Fermion Correlation Inequalities

Derived from G- and P-Conditions

Volker Bach, Hans Konrad Knörr, Edmund Menge

Received: March 14, 2012 Revised: May 7, 2012

Communicated by Heinz Siedentop

Abstract. It is shown in this paper that the G-Condition and the P-Condition from representability imply the fermion correlation estimate from [1] which, in turn, is known to yield a nontrivial bound on the accuracy of the Hartree–Fock approximation for large Coulomb systems.

2010 Mathematics Subject Classification: Primary: 81V70; Secondary: 81V45 and 81V55 Keywords and Phrases: Fermion correlation estimate, representability conditions

1 Introduction

The dynamics of N electrons in an atom (K = 1) or molecule (K ≥ 2) with K nuclei of charges \underline{Z} := (Z_{1}, Z_{2}, …, Z_{K}), fixed at positions \underline{R} := (R_{1}, R_{2}, …, R_{K}), is generated by the Hamiltonian

H^{(N)}(\underline{Z}, \underline{R}) := ∑_{n=1}^{N} ( − Δ_{x_{n}} − ∑_{j=1}^{K} \frac{Z_{j}}{|x_{n} − R_{j}|}) + ∑_{1≤n<m≤N} \frac{1}{|x_{n} − x_{m}|} (1)

to lowest order in the Born–Oppenheimer approximation. H^{(N)}(\underline{Z}, \underline{R}) ≡ H^{(N)} is a semibounded, self-adjoint operator defined on a suitable dense domain D^{(N)} in the Hilbert space F_{f}^{(N)}[𝔥] of antisymmetric N-electron wave functions, cf. (18) below. Basic quantities of interest are the ground state energy

E_{gs}^{(N)}(\underline{Z}, \underline{R}) := inf \{σ \{H^{(N)}(\underline{Z}, \underline{R})\}\} ,

whose variational characterization

E_{gs}^{(N)}(\underline{Z}, \underline{R}) = inf \{⟨Ψ^{(N)}| H^{(N)}Ψ^{(N)}⟩ | Ψ^{(N)} ∈ D^{(N)} ∩ F_{f}^{(N)}[𝔥], ∥Ψ^{(N)}∥ = 1\}

(2)

is given by the Rayleigh–Ritz principle, and corresponding ground states Ψ_{gs}^{(N)}, i.e., normalized solutions of the stationary Schrödinger equation

H^{(N)}(\underline{Z}, \underline{R})Ψ_{gs}^{(N)} = E_{gs}^{(N)}(\underline{Z}, \underline{R})Ψ_{gs}^{(N)}.

The Hartree–Fock (HF) variational principle is an important method to obtain approximations to both, the ground state energy and ground states. The HF energy E_{hf}^{(N)}(\underline{Z}, \underline{R}) is defined by restricting the variation in (2) to SD^{(N)}[𝔥],

E_{hf}^{(N)}(\underline{Z}, \underline{R})

= inf {⟨Φ^{(N)}| H^{(N)}Φ^{(N)}⟩ | Φ^{(N)} ∈ D^{(N)} ∩ SD^{(N)}[𝔥], ∥Φ^{(N)}∥ = 1} ,(3)

where SD^{(N)}[𝔥] ⊆ F_{f}^{(N)}[𝔥] denotes the set of Slater determinants, i.e., the set of all antisymmetrized product vectors φ_{1} ∧ ⋯ ∧ φ_{N}. Since the variation in (3), compared to (2), is restricted, we clearly have

E_{hf}^{(N)}(\underline{Z}, \underline{R}) ≥ E_{gs}^{(N)}(\underline{Z}, \underline{R}).

A lower bound to the ground state energy by the HF energy minus an error which is small in the large-Z limit was obtained by one of us in [1, 2]. In the case of a neutral atom, i.e., N = Z := Z_{1} and R_{1} = 0, the resulting estimate was

E_{gs}^{(Z)}(Z) ≥ E_{hf}^{(Z)}(Z) − O(Z^{(5/3)−ε}),  (4)

for some ε > 0. The error term O(Z^{(5/3)−ε}) is small compared to all three contributions to E_{hf}^{(Z)}(Z), namely, the kinetic, the classical electrostatic, and the exchange energy which are at least of size cZ^{5/3} in magnitude for some constant c > 0. A key inequality derived in [1] that eventually lead to (4) is the fermion correlation estimate

tr_{2} \{(X ⊗ X)Γ^{(T)}\} ≥ − tr_{1} \{Xγ\} min \{1; const · \sqrt{tr_{1} \{X (γ − γ^{2})\}}\} (5)

Documenta Mathematica 17 (2012) 451–481

where X = X^{∗} = X^{2} is an orthogonal projection, Γ^{(T)} := Γ − (1 − Ex)(γ ⊗ γ), Γ ≡ Γ_{Φ^{(N)}} is the two-particle- and γ ≡ γ_{Φ^{(N)}} the one-particle density matrix of a normalized N-electron state Φ^{(N)} ∈ F_{f}^{(N)}[𝔥]. The purpose of the present paper is to give an alternative derivation of (5) by using ideas originating from the theory of N-representability. More precisely, we show that (5) follows already from the G-Condition and the P-Condition specified by Garrod and Percus [8] and Coleman [4]. Observing that the Rayleigh–Ritz principle (2) can be rewritten as a variation over all N-representable two-particle density matrices Γ, we consequently obtain (4) from relaxing the requirement of N-representability of Γ to merely requiring Γ to fulfill the G-Condition and the P-Condition:

Theorem 1.1. The G-Condition and the P-Condition imply (5).

We note that (5) was also derived by Graf and Solovej in [13] by a different method that, in retrospective, resembles the application of Garrod and Percus’ G-Condition. In fact, one part of the derivation in [13] follows already from the G-Condition. A main difference to using representability methods, however, lies in the use of operator inequalities in [13] which are necessarily formulated on the N-particle Hilbert space, as opposed to the one- or two-particle Hilbert spaces in the presented work.

In future work we plan to sharpen this result by making additional use of Erdahl’s T_{1}- and T_{2}-Conditions [6, 5] which have recently lead to very good numerical results in quantum chemistry computations [3, 10, 11], as well as Coleman’s Q-Condition which was also given in [4] but is not necessary for the derivation of our present result. Furthermore, similar representability conditions also exist for bosons [10]. There we like to adress the question whether analogous results can also be obtained.

Acknowledements. We would like to thank Gero Friesecke, Peter Müller and Heinz Siedentop for fruitful remarks and discussions. H. K. K. and E. M. were partially supported by the MPGC Mainz and the ESI.

2 Density Matrices and Reduced Density Matrices

2.1 Fock space, Creation and Annihilation Operators

Let 𝔥 be a separable complex Hilbert space which we henceforth refer to as the one-particle Hilbert space. The fermion Fock space F ≡ F_{f}[𝔥] is defined to be the orthogonal sum

F_{f}[𝔥] := ⨁_{N=0}^{∞} F_{f}^{(N)}[𝔥],

where

F_{f}^{(N)}[𝔥] :=⋀^{N} 𝔥 := A_{N}( ⨂^{N} 𝔥)

is the antisymmetric tensor product of N copies of 𝔥, for N ≥ 1, and F_{f}^{(0)}[𝔥] := ℂ · Ω, with Ω being the normalized vacuum vector. Here, A_{N} is the orthogonal

projection from ⨂^{N} 𝔥 onto ⋀^{N} 𝔥 uniquely defined by

A_{N} (φ_{1} ⊗ ⋯ ⊗ φ_{N}) := \frac{1}{N!} ∑_{π∈S_{N}} (−1)^{π} φ_{π(1)} ⊗ ⋯ ⊗ φ_{π(N)}

=: \frac{1}{\sqrt{N!}}φ_{1} ∧ ⋯ ∧ φ_{N},

for φ_{1}, …, φ_{N} ∈ 𝔥. It is convenient to introduce creation operators c^{∗}(f) ∈ B(F) for any f ∈ 𝔥 by

c^{∗}(f)Ω := f,  (6)
c^{∗}(f) (φ_{1} ∧ ⋯ ∧ φ_{N}) := f ∧ φ_{1} ∧ ⋯ ∧ φ_{N}  (7)

for φ_{1}, …, φ_{N} ∈ 𝔥, and extension by linearity and continuity. By induction and (6)-(7)

φ_{1} ∧ φ_{2} ∧ ⋯ ∧ φ_{N} = c^{∗}(φ_{1})c^{∗}(φ_{2}) ⋯ c^{∗}(φ_{N})Ω  (8)

for all φ_{1}, φ_{2}, …, φ_{N} ∈ 𝔥. If {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 is an orthonormal basis (ONB) of 𝔥, then for any N ∈ ℕ

{c^{∗}(φ_{k_{1}}) ⋯ c^{∗}(φ_{k_{N}})Ω| 1 ≤ k_{1} < k_{2} < ⋯ < k_{N}} ⊆ F_{f}^{(N)}[𝔥]  (9)

is an ONB of F_{f}^{(N)}[𝔥], and

{c^{∗}(φ_{k_{1}}) ⋯ c^{∗}(φ_{k_{N}})Ω| N ∈ ℕ_{0}, 1 ≤ k_{1} < k_{2} < ⋯ < k_{N}} ⊆ F  (10)

is an ONB of F. The adjoint operators c(f) := (c^{∗}(f))^{∗} ∈ B(F), with f ∈ 𝔥, are the annihilation operators. Note that, while f ↦ c^{∗}(f) is linear, f ↦ c(f) is antilinear. Together with the creation operators they fulfill the canonical anticommutation relations (CAR), i.e.,

∀f, g ∈ 𝔥 : {c(f), c^{∗}(g)} = ⟨f|g⟩_{𝔥} · 𝟙_{F}, {c^{∗}(f), c^{∗}(g)} = 0,  (11)

where {A, B} := AB + BA denotes the anticommutator. Moreover,

∀f ∈ 𝔥 : c(f)Ω = 0,  (12)

and {c^{∗}(f), c(f)| f ∈ 𝔥} is completely determined by (6), (11) and (12), i.e., (7)-(10) follow from (6), (11) and (12). The creation and annihilation operators introduced here are a specific representation of the (abstract) CAR (11), namely the Fock representation. For φ_{k} being any element of a given ONB {φ_{k}}_{k=1}^{∞} ⊆ 𝔥, we write

c_{k}^{∗} ≡ c^{∗}(φ_{k}), c_{k} ≡ c(φ_{k}).

