Concurrence (quantum computing)

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

Definition

The concurrencedoes not have a definition

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

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.

A generalized version of concurrence for multiparticle pure states in arbitrary dimensions is defined as:

${\mathcal {C}}_{\mathcal {M}}(\rho )={\sqrt {2(1-{\text{Tr}}\rho _{\mathcal {M}}^{2})}}$

in which $\rho _{\mathcal {M}}$ is the reduced density matrix across the bipartition ${\mathcal {M}}$ 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 $\lambda _{i}$'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.