Hilbert-Schmidt Estimates for Fermionic 2-Body Operators

We prove that the 2-body operator $\gamma_2^\Psi$ of a fermionic $N$-particle state $\Psi$ obeys $||\gamma_2^\Psi||_{HS} \leq \sqrt{5} N$, which complements the bound of Yang that $||\gamma_2^\Psi||_{op} \leq N$. This estimate furthermore resolves a conjecture of Carlen-Lieb-Reuvers (arXiv:1403.3816) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert-Schmidt norm of the truncated 2-body operator $\gamma_2^{\Psi,T}$ obeys the inequality $||\gamma_2^{\Psi,T}||_{HS} \leq \sqrt{5 N \, \mathrm{tr}(\gamma_1^\Psi (1 - \gamma_1^\Psi))}$.


Introduction
Let (h, •, • ) be a (separable) Hilbert space and consider the space of fermionic N -particle states, N h.Given a normalized state Ψ ∈ N h, one defines the 1-and 2-body operators associated to Ψ, γ Ψ 1 : h → h and γ Ψ 2 : h ⊗ h → h ⊗ h, by We recall some well-known properties of γ Ψ 1 and γ Ψ 2 : Firstly, they are non-negative.This is obvious in the case of γ Ψ 1 , and for γ Ψ 2 this follows by noting that if (u k ) ∞ k=1 is an orthonormal basis for h, then a general tensor Φ ∈ h ⊗ h can be written as for Φ k,l = Φ, u k ⊗ u l , from which it readily follows that the inner product Φ, γ Ψ 2 Φ is given by where c * k = c * (u k ) and c k = c(u k ) denotes the creation and annihilation operators associated to the basis (u k ) ∞ k=1 .
Secondly, they are trace-class, with which follows from the identity ∞ k=1 c * k c k = M id M h , which holds on M h for any M ∈ N.This combined with their non-negativity implies the trivial estimates γ Ψ 1 op ≤ N and γ Ψ 2 op ≤ N (N − 1).In the bosonic case these are in fact optimal, but this is not so for the fermionic case: For γ Ψ 1 one may note that where we simply added the non-negative quantity c * (ϕ)Ψ 2 and applied the CAR.This shows that γ Ψ 1 op ≤ 1, which is usually described as a consequence of the Fermi exclusion principle, which informally speaking prohibits more than one fermion from occupying any given oneparticle state.In terms of the operators c k this bound can equivalently be phrased as which we can also use to improve our bound on γ Ψ 2 op as follows: By the triangle and Cauchy-Schwarz inequalities, we can apply this to equation (1.5) for which implies that γ Ψ 2 op ≤ N -a bound which Yang proved was optimal 1 in the classic paper [1].

Main Results
In this paper we present two results on γ Ψ 2 .The first result concerns γ Ψ 2 HS : Given the identity γ Ψ 2 tr = tr γ Ψ 2 = N (N − 1) and the general inequality γ Ψ 2 op ≤ N , one can deduce that the Hilbert-Schmidt norm of γ Ψ 2 must obey and since γ Ψ 2 tr = N (N − 1) is an identity and γ Ψ 2 op ≤ N is optimal, one might suppose that this is at least nearly optimal.Below we will however prove that this is far from the case, as the following holds: This result is interesting in two respects: Firstly, it is of the same order (with respect to N ) as the optimal bound γ Ψ 2 op ≤ N , even though γ Ψ 2 tr = N (N − 1).This informally implies that although it is possible for γ Ψ 2 to have eigenvalues of order N , it can not have "too many" large eigenvalues. 1In fact he proved the stronger statement that γ Ψ 2 op ≤ M −N+2