An important unbounded, self-adjoint and positive operator on F is the number operator ℕ̂ defined by

ℕ̂ (c^{∗}(f_{1}) ⋯ c^{∗}(f_{N})Ω) := N · c^{∗}(f_{1}) ⋯ c^{∗}(f_{N})Ω

for any f_{1}, …, f_{N} ∈ 𝔥. It is not difficult to see that

ℕ̂ = ∑_{k=1}^{∞} c_{k}^{∗}c_{k}

as a quadratic form, for any ONB {φ_{k}}_{k=1}^{∞} ⊆ 𝔥.

2.2 Density Matrices

A positive trace class operator ρ ∈ L_{+}^{1}(F) of unit trace, tr_{F} {ρ} = 1, is called density matrix. Given a density matrix ρ, the map A ↦ tr_{F} {ρ A} defines a state, i.e., a normalized, linear, and positive functional on B(F) ∋ A. If Ψ ∈ F is a normalized vector, then |Ψ⟩ ⟨Ψ| is a density matrix (of rank one) called pure state. In this paper we study fermion systems with a repulsive interaction and whose dynamics preserve the particle number. For this reason we restrict our attention to density matrices which commute with the particle number operator and have a finite squared particle number expectation value,

ρ = ⨁_{N=0}^{∞} ρ^{(N)}, and ⟨ℕ̂^{2}⟩_{ρ} < ∞,  (13)

where here and henceforth we denote for any A ∈ B(𝔥)

⟨A⟩_{ρ} := tr_{F} \{ρ^{\frac{1}{2}} Aρ^{\frac{1}{2}} \} .

Note that, if m ≠ n, m, n ≥ 0, then tr_{F} {ρ c^{∗}(f_{1}) ⋯ c^{∗}(f_{m})c(g_{1}) ⋯ c(g_{n})} = 0 for any choice of f_{1}, …, f_{m}, g_{1}, …, g_{n} ∈ 𝔥, due to (13).

2.3 Reduced Density Matrices

Given a density matrix ρ ∈ L_{+}^{1}(F) subject to (13), we introduce two bounded operators, γ_{ρ} ∈ B(𝔥) and Γ_{ρ} ∈ B(𝔥 ⊗ 𝔥), by

∀f, g ∈ 𝔥 : ⟨f| γ_{ρ}g⟩ := tr_{F} {ρ c^{∗}(g)c(f)}  (14)

and

∀f_{1}, f_{2}, g_{1}, g_{2} ∈ 𝔥 : ⟨f_{1} ⊗ f_{2}| Γ_{ρ}(g_{1} ⊗ g_{2})⟩ := tr_{F} {ρ c^{∗}(g_{2})c^{∗}(g_{1})c(f_{1})c(f_{2})} . (15)

γ_{ρ} is called the one-particle density matrix (1-pdm) and Γ_{ρ} the two-particle density matrix (2-pdm) corresponding to ρ. For any ONB {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 we define the exchange operator Ex ∈ B (𝔥 ⊗ 𝔥) by

Ex := ∑_{k,l=1}^{∞} |φ_{k} ⊗ φ_{l}⟩ ⟨φ_{l} ⊗ φ_{k}| ,  (16)

such that Ex (f ⊗ g) = g ⊗ f. Then the CAR leads to the antisymmetry property of Γ_{ρ}:

Ex Γ_{ρ} = −Γ_{ρ} = Γ_{ρ} Ex.  (17)

The following properties of the 1-pdm and the 2-pdm are easily proven (we denote tr_{1} := tr_{𝔥} and tr_{2} := tr_{𝔥⊗𝔥}):

Lemma 2.1. Let ρ ∈ L_{+}^{1}(F) be a density matrix obeying (13). Then the following assertions hold true:

i) γ_{ρ} ∈ L_{+}^{1}(𝔥), 0 ≤ γ_{ρ} ≤ 𝟙, tr_{1} {γ_{ρ}} = ⟨ℕ̂⟩_{ρ}, Γ_{ρ} ∈ L_{+}^{1}(𝔥 ⊗ 𝔥),

0 ≤ Γ_{ρ} ≤ ⟨ℕ̂⟩_{ρ}, and tr_{2} {Γ_{ρ}} = ⟨ℕ̂(ℕ̂ − 1)⟩_{ρ}.

ii) If Ran {ρ} ⊆ F_{f}^{(N)}, then, for all f, g ∈ 𝔥,

⟨f| γ_{ρ}g⟩ = \frac{1}{N − 1} ∑_{k=1}^{∞} ⟨f ⊗ φ_{k}| Γ_{ρ}(g ⊗ φ_{k})⟩ ,

where {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 is an ONB.

iii) Furthermore,

ρ = |c^{∗}(φ_{1}) ⋯ c^{∗}(φ_{N})Ω⟩ ⟨c^{∗}(φ_{1}) ⋯ c^{∗}(φ_{N})Ω| ⇔ γ_{ρ} = ∑_{i=1}^{N} |φ_{i}⟩ ⟨φ_{i}|

and, in this case,

Γ_{ρ} = (1 − Ex) (γ_{ρ} ⊗ γ_{ρ}) .

2.4 Hamiltonian and Ground State Energy

Recall from (1) that the Hamiltonian of an atom or molecule is given by

H^{(N)}(\underline{Z}, \underline{R}) := ∑_{n=1}^{N} (−Δ_{x_{n}} − ∑_{k=1}^{K} \frac{Z_{k}}{|x_{n} − R_{k}|}) + ∑_{1≤n<m≤N} \frac{1}{|x_{n} − x_{m}|}. (18)

Choosing an ONB {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 = L^{2} (ℝ^{3} × ℤ_{2}) such that {φ_{k}}_{k=1}^{∞} ⊆ H^{2} (ℝ^{3} × ℤ_{2}), where H^{2} (ℝ^{3} × ℤ_{2}) denotes the Sobolev space, we define

h_{kl} := ⟨φ_{k}| (−Δ_{x} − ∑_{k=1}^{K} \frac{Z_{k}}{|x − R_{k}|}) φ_{l}⟩ ,

V_{kl;mn} := ⟨φ_{k} ⊗ φ_{l}|\frac{1}{|x − y|} (φ_{m} ⊗ φ_{n})⟩ ,

and

ℍ̂ := ∑_{k,l=1}^{∞} h_{kl} c_{k}^{∗}c_{l} + ∑_{k,l,m,n=1}^{∞} V_{kl;mn} c_{l}^{∗}c_{k}^{∗}c_{m}c_{n}.

Stability of matter ensures that ℍ̂+μℕ̂ is a semibounded self-adjoint operator, provided μ < ∞ is sufficiently large. Moreover, the Hamiltonian of an atom or molecule can be viewed as

H^{(N)}(\underline{Z}, \underline{R}) = ℍ̂|_{F_{f}^{(N)}[𝔥]},

i.e., H^{(N)}(\underline{Z}, \underline{R}) is the restriction of ℍ̂ to F_{f}^{(N)} [𝔥]. The ground state energy can now be reexpressed as

E_{gs}^{(N)}(\underline{Z}, \underline{R}) = inf \{tr_{F} \{ρ^{\frac{1}{2}} ℍ̂ρ^{\frac{1}{2}} \} | ρ ∈ L_{+}^{1}(F), ℕ̂ ρ = Nρ, tr_{F} \{ρ\} = 1\}
= inf {E (γ_{ρ}, Γ_{ρ}) | ρ ∈ L_{+}^{1}(F), ℕ̂ ρ = Nρ, tr_{F} {ρ} = 1} ,

where the energy functional is defined as

E (γ_{ρ}, Γ_{ρ}) := tr_{1} \{hγ_{ρ}\} + \frac{1}{2} tr_{2} \{V Γ_{ρ}\} .

We call (γ, Γ) ∈ B(𝔥)×B(𝔥⊗𝔥) \underline{N-representable} if there exists a density matrix ρ ∈ L_{+}^{1}(F) with ℕ̂ ρ = Nρ and tr_{F} {ρ} = 1 such that γ = γ_{ρ} and Γ = Γ_{ρ}. Using the notion of N-representability, the ground state energy can be rewritten as

E_{gs}^{(N)} = inf {E (γ, Γ) | (γ, Γ) is N-representable} .

