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

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

𝒩(ρ)||ρΓA||112

where:

  • ρΓA is the partial transpose of ρ with respect to subsystem A
  • ||X||1=Tr|X|=TrXX is the trace norm or the sum of the singular values of the operator X.

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

𝒩(ρ)=|λi<0λi|=i|λi|λi2

where λi are all of the eigenvalues.

Properties

𝒩(ipiρi)ipi𝒩(ρi)
𝒩(P(ρ))𝒩(ρ)

where P(ρ) is an arbitrary LOCC operation over ρ

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

EN(ρ)log2||ρΓA||1

where ΓA is the partial transpose operation and ||||1 denotes the trace norm.

It relates to the negativity as follows:[1]

EN(ρ):=log2(2𝒩+1)

Properties

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: EN(ρσ)=EN(ρ)+EN(σ)
  • is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces H1,H2, (typically with increasing dimension) we can have a sequence of quantum states ρ1,ρ2, which converges to ρn1,ρn2, (typically with increasing ni) in the trace distance, but the sequence EN(ρ1)/n1,EN(ρ2)/n2, does not converge to EN(ρ).
  • is an upper bound to the distillable entanglement

References

  • This page uses material from Quantiki licensed under GNU Free Documentation License 1.2
  1. 1.0 1.1 K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. A 58 (2): 883–92. doi:10.1103/PhysRevA.58.883. Bibcode1998PhRvA..58..883Z. 
  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. doi:10.1103/PhysRevA.65.032314. Bibcode2002PhRvA..65c2314V. 
  4. M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 95 (9): 090503. doi:10.1103/PhysRevLett.95.090503. PMID 16197196. Bibcode2005PhRvL..95i0503P.