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. It has shown to be an entanglement monotone 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. 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