458 Bach, Knörr, Menge

By Lemma 2.1, we have that

E_{hf}^{(N)} = inf {E(γ, (1 − Ex)(γ ⊗ γ))| γ = γ^{∗} = γ^{2}, tr_{1} {γ} = N} , and Lieb’s variational principle [9, 1] ensures that actually

E_{hf}^{(N)} = inf {E(γ, (1 − Ex)(γ ⊗ γ))| 0 ≤ γ ≤ 1, tr_{1} {γ} = N} .

3 G-, P- and Q-Conditions In this section we derive necessary conditions on (γ, Γ) to be N-representable. To this end, we assume N ∈ ℕ, γ ∈ L^{1}(𝔥) with 0 ≤ γ ≤ 1 and tr_{1} {γ} = N, Γ ∈ L^{1}(𝔥 ⊗ 𝔥), Ex Γ = Γ Ex = − Γ, and we call (γ, Γ) admissible in this case. (P) (γ, Γ) fulfills the P-Condition

:⇔ Γ ≥ 0.  (19)

(G) (γ, Γ) fulfills the G-Condition

:⇔ ∀A ∈ B(𝔥) : tr_{2} {(A^{∗} ⊗ A) (Γ + Ex (γ ⊗ 𝟙))} ≥ |tr_{1} {Aγ}|^{2} . (20)

(Q) (γ, Γ) fulfills the Q-Condition

:⇔ Γ + (1 − Ex)(𝟙 ⊗ 𝟙 − γ ⊗ 𝟙 − 𝟙 ⊗ γ) ≥ 0.  (21)

Our main result of this section is Theorem 3.1. Let ρ ∈ L^{1}(F) (not necessarily positive) such that tr_{F} {ρ} = 1, tr_{F} \{|ρ|^{\frac{1}{2}} ℕ̂^{2} |ρ|^{\frac{1}{2}} \} < ∞, and that ρ preserves the particle number, i.e., [ℕ̂, ρ] = 0. Define γ_{ρ} and Γ_{ρ} by (14) and (15), respectively, and let {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 be an ONB. Then the following two statements are equivalent.

(i) If P_{r} ∈ B(F) is a polynomial in {c_{k}^{∗}, c_{k}}_{k=1}^{∞} of degree r ≤ 2, then

tr_{F} {ρ P_{r}^{∗}P_{r}} ≥ 0.  (22)

(ii) (γ_{ρ}, Γ_{ρ}) is admissible and fulfills the G-, P- and Q-Conditions. Before we turn to the proof of Theorem 3.1, we establish its finite-dimensional analogue in Lemma 3.2 below. Theorem 3.1 then follows from Lemma 3.2 by a limiting argument. Lemma 3.2. Let ρ ∈ L^{1}(F) (not necessarily positive) such that tr_{F} {ρ} = 1, tr_{F} \{|ρ|^{\frac{1}{2}} ℕ̂^{2} |ρ|^{\frac{1}{2}} \} < ∞, and that ρ preserves the particle number, i.e., [ℕ̂, ρ] = 0. Define γ_{ρ} and Γ_{ρ} by (14) and (15), respectively, and let {φ_{k}}_{k=1}^{∞} ⊆ 𝔥 be an ONB. Then the following statements are equivalent.

Documenta Mathematica 17 (2012) 451–481

Correlation Inequalities from G and P 459

(i) If P_{r} ∈ B(F) is a polynomial in {c_{k}^{∗}, c_{k}}_{k=1}^{∞} of degree r ≤ 2, then

tr_{F} {ρ P_{r}^{∗}P_{r}} ≥ 0.  (23)

(ii) For any ϕ ∈ span {φ_{k}| k ∈ ℕ}, Ψ ∈ span {φ_{k} ⊗ φ_{l}| k, l ∈ ℕ}, we have

0 ≤ ⟨ϕ| γ_{ρ}ϕ⟩ ≤ 1,  (24)
⟨Ψ| Γ_{ρ}Ψ⟩ ≥ 0,  (25)
⟨Ψ| (Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ})) Ψ⟩ ≥ 0,  (26)

and, for all A := ∑_{k,l=1}^{M} α_{kl} |φ_{k}⟩ ⟨φ_{l}|, M < ∞, (α_{kl})_{k,l=1}^{M} ∈ ℂ^{M×M},

tr_{2} {(A^{∗} ⊗ A) (Γ_{ρ} + Ex (γ_{ρ} ⊗ 𝟙))} ≥ |tr_{1} {Aγ_{ρ}}|^{2} .  (27)

Proof. First we show (i) ⇒ (ii). The properties (24)-(27) of (γ_{ρ}, Γ_{ρ}) can be checked by suitable choices of P_{r}.

a) The first inequality of (24) follows by choosing P_{1} := ∑_{i=1}^{M} \overline{α}_{i}c_{i}, where

α_{i} ∈ ℂ and M < ∞:

0 ≤ tr_{F} \{ρ P_{1}^{∗}P_{1}\} = ∑_{i,j=1}^{M} α_{i}\overline{α}_{j}tr_{F} \{ρ c_{i}^{∗}c_{j}\}
= ∑_{j=1}^{M} ∑_{i=1}^{M} ⟨α_{j}φ_{j}| γ_{ρ}(α_{i}φ_{i})⟩ = ⟨ϕ_{M} | γ_{ρ}ϕ_{M}⟩ , (28)

with ϕ_{M} := ∑_{i=1}^{M} α_{i}φ_{i} ∈ span {φ_{k}| k ∈ ℕ}. The second inequality derives from the CAR and P_{1} := ∑_{i=1}^{M} α_{i}c_{i}^{∗}:

0 ≤ tr_{F} \{ρ P_{1}^{∗}P_{1}\} = ∑_{i,j=1}^{M} \overline{α}_{i}α_{j}tr_{F} \{ρ c_{i}c_{j}^{∗}\}

= ∑_{i,j=1}^{M} \overline{α}_{i}α_{j}tr_{F} \{ρ (δ_{ij} − c_{j}^{∗}c_{i})\}

= ∑_{i=1}^{M} ∑_{j=1}^{M} ⟨α_{i}φ_{i}| (𝟙 − γ_{ρ})(α_{j}φ_{j})⟩

= ⟨ϕ_{M} | (𝟙 − γ_{ρ})ϕ_{M}⟩ .  (29)

Documenta Mathematica 17 (2012) 451–481

b) Property (27) is obtained by choosing P_{2} := μ + \frac{1}{2} ∑_{k,l=1}^{M} α_{kl} (c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗})

with μ, α_{kl} ∈ ℂ, M < ∞ and calculating tr_{F} {ρ P_{2}^{∗}P_{2}}:

0 ≤ tr_{F}\{ρ(μ + \frac{1}{2} ∑_{k,l=1}^{M} α_{kl}(c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗}))^{∗}

× (μ + \frac{1}{2} ∑_{m,n=1}^{M} α_{mn}(c_{m}^{∗}c_{n} − c_{n}c_{m}^{∗}))\}

= tr_{F}\{ρ(\frac{1}{2} ∑_{k,l=1}^{M} α_{kl}(c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗}))^{∗}(\frac{1}{2} ∑_{m,n=1}^{M} α_{mn}(c_{m}^{∗}c_{n} − c_{n}c_{m}^{∗}))\}

+ 2 ℜ𝔢\{\overline{μ} tr_{F}\{\frac{1}{2} ρ ∑_{k,l=1}^{M} α_{kl}(c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗})\}\} + |μ|^{2} .

Now we expand the brackets and use the CAR to reorder the annihilation and creation operators:

0 ≤ ∑_{k,l,m,n=1}^{M} \overline{α}_{kl}α_{mn}tr_{F}\{ρ( − c_{l}^{∗}c_{m}^{∗}c_{k}c_{n} + δ_{km}c_{l}^{∗}c_{n} − \frac{1}{2}δ_{kl}c_{m}^{∗}c_{n}

− \frac{1}{2}δ_{mn}c_{l}^{∗}c_{k} + \frac{1}{4}δ_{kl}δ_{mn})\}

+ 2 ℜ𝔢\{\overline{μ} ∑_{k,l=1}^{M} α_{kl}tr_{F} \{ρ(c_{k}^{∗}c_{l} − \frac{1}{2}δ_{kl})\} \} + |μ|^{2} .

Bearing α_{kl} = ⟨φ_{k}| Aφ_{l}⟩ and tr_{F} {ρ} = 1 in mind, we derive from the definitions of Γ_{ρ} and γ_{ρ} that

0 ≤ ∑_{k,l,m,n=1}^{M} ⟨φ_{k} ⊗ φ_{n}|( − Γ_{ρ} + 𝟙 ⊗ γ_{ρ} − \frac{1}{2}Ex (γ_{ρ} ⊗ 𝟙) − \frac{1}{2}Ex (𝟙 ⊗ γ_{ρ})

+ \frac{1}{4}Ex (𝟙 ⊗ 𝟙) )(φ_{m} ⊗ φ_{l})⟩

× ⟨φ_{l} ⊗ φ_{m}| (A^{∗} ⊗ A) (φ_{k} ⊗ φ_{n})⟩

+ 2 ℜ𝔢 \{\overline{μ} ∑_{k,l=1}^{M} ⟨φ_{k}| Aφ_{l}⟩ ⟨φ_{l}| γ_{ρ}φ_{k}⟩ − \frac{\overline{μ}}{2} ∑_{k=1}^{M} ⟨φ_{k}| Aφ_{k}⟩\} + |μ|^{2} .

