Concurrence (quantum computing)

Definition

The concurrence is an entanglement monotone defined for a mixed state of two qubits as:

in which λ 1 , . . . , λ 4 are the eigenvalues, in decreasing order, of the Hermitian matrix

with

the spin-flipped state of ρ , σ y a Pauli spin matrix, and the eigenvalues listed in decreasing order.

Other formulations

Alternatively, the λ i 's represent the square roots of the eigenvalues of the non-Hermitian matrix ρ ρ ~ . Note that each λ i is a non-negative real number. From the concurrence, the entanglement of formation can be calculated.

Properties

For pure states, the concurrence is a polynomial S L ( 2 , C ) ⊗ 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.