Negativity (quantum mechanics) explained

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: l{N}(\rho) \equiv (||\rho ^{\Gamma _{A}} || _{1-1})/(2)

where:

\rho ^{\Gamma _{A}} is the partial transpose of \rho with respect to subsystem A ||X|| _{1} = Tr |X| = 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}} :l{N}(\rho) = \left|\sum _{λ _{i} < 0} λ _{i} \right| = \sum _{i} (| λ _{i} | -λ _{i})/(2) where λ _{i} are all of the eigenvalues.

Properties

• Is a convex function of

\rho :l{N}(\sum _{i} p _{i} \rho _{i} ) \le \sum _{i} p _{i} l{N}(\rho _{i} )

• Is an entanglement monotone:

l{N}(P(\rho)) \le l{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.^[4] It is defined as

E _{N(\rho)} \equiv log _{2} ||\rho ^{\Gamma _{A}} || _{1} where \Gamma _{A} is the partial transpose operation and || ⋅ || _{1} denotes the trace norm.

It relates to the negativity as follows:^[1]

E _{N(\rho)} : = log _{2(} 2 l{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} ⊗ \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 ^{⊗ n _{1}} , \rho ^{⊗ 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

Notes and References

1. K. Zyczkowski . P. Horodecki . A. Sanpera . M. Lewenstein . Volume of the set of separable states. Phys. Rev. A. 1998. 58. 2 . 883–92. 1998PhRvA..58..883Z . 10.1103/PhysRevA.58.883. quant-ph/9804024. 119391103 .
2. J. Eisert. Entanglement in quantum information theory. 2001. University of Potsdam. quant-ph/0610253. 2006PhDT........59E.
3. G. Vidal . R. F. Werner . A computable measure of entanglement. Phys. Rev. A. 2002. 65. 3 . 032314. 2002PhRvA..65c2314V. 10.1103/PhysRevA.65.032314. quant-ph/0102117. 32356668 .
4. M. B. Plenio. The logarithmic negativity: A full entanglement monotone that is not convex. Phys. Rev. Lett.. 2005. 95. 9 . 090503. 2005PhRvL..95i0503P. 10.1103/PhysRevLett.95.090503. 16197196 . quant-ph/0505071. 20691213 .