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Topic 11 of 12

Density Matrices & Open Systems

A pure quantum state $|\psi\rangle$ describes a system with complete information. But real quantum systems interact with their environment, and we often have incomplete knowledge β€” these are mixed states, described by density matrices. The density matrix formalism is essential for understanding decoherence, entanglement structure, and the transition from quantum to classical behavior.

Key Concepts

Density Operator
A density operator ρ\rho is a positive semi-definite Hermitian matrix with Tr(ρ)=1\text{Tr}(\rho) = 1. For a pure state ∣ψ⟩|\psi\rangle, we have ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle\psi|. Mixed states are convex combinations ρ=βˆ‘ipi∣ψi⟩⟨ψi∣\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i| representing classical uncertainty over quantum states.
Pure vs Mixed States
A state is pure if Tr(ρ2)=1\text{Tr}(\rho^2) = 1 and mixed if Tr(ρ2)<1\text{Tr}(\rho^2) < 1. The purity γ=Tr(ρ2)\gamma = \text{Tr}(\rho^2) ranges from 1/d1/d (maximally mixed, dd-dimensional system) to 1 (pure state). Equivalently, a state is pure iff ρ2=ρ\rho^2 = \rho (it is a projector).
Von Neumann Entropy
For a state ρ\rho, the von Neumann entropy is S(ρ)=βˆ’Tr(ρlog⁑2ρ)=βˆ’βˆ‘iΞ»ilog⁑2Ξ»iS(\rho) = -\text{Tr}(\rho \log_2 \rho) = -\sum_i \lambda_i \log_2 \lambda_i where Ξ»i\lambda_i are the eigenvalues. S=0S = 0 for pure states and S=log⁑2dS = \log_2 d for the maximally mixed state in dd dimensions. It quantifies quantum uncertainty and entanglement.
Partial Trace
For a bipartite system ABAB, the reduced density matrix of subsystem AA is ρA=TrB(ρAB)\rho_A = \text{Tr}_B(\rho_{AB}) β€” tracing out the BB degrees of freedom. This is how subsystems are described when we ignore (or cannot access) part of the system. For entangled ∣Φ+⟩|\Phi^+\rangle, tracing out one qubit gives ρ=I/2\rho = I/2 (maximally mixed).
Decoherence
Interaction of a quantum system with its environment causes decoherence: off-diagonal elements of ρ\rho (coherences) decay exponentially over the decoherence time T2T_2. The system transitions from a superposition ∣ψ⟩⟨ψ∣|\psi\rangle\langle\psi| to a classical mixture βˆ‘i∣ci∣2∣i⟩⟨i∣\sum_i |c_i|^2 |i\rangle\langle i|. This is why we don't see quantum superpositions in everyday life.
Lindblad Master Equation
The Lindblad equation ρ˙=βˆ’i[H,ρ]+βˆ‘kΞ³k(LkρLkβ€ βˆ’12Lk†LkΟβˆ’12ρLk†Lk)\dot{\rho} = -i[H,\rho] + \sum_k \gamma_k(L_k\rho L_k^\dagger - \frac{1}{2}L_k^\dagger L_k\rho - \frac{1}{2}\rho L_k^\dagger L_k) describes open quantum system dynamics. The LkL_k are jump operators modeling noise channels (amplitude damping, dephasing). It generalizes the SchrΓΆdinger equation to include dissipation.