Correlation Inequalities from G and P 461

We can now perform the summations and arrive at

0 ≤ tr_{2}{Ex (A^{∗} ⊗ A) ( − Γ_{ρ} + 𝟙 ⊗ γ_{ρ}

+ \frac{1}{2}Ex(\frac{1}{2}𝟙 ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ} − γ_{ρ} ⊗ 𝟙))\}

+ 2 ℜ𝔢 \{\overline{μ} tr_{1} \{Aγ_{ρ}\} − \frac{\overline{μ}}{2} tr_{1} \{A\}\} + |μ|^{2}

= tr_{2}\{(A^{∗} ⊗ A)(Γ_{ρ} + (𝟙 ⊗ γ_{ρ})Ex + \frac{1}{4}𝟙 ⊗ 𝟙 − \frac{1}{2}(𝟙 ⊗ γ_{ρ} + γ_{ρ} ⊗ 𝟙))\}

+ 2 ℜ𝔢 \{\overline{μ} tr_{1} \{Aγ_{ρ}\} − \frac{\overline{μ}}{2} tr_{1} \{A\}\} + |μ|^{2} .  (30)

Defining s ∈ ℂ by μ =: (s + \frac{1}{2}) tr_{1} \{A\}, the inequality can be rewritten as

0 ≤ tr_{F}\{ρ(μ + \frac{1}{2} ∑_{k,l=1}^{M} α_{kl}(c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗}))^{∗}

× (μ + \frac{1}{2} ∑_{m,n=1}^{M} α_{mn}(c_{m}^{∗}c_{n} − c_{n}c_{m}^{∗}))\}

= tr_{2}\{(A^{∗} ⊗ A)(Γ_{ρ} + |s|^{2} 𝟙 ⊗ 𝟙 + \overline{s} 𝟙 ⊗ γ_{ρ} + s γ_{ρ} ⊗ 𝟙 + (𝟙 ⊗ γ_{ρ})Ex)\}. (31)

This inequality is valid for all s. We first assume that tr_{1} {A} = ∑_{i=1}^{M} α_{ii} ≠ 0, then the choice s := − \begin{matrix}tr & 1 & \{Aγ\} \\ tr & 1 & \{A\} \end{matrix} optimizes the inequality. The conclusion is (27):

tr_{2} {(A^{∗} ⊗ A) (Γ_{ρ} + Ex (γ_{ρ} ⊗ 𝟙))} − |tr_{1} {Aγ_{ρ}}|^{2} ≥ 0.  (32)

Note that (31) and (32) are equivalent because (32) implies (31) by

tr_{2} {(A^{∗} ⊗ A) (Γ_{ρ} + Ex (γ_{ρ} ⊗ 𝟙))} ≥ |tr_{1} {Aγ_{ρ}}|^{2}

≥ −tr_{2} \{(A^{∗} ⊗ A) (|s|^{2} 𝟙 ⊗ 𝟙 + \overline{s} 𝟙 ⊗ γ_{ρ} + s γ_{ρ} ⊗ 𝟙)\} .

Conversely, if tr_{1} {A} = 0, the choice μ = − tr_{1} {Aγ} in (30) leads directly to (32).

Documenta Mathematica 17 (2012) 451–481

462 Bach, Knörr, Menge c) Inserting P_{2} := ∑_{k,l=1}^{M} \overline{α}_{kl}c_{k}c_{l} into (23) yields inequality (25):

0 ≤ tr_{F}\{ρ( ∑_{k,l=1}^{M} \overline{α}_{kl}c_{k}c_{l})^{∗}( ∑_{m,n=1}^{M} \overline{α}_{mn}c_{m}c_{n})\}

= ∑_{k,l,m,n=1}^{M} α_{kl}\overline{α}_{mn}tr_{F} \{ρ c_{l}^{∗}c_{k}^{∗}c_{m}c_{n}\} .

By the definition of Γ_{ρ} one finds

0 ≤ ∑_{k,l,m,n=1}^{M} α_{kl}\overline{α}_{mn} ⟨φ_{m} ⊗ φ_{n}| Γ_{ρ}(φ_{k} ⊗ φ_{l})⟩
= ⟨Ψ_{M} | Γ_{ρ}Ψ_{M}⟩ ,  (33)

where Ψ_{M} := ∑_{i,j=1}^{M} α_{ij} (φ_{i} ⊗ φ_{j}) ∈ span {φ_{k} ⊗ φ_{l}| k, l ∈ ℕ}.

d) Inequality (26) follows from (23) by choosing P_{2} := ∑_{k,l=1}^{M} α_{kl}c_{k}^{∗}c_{l}^{∗}:

0 ≤ tr_{F}{ρ( ∑_{k,l=1}^{M} α_{kl}c_{k}^{∗}c_{l}^{∗})^{∗}( ∑_{m,n=1}^{M} α_{mn}c_{m}^{∗}c_{n}^{∗})}

= ∑_{k,l,m,n=1}^{M} \overline{α}_{kl}α_{mn}tr_{F} \{ρ c_{l}c_{k}c_{m}^{∗}c_{n}^{∗}\} .

By normal-ordering using the CAR, one establishes the required relationship to Γ_{ρ} and γ_{ρ}:

0 ≤ ∑_{k,l,m,n=1}^{M} \overline{α}_{kl}α_{mn}tr_{F}\{ρ (c_{m}^{∗}c_{n}^{∗}c_{l}c_{k} − δ_{ln}c_{m}^{∗}c_{k} + δ_{kn}c_{m}^{∗}c_{l} + δ_{lm}c_{n}^{∗}c_{k}
− δ_{km}c_{n}^{∗}c_{l} − δ_{lm}δ_{kn} + δ_{km}δ_{ln})}

= ⟨ ∑_{k,l=1}^{M} α_{kl}(φ_{l} ⊗ φ_{k})|(Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ}))

× ( ∑_{m,n=1}^{M} α_{mn}(φ_{n} ⊗ φ_{m})) ⟩

= ⟨Ψ_{M} |(Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ}))Ψ_{M}⟩.  (34)

Next we prove (ii) ⇒ (i). Thus, we assume (24)-(27).

Documenta Mathematica 17 (2012) 451–481

e) A general polynomial of degree r ≤ 1 is of the form P_{1} = ∑_{k=1}^{M}(α_{k}c_{k}^{∗} +

β_{k}c_{k}) + μ with μ, α_{k}, β_{k} ∈ ℂ. This means we have to consider

tr_{F} {ρ P_{1}^{∗}P_{1}} = ∑_{k,l=1}^{M} tr_{F} {ρ (α_{k}c_{k}^{∗} + β_{k}c_{k} + μ)^{∗}(α_{l}c_{l}^{∗} + β_{l}c_{l} + μ)} . (35)

We expand the product on the right side of (35) and compute the traces, taking into account that tr_{F} {ρ c_{i}^{∗}} = tr_{F} {ρ c_{i}} = tr_{F} {ρ c_{i}^{∗}c_{j}^{∗}} = tr_{F} {ρ c_{i}c_{j}} = 0 for every i, j since ρ preserves the particle number. Therefore, only three terms in (35) are non-vanishing,

tr_{F} {ρ P_{1}^{∗}P_{1}} = ∑_{k,l=1}^{M} tr_{F} {ρ ((α_{k}c_{k}^{∗})^{∗} (α_{l}c_{l}^{∗}) + (β_{k}c_{k})^{∗} (β_{k}c_{k}))} + |μ|^{2} ,

where we additionally use tr_{F} {ρ} = 1. The sum over the terms in braces is non-negative due to (28) and (29). The conclusion is tr_{F} {ρ P_{1}^{∗}P_{1}} ≥ 0.

f) For r ≤ 2 we have a general polynomial given by

P_{2} = ν + ∑_{k=1}^{M} (α_{k}c_{k}^{∗} + β_{k}c_{k}) + ∑_{k,l=1}^{M} α_{kl}c_{k}^{∗}c_{l}^{∗} + ∑_{k,l=1}^{M} β_{kl}c_{k}c_{l}

+ ∑_{k,l=1}^{M} κ_{kl}c_{k}^{∗}c_{l} + ∑_{k,l=1}^{M} η_{kl}c_{k}c_{l}^{∗},

where ν, α_{k}, β_{k}, α_{kl}, β_{kl}, κ_{kl}, η_{kl} ∈ ℂ, for all 1 ≤ k, l ≤ M. Using the CAR, we rewrite P_{2} as

P_{2} = P_{1} + P_{2,α} + P_{2,β} + P_{2,η},

where

P_{1} := μ + ∑_{k=1}^{M} (α_{k}c_{k}^{∗} + β_{k}c_{k}) ,  P_{2,α} := ∑_{k,l=1} α_{kl}c_{k}^{∗}c_{k}^{∗},

P_{2,β} := ∑_{k=1}^{M} β_{kl}c_{k}c_{l},  P_{2,θ} := ∑_{k,l=1}^{M} θ_{kl} (c_{k}^{∗}c_{l} − c_{l}c_{k}^{∗}) ,

and

μ := ν + \frac{1}{2} ∑_{k=1}^{M} (κ_{kk} + η_{kk}) , θ_{kl} := \frac{1}{2} (κ_{kl} − η_{lk}) .

