Concurrence (quantum computing)

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

Definition[edit]

The concurrence is an entanglement monotone 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 , a Pauli spin matrix, and the eigenvalues listed in decreasing order.

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 formulations[edit]

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.

Properties[edit]

For pure states, the concurrence 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 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[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. ^ Rungta, Pranaw (2001-01-01). "Universal state inversion and concurrence in arbitrary dimensions". Physical Review A. 64 (4). doi:10.1103/PhysRevA.64.042315. 
  6. ^ Bhaskara, Vineeth S.; Panigrahi, Prasanta K. (2017-05-01). "Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product". Quantum Information Processing. 16 (5): 118. doi:10.1007/s11128-017-1568-0. ISSN 1570-0755. 
  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