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

Definition

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

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

where:

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

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

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

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

Properties

• Is a convex function of $\rho$:

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

• Is an entanglement monotone:

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

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

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

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

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

It relates to the negativity as follows:

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

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

References

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