Symmetric logarithmic derivative
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The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.
Definition
[edit]Let and be two operators, where is Hermitian and positive semi-definite. In most applications, and fulfill further properties, that also is Hermitian and is a density matrix (which is also trace-normalized), but these are not required for the definition.
The symmetric logarithmic derivative is defined implicitly by the equation[1][2]
where is the commutator and is the anticommutator. Explicitly, it is given by[3]
where and are the eigenvalues and eigenstates of , i.e. and .
Formally, the map from operator to operator is a (linear) superoperator.
Properties
[edit]The symmetric logarithmic derivative is linear in :
The symmetric logarithmic derivative is Hermitian if its argument is Hermitian:
The derivative of the expression w.r.t. at reads
where the last equality is per definition of ; this relation is the origin of the name "symmetric logarithmic derivative". Further, we obtain the Taylor expansion
- .
References
[edit]- ^ Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters. 72 (22). American Physical Society (APS): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN 0031-9007. PMID 10056200.
- ^ Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance". Annals of Physics. 247 (1): 135–173. arXiv:quant-ph/9507004. Bibcode:1996AnPhy.247..135B. doi:10.1006/aphy.1996.0040. S2CID 358923.
- ^ Paris, Matteo G. A. (21 November 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information. 07 (supp01): 125–137. arXiv:0804.2981. doi:10.1142/S0219749909004839. S2CID 2365312.