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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 shown to be an entanglement monotone[2][3] and hence a proper measure of entanglement.

Definition

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

where:

  • is the partial transpose of with respect to subsystem
  • is the trace norm or the sum of the singular values of the operator .

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

where are all of the eigenvalues.

Properties

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where is an arbitrary LOCC operation over

Logarithmic negativity

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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

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The logarithmic negativity

  • can be zero even if the state is entangled (if the state is PPT entangled).
  • does not reduce to the entropy of entanglement on pure states like most other entanglement measures.
  • is additive on tensor products:
  • is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces (typically with increasing dimension) we can have a sequence of quantum states which converges to (typically with increasing ) in the trace distance, but the sequence does not converge to .
  • is an upper bound to the distillable entanglement

References

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  • This page uses material from Quantiki licensed under GNU Free Documentation License 1.2
  1. ^ a b 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.