Calculated Result
Density Matrix Calculator
This calculator builds a two-level mixed-state density matrix using two weighted pure states. It also evaluates entropy, purity, and magnetization.
Density Matrix
| Element | Value |
|---|---|
| ρ₁₁ | - |
| ρ₁₂ | - |
| ρ₂₁ | - |
| ρ₂₂ | - |
Eigenvalues and Graph
| Quantity | Value |
|---|---|
| λ₁ | - |
| λ₂ | - |
Example Data Table
| Case | p₁ | p₂ | |ψ₁⟩ | |ψ₂⟩ | Trace | Purity |
|---|---|---|---|---|---|---|
| Pure ground state | 1.00 | 0.00 | [1, 0] | [0, 1] | 1.00 | 1.00 |
| Balanced mixture | 0.50 | 0.50 | [1, 0] | [0, 1] | 1.00 | 0.50 |
| Biased mixture | 0.65 | 0.35 | [1, 0] | [0, 1] | 1.00 | 0.55 |
Formula Used
The calculator models a two-level mixed state with two weighted pure states. Each state contributes an outer product scaled by its probability.
|ψ⟩ = [a, b]ᵀ, where a and b may be complex.
Tr(ρ) = ρ₁₁ + ρ₂₂
Purity = Tr(ρ²)
S = -Tr(ρ ln ρ) = -Σ λᵢ ln(λᵢ)
M = Tr(ρ μσz) = μ(ρ₁₁ - ρ₂₂)
Entropy is computed from the eigenvalues of the density matrix. The natural logarithm is used, so entropy is reported in nats.
How to Use This Calculator
- Enter probabilities for the two states. They should sum to one.
- Enter the real and imaginary parts of amplitudes a and b for each state.
- Provide the magnetic moment if you want magnetization.
- Click Calculate Density Matrix to generate outputs.
- Review the density matrix, eigenvalues, entropy, purity, and magnetization.
- Use the CSV or PDF buttons to export results.
Requested Answers
Calculate entropy s in terms of the density matrix operator
The entropy of a quantum state is the von Neumann entropy. It is written as S = -Tr(ρ ln ρ). In practice, diagonalize the density matrix, find eigenvalues λᵢ, then compute S = -Σ λᵢ ln λᵢ. Zero eigenvalues contribute zero because the limiting term vanishes.
How to calculate density matrix
Write each normalized state vector |ψᵢ⟩, form the outer product |ψᵢ⟩⟨ψᵢ|, multiply by its probability pᵢ, then sum all weighted outer products. The result is ρ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|. For a pure state, the density matrix reduces to ρ = |ψ⟩⟨ψ|.
Calculating magnetization using density matrix
Choose the magnetization operator M̂ for the system, then compute the expectation value using ⟨M⟩ = Tr(ρM̂). For a two-level spin model aligned with z, M̂ = μσz. The magnetization becomes μ(ρ₁₁ − ρ₂₂), so population imbalance directly controls the result.
FAQs
1. What is a density matrix?
A density matrix describes a quantum state using probabilities and amplitudes. It handles both pure states and mixed states, making it useful for spectroscopy, magnetic systems, quantum chemistry, and statistical ensembles.
2. Why must the trace equal one?
The trace represents total probability. A valid physical density matrix has Tr(ρ) = 1 because all state probabilities together must account for the full system.
3. What does purity tell me?
Purity measures how mixed the state is. A value of 1 indicates a pure state, while smaller values indicate a stronger mixture. For a two-level system, the minimum purity is 0.5.
4. What is von Neumann entropy?
Von Neumann entropy measures uncertainty in a quantum state. It is zero for a pure state and increases as the state becomes more mixed. This calculator reports entropy using natural logarithms.
5. Can amplitudes be complex numbers?
Yes. Quantum state amplitudes are often complex. This calculator accepts real and imaginary parts separately, then builds the correct outer products and matrix elements.
6. Why are eigenvalues important here?
Eigenvalues reveal whether the density matrix is physically meaningful and are needed for entropy. Valid states should have non-negative eigenvalues, and their sum should equal one.
7. How is magnetization obtained?
Magnetization is the expectation value of a magnetization operator. This page uses a simple two-level z-axis model, where magnetization depends on diagonal populations and the chosen magnetic moment.
8. Can I use this for chemistry studies?
Yes. It helps with introductory density matrix analysis relevant to quantum chemistry, spin systems, ensemble descriptions, and educational demonstrations of mixed-state behavior.