Concurrence (quantum computing)

In quantum information science, the concurrence is a state invariant involving qubits.

Definition

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

${\mathcal {C}}(\rho )\equiv \max(0,\lambda _{1}-\lambda _{2}-\lambda _{3}-\lambda _{4})$

in which $\lambda _{1},...,\lambda _{4}$ are the eigenvalues, in decreasing order, 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 $\rho$, $\sigma _{y}$ a Pauli spin matrix, and the eigenvalues listed in decreasing order.

Other formulations

Alternatively, the $\lambda _{i}$'s represent the square roots of the eigenvalues of the non-Hermitian matrix $\rho {\tilde {\rho }}$. Note that each $\lambda _{i}$ 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 $SL(2,\mathbb {C} )^{\otimes 2}$ 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.