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.
where are all of the eigenvalues.
Properties
- Is a convex function of :
- Is an entanglement monotone:
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.
- 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 we can have a sequence of quantum states which converges to in the trace distance, but the sequence does not converge to.
- is an upper bound to the distillable entanglement