Landauer formula

In quantum mechanics, negativity is a measure of quantum entanglement which is easy to compute.

Definition

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

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

where:

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

$\mathcal{𝒩}(\rho )=\sum _i\frac{\left|{\lambda }_i\right|-{\lambda }_i}{2}$

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

Properties

• Is a convex function of $\rho$:

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

• Is an entanglement monotone:

$\mathcal{𝒩}(P({\rho }_i))\le \mathcal{𝒩}({\rho }_i)$

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

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

where ${\mathrm{\Gamma }}_A$ is the partial transpose operation and $\left|\right|\cdot |{|}_1$ 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 $H_1,H_2,\dots$ (typically with increasing dimension) we can have a sequence of quantum states ${\rho }_1,{\rho }_2,\dots$ which converges to ${\rho }^{\otimes n_1},{\rho }^{\otimes n_2},\dots$ (typically with increasing $n_i$) in the trace distance, but the sequence $E_N({\rho }_1)/n_1,E_N({\rho }_2)/n_2,\dots$ does not converge to $E_N(\rho )$.
• is an upper bound to the distillable entanglement

References

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

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