Negativity (quantum mechanics)

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In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute. It is a measure deriving from the PPT criterion for separability. [1] It has been shown to be an entanglement monotone [2] [3] and hence a proper measure of entanglement.

Contents

Definition

The negativity of a subsystem can be defined in terms of a density matrix as:

where:

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of :

where are all of the eigenvalues.

Properties

where is an arbitrary LOCC operation over

Logarithmic negativity

The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement. [4] It is defined as

where is the partial transpose operation and denotes the trace norm.

It relates to the negativity as follows: [1]

Properties

The logarithmic negativity

References

  1. 1 2 K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. A. 58 (2): 883–92. arXiv: quant-ph/9804024 . Bibcode:1998PhRvA..58..883Z. doi:10.1103/PhysRevA.58.883. S2CID   119391103.
  2. J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam. arXiv: quant-ph/0610253 . Bibcode:2006PhDT........59E.
  3. G. Vidal; R. F. Werner (2002). "A computable measure of entanglement". Phys. Rev. A. 65 (3) 032314. arXiv: quant-ph/0102117 . Bibcode:2002PhRvA..65c2314V. doi:10.1103/PhysRevA.65.032314. S2CID   32356668.
  4. M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 95 (9) 090503. arXiv: quant-ph/0505071 . Bibcode:2005PhRvL..95i0503P. doi:10.1103/PhysRevLett.95.090503. PMID   16197196. S2CID   20691213.