Then

tr_{F} {ρ P_{2}^{∗}P_{2}}

= tr_{F} {ρ (P_{1}^{∗} + P_{2,α}^{∗} + P_{2,β}^{∗} + P_{2,η}^{∗}) (P_{1} + P_{2,α} + P_{2,β} + P_{2,θ})}

= tr_{F} {ρ P_{1}^{∗}P_{1}} + tr_{F} {ρ P_{2,α}^{∗}P_{2,α}} + tr_{F} {ρ P_{2,β}^{∗}P_{2,β}}
+ tr_{F} {ρ P_{2,θ}^{∗}P_{2,θ}} ,

where we use that tr_{F} {ρ P_{a}^{∗}P_{b}} = 0 whenever a ≠ b, since ρ conserves the particle number. Now, e) implies tr_{F} {ρ P_{1}^{∗}P_{1}} ≥ 0, (34) yields tr_{F} {ρ P_{2,α}^{∗}P_{2,α}} ≥ 0 (see d)), (33) yields tr_{F} {ρ P_{2,β}^{∗}P_{2,β}} ≥ 0

(see c)), and tr_{F} {ρ P_{2,θ}^{∗}P_{2,θ}} ≥ 0 follows from (32), see b). Hence, tr_{F} {ρ P_{2}^{∗}P_{2}} ≥ 0.

Lemma 3.2 is the algebraic part of the proof of Theorem 3.1. To conclude, we have to extend the proof to infinite dimensions.

Proof of Theorem 3.1. Since (ii) contains (24)-(27) of Lemma 3.2, the implication (ii) ⇒ (i) is obvious. For (i) ⇒ (ii), let ϕ, ψ and Φ, Ψ be normalized vectors in 𝔥 and 𝔥 ⊗ 𝔥, respectively, and set α_{i} := ⟨φ_{i}|ϕ⟩, β_{i} := ⟨φ_{i}|ψ⟩ and α_{ij} := ⟨φ_{i} ⊗ φ_{j}|Φ⟩, β_{ij} := ⟨φ_{i} ⊗ φ_{j}|Ψ⟩ for all i, j ∈ ℕ. For M ∈ ℕ, we define

the orthogonal projection P_{M} := ∑_{k=1}^{M} |φ_{k}⟩ ⟨φ_{k}| and set ϕ_{M} := P_{M}ϕ = ∑_{i=1}^{M} α_{i}φ_{i}

and Ψ_{M} := (P_{M} ⊗ P_{M}) Ψ = ∑_{i,j=1}^{M} β_{ij} (φ_{i} ⊗ φ_{j}). The admissibility and the G-,

P-, and Q-Condititions follow from Lemma 3.2 as follows:

a) For the 1-pdm we have ∥γ_{ρ}∥_{op} < ∞, since

|⟨ϕ| γ_{ρ}ψ⟩| = |tr_{F} {ρ c^{∗}(ψ)c(ϕ)}|
≤ tr_{F} {|ρ|} ∥ψ∥∥ϕ∥ = tr_{F} {|ρ|} < ∞

by the Cauchy–Schwarz inequality and c^{∗}(ϕ)c(ϕ) ≤ 𝟙 ⟨ϕ|ϕ⟩. Afterwards, we infer by the triangle inequality

|⟨ϕ| γ_{ρ}ϕ⟩ − ⟨ϕ_{M}| γ_{ρ}ϕ_{M}⟩| = |⟨ϕ − ϕ_{M}| γ_{ρ}ϕ⟩ + ⟨ϕ_{M}| γ_{ρ}(ϕ − ϕ_{M})⟩|
≤ |⟨ϕ − ϕ_{M}| γ_{ρ}ϕ⟩| + |⟨ϕ_{M}| γ_{ρ}(ϕ − ϕ_{M})⟩|
≤ ∥ϕ − ϕ_{M}∥ (∥ϕ∥ + ∥ϕ_{M}∥) ∥γ_{ρ}∥_{op}
≤ 2 ∥ϕ − ϕ_{M}∥∥γ_{ρ}∥_{op}.  (36)

As M → ∞, ∥ϕ − ϕ_{M}∥ vanishes and we conclude with ⟨ϕ_{M}| γ_{ρ}ϕ_{M}⟩ ≥ 0 that also ⟨ϕ| γ_{ρ}ϕ⟩ ≥ 0 for all ϕ ∈ 𝔥. The same argument with γ_{ρ} replaced by 𝟙 − γ_{ρ} leads to ⟨ϕ| (𝟙 − γ_{ρ})ϕ⟩ ≥ 0 for all ϕ ∈ 𝔥. b) γ_{ρ} ∈ L^{1}(𝔥) follows by monotone convergence since γ_{ρ} ≥ 0. For any ONB

{ψ_{i}}_{i=1}^{∞} ⊆ 𝔥 and ϵ > 0, we have

∑_{k=1}^{∞} ⟨ψ_{k}| γ_{ρ}ψ_{k}⟩ ≤ sup_{M} { ∑_{k=1}^{M} tr_{F} {ρ c_{k}^{∗}c_{k}} }

= sup_{M} {tr_{F}{ρ ∑_{k=1}^{M} c_{k}^{∗}c_{k}}}

= (N + ϵ) sup_{M} \{tr_{F}\{ρ (\frac{1}{ℕ̂ + ϵ})^{\frac{1}{2}} ( ∑_{k=1}^{M} c_{k}^{∗}c_{k}) (\frac{1}{ℕ̂ + ϵ})^{\frac{1}{2}} \}\}

≤ (N + ϵ) tr_{F} {|ρ|} < ∞,

since ℕ̂ ρ = Nρ and ∥(\frac{1}{ℕ̂+ϵ})^{\frac{1}{2}} ( ∑_{k=1}^{M} c_{k}^{∗}c_{k}) (\frac{1}{ℕ̂+ϵ})^{\frac{1}{2}} ∥_{op} ≤ 1.

c) Thanks to b) we can compute ∑_{k=1}^{∞} ⟨φ_{k}| γ_{ρ}φ_{k}⟩ using monotone convergence

and ℕ̂ ρ = Nρ. This gives the trace of γ_{ρ}.

tr_{1} {γ_{ρ}} := ∑_{k=1}^{∞} ⟨φ_{k}| γ_{ρ}φ_{k}⟩ = ∑_{k=1}^{∞} tr_{F} {ρ c_{k}^{∗}c_{k}}

= tr_{F}{ρ ∑_{k=1}^{∞} c_{k}^{∗}c_{k}} = tr_{F} {ρ ℕ̂} = N.

d) For any basis of 𝔥, the identities Ex Γ_{ρ} = Γ_{ρ} Ex = − Γ_{ρ} are a consequence of the definition of Γ_{ρ} and the CAR.

e) We conclude from the definition of Γ_{ρ} by the Cauchy–Schwarz inequality

|⟨Ψ| Γ_{ρ}Φ⟩| = |tr_{F}{ρ ( ∑_{m,n=1}^{∞} β_{mn}c_{n}c_{m})^{∗}( ∑_{k,l=1}^{∞} α_{kl}c_{l}c_{k})}|
≤ tr_{F} {|ρ|} ∥Ψ∥∥Φ∥ = tr_{F} {|ρ|} < ∞.

Therefore, we have ∥Γ_{ρ}∥_{op} < ∞. Afterwards, we infer analogously to (36)

|⟨Ψ| Γ_{ρ}Ψ⟩ − ⟨Ψ_{M}| Γ_{ρ}Ψ_{M}⟩| ≤ 2 ∥Ψ − Ψ_{M}∥∥Γ_{ρ}∥_{op},

which tends to zero as M → ∞ due to the definition of Ψ_{M}. With ⟨Ψ_{M}| Γ_{ρ}Ψ_{M}⟩ ≥ 0 from (33), this means ⟨Ψ| Γ_{ρ}Ψ⟩ ≥ 0 for all Ψ ∈ 𝔥 ⊗ 𝔥.

f) To prove that Γ_{ρ} ∈ L^{1}(𝔥 ⊗ 𝔥) we show that there is an ONB {ψ_{i}}_{i=1}^{∞} ⊆ 𝔥

such that ∑_{k,l=1}^{∞} ⟨ψ_{k} ⊗ ψ_{l}| Γ_{ρ}(ψ_{k} ⊗ ψ_{l})⟩ is finite, again using Γ_{ρ} ≥ 0. For any ϵ > 0 we have, using ℕ̂ ρ = Nρ and monotone convergence,

∑_{k,l=1}^{∞} ⟨ψ_{k} ⊗ ψ_{l}| Γ_{ρ}(ψ_{k} ⊗ ψ_{l})⟩ ≤ sup_{M} { ∑_{k,l=1}^{M} tr_{F} {ρ c_{l}^{∗}c_{k}c_{k}^{∗}c_{l}} }

= sup_{M} {tr_{F}{ρ [( ∑_{k=1}^{M} c_{k}^{∗}c_{k})^{2} − ( ∑_{k=1}^{M} c_{k}^{∗}c_{k})]}}

