Concurrence (quantum computing)

In quantum computing, the concurrence is an entanglement monotone defined for a mixed state of two qubits as .

$\mathcal{C}(\rho)\equiv\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$

in which $\lambda_1,...,\lambda_4$ are the eigenvalues of the Hermitian matrix

$R = \sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$

with

$\tilde{\rho} = (\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$

the spin-flipped state of $\rho$, $\sigma_y$ a Pauli spin matrix, and the eigenvalues listed in decreasing order. Alternatively, the $\lambda_{i}$'s represent the square roots of the eigenvalues of the non-Hermitian matrix $\rho\tilde{\rho}$. From the concurrence, the entanglement of formation can be calculated.

For pure states, the concurrence is a polynomial $SL(2,\mathbb{C})^{\otimes 2}$ invariant in the state's coefficients. For mixed states, the concurrence can be defined by convex roof extension.

For the concurrence, there is monogamy of entanglement, that is, the concurrence of a qubit with the rest of the system cannot ever exceed the sum of the concurrences of qubit pairs which it is part of.