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 been 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  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

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* 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

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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

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:[1]

E N ( ρ ) := log 2 ⁡ ( 2 N + 1 ) {\displaystyle E_{N}(\rho ):=\log _{2}(2{\mathcal {N}}+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:  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

## 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.