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 been shown to be an entanglement monotone and hence a proper measure of entanglement.

Definition

The negativity of a subsystem can be defined in terms of a density matrix as:

where:

• is the partial transpose of with respect to subsystem
• is the trace norm or the sum of the singular values of the operator .

An alternative and equivalent definition is the absolute sum of the negative eigenvalues of :

where are all of the eigenvalues.

Properties

• Is a convex function of :

• Is an entanglement monotone:

where is an arbitrary LOCC operation over

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

where is the partial transpose operation and denotes the trace norm.

It relates to the negativity as follows:

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:
• is not asymptotically continuous. That means that for a sequence of bipartite Hilbert spaces (typically with increasing dimension) we can have a sequence of quantum states which converges to (typically with increasing ) in the trace distance, but the sequence does not converge to .
• is an upper bound to the distillable entanglement

References

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