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.

## Contents

* 1 Definition
* 1.1 Properties
* 2 Logarithmic negativity
* 2.1 Properties
* 3 References

## Definition

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

[math]\displaystyle{ \mathcal{N}(\rho) \equiv \frac{||\rho^{\Gamma_A}||_1-1}{2} }[/math]

where:

* [math]\displaystyle{ \rho^{\Gamma_A} }[/math] is the partial transpose of [math]\displaystyle{ \rho }[/math] with respect to subsystem [math]\displaystyle{ A }[/math]
* [math]\displaystyle{ ||X||_1 = \text{Tr}|X| = \text{Tr} \sqrt{X^\dagger X} }[/math] is the trace norm or the sum of the singular values of the operator [math]\displaystyle{ X }[/math].

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of [math]\displaystyle{ \rho^{\Gamma_A} }[/math]:

[math]\displaystyle{ \mathcal{N}(\rho) = \left|\sum_{\lambda_i \lt 0} \lambda_i \right| = \sum_i \frac{|\lambda_{i}|-\lambda_{i}}{2} }[/math]

where [math]\displaystyle{ \lambda_i }[/math] are all of the eigenvalues.

### Properties

* Is a convex function of [math]\displaystyle{ \rho }[/math]:

[math]\displaystyle{ \mathcal{N}(\sum_{i}p_{i}\rho_{i}) \le \sum_{i}p_{i}\mathcal{N}(\rho_{i}) }[/math]

* Is an entanglement monotone:

[math]\displaystyle{ \mathcal{N}(P(\rho)) \le \mathcal{N}(\rho) }[/math]

where [math]\displaystyle{ P(\rho) }[/math] is an arbitrary LOCC operation over [math]\displaystyle{ \rho }[/math]

## 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

[math]\displaystyle{ E_N(\rho) \equiv \log_2 ||\rho^{\Gamma_A}||_1 }[/math]

where [math]\displaystyle{ \Gamma_A }[/math] is the partial transpose operation and [math]\displaystyle{ || \cdot ||_1 }[/math] denotes the trace norm.

It relates to the negativity as follows:[1]

[math]\displaystyle{ E_N(\rho) := \log_2( 2 \mathcal{N} +1) }[/math]

### 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: [math]\displaystyle{ E_N(\rho \otimes \sigma) = E_N(\rho) + E_N(\sigma) }[/math]
* is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces [math]\displaystyle{ H_1, H_2, \ldots }[/math] (typically with increasing dimension) we can have a sequence of quantum states [math]\displaystyle{ \rho_1, \rho_2, \ldots }[/math] which converges to [math]\displaystyle{ \rho^{\otimes n_1}, \rho^{\otimes n_2}, \ldots }[/math] (typically with increasing [math]\displaystyle{ n_i }[/math]) in the trace distance, but the sequence [math]\displaystyle{ E_N(\rho_1)/n_1, E_N(\rho_2)/n_2, \ldots }[/math] does not converge to [math]\displaystyle{ E_N(\rho) }[/math].
* is an upper bound to the distillable entanglement

## References

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

1. ↑ 1.0 1.1 K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. A 58 (2): 883–92. doi:10.1103/PhysRevA.58.883. Bibcode: 1998PhRvA..58..883Z.
2. ↑ J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam. arXiv:quant-ph/0610253. Bibcode:2006PhDT........59E.
3. ↑ G. Vidal; R. F. Werner (2002). "A computable measure of entanglement". Phys. Rev. A 65 (3): 032314. doi:10.1103/PhysRevA.65.032314. Bibcode: 2002PhRvA..65c2314V.
4. ↑ M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 95 (9): 090503. doi:10.1103/PhysRevLett.95.090503. PMID 16197196. Bibcode: 2005PhRvL..95i0503P.

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