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

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

${\displaystyle {\mathcal {N}}(\rho )\equiv {\frac {||\rho ^{\Gamma _{A}}||_{1}-1}{2}}}$

where:

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

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

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

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

Properties

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

${\displaystyle {\mathcal {N}}(\sum _{i}p_{i}\rho _{i})\leq \sum _{i}p_{i}{\mathcal {N}}(\rho _{i})}$

• Is an entanglement monotone:

${\displaystyle {\mathcal {N}}(P(\rho ))\leq {\mathcal {N}}(\rho )}$

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

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

${\displaystyle E_{N}(\rho )\equiv \log _{2}||\rho ^{\Gamma _{A}}||_{1}}$

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

It relates to the negativity as follows:^[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: ${\displaystyle E_{N}(\rho \otimes \sigma )=E_{N}(\rho )+E_{N}(\sigma )}$
• is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces ${\displaystyle H_{1},H_{2},\ldots }$ (typically with increasing dimension) we can have a sequence of quantum states ${\displaystyle \rho _{1},\rho _{2},\ldots }$ which converges to ${\displaystyle \rho ^{\otimes n_{1}},\rho ^{\otimes n_{2}},\ldots }$ (typically with increasing ${\displaystyle n_{i}}$) in the trace distance, but the sequence ${\displaystyle E_{N}(\rho _{1})/n_{1},E_{N}(\rho _{2})/n_{2},\ldots }$ does not converge to ${\displaystyle E_{N}(\rho )}$.
• is an upper bound to the distillable entanglement