state_props.concurrence
Concurrence property.
Module Contents
Functions
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Calculate the concurrence of a bipartite state [1]. |
- state_props.concurrence.concurrence(rho)
Calculate the concurrence of a bipartite state [1].
The concurrence of a bipartite state \(\rho\) is defined as
\[\max(0, \lambda_1 - \lambda_2 - \lambda_3 - \lambda_4),\]where \(\lambda_1, \ldots, \lambda_4\) are the square roots of the eigenvalues in decreasing order of the matrix
\[\rho\tilde{\rho} = \rho \sigma_y \otimes \sigma_y \rho^* \sigma_y \otimes \sigma_y.\]Concurrence can serve as a measure of entanglement.
Examples
Consider the following Bell state:
\[u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right).\]The concurrence of the density matrix \(\rho = u u^*\) defined by the vector \(u\) is given as
\[\mathcal{C}(\rho) \approx 1.\]The following example calculates this quantity using the
toqito
package.>>> import numpy as np >>> from toqito.states import basis >>> from toqito.state_props import concurrence >>> e_0, e_1 = basis(2, 0), basis(2, 1) >>> e_00, e_11 = np.kron(e_0, e_0), np.kron(e_1, e_1) >>> u_vec = 1 / np.sqrt(2) * (e_00 + e_11) >>> rho = u_vec * u_vec.conj().T >>> concurrence(rho) 0.9999999999999998
Consider the concurrence of the following product state
\[v = |0\rangle \otimes |1 \rangle.\]As this state has no entanglement, the concurrence is zero.
>>> import numpy as np >>> from toqito.states import basis >>> from toqito.state_props import concurrence >>> e_0, e_1 = basis(2, 0), basis(2, 1) >>> v_vec = np.kron(e_0, e_1) >>> sigma = v_vec * v_vec.conj().T >>> concurrence(sigma) 0
References
- Raises:
ValueError – If system is not bipartite.
- Parameters:
rho (numpy.ndarray) – The bipartite system specified as a matrix.
- Returns:
The concurrence of the bipartite state \(\rho\).
- Return type:
float