• Images
• Videos
• Documents
• Books
• Articles

Definition

The negativity of a subsystem A {\displaystyle A} can be defined in terms of a density matrix ρ {\displaystyle \rho } as:

N ( ρ ) ≡ | | ρ Γ A | | 1 − 1 2 {\displaystyle {\mathcal {N}}(\rho )\equiv {\frac {||\rho ^{\Gamma _{A}}||_{1}-1}{2}}}

where:

• ρ Γ A {\displaystyle \rho ^{\Gamma _{A}}} is the partial transpose of ρ {\displaystyle \rho } with respect to subsystem A {\displaystyle A}
• | | X | | 1 = Tr | X | = Tr X † X {\displaystyle ||X||_{1}={\text{Tr}}|X|={\text{Tr}}{\sqrt {X^{\dagger }X}}} is the trace norm or the sum of the singular values of the operator X {\displaystyle X} .

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of ρ Γ A {\displaystyle \rho ^{\Gamma _{A}}} :

N ( ρ ) = | ∑ λ i < 0 λ i | = ∑ i | λ i | − λ i 2 {\displaystyle {\mathcal {N}}(\rho )=\left|\sum _{\lambda _{i}<0}\lambda _{i}\right|=\sum _{i}{\frac {|\lambda _{i}|-\lambda _{i}}{2}}}

where λ i {\displaystyle \lambda _{i}} are all of the eigenvalues.

Properties

• Is a convex function of ρ {\displaystyle \rho } :

N ( ∑ i p i ρ i ) ≤ ∑ i p i N ( ρ i ) {\displaystyle {\mathcal {N}}(\sum _{i}p_{i}\rho _{i})\leq \sum _{i}p_{i}{\mathcal {N}}(\rho _{i})}

• Is an entanglement monotone:

N ( P ( ρ ) ) ≤ N ( ρ ) {\displaystyle {\mathcal {N}}(P(\rho ))\leq {\mathcal {N}}(\rho )}

where P ( ρ ) {\displaystyle P(\rho )} is an arbitrary LOCC operation over ρ {\displaystyle \rho }

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

E N ( ρ ) ≡ log 2 ⁡ | | ρ Γ A | | 1 {\displaystyle E_{N}(\rho )\equiv \log _{2}||\rho ^{\Gamma _{A}}||_{1}}

where Γ A {\displaystyle \Gamma _{A}} is the partial transpose operation and | | ⋅ | | 1 {\displaystyle ||\cdot ||_{1}} denotes the trace norm.

It relates to the negativity as follows:⁵

E N ( ρ ) := log 2 ⁡ ( 2 N + 1 ) {\displaystyle E_{N}(\rho ):=\log _{2}(2{\mathcal {N}}+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: E N ( ρ ⊗ σ ) = E N ( ρ ) + E N ( σ ) {\displaystyle E_{N}(\rho \otimes \sigma )=E_{N}(\rho )+E_{N}(\sigma )}
• is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces H 1 , H 2 , … {\displaystyle H_{1},H_{2},\ldots } (typically with increasing dimension) we can have a sequence of quantum states ρ 1 , ρ 2 , … {\displaystyle \rho _{1},\rho _{2},\ldots } which converges to ρ ⊗ n 1 , ρ ⊗ n 2 , … {\displaystyle \rho ^{\otimes n_{1}},\rho ^{\otimes n_{2}},\ldots } (typically with increasing n i {\displaystyle n_{i}} ) in the trace distance, but the sequence E N ( ρ 1 ) / n 1 , E N ( ρ 2 ) / n 2 , … {\displaystyle E_{N}(\rho _{1})/n_{1},E_{N}(\rho _{2})/n_{2},\ldots } does not converge to E N ( ρ ) {\displaystyle E_{N}(\rho )} .
• is an upper bound to the distillable entanglement

• This page uses material from Quantiki licensed under GNU Free Documentation License 1.2

References

1. 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. /wiki/ArXiv_(identifier) ↩

2. J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam. arXiv:quant-ph/0610253. Bibcode:2006PhDT........59E. /wiki/ArXiv_(identifier) ↩

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. /wiki/ArXiv_(identifier) ↩

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. /wiki/ArXiv_(identifier) ↩

5. 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. /wiki/ArXiv_(identifier) ↩