Negativity (quantum mechanics)

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.

## Contents

* 1 Definition
* 1.1 Properties
* 2 Logarithmic negativity
* 2.1 Properties
* 3 References

## Definition

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

𝒩(ρ)≡||ρΓA||1−12

where:

* ρΓA is the partial transpose of ρ with respect to subsystem A
* ||X||1=Tr|X|=TrX†X 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

* Is a convex function of ρ:

𝒩(∑ipiρi)≤∑ipi𝒩(ρi)

* Is an entanglement monotone:

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(ρ⊗σ)=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 ρ⊗n1,ρ⊗n2,… (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. Bibcode: 1998PhRvA..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. Bibcode: 2002PhRvA..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. Bibcode: 2005PhRvL..95i0503P.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Negativity (quantum mechanics). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Negativity_(quantum_mechanics)&oldid=206841"