M
N when M = dim(h) < ∞ and M and N are even, and characterized the optimizers in this case.
Secondly, it is easy to compute that for a Slater state Ψ = u 1 ∧• • •∧u N , γ Ψ 2 HS = 2N (N − 1), which is also O(N ).This suggests that an O(N ) behavior of the Hilbert-Schmidt norm of fermionic 2-body operators might be a general feature.This bound also resolves a conjecture of Carlen-Lieb-Reuvers concerning the entropy of the trace-normalized 2-body operator In [6] they conjectured (Conjecture 2.6) that the entropy of this should always obey a bound of the form (1.11) As noted in [6], Jensen's inequality for the convex function x → − log(x) implies that for any density operator γ, so as a consequence of Theorem 1 we find that

Truncated 2-Body Operators
Our second result concerns the truncated 2-body operator: It is well-known that Ψ is a Slater state if and only if γ Ψ 1 is a projection, i.e. γ Ψ 1 2 = γ Ψ 1 , and that in this case the 2-body operator can be expressed in terms of γ Ψ 1 as where Ex : For a general state Ψ one would expect that if Ψ is "nearly Slater", in an appropriate sense, then the same expression should be approximately valid for γ Ψ 2 -this leads to the Hartree-Fock functional as an approximation to the energy of a fermionic system, for instance.
A way to make this precise is to consider the truncated 2-body operator γ Ψ,T 2 and establish that this can be controlled in terms of tr , which can be viewed as a measure of the "Slaterness" of the state Ψ.A result of this form was first derived by Bach in [3], where he proved the following (see also Bach's paper [4] and Graf-Solovej's paper [5] for related generalizations): Theorem (Bach [3]).Let X : h → h be an orthogonal projection.Then for any normalized This can be seen as a sort of lower bound for γ Ψ,T

2
. Below we will complement this result by proving the following Hilbert-Schmidt bound for γ Ψ,T 2 : Theorem 2. For any N ∈ N and Ψ ∈ N h it holds that We remark that at worst tr = N , in which case this bound essentially reduces to that of Theorem 1 (up to the quantity (1 − Ex)γ Ψ 1 ⊗ γ Ψ 1 HS ), but in view of that theorem this bound is non-trivial for any Ψ with tr γ

Acknowledgements
I am grateful to Phan Thành Nam for informing me of the application of Theorem 1 to the conjecture of Carlen-Lieb-Reuvers, and for providing me with valuable feedback during the writing of this paper.
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 470903074 -TRR 352.

Proof of the Theorems
To prove Theorem 1 we first observe that by the characterization T HS = sup A |tr(AT )|, where the supremum is over all Hilbert-Schmidt operators A with A HS = 1, it suffices to estimate a quantity of the form tr Aγ Ψ 2 .
To do this we begin by writing The Cauchy-Schwarz inequality thus allows us to estimate where we recognized that ∞ n=1 c n Ψ 2 = Ψ, N Ψ = N , and used that as in the proof of the inequality γ Ψ 1 ≤ 1, we are free to add the quantity to obtain the anticommutator of ∞ k,l,m=1 A k,l,m,n c * m c l c k with its adjoint.The reason for this last step is an argument which was recently used in the study of the correlation energy of Fermi gases in [7], although it has since been learned this was first used by Bell in [2]: Although the CAR does not imply a useful anticommutator identity for sums of an even number of creation and annihilation operators, it does for sums of an odd number of such operators.
In the present case we note the following identity: Proof: By applying the CAR we find that for any k, l, m, k and by standard commutator identities Consequently T n is given by The claim now follows by renaming indices and using the assumed antisymmetry of A k,l,m,n .
With this identity we now obtain the central estimate of this paper: Proof: We begin by noting that as It then follows from the previous proposition that and the same argument we applied in equation (1.9) now shows that for any HS , and as also

Proof of Theorem 2
We note the following identity for γ Ψ,T 2 : Proposition 5.For any ϕ 1 , ϕ 2 , ψ 1 , ψ 2 ∈ h it holds that Proof: By the CAR we have that Theorem 2 can now be concluded in the following form: Now we note that by the CAR for the claim.