a physical system is described by a density operator (a positive operator with unit trace) on its state space (a Hilbert space). A system with probability p_i of being in the state ρ_i has density operator ρ = ^P_i p_iρ_i.

Postulate 2 The evolution of a physical sys- tem is unitary:

ρ⁰ = UρU^†.

Postulate 3 Quantum measurements are de-

and the new state is

M^†_mρM_m tr ^M^†_mM_mρ^

.

Postulate 4 The state of the composite sys- tem of systems 1 through n is the density op- erator

ρ₁ ⊗ ρ₂ ⊗ . . . ⊗ ρ_n =

n O

i=1

ρ_i acting on

H₁ ⊗ H₂ ⊗ . . . ⊗ H_n =

n O

i=1

H_i.

Density Operators in the Bloch “Ball”

For the Bloch sphere state

ˆn = (sin θ cos ϕ, sin θ sin ϕ, cos θ)

(spinup in the (θ, ϕ) direction), the density ma- trix is

ρ(ˆn) = |ˆnihˆn|

= cos ₂^θ e^iϕ sin ₂^θ

!



cos ₂^θ e^−iϕsin ₂^θ



= cos^{2 θ}₂ e^−iϕsin ₂^θ cos ₂^θ e^iϕ sin ₂^θ cos ₂^θ sin² ₂^θ

!

= 1

21 ^{+ 1} 2

cos θ e^−iϕ sin θ e^iϕ sin θ − cos θ

!

= 1

21 ^{+ 1}

2 sin θ cos ϕ 0 1 1 0

!

! !!

The density matrix for the pure state |ψ(θ, ϕ)i = cos ^θ

2|0i + e^iϕ sin ^θ

2|1i is ρ(~n) = 1

2(1 + ~n · ~σ).

Since density operators are positive, detρ = 1

4(1 − ~n²) ≥ 0,

So ~n² ≤ 0 are all valid states. For density matrices, the Bloch sphere becomes a “ball”.

The density operator is pure if and only if its Bloch “ball” representation is a unit vector.

Ambiguity of the Ensemble Representation

For two ensembles of pure states {p_i, |ψ_ii} and {q_j, |ϕ_ji} if

√p_i|ψ_ii = ^X

j

u_ij√

q_j|ϕ_ji for some unitary matrix u_ij, then

ρ = ^X

i

p_i|ψ_iihψ_i| = ^X

j

q_j|ϕ_jihϕ_j|.

A vector inside the Bloch “ball” can be writ- ten as the sum of unit vectors in infinite ways.

A density operator can also be formed by the