Schmidt Rank and Entanglement

Theory

In quantum mechanics, a pure state of a bipartite system can be expressed in terms of a product basis. For two qubits, any pure state can be written as ∑ C_(ij)|ij⟩.

By performing a Singular Value Decomposition (SVD) on the coefficient matrix C, we obtain C = U Σ V^T, where Σ is a diagonal matrix of non-negative real numbers called Schmidt coefficients, and U, V are orthogonal matrices (representing local basis changes). The number of non-zero Schmidt coefficients is the Schmidt rank. A state is entangled if and only if its Schmidt rank is strictly greater than 1.

The eigenvalues of the reduced density matrices (obtained by partial trace) are the squares of the Schmidt coefficients. We can measure entanglement via entanglement entropy S = -∑ \lambda_i \log_2 \lambda_i, purity ∑ \lambda_i², and concurrence, which is given by 2|det C| for this specific pure state class.

Worked Example

Consider the state generated with Schmidt angle θ = 30^(\circ) and local rotation α = 0^(\circ). You can observe this exact configuration directly in the module.

The computation follows these steps:

1. The matrix C = [[cos 30^(\circ), 0], [], [0, sin 30^(\circ)]].
2. Schmidt coefficients are cos 30^(\circ) and sin 30^(\circ). The reduced density matrix eigenvalues are \lambda_1 = cos² 30^(\circ) = 0.75 and \lambda_2 = sin² 30^(\circ) = 0.25.
3. Entanglement entropy: -0.75 \log_2 0.75 - 0.25 \log_2 0.25 ≈ 0.811 bits.
4. Concurrence: sin(2 × 30^(\circ)) = sin 60^(\circ) ≈ 0.866.
5. Since there are two non-zero eigenvalues, the expected outcome for the Schmidt rank is exactly 2.

Experiment

Use the interactive module to explore how the local rotation changes basis probabilities without affecting entanglement entropy or concurrence, whereas changing the Schmidt angle fundamentally alters entanglement.

1. Select the Schmidt angle degrees option 30.
2. Select the Local rotation degrees option 0.
3. Select the Basis probe option 00.
4. Observe the resulting Schmidt rank which becomes exactly 2.

Open the interactive module to visualize these transformations.

Limitations

This model focuses exclusively on real-coefficient pure states of two qubits. For mixed states, the direct SVD approach to the coefficient matrix is insufficient to characterize entanglement. Furthermore, states with complex coefficients require unitary instead of orthogonal rotations.

References

The primary references for this quantum information theory model include:

• Preskill, J. (1998). Lecture Notes for Physics 229: Quantum Information and Computation, Chapter 2.