Quantum Entanglement Entropy Calculator
Quantum Entanglement Entropy Calculator computes von Neumann entropy for a two-qubit system, aiding quantum information and quantum computing studies.
Formulas Used in Quantum Entanglement Entropy Calculator
The calculator computes the von Neumann entanglement entropy for a two-qubit system:
Two-Qubit State:
\\[ |\psi\rangle = a |00\rangle + b |01\rangle + c |10\rangle + d |11\rangle \\]Reduced Density Matrix (Trace over Qubit B):
\\[ \rho_A = \begin{pmatrix} |a|^2 + |b|^2 & a c^* + b d^* \\ c a^* + d b^* & |c|^2 + |d|^2 \end{pmatrix} \\]Von Neumann Entropy:
\\[ S(\rho_A) = -\sum_i \lambda_i \log_2 \lambda_i \\]Eigenvalues of \\( \rho_A \\):
\\[ \lambda_{\pm} = \frac{1 \pm \sqrt{1 – 4 \det(\rho_A)}}{2} \\] \\[ \det(\rho_A) = (|a|^2 + |b|^2)(|c|^2 + |d|^2) – |a c^* + b d^*|^2 \\]Where:
- \\( a, b, c, d \\): Complex coefficients (\\( |a|^2 + |b|^2 + |c|^2 + |d|^2 = 1 \\))
- \\( \rho_A \\): Reduced density matrix of subsystem A
- \\( \lambda_i \\): Eigenvalues of \\( \rho_A \\)
- \\( S \\): Entanglement entropy (bits)
Example Calculations
Example 1: Maximally Entangled State (Bell State)
Input: a_r = 0.707, a_i = 0, b_r = 0, b_i = 0, c_r = 0, c_i = 0, d_r = 0.707, d_i = 0
Result: Reduced Density Matrix: [[0.5, 0], [0, 0.5]], Eigenvalues: [0.5, 0.5], Entropy: 1 bits
Example 2: Separable State
Input: a_r = 1, a_i = 0, b_r = 0, b_i = 0, c_r = 0, c_i = 0, d_r = 0, d_i = 0
Result: Reduced Density Matrix: [[1, 0], [0, 0]], Eigenvalues: [1, 0], Entropy: 0 bits
Example 3: Partially Entangled State
Input: a_r = 0.8, a_i = 0, b_r = 0, b_i = 0, c_r = 0, c_i = 0, d_r = 0.6, d_i = 0
Result: Reduced Density Matrix: [[0.64, 0.48], [0.48, 0.36]], Eigenvalues: [0.951, 0.049], Entropy: 0.31 bits