Negativity (quantum mechanics)

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.

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]

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

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