One-particle reduced density matrix

The interesting fact about this object is a negative one about its neighbour. Given a matrix $\gamma_{pq}$, when is it the one-body reduced density matrix of some antisymmetric $N$-fermion state? That is the $N$-representability question, and Liu, Christandl and Verstraete record that Coulson posed the problem and that its fermionic form takes its name from Coleman.

For the one-body case their paper is explicit, and the sentence is worth having exactly: "checking consistency of 2-body reduced density operators of fermionic states is so hard, while checking consistency of 1-body reduced density operators is simple" — and, they add, consistency in that case "can be decided … based solely on the eigenvalues of the reduced density operators."

The reason is structural rather than lucky. The paper gives it: the extreme points of the convex set of one-body density operators $\langle a^\dagger_i a_j\rangle$ are ground states of Hamiltonians containing only bilinear terms in $a^\dagger_i$ and $a_j$ — free fermions — and those diagonalize easily. So the one-body consistency problem inherits the tractability of the free-fermion problem, which is exactly why this object is a workable state descriptor and a workable orbital-gradient input.

The neighbouring object is intractable, and the precise claim matters. The paper's result is that deciding $N$-representability of the two-body reduced density matrix is QMA-complete, and hence NP-hard. Two qualifications that a shortened version of this sentence would lose: the result classifies the problem's difficulty rather than solving it, and it is about the 2-RDM specifically — the 1-RDM case remains the easy one described above. The paper separately notes that restricting to the diagonal elements of the 2-RDM leaves an NP-hard problem.

One claim this record refuses to make. The explicit one-body condition often quoted — occupation numbers in $[0,1]$ summing to $N$ — is not stated in the paper cited here, which says only that eigenvalues suffice. It may well be correct and it is attributed elsewhere to Coleman's 1963 paper, which is behind a paywall and was not read. So it is left out. Citing this source for a condition it does not state would be the exact failure the catalog's sourcing rule exists to prevent, and "decidable from the eigenvalues" is the true claim and is enough.

Why it does not join the map. It is measured. observable-estimation names the operator being measured in its contract prose, where a parameter lives and a state does not — this record is one of the seventeen the shelf's observable abstention covers.