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.[1] It has shown to be an entanglement monotone [2][3] 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
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): ||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 .
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.[4] It is defined as
where is the partial transpose operation and denotes the trace norm.
It relates to the negativity as follows:[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:
- 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 Quantwiki licensed under GNU Free Documentation License 1.2