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 are the eigenvalues, in decreasing order, of the Hermitian matrix with 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. A generalized version of concurrence for multiparticle pure states in arbitrary dimensions (including the case of continuous-variables in infinite dimensions) is defined as: in which is the reduced density matrix (or its continuous-variable analogue) 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. Other formulations 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. Properties 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 tangle of a qubit with the rest of the system cannot ever exceed the sum of the tangles of qubit pairs which it is part of. References Theoretical computer science Quantum information science