In quantum information science, the concurrence is a state invariant involving qubits.
in which are the eigenvalues, in decreasing order, of the Hermitian matrix
the spin-flipped state of and a Pauli spin matrix. The complex conjugation is taken in the eigenbasis of the Pauli matrix . Also, here, for a positive semidefinite matrix A, denotes a positive semidefinite matrix B such that . Note that B is a unique matrix so defined.
in which is the reduced density matrix across the bipartition of the pure state, and it measures how much the complex amplitudes deviate from the constraints required for tensor separability. The faithful nature of the measure admits necessary and sufficient conditions of separability for pure states.
Alternatively, the 's represent the square roots of the eigenvalues of the non-Hermitian matrix . Note that each is a non-negative real number. From the concurrence, the entanglement of formation can be calculated.
For pure states, the square of the concurrence (also known as the tangle) is a polynomial invariant in the state's coefficients. For mixed states, the concurrence can be defined by convex roof extension.
For the tangle, 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.
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