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. It has been shown to be an entanglement monotone and hence a proper measure of entanglement.

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

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})}

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. 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:

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 licensed under GNU Free Documentation License 1.2