Key Equations

Density Matrix of Pure State
ρ=∣ψ⟩⟨ψ∣,Tr(ρ)=1,ρ2=ρ\rho = |\psi\rangle\langle\psi|, \quad \text{Tr}(\rho) = 1, \quad \rho^2 = \rho
Pure states are rank-1 projectors with purity Tr(ρ²) = 1.
Von Neumann Entropy
S(ρ)=βˆ’Tr(ρlog⁑2ρ)=βˆ’βˆ‘iΞ»ilog⁑2Ξ»iS(\rho) = -\text{Tr}(\rho\log_2\rho) = -\sum_i \lambda_i \log_2 \lambda_i
Quantum analogue of Shannon entropy. Zero for pure states, logβ‚‚d for maximally mixed d-dim state.
Purity
γ=Tr(ρ2)∈[1d,1]\gamma = \text{Tr}(\rho^2) \in \left[\frac{1}{d}, 1\right]
Purity equals 1 for pure states, 1/d for the maximally mixed state in d dimensions.
Partial Trace
ρA=TrB(ρAB)=βˆ‘j⟨jB∣ρAB∣jB⟩\rho_A = \text{Tr}_B(\rho_{AB}) = \sum_j \langle j_B|\rho_{AB}|j_B\rangle
Reduced density matrix: trace over the environment/partner system to get the marginal state.
Maximally Mixed Qubit
ρ=I2=12(1001),S=1 bit,γ=12\rho = \frac{I}{2} = \frac{1}{2}\begin{pmatrix}1&0\\0&1\end{pmatrix}, \quad S = 1 \text{ bit}, \quad \gamma = \frac{1}{2}
Equal mixture of |0⟩ and |1⟩: maximum entropy, minimum purity.
Worked Example

Von Neumann Entropy Calculation

Problem

Compute the von Neumann entropy of ρ=34∣0⟩⟨0∣+14∣1⟩⟨1∣\rho = \frac{3}{4}|0\rangle\langle 0| + \frac{1}{4}|1\rangle\langle 1|.

Solution

ρ\rho is already diagonal β€” eigenvalues are Ξ»1=3/4\lambda_1 = 3/4 and Ξ»2=1/4\lambda_2 = 1/4.

Apply the von Neumann entropy formula:

S(ρ)=βˆ’34log⁑2 ⁣34βˆ’14log⁑2 ⁣14S(\rho) = -\frac{3}{4}\log_2\!\frac{3}{4} - \frac{1}{4}\log_2\!\frac{1}{4}
=βˆ’34(βˆ’0.415)βˆ’14(βˆ’2)=0.311+0.500=0.811Β bits= -\frac{3}{4}(-0.415) - \frac{1}{4}(-2) = 0.311 + 0.500 = 0.811 \text{ bits}
Answer S(ρ) β‰ˆ 0.811 bits (between 0 for pure and 1 for maximally mixed)
Practice

Exercises

6 problems
1 of 6

For the maximally mixed qubit ρ=I/2\rho = I/2, what is the purity Tr(ρ2)\text{Tr}(\rho^2)?

2 of 6

Von Neumann entropy of a pure state ∣0⟩⟨0∣|0\rangle\langle 0| (in bits)?

bits
3 of 6

Von Neumann entropy of the maximally mixed qubit ρ=I/2\rho = I/2 (in bits)?

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4 of 6

For a pure state ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle\psi|, what is Tr(ρ2)\text{Tr}(\rho^2)?

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5 of 6

For ρ=34∣0⟩⟨0∣+14∣1⟩⟨1∣\rho = \frac{3}{4}|0\rangle\langle 0| + \frac{1}{4}|1\rangle\langle 1|, what is Tr(ρ2)\text{Tr}(\rho^2)?

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6 of 6

Von Neumann entropy of ρ=34∣0⟩⟨0∣+14∣1⟩⟨1∣\rho = \frac{3}{4}|0\rangle\langle 0| + \frac{1}{4}|1\rangle\langle 1| in bits (to 3 decimal places)?

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Key Takeaways

  • Density matrix ρ\rho: positive semi-definite, Tr(ρ)=1\text{Tr}(\rho)=1; pure iff Tr(ρ2)=1\text{Tr}(\rho^2)=1.
  • Von Neumann entropy S=βˆ’Tr(ρlog⁑2ρ)S=-\text{Tr}(\rho\log_2\rho): zero for pure states, log⁑2d\log_2 d for maximally mixed.
  • Partial trace ρA=TrB(ρAB)\rho_A = \text{Tr}_B(\rho_{AB}) gives the reduced state of a subsystem.
  • Decoherence destroys off-diagonal elements of ρ\rho (quantum coherences) over timescale T2T_2.
  • The Lindblad master equation governs open quantum system evolution including dissipation and dephasing.
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