Negativity (quantum mechanics)

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 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. 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

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

References

1. (1998). "Volume of the set of separable states". Phys. Rev. A.
2. J. Eisert. (2001). "Entanglement in quantum information theory". University of Potsdam.
3. (2002). "A computable measure of entanglement". Phys. Rev. A.
4. M. B. Plenio. (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett..