Concurrence (quantum computing)
In quantum information science, the concurrence is a state invariant involving qubits. Definition. The concurrence is an entanglement monotone (a way of measuring entanglement) defined for a mixed state of two qubits as: in which formula_2 are the eigenvalues, in decreasing order, of the Hermitian matrix with the spin-flipped state of formula_5 and formula_6 a Pauli spin matrix. The complex conjugation formula_7 is taken in the eigenbasis of the Pauli matrix formula_8. Also, here, for a positive semidefinite matrix A, formula_9 denotes a positive semidefinite matrix B such that formula_10. Note that B is a unique  matrix so defined. A generalized version of concurrence for multiparticle pure states in arbitrary dimensions is defined as: in which formula_12 is the reduced density matrix across the bipartition formula_13 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. Other formulations. Alternatively, the formula_14's represent the square roots of the eigenvalues of the non-Hermitian matrix formula_15. Note that each formula_14 is a non-negative real number. From the concurrence, the entanglement  of formation can be calculated. Properties. For pure states, the "square" of the concurrence (also known as the "tangle") is a polynomial formula_17 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.