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.

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 )=\sum _{\lambda _{i}<0}|\lambda _{i}|=\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 _{i}))\leq {\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.[4] 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:[1]

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 Quantwiki licensed under GNU Free Documentation License 1.2

K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. A. 58: 883–92. arXiv:quant-ph/9804024. Bibcode:1998PhRvA..58..883Z. doi:10.1103/PhysRevA.58.883.
J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam. arXiv:quant-ph/0610253. Bibcode:2006PhDT........59E.
G. Vidal; R. F. Werner (2002). "A computable measure of entanglement". Phys. Rev. A. 65: 032314. arXiv:quant-ph/0102117. Bibcode:2002PhRvA..65c2314V. doi:10.1103/PhysRevA.65.032314.
M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 95: 090503. arXiv:quant-ph/0505071. Bibcode:2005PhRvL..95i0503P. doi:10.1103/PhysRevLett.95.090503.

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[ref1-1] K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. A. 58: 883–92. arXiv:quant-ph/9804024. Bibcode:1998PhRvA..58..883Z. doi:10.1103/PhysRevA.58.883.

[ref2-2] J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam. arXiv:quant-ph/0610253. Bibcode:2006PhDT........59E.

[ref3-3] G. Vidal; R. F. Werner (2002). "A computable measure of entanglement". Phys. Rev. A. 65: 032314. arXiv:quant-ph/0102117. Bibcode:2002PhRvA..65c2314V. doi:10.1103/PhysRevA.65.032314.

[4] M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 95: 090503. arXiv:quant-ph/0505071. Bibcode:2005PhRvL..95i0503P. doi:10.1103/PhysRevLett.95.090503.