Concurrence (quantum computing)

In quantum computing, the concurrence is an entanglement monotone defined for a mixed state of two qubits as ^{[1] [2] [3] [4]}.

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

in which λ₁,...,λ₄ 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 ρ, σ_y a Pauli spin matrix, and the eigenvalues listed in decreasing order. Alternatively, the λ_i's represent the square roots of the eigenvalues of the non-Hermitian matrix $\rho\tilde{\rho}$.^[2] 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^[5]. For mixed states, the concurrence can be defined by convex roof extension^[3].

For the concurrence, there is monogamy of entanglement^[6][7], 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.

References

1. ^ Scott Hill and William K. Wootters, Entanglement of a Pair of Quantum Bits, 1997.
2. ^ ^{a b} William K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits 1998.
3. ^ ^{a b} Roland Hildebrand, Concurrence revisited, 2007
4. ^ Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, Quantum entanglement, 2009
5. ^ D. Ž. Ðoković and A. Osterloh, On polynomial invariants of several qubits, 2009
6. ^ Valerie Coffman, Joydip Kundu, and William K. Wootters, Distributed entanglement, 2000
7. ^ Tobias J. Osborne and Frank Verstraete, General Monogamy Inequality for Bipartite Qubit Entanglement, 2006