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:[1][2][3][4]

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

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

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

the spin-flipped state of ${\displaystyle \rho }$ and ${\displaystyle \sigma _{y}}$ a Pauli spin matrix. The complex conjugation ${\displaystyle {}^{*}}$ is taken in the eigenbasis of the Pauli matrix ${\displaystyle \sigma _{z}}$.

A generalized version of concurrence for multiparticle pure states in arbitrary dimensions is defined as:[5][6]

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

in which ${\displaystyle \rho _{\mathcal {M}}}$ is the reduced density matrix 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.

Other formulations

Alternatively, the ${\displaystyle \lambda _{i}}$'s represent the square roots of the eigenvalues of the non-Hermitian matrix ${\displaystyle \rho {\tilde {\rho }}}$.[2] Note that each ${\displaystyle \lambda _{i}}$ 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 ${\displaystyle SL(2,\mathbb {C} )^{\otimes 2}}$ invariant in the state's coefficients.[7] For mixed states, the concurrence can be defined by convex roof extension.[3]

For the concurrence, there is monogamy of entanglement,[8][9] 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. William K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits 1998.
3. Roland Hildebrand, Concurrence revisited, 2007
4. Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, Karol Horodecki, Quantum entanglement, 2009
5. P. Rungta; V. Bužek; C. M. Caves; M. Hillery; G. J. Milburn (2001). "Universal state inversion and concurrence in arbitrary dimensions". Phys. Rev. A. 64: 042315. arXiv:quant-ph/0102040. Bibcode:2001PhRvA..64d2315R. doi:10.1103/PhysRevA.64.042315.
6. Bhaskara, Vineeth S.; Panigrahi, Prasanta K. (2017). "Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product". Quantum Information Processing. 16 (5): 118. arXiv:1607.00164. Bibcode:2017QuIP...16..118B. doi:10.1007/s11128-017-1568-0.
7. D. Ž. Ðoković and A. Osterloh, On polynomial invariants of several qubits, 2009
8. Valerie Coffman, Joydip Kundu, and William K. Wootters, Distributed entanglement, 2000
9. Tobias J. Osborne and Frank Verstraete, General Monogamy Inequality for Bipartite Qubit Entanglement, 2006