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. 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:
:$\mathcal(\rho) \equiv \frac$

where:
*$\rho^$ is the partial transpose of $\rho$ with respect to subsystem $A$
*$, , X, , _1 = \text, X, = \text \sqrt$ 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 $\rho^$:
:$\mathcal(\rho) = \left, \sum_ \lambda_i \ = \sum_i \frac$
where $\lambda_i$ are all of the eigenvalues.

Properties

* Is a convex function of $\rho$:
:$\mathcal(\sum_p_\rho_) \le \sum_p_\mathcal(\rho_)$
* Is an entanglement monotone:
:$\mathcal(P(\rho_)) \le \mathcal(\rho_)$
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
:$E_N(\rho) \equiv \log_2 , , \rho^, , _1$
where $\Gamma_A$ is the partial transpose operation and $, , \cdot , , _1$ denotes the trace norm.

It relates to the negativity as follows:

:$E_N(\rho) := \log_2( 2 \mathcal +1)$

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 $\rho^, \rho^, \ldots$ (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 fro
Quantiki
licensed under GNU Free Documentation License 1.2
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