Negativity (quantum mechanics)#Logarithmic negativity

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.{{cite journal|author1=K. Zyczkowski |author2=P. Horodecki |author3=A. Sanpera |author4=M. Lewenstein |title=Volume of the set of separable states|journal=Phys. Rev. A|year=1998|volume=58|issue=2 |pages=883–92|bibcode = 1998PhRvA..58..883Z |doi = 10.1103/PhysRevA.58.883|arxiv = quant-ph/9804024|s2cid=119391103 }} It has been shown to be an entanglement monotone{{cite thesis|author=J. Eisert|title=Entanglement in quantum information theory|year=2001|publisher=University of Potsdam|arxiv=quant-ph/0610253|bibcode=2006PhDT........59E}}{{cite journal|author1=G. Vidal |author2=R. F. Werner |title=A computable measure of entanglement|journal=Phys. Rev. A|year=2002|volume=65|issue=3 |pages=032314|bibcode = 2002PhRvA..65c2314V|doi=10.1103/PhysRevA.65.032314|arxiv = quant-ph/0102117|s2cid=32356668 }} 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 $\rho$ :

: $\mathcal{N}(\sum_{i}p_{i}\rho_{i}) \le \sum_{i}p_{i}\mathcal{N}(\rho_{i})$

Is an entanglement monotone:

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

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.{{cite journal|author=M. B. Plenio|title=The logarithmic negativity: A full entanglement monotone that is not convex|journal=Phys. Rev. Lett.|year=2005|volume=95|issue=9 |pages=090503|bibcode = 2005PhRvL..95i0503P|doi=10.1103/PhysRevLett.95.090503|pmid=16197196 |arxiv = quant-ph/0505071|s2cid=20691213 }}

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

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Category:Quantum information science