Concurrence (quantum computing)

Definition

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

${\displaystyle \mathcal{C}(\rho )\equiv max(0,{\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4)}$

in which ${\displaystyle {\lambda }_1,...,{\lambda }_4}$ are the eigenvalues, in decreasing order, of the Hermitian matrix

${\displaystyle R=\sqrt{\sqrt{\rho }\tilde{\rho }\sqrt{\rho }}}$

with

${\displaystyle \tilde{\rho }=({\sigma }_y\otimes {\sigma }_y){\rho }^{\ast }({\sigma }_y\otimes {\sigma }_y)}$

the spin-flipped state of ${\displaystyle \rho }$ and ${\displaystyle {\sigma }_y}$ a Pauli spin matrix. The complex conjugation ${\displaystyle {}^{\ast }}$ is taken in the eigenbasis of the Pauli matrix ${\displaystyle {\sigma }_z.}$ Also, here, for a positive semidefinite matrix ${\displaystyle A}$, ${\displaystyle \sqrt{A}}$ denotes a positive semidefinite matrix ${\displaystyle B}$ such that ${\displaystyle B^2=A}$. Note that ${\displaystyle B}$ 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:

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

in which ${\displaystyle {\rho }_{\mathcal{M}}}$ is the reduced density matrix (or its continuous-variable analogue) across the bipartition ${\displaystyle \mathcal{M}}$ 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.