Concurrence (quantum informatics)

The Concurrence (English for “participation”, “agreement”, “meeting”) describes in quantum informatics a measure of the entanglement of two qubits . The concurrence is exactly zero if a state is separable and is equal to one for maximally entangled states.

definition

The concurrence is defined as a function of the density matrix of a state ${\displaystyle \rho }$

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

Here the eigenvalues ​​are in descending order of the Hermitian matrix ${\displaystyle \lambda _{1},...,\lambda _{4}}$

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

With

${\displaystyle {\tilde {\rho }}=(\sigma _{y}\otimes \sigma _{y})\rho ^{*}(\sigma _{y}\otimes \sigma _{y})}$

the spin-flipped state of and the Pauli matrix . Alternatively, they represent the roots of the eigenvalues ​​of the non-Hermitian matrix . All of them are non-negative real numbers. ${\displaystyle \rho }$ ${\displaystyle \sigma _{y}}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle \rho {\tilde {\rho }}}$ ${\displaystyle \lambda _{i}}$

For a pure state the definition is simplified to ${\displaystyle |\psi \rangle =\alpha |0,0\rangle +\beta |0,1\rangle +\gamma |1,0\rangle +\delta |1,1\rangle }$

${\displaystyle {\mathcal {C}}(|\psi \rangle )=2|\alpha \delta -\beta \gamma |\ .}$

properties

The formation entanglement for bipartite states can be calculated from the concurrence by means of a monotone mapping. The formation entanglement is also defined for qubit states of higher dimensions.

For pure states the concurrence is a polynomial invariant among the coefficients of the state. For mixed states the concurrence can be defined as a convex continuation. ${\displaystyle SL(2,\mathbb {C} )^{\otimes 2}}$

The concurrence of a qubit with the rest of a system cannot exceed the sum of the concurrences of the qubit pairs to which it belongs.

generalization

For multi-qubit states with more than two dimensions, the definition can be generalized to generalized concurrence . The generalization is not clear-cut.

Individual evidence

1. ^ Scott Hill, William K. Wootters: Entanglement of a Pair of Quantum Bits . 1997, arxiv : quant-ph / 9703041 .
2. ^ ^{A b} William K. Wootters: Entanglement of Formation of an Arbitrary State of Two Qubits . 1998. doi: 10.1103 / PhysRevLett.80.2245
3. ↑ ^{a b} Roland Hildebrand: Concurrence revisited . 2007, doi: 10.1063 / 1.2795840
4. ^ Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki: Quantum entanglement . 2009, doi: 10.1103 / RevModPhys.81.865
5. ↑ D. Ž. Ðoković, A. Osterloh: On polynomial invariants of several qubits . 2009, doi: 10.1063 / 1.3075830
6. ↑ Valerie Coffman, Joydip Kundu, William K. Wootters: entanglement Distributed . 2000, doi: 10.1103 / PhysRevA.61.052306
7. ↑ Tobias J. Osborne, Frank Verstraete: General Monogamy Inequality for Bipartite Qubit Entanglement . 2006, doi: 10.1103 / PhysRevLett.96.220503
8. ↑ Shao-Ming Fei et al .: Entanglement of formation for a class of quantum states . 2003 arxiv : quant-ph / 0304095 .