Calculator
Use a real-valued coefficient matrix. Enter one matrix row per line, with values separated by spaces or commas.
Example data table
| Example state | Dimensions | Coefficient matrix | Normalized Schmidt coefficients | Rank | Entropy |
|---|---|---|---|---|---|
| Bell pair | 2 × 2 | [0.7071, 0; 0, 0.7071] | [0.7071, 0.7071] | 2 | 1.0000 (base 2) |
| Product state | 2 × 2 | [1, 0; 0, 0] | [1, 0] | 1 | 0.0000 |
| Partially entangled qutrit state | 3 × 3 | [0.8, 0, 0; 0, 0.5, 0; 0, 0, 0.3317] | [0.8, 0.5, 0.3317] | 3 | 1.3499 (base 2) |
Formula used
State expansion: |ψ⟩ = Σi,j aij|i⟩A|j⟩B
Coefficient matrix: C = [aij]
Schmidt form: |ψ⟩ = Σk sk|uk⟩A ⊗ |vk⟩B
Reduced states: ρA = CCT, ρB = CTC for a normalized real matrix.
Entanglement entropy: S = −Σk λk log(λk), where λk = sk2.
The calculator diagonalizes the symmetric matrix CCT. Its nonnegative eigenvalues equal the squared singular values of C. After normalizing by the total squared norm, the singular values become Schmidt coefficients for the corresponding pure state.
Purity is computed as Tr(ρA2) = Σλk2. Linear entropy is 1 − purity. The participation ratio equals 1 / purity and estimates how many Schmidt modes contribute effectively.
How to use this calculator
- Choose subsystem dimensions for A and B. The matrix must match those dimensions exactly.
- Paste the coefficient matrix with one row per line. Separate entries using spaces or commas.
- Set the display precision, entropy base, and numerical threshold for rank detection.
- Press Compute decomposition. Results appear above the form, directly below the page header.
- Review the normalized Schmidt coefficients, reduced density matrices, entropy, purity, and mode vectors.
- Use the export buttons to save a CSV summary or generate a PDF report of the current output.
FAQs
1) What does the Schmidt rank tell me?
It counts the nonzero Schmidt coefficients above the selected threshold. Rank one indicates a separable pure state. Higher rank signals entanglement across the chosen bipartition.
2) Why are normalized Schmidt coefficients shown?
Quantum-state coefficients must satisfy total probability one. The tool rescales singular values using the matrix norm so the displayed Schmidt coefficients correspond to a normalized pure state.
3) Can I use rectangular matrices?
Yes. Bipartite subsystems may have different dimensions, such as 2 × 3 or 3 × 5. The Schmidt rank never exceeds the smaller subsystem dimension.
4) What if some entries are zero?
Zero entries are fine. They simply reduce contributions to the coefficient matrix. Entire zero rows or columns can lower rank and may reveal a product-like structure.
5) What entropy base should I choose?
Base 2 gives entanglement in bits, the natural logarithm gives nats, and base 10 gives common-log units. The physics is unchanged; only the numeric scale differs.
6) Why include purity and linear entropy?
They summarize mixedness of the reduced subsystem. Purity equals one for product states and decreases as entanglement spreads weight across more Schmidt modes.
7) Does this page support complex amplitudes?
This implementation is designed for real-valued coefficient matrices. For complex amplitudes, separate handling of phases and Hermitian conjugation would be required.
8) When is concurrence displayed?
Concurrence appears only for 2 × 2 pure states. It provides an additional entanglement measure that ranges from zero for product states to one for maximally entangled states.