Density Matrix Calculator

Density Matrix Calculator computes density matrix, purity, and entropy for a spin-1/2 quantum system, aiding quantum mechanics and quantum computing analysis.

Spin-Up Coefficient Real Part (α_r):

Spin-Up Coefficient Imag. Part (α_i):

Spin-Down Coefficient Real Part (β_r):

Spin-Down Coefficient Imag. Part (β_i):

Formulas Used in Density Matrix Calculator

The calculator uses the following formulas for a spin-1/2 system:

Density Matrix:

\\[ \rho = |\psi\rangle\langle\psi| = \begin{pmatrix} |\alpha|^2 & \alpha \beta^* \\ \beta \alpha^* & |\beta|^2 \end{pmatrix} \\]

Purity:

\\[ \text{Tr}(\rho^2) = |\alpha|^4 + |\beta|^4 + 2 |\alpha|^2 |\beta|^2 \\]

Von Neumann Entropy:

\\[ S = -\text{Tr}(\rho \ln \rho) = -\sum_i \lambda_i \ln_2 \lambda_i \\]

Where:

\\( \rho \\): Density matrix
\\( \alpha = a_r + i a_i, \beta = b_r + i b_i \\): Spin-up and spin-down coefficients
\\( |\alpha|^2 + |\beta|^2 = 1 \\): Normalization condition
\\( \text{Tr}(\rho^2) \\): Purity (1 for pure states, < 1 for mixed states)
\\( \lambda_i \\): Eigenvalues of \\( \rho \\)
\\( S \\): Entropy in bits (base-2 logarithm)

Example Calculations

Example 1: Pure State, Equal Superposition

Input: α_r = 0.707, α_i = 0, β_r = 0.707, β_i = 0

\\[ \rho = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \\] \\[ \text{Tr}(\rho^2) = 0.5^2 + 0.5^2 + 2 \cdot 0.5 \cdot 0.5 = 1 \\] \\[ \lambda_{\pm} = \frac{1 \pm \sqrt{1 – 4 \cdot 0.5 \cdot 0.5 \cdot (1 – 0.5^2)}}{2} = \{1, 0\} \\] \\[ S = – (1 \cdot \log_2 1 + 0 \cdot \log_2 0) = 0 \ \text{bits} \\]

Result: Density Matrix: [[0.5, 0.5], [0.5, 0.5]], Purity: 1, Entropy: 0 bits

Example 2: Pure State, Spin-Up

Input: α_r = 1, α_i = 0, β_r = 0, β_i = 0

\\[ \rho = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \\] \\[ \text{Tr}(\rho^2) = 1^2 + 0^2 + 2 \cdot 1 \cdot 0 = 1 \\] \\[ \lambda_{\pm} = \{1, 0\} \\] \\[ S = – (1 \cdot \log_2 1 + 0 \cdot \log_2 0) = 0 \ \text{bits} \\]

Result: Density Matrix: [[1, 0], [0, 0]], Purity: 1, Entropy: 0 bits

Example 3: Non-Normalized State (Normalized Internally)

Input: α_r = 0.8, α_i = 0.4, β_r = 0.4, β_i = 0.2

\\[ |\alpha|^2 = 0.8^2 + 0.4^2 = 0.8, \quad |\beta|^2 = 0.4^2 + 0.2^2 = 0.2 \\] \\[ \text{Norm} = 0.8 + 0.2 = 1 \\] \\[ \rho = \begin{pmatrix} 0.8 & 0.32 – 0.16i \\ 0.32 + 0.16i & 0.2 \end{pmatrix} \\] \\[ \text{Tr}(\rho^2) \approx 0.84 \\] \\[ \lambda_{\pm} \approx \{0.95, 0.05\} \\] \\[ S \approx – (0.95 \cdot \log_2 0.95 + 0.05 \cdot \log_2 0.05) \approx 0.29 \ \text{bits} \\]

Result: Density Matrix: [[0.8, 0.32-0.16i], [0.32+0.16i, 0.2]], Purity: 0.84, Entropy: 0.29 bits

Formulas Used in Density Matrix Calculator

Example Calculations

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