≤ sup_{M} \{ (N^{2} + ϵ) tr_{F}\{|ρ| (\frac{1}{ℕ̂^{2} + ϵ})^{\frac{1}{2}} ( ∑_{k=1}^{M} c_{k}^{∗}c_{k})^{2} (\frac{1}{ℕ̂^{2} + ϵ})^{\frac{1}{2}} \}\}

≤ (N^{2} + ϵ) tr_{F} {|ρ|} < ∞

since ∑_{k=1}^{M} c_{k}^{∗}c_{k} ≥ 0 and, due to ( ∑_{k=1}^{M} c_{k}^{∗}c_{k})^{2} ≤ ℕ̂^{2},

∥(\frac{1}{ℕ̂^{2} + ϵ})^{\frac{1}{2}} ( ∑_{k=1}^{M} c_{k}^{∗}c_{k})^{2} (\frac{1}{ℕ̂^{2} + ϵ})^{\frac{1}{2}} ∥_{op} ≤ 1.

g) To check (32) for any bounded A (not necessarily of finite rank) we abbreviate Λ_{G} := Γ_{ρ} + Ex (γ_{ρ} ⊗ 𝟙) − γ_{ρ} ⊗ γ_{ρ} and set A_{M} := P_{M}AP_{M}. Clearly, A_{M} is of finite rank and we observe

|tr_{2} {(A^{∗} ⊗ A) Λ_{G}} − tr_{2} {(A_{M}^{∗} ⊗ A_{M}) Λ_{G}}|

= |tr_{2} {[(A − A_{M})^{∗} ⊗ A + A_{M}^{∗} ⊗ (A − A_{M})] Λ_{G}}|
= |tr_{2}{[ (P_{M}A^{∗}P_{M}^{⊥} + P_{M}^{⊥}A^{∗}P_{M} + P_{M}^{⊥}A^{∗}P_{M}^{⊥}) ⊗ A
+ A_{M}^{∗} ⊗ (P_{M}^{⊥}AP_{M} + P_{M}AP_{M}^{⊥} + P_{M}^{⊥}AP_{M}^{⊥}) ]Λ_{G}}| (37)

using P_{M}^{⊥} := 𝟙 − P_{M}. For |tr_{2} {(P_{M}^{⊥}A^{∗}P_{M} ⊗ A) Λ_{G}}|, for instance, we find

|tr_{2} {(P_{M}^{⊥}A^{∗}P_{M} ⊗ A) Λ_{G}}|
= |tr_{2} {(P_{M}^{⊥}A^{∗}P_{M} ⊗ A) Γ_{ρ}} + tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}A}
− tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}} tr_{1} {Aγ_{ρ}} |
≤ |tr_{2} {(P_{M}^{⊥}A^{∗}P_{M} ⊗ A) Γ_{ρ}}| + |tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}A}|
+ |tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}} tr_{1} {Aγ_{ρ}}| .  (38)

Since Γ_{ρ} ≥ 0 due to the P-Condition (see e)), P_{M}^{⊥}, P_{M} ≥ 0 and ℕ̂ ρ = Nρ we have on the one hand

|tr_{2} {(P_{M}^{⊥}A^{∗}P_{M} ⊗ A) Γ_{ρ}}| ≤ ∥A∥_{op}^{2} tr_{2} {(P_{M}^{⊥} ⊗ 𝟙) Γ_{ρ}}
= (N − 1) ∥A∥_{op}^{2} tr_{1} {P_{M}^{⊥}γ_{ρ}} ,

and on the other hand with 0 ≤ γ_{ρ} ≤ 𝟙 and P_{M} ≤ 𝟙

|tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}A}| ≤ ∥A∥_{op}^{2} tr_{1} {P_{M}^{⊥}γ_{ρ}} ,

and

|tr_{1} {P_{M}^{⊥}A^{∗}P_{M}γ_{ρ}} tr_{1} {Aγ_{ρ}}| ≤ N ∥A∥_{op}^{2} tr_{1} {P_{M}^{⊥}γ_{ρ}} .

Note that tr_{1} {P_{M}^{⊥}γ_{ρ}} = ∑_{k=M+1}^{∞} ⟨φ_{k}| γ_{ρ}φ_{k}⟩ → 0, as M → ∞, since

∑_{k=1}^{∞} ⟨φ_{k}| γ_{ρ}φ_{k}⟩ = tr_{1} {γ_{ρ}} = N is convergent. Analogously, one finds

that all terms on the right hand side of (37) tend to zero, as M → ∞. This, in turn, implies tr_{2} {(A^{∗} ⊗ A) Λ_{G}} ≥ 0 for any bounded A since tr_{2} {(A_{M}^{∗} ⊗ A_{M}) Λ_{G}} ≥ 0 due to (32).

h) Again by the Cauchy–Schwarz inequality we find

|⟨Ψ|(Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ}))Φ⟩|

= |tr_{F}\{ρ ( ∑_{m,n=1}^{∞} \overline{β}_{mn}c_{m}c_{n})( ∑_{k,l=1}^{∞} \overline{α}_{kl}c_{k}c_{l})^{∗}\}|

≤ tr_{F} {|ρ|} ∥Ψ∥∥Φ∥

and, therefore, ∥Γ_{ρ} +(1−Ex)(𝟙⊗𝟙−γ_{ρ} ⊗𝟙−𝟙⊗γ_{ρ})∥_{op} < ∞. Following (36) with Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ}) instead of Γ_{ρ}, we arrive at

⟨Ψ|(Γ_{ρ} + (1 − Ex)(𝟙 ⊗ 𝟙 − γ_{ρ} ⊗ 𝟙 − 𝟙 ⊗ γ_{ρ}))Ψ⟩ ≥ 0 ∀Ψ ∈ 𝔥 ⊗ 𝔥.

(γ_{ρ}, Γ_{ρ}) obeys the P-, G-, and Q-Conditions by e), g) and h). The admissibility is ensured in a) to d), and f).

A simple consequence of Theorem 3.1 is

Corollary 3.3. Let N ∈ ℕ and assume that (γ, Γ) is N-representable. Then (γ, Γ) is admissible and fulfills the G-, P- and Q-Conditions.

Proof. Since (γ, Γ) is N-representable, there exists a density matrix ρ ∈ L_{+}^{1}(F) with (γ, Γ) ≡ (γ_{ρ}, Γ_{ρ}). By the last theorem, (γ, Γ) then is admissible and fulfills the G-, P- and Q-Conditions.

Remark 3.4. The G-Condition (27) seems to be asymmetric in terms of γ_{ρ}. However, since tr_{2} {(A ⊗ B) Γ_{ρ}} = tr_{2} {(B ⊗ A) Γ_{ρ}}, it is easy to show that also tr_{2} {(A^{∗} ⊗ A) (Γ_{ρ} + Ex (𝟙 ⊗ γ_{ρ}))} ≥ |tr_{1} {Aγ_{ρ}}|^{2} holds. Thus, we have a symmetrized, but weaker, G-Condition given by

tr_{2} \{(A^{∗} ⊗ A) (Γ_{ρ} + \frac{1}{2}Ex (𝟙 ⊗ γ_{ρ} + γ_{ρ} ⊗ 𝟙))\} ≥ |tr_{1} \{Aγ_{ρ}\}|^{2} .

4 Correlation inequalities from G- and P-Conditions

In [1], a lower bound on the difference of the ground state functional E (γ, Γ) and the Hartree–Fock functional E (γ, (1 − Ex (γ ⊗ γ))), i.e.,

tr_{2} {V Γ^{(T)}} = tr_{2} {V (Γ − (1 − Ex) (γ ⊗ γ))} ,  (39)

is derived using the decomposition of the potential V according to Fefferman and de la Llave [7]. It turns out that this decomposition is also useful to derive lower bounds only by means of N-representability. The main result of this section is the following theorem.

Theorem 4.1. Let X = X^{∗} = X^{2} ∈ B(𝔥) be an orthogonal projection on 𝔥. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

tr_{2} {(X ⊗ X) Γ^{(T)}} ≥ − tr_{1} {Xγ} min {1; 38 tr_{1} {X (γ − γ^{2})}

+ 4[tr_{1} \{X (γ − γ^{2})\} (2 + 8 tr_{1} \{X (γ − γ^{2})\}^{2} )]^{\frac{1}{2}} \}. (40)

Proof. The proof is carried out in several parts in the following sections. The first inequality is derived in Theorem 4.4. The second inequality follows from Theorems 4.15, 4.17 and 4.19.

In order to apply Theorem 4.1 to (39), the potential V on 𝔥 ⊗ 𝔥 is decomposed into an integral of a tensor product of two copies of the one-particle operator X. This decomposition is called Fefferman–de la Llave identity.

Lemma 4.2. For all x , y ∈ ℝ^{3}, x ≠ y, one has

\frac{1}{|x − y|} = ∫_{0}^{∞} \frac{dr}{πr^{5}} ∫_{ℝ^{3}} d^{3}z χ_{B(z,r)} (x) χ_{B(z,r)} (y) ,  (41)

where χ_{B(z,r)} is the characteristic function of the ball B (z, r) := {x ∈ ℝ^{3} | |x − z| ≤ r} of radius r > 0 centered at z ∈ ℝ^{3}.

