Concurrence (quantum computing)

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

DefinitionEdit

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

 

in which   are the eigenvalues, in decreasing order, of the Hermitian matrix

 

with

 

the spin-flipped state of   and   a Pauli spin matrix. The complex conjugation   is taken in the eigenbasis of the Pauli matrix  . Also, here, for a positive semidefinite matrix A,   denotes a positive semidefinite matrix B such that  . Note that B is a unique matrix so defined.

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

 

in which   is the reduced density matrix across the bipartition   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 formulationsEdit

Alternatively, the  's represent the square roots of the eigenvalues of the non-Hermitian matrix  .[2] Note that each   is a non-negative real number. From the concurrence, the entanglement of formation can be calculated.

PropertiesEdit

For pure states, the square of the concurrence (also known as the tangle) is a polynomial   invariant in the state's coefficients.[7] For mixed states, the concurrence can be defined by convex roof extension.[3]

For the tangle, 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.

ReferencesEdit

  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: 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