Properties of reduced density matrices

We will review here some definitions and properties of reduced density matrices for fermionic systems. The focus will be on the general density operator p and the corresponding Liouville equation 

The historic development has been accounted for in many reviews (see Refs. [9,13]). Important theorems regarding fermionic behaviour was developed by Yang [14], Coleman [15] and Sasaki [16], for a recent review on reduced density matrices and the famous N-representability problem, see Ref. [17]. 

The N particle (and its p-reduced companions) representable density matrix can be defined as follows 

Note that the definition (A2) normalizes p = to the number of pairings of N fermions [13], Other normalizations [14,15] also exists, i.e. 

In quantum chemistry file Lowdin normalization appears natural, but other choices are often made depending on the circumstances. 

---

Appendix A. Properties of reduced density matrices References... 

---