The proof of the decomposition can be found in the original work of Fefferman and de la Llave in [7]. In [12], Hainzl and Seiringer have derived sufficient conditions on a pair potential V : ℝ^{n} → ℝ so as to admit a decomposition of the form (41). Remark 4.3. The multiplication operator corresponding to χ_{B(z,r)} is denoted by X_{r,z} ≡ X. Clearly,

B(𝔥) ∋ X = X^{∗} = X^{2}  (42)

is an orthogonal projection.

Instead of (39) we consider from now on

tr_{2} {(X ⊗ X) Γ^{(T)}} = tr_{2} {(X ⊗ X) (Γ − (1 − Ex) (γ ⊗ γ))} .  (43)

A first estimation of this quantity is immediately obtained by applying the G-Condition directly on tr_{2} {(X ⊗ X) Γ}. This yields the first inequality of (40).

Theorem 4.4. Let X be as in (42). Assume that (γ, Γ) is admissible and fulfills the G-Condition. Then

tr_{2} {(X ⊗ X) Γ^{(T)}} ≥ − tr_{1} {Xγ} .

Proof. As mentioned, we apply the G-Condition (20) with A^{∗} = A := X directly on tr_{2} {(X ⊗ X) Γ}. The HF-part is carried out:

tr_{2} {(X ⊗ X) (Γ − (1 − Ex) (γ ⊗ γ))}
≥ (tr_{1} {Xγ})^{2} − tr_{1} {Xγ} − (tr_{1} {Xγ})^{2} + tr_{1} {XγXγ}
≥ − tr_{1} {Xγ} .  (44)

The last inequality follows from tr_{1} {XγXγ} = tr_{1} {XγXγX} ≥ 0.

The goal of the next sections is an estimation of (43) in terms of tr_{1} {X (γ − γ^{2})}.

4.1 Preparation

A crucial step in [1] is the decomposition of the spectrum of γ into eigenvalues which are larger than \frac{1}{2}, and those which are smaller or equal \frac{1}{2}. Following this step, the decomposition is denoted by two orthogonal projections, P and P^{⊥} (a comparable strategy was also used by Graf and Solovej in [13]). The first one, P, projects on the space which is spanned by the eigenvectors of γ corresponding to eigenvalues larger than \frac{1}{2}. The second one treats the eigenvectors with eigenvalues smaller or equal \frac{1}{2}. Furthermore, the eigenvectors of γ, {φ_{i} | γφ_{i} = λ_{i}φ_{i}}_{i=1}^{∞}, are used as an ONB of 𝔥 which we mainly refer to. In this basis the two projections can be defined straightforwardly.

Definition 4.5. On 𝔥, the orthogonal projections P and P^{⊥} are defined by

P := 𝟙 [γ > \frac{1}{2}] = ∑_{k> \frac{1}{2}} |φ_{k}⟩ ⟨φ_{k}| and P^{⊥} := 𝟙 [γ ≤ \frac{1}{2}] = ∑_{k≤ \frac{1}{2}} |φ_{k}⟩ ⟨φ_{k}| . (45)

Here, the summation over ”k > \frac{1}{2}” denotes the summation over \{k | λ_{k} > \frac{1}{2}\} and for ”k ≤ \frac{1}{2}” analogously. Obviously,

P + P^{⊥} = 𝟙, P P^{⊥} = P^{⊥}P = 0, P γ = γP and P^{⊥}γ = γP^{⊥}.

Moreover, the projections are bounded from above.

Lemma 4.6. For P and P^{⊥} defined in (45)

P ≤ 2γ and P^{⊥} ≤ 2 (𝟙 − γ)  (46)

hold true.

Note that, since rk {P } ≤ 2N, P is of finite rank and, hence, trace class.

Proof. Using the definition of the projections together with 0 ≤ γ ≤ 𝟙, one finds for P:

P = ∑_{k> \frac{1}{2}} |φ_{k}⟩ ⟨φ_{k}| ≤ ∑_{k> \frac{1}{2}} 2λ_{k} |φ_{k}⟩ ⟨φ_{k}| ≤ 2 ∑_{k=1}^{∞} λ_{k} |φ_{k}⟩ ⟨φ_{k}| = 2γ,

and for P^{⊥}:

P^{⊥} = ∑_{k< \frac{1}{2}} |φ_{k}⟩ ⟨φ_{k}| ≤ ∑_{k< \frac{1}{2}} 2 (1 − λ_{k}) |φ_{k}⟩ ⟨φ_{k}| ≤ 2 (𝟙 − γ) .

Thanks to P^{⊥} +P = 𝟙 we can expand tr_{2} {(X ⊗ X) Γ} into three parts to have expressions on which we can apply the conditions on (γ, Γ). We denote this three parts by Main Part (MP), Remainder (R) and Main Error Term (MET).

Lemma 4.7. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Then

tr_{2} {(X ⊗ X) Γ}
= tr_{2} {(P XP ⊗ P XP) Γ} + 4 ℜ𝔢 {tr_{2} {(P XP ⊗ P^{⊥}XP ) Γ}}
+ 2 tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ}

+ tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 4 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP^{⊥}) Γ}}

+ 2 tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP ) Γ}
+ 2 ℜ𝔢 {tr_{2} {(P XP^{⊥} ⊗ P XP^{⊥}) Γ}} .  (47)

Proof. After replacing the identity operator on each side of X in each factor of the tensor product X ⊗ X by (P + P^{⊥}), one can expand the r.h.s. of

tr_{2} {(X ⊗ X) Γ}
= tr_{2} {((P + P^{⊥}) X (P + P^{⊥}) ⊗ (P + P^{⊥}) X (P + P^{⊥})) Γ} .

Using tr_{2} {(A ⊗ B) Γ} = tr_{2} {(B ⊗ A) Γ}, which is a consequence of Ex Γ Ex = Γ, one arrives at the assertion after rearranging.

Afterwards, we collect the terms of (47) in a suitable way. Note that compared to [1] the definitions of the Main Part and the Remainder are slightly changed.

Definition 4.8. The term

T_{MP} := tr_{2} {(P XP ⊗ P XP) Γ} + 4 ℜ𝔢 {tr_{2} {(P XP ⊗ P^{⊥}XP ) Γ}}
+ 4 tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ}

is called Main Part,

T_{R} := tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 4 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP^{⊥}) Γ}}
+ 2 tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP ) Γ} − 2 tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ}

is called Remainder, and

T_{MET} := 2 ℜ𝔢 {tr_{2} {(P XP^{⊥} ⊗ P XP^{⊥}) Γ}}

is called Main Error Term.

One estimate is used more than once when considering the terms in the Remainder and Main Error Term. This estimate requires the following lemma.

Lemma 4.9. Let {ψ_{i}}_{i=1}^{∞} ⊆ 𝔥 be an ONB, Q = Q^{∗} = Q^{2}, Q^{⊥} := 𝟙 − Q, and Y = Y^{∗} = Y^{2} ∈ B(𝔥) orthogonal projections. For r, s ∈ ℕ define

B (r, s) := |QY ψ_{r}⟩ ⟨Q^{⊥}Y ψ_{s}| ∈ B(𝔥).  (48)

Then one has

∑_{r,s=1}^{∞} tr_{2} {(B^{∗} (r, s) ⊗ B (r, s)) (Γ + Ex (γ ⊗ 𝟙))}
= tr_{2} {(QY Q ⊗ Q^{⊥}Y Q^{⊥}) (−Γ + 𝟙 ⊗ γ)} .  (49)

Proof. Denoting K := ∑_{r,s=1}^{∞} tr_{2} {(B^{∗} (r, s) ⊗ B (r, s)) (Γ + Ex (γ ⊗ 𝟙))}, we cal-

culate the trace using the ONB {ψ_{i}}_{i=1}^{∞} ⊆ 𝔥.

K = ∑_{r,s=1}^{∞} ∑_{k,l,m,n=1}^{∞} ⟨ψ_{k} ⊗ ψ_{l}| (B^{∗} (r, s) ⊗ B (r, s)) (ψ_{m} ⊗ ψ_{n})⟩
× ⟨ψ_{m} ⊗ ψ_{n}| (Γ + Ex (γ ⊗ 𝟙)) (ψ_{k} ⊗ ψ_{l})⟩ .

In the next step, the definition of B (r, s) together with (A)_{ij} := ⟨ψ_{i}| Aψ_{j}⟩, for any A ∈ B(𝔥), can be used to write

K = ∑_{r,s=1}^{∞} ∑_{k,l,m,n=1}^{∞} (Y Q)_{rm} (Q^{⊥}Y )_{ks} (QY)_{lr} (Y Q^{⊥})_{sn}
× ⟨ψ_{m} ⊗ ψ_{n}| (Γ + Ex (γ ⊗ 𝟙)) (ψ_{k} ⊗ ψ_{l})⟩ .

Performing the summation over r and s leads to

K = ∑_{k,l,m,n=1}^{∞} (QY Q)_{lm} (Q^{⊥}Y Q^{⊥})_{kn}
× ⟨ψ_{m} ⊗ ψ_{n}| (Γ + Ex (γ ⊗ 𝟙)) (ψ_{k} ⊗ ψ_{l})⟩

= ∑_{k,l,m,n=1}^{∞} ⟨ψ_{l} ⊗ ψ_{k}| (QY Q ⊗ Q^{⊥}Y Q^{⊥}) (ψ_{m} ⊗ ψ_{n})⟩

× ⟨ψ_{m} ⊗ ψ_{n}| (Γ + Ex (γ ⊗ 𝟙)) (ψ_{k} ⊗ ψ_{l})⟩ .

