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
- 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
- 1 2 K. Zyczkowski; P. Horodecki; A. Sanpera; M. Lewenstein (1998). "Volume of the set of separable states". Phys. Rev. 883. A 58. arXiv:quant-ph/9804024. Bibcode:1998PhRvA..58..883Z. doi:10.1103/PhysRevA.58.883. Retrieved 24 January 2015.
- ↑ J. Eisert (2001). Entanglement in quantum information theory (Thesis). University of Potsdam.
- ↑ G. Vidal; R. F. Werner (2002). "A computable measure of entanglement". Phys. Rev. 032314. A 65. arXiv:quant-ph/0102117. Bibcode:2002PhRvA..65c2314V. doi:10.1103/PhysRevA.65.032314. Retrieved 24 March 2012.
- ↑ M. B. Plenio (2005). "The logarithmic negativity: A full entanglement monotone that is not convex". Phys. Rev. Lett. 090503. 95. arXiv:quant-ph/0505071. Bibcode:2005PhRvL..95i0503P. doi:10.1103/PhysRevLett.95.090503. Retrieved 24 March 2012.