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 shown to be an entanglement monotone^[2][3] and hence a proper measure of entanglement.

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

${\displaystyle {𝒩}(\rho )\equiv \frac{||{\rho }^{{\mathrm{\Gamma }}_A}|{|}_1-1}{2}}$

where:

• ${\displaystyle {\rho }^{{\mathrm{\Gamma }}_A}}$ is the partial transpose of ${\displaystyle \rho }$ with respect to subsystem ${\displaystyle A}$
• ${\displaystyle ||X|{|}_1=\text{Tr}|X|=\text{Tr}\sqrt{X^{†}X}}$ 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 ${\displaystyle {\rho }^{{\mathrm{\Gamma }}_A}}$:

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

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

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

${\displaystyle {𝒩}(\sum _i{p}_i{\rho }_i)\le \sum _i{p}_i{𝒩}({\rho }_i)}$

• Is an entanglement monotone:

${\displaystyle {𝒩}(P(\rho ))\le {𝒩}(\rho )}$

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

The logarithmic negativity is an entanglement measure which is easily computable and an upper bound to the distillable entanglement.^[4] It is defined as

${\displaystyle E_N(\rho )\equiv {\log }_2||{\rho }^{{\mathrm{\Gamma }}_A}|{|}_1}$

where ${\displaystyle {\mathrm{\Gamma }}_A}$ is the partial transpose operation and ${\displaystyle ||\cdot |{|}_1}$ denotes the trace norm.

It relates to the negativity as follows:^[1]

${\displaystyle E_N(\rho ):={\log }_2(2{𝒩}+1)}$

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 ${\displaystyle H_1,H_2,\dots }$ (typically with increasing dimension) we can have a sequence of quantum states ${\displaystyle {\rho }_1,{\rho }_2,\dots }$ which converges to ${\displaystyle {\rho }^{\otimes n_1},{\rho }^{\otimes n_2},\dots }$ (typically with increasing ${\displaystyle n_i}$) in the trace distance, but the sequence ${\displaystyle E_N({\rho }_1)/n_1,E_N({\rho }_2)/n_2,\dots }$ does not converge to ${\displaystyle E_N(\rho )}$.
• is an upper bound to the distillable entanglement

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

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