Concurrence (quantum computing)

A state invariant involving qubits, in quantum information science

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

## Definition

[edit]

The concurrence is an entanglement monotone (a way of measuring entanglement) defined for a mixed state of two qubits as:[1][2][3][4]

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

in which  λ 1 , . . . , λ 4 {\displaystyle \lambda _{1},...,\lambda _{4}} are the eigenvalues, in decreasing order, of the Hermitian matrix

R = ρ ρ ~ ρ {\displaystyle R={\sqrt {{\sqrt {\rho }}{\tilde {\rho }}{\sqrt {\rho }}}}}

with

ρ ~ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) {\displaystyle {\tilde {\rho }}=(\sigma _{y}\otimes \sigma _{y})\rho ^{*}(\sigma _{y}\otimes \sigma _{y})}

the spin-flipped state of  ρ {\displaystyle \rho } and  σ y {\displaystyle \sigma _{y}} a Pauli spin matrix. The complex conjugation  ∗ {\displaystyle {}^{*}} is taken in the eigenbasis of the Pauli matrix  σ z . {\displaystyle \sigma _{z}.} Also, here, for a positive semidefinite matrix  A {\displaystyle A} ,  A {\displaystyle {\sqrt {A}}} denotes a positive semidefinite matrix  B {\displaystyle B} such that  B 2 = A {\displaystyle B^{2}=A} . Note that  B {\displaystyle B} is a unique matrix so defined.

A generalized version of concurrence for multiparticle pure states in arbitrary dimensions[5][6] (including the case of continuous-variables in infinite dimensions[7]) is defined as:

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

in which  ρ M {\displaystyle \rho _{\mathcal {M}}} is the reduced density matrix (or its continuous-variable analogue[7]) across the bipartition  M {\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

[edit]

Alternatively, the  λ i {\displaystyle \lambda _{i}} 's represent the square roots of the eigenvalues of the non-Hermitian matrix  ρ ρ ~ {\displaystyle \rho {\tilde {\rho }}} .[2] Note that each  λ i {\displaystyle \lambda _{i}} is a non-negative real number. From the concurrence, the entanglement of formation can be calculated.

## Properties

[edit]

For pure states, the square of the concurrence (also known as the tangle) is a polynomial  S L ( 2 , C ) ⊗ 2 {\displaystyle SL(2,\mathbb {C} )^{\otimes 2}} invariant in the state's coefficients.[8] For mixed states, the concurrence can be defined by convex roof extension.[3]

For the tangle, there is monogamy of entanglement,[9][10] that is, the tangle of a qubit with the rest of the system cannot ever exceed the sum of the tangles of qubit pairs which it is part of.

## References

[edit]

1. ^ Scott Hill and William K. Wootters, Entanglement of a Pair of Quantum Bits, 1997.
2. ^ a b William K. Wootters, Entanglement of Formation of an Arbitrary State of Two Qubits 1998.
3. ^ a b 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 (4): 042315. arXiv:quant-ph/0102040. Bibcode:2001PhRvA..64d2315R. doi:10.1103/PhysRevA.64.042315. S2CID 12594864.
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. S2CID 43754114.
7. ^ a b Swain, S. Nibedita; Bhaskara, Vineeth S.; Panigrahi, Prasanta K. (27 May 2022). "Generalized entanglement measure for continuous-variable systems". Physical Review A. 105 (5): 052441. arXiv:1706.01448. Bibcode:2022PhRvA.105e2441S. doi:10.1103/PhysRevA.105.052441. S2CID 239885759. Retrieved 27 May 2022.
8. ^ D. Ž. Ðoković and A. Osterloh, On polynomial invariants of several qubits, 2009
9. ^ Valerie Coffman, Joydip Kundu, and William K. Wootters, Distributed entanglement, 2000
10. ^ Tobias J. Osborne and Frank Verstraete, General Monogamy Inequality for Bipartite Qubit Entanglement, 2006