The summation over m and n can also be carried out. Finally, the summation over k and l gives

K = ∑_{k,l=1}^{∞} ⟨ψ_{k} ⊗ ψ_{l}| (Ex (QY Q ⊗ Q^{⊥}Y Q^{⊥}) (Γ + Ex (γ ⊗ 𝟙))) (ψ_{k} ⊗ ψ_{l})⟩
= tr_{2} {(QY Q ⊗ Q^{⊥}Y Q^{⊥}) (− Γ + 𝟙 ⊗ γ)} ,

using the cyclicity of the trace, Γ Ex = − Γ, and Ex (𝟙 ⊗ γ) Ex = γ ⊗ 𝟙.

Remark 4.10. By changing the definition of B (r, s) it is also possible to treat, for example, QY Q ⊗ QY Q similarly. However, it is important to notice that ∑_{r,s} B^{∗} (r, s) ⊗ B (r, s) is, in general, indefinite. In fact, the trace of B (r, s) is

vanishing in our case and so is the trace of B^{∗} (r, s)⊗B (r, s). Hence, B^{∗} (r, s)⊗ B (r, s) is either indefinite or zero. Furthermore, since P^{⊥} and P commute with γ, we also have tr_{1} {B (r, s) γ} = 0.

A consequence of Lemma 4.9 is a key inequality for proving the estimate on the Remainder and the Main Error Term. This inequality is given in (50).

Lemma 4.11. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G-Condition. Then

tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} ≤ 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} . (50)

Proof. First, we observe that (49) with Y = X and Q = P and the G-Condition immediately lead to

0 ≤ ∑_{p,q} tr_{2} {(B^{∗} (r, s) ⊗ B (r, s)) (Γ + Ex (γ ⊗ 𝟙))}
= tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) (− Γ + (𝟙 ⊗ γ))} .

Consequently,

tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} ≤ tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) (𝟙 ⊗ γ)}
= tr_{1} {P X} tr_{1} {P^{⊥}Xγ} .

Secondly, we permute the arguments in the trace cyclically and use that γ is trace class and P X and P^{⊥}X are bounded. Then we use (46) to estimate the projections:

tr_{1} \{P X\} tr_{1} \{P^{⊥}Xγ\} = tr_{1} \{XP X\} tr_{1} \{X\sqrt{γ}P^{⊥}\sqrt{γ}X\}
≤ 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})}

since γP^{⊥} = \sqrt{γ}P^{⊥}\sqrt{γ}.

Remark 4.12. ¿From tr_{2} {(A ⊗ B) Γ} = tr_{2} {(B ⊗ A) Γ}, for A, B ∈ B(𝔥), one directly can conclude

tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP ) Γ} ≤ 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .  (51)

4.2 Estimation of the Remainder

Now we consider the Remainder of (47):

T_{R} := tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 4 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP^{⊥}) Γ}}
+ 2 tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP ) Γ} − 2 tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ} .

The first three terms, summed up in T_{R_{1}} and the last term, called T_{R_{2}}, are treated separately to derive a lower bound.

Lemma 4.13. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

T_{R_{1}} := tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 2 tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP ) Γ}
+ 4 ℜ𝔢 {tr_{2} {(P^{⊥}XP^{⊥} ⊗ P XP^{⊥}) Γ}}
≥ − 8 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

Proof. First, we use ℜ𝔢 (ζ) ≥ − |ζ|, for any complex number ζ, and P^{⊥}XP ⊗ P^{⊥}XP^{⊥} = (P^{⊥}X ⊗ P^{⊥}X) (XP ⊗ XP^{⊥}) to infer

T_{R_{1}} = tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ}
+ 4 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP^{⊥}) Γ}}
≥ tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ}
− 4 |tr_{2} {(P^{⊥}X ⊗ P^{⊥}X) (XP ⊗ XP^{⊥}) Γ}| .

Then we use that (A, B) := tr_{2} {A^{∗}B Γ} defines a positive semidefinite Hermitian form on B(𝔥 ⊗ 𝔥), due to Γ ≥ 0, which is the P-Condition. Hence, the Cauchy–Schwarz inequality

|(A, B)| ≥ (A, A)^{\frac{1}{2}} (B, B)^{\frac{1}{2}}  (52)

holds, and we arrive at

T_{R_{1}} ≥ tr_{2} {(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ} + 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ}

− 4 (tr_{2} \{(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ\})^{\frac{1}{2}} (tr_{2} \{(P XP ⊗ P^{⊥}XP^{⊥}) Γ\})^{\frac{1}{2}} .

As x^{2} − 4bx ≥ − 4b^{2}, for x := (tr_{2} \{(P^{⊥}XP^{⊥} ⊗ P^{⊥}XP^{⊥}) Γ\})^{\frac{1}{2}} and b :=

(tr_{2} \{(P XP ⊗ P^{⊥}XP^{⊥}) Γ\})^{\frac{1}{2}} , one then easily concludes

T_{R_{1}} ≥ − 4 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} + 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ}
= − 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} .

The proof is completed by using (50):

T_{R_{1}} ≥ − 2 tr_{2} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} ≥ − 8 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

The estimate on T_{R_{2}} := − 2 tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Γ} is addressed in the next lemma.

Lemma 4.14. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

T_{R_{2}} = − 2 tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Γ} ≥ − 8 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} . (53)

Proof. First, the left side is estimated by its absolute value. Then the Cauchy– Schwarz inequality (52) is used:

− 2 tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Γ}

≥ − 2 |tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Γ}|

≥ − 2 (tr_{2} \{(P^{⊥}XP^{⊥} ⊗ P XP ) Γ\})^{\frac{1}{2}} (tr_{2} \{(P XP ⊗ P^{⊥}XP^{⊥}) Γ\})^{\frac{1}{2}}

= − 2 tr_{1} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ} .  (54)

The assertion (53) follows again from (50).

Summing up the results, we obtain the following estimate of the Remainder directly from Lemmata 4.13 and 4.14.

Theorem 4.15. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

T_{R} ≥ − 16 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

4.3 Estimation of the Main Error Term

The main error in Theorem 4.1 results from estimating T_{MET} in (47),

T_{MET} = 2 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) Γ}} .  (55)

A key observation is that terms of the form A ⊗ B ∈ B(𝔥 ⊗ 𝔥) or Ex (A ⊗ B) ∈ B(𝔥⊗𝔥) can be added to Γ in (55) without changing the value of T_{MET}, provided A and B commute with P^{⊥} and P. Therefore, one can also consider

T_{MET} = 2 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))}} .

This expression can now be estimated using a variant of the Cauchy–Schwarz inequality given in the next lemma.

Lemma 4.16. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G-Condition. Then

ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))}}
≥ − (tr_{2} \{(P XP^{⊥} ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}}
× (tr_{2} \{(P^{⊥}XP ⊗ P XP^{⊥}) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}} . (56)

Proof. We define

(A, B) := tr_{2} {(A^{∗} ⊗ B) (Γ + Ex (γ ⊗ 𝟙))}

on B(𝔥) × B(𝔥) and observe that, because

tr_{2} \{(B^{∗} ⊗ A) Γ\} = tr_{2} \{(A ⊗ B^{∗}) Γ\} = \overline{tr_{2} \{(A^{∗} ⊗ B) Γ\}}

and

tr_{2} \{(B^{∗} ⊗ A) Ex (γ ⊗ 𝟙)\} = tr_{1} \{B^{∗}A γ\} = \overline{tr_{1} \{A^{∗}B γ\}}
= \overline{tr_{2} \{(A^{∗} ⊗ B) Ex (γ ⊗ 𝟙)\}},

(· , ·) defines a Hermitian form on B(𝔥) × B(𝔥). Furthermore, (· , ·) is positive semidefinite, since

(A, A) = tr_{2} {(A^{∗} ⊗ A) (Γ + Ex (γ ⊗ 𝟙))}
≥ |tr_{1} {Aγ}|^{2} ≥ 0

thanks to the G-Condition. Hence, the Cauchy–Schwarz inequality |(A, B)| ≤ (A, A)^{\frac{1}{2}} (B, B)^{\frac{1}{2}} holds true. Applying this with A^{∗} := B := P^{⊥}XP, we arrive at the asserted estimate (56):

ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))}}
= ℜ𝔢 \{(A^{∗}, A)\} ≥ − |(A^{∗}, A)| ≥ − (A^{∗}, A^{∗})^{\frac{1}{2}} (A, A)^{\frac{1}{2}}
= − (tr_{2} \{(P XP^{⊥} ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}}

× (tr_{2} \{(P^{⊥}XP ⊗ P XP^{⊥}) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}} .

Now the Main Error Term can be estimated.

Theorem 4.17. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

T_{MET} ≥ − 2 tr_{1} \{Xγ\} [8 tr_{1} \{X (γ − γ^{2})\} (1 + 4 tr_{1} \{X (γ − γ^{2})\} )]^{\frac{1}{2}} .

Proof. We rewrite T_{MET} adding the necessary exchange term to allow for an application of (56):

T_{MET} = 2 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) Γ}}
= 2 ℜ𝔢 {tr_{2} {(P^{⊥}XP ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))}}
≥ − 2 (tr_{2} \{(P XP^{⊥} ⊗ P^{⊥}XP ) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}}
× (tr_{2} \{(P^{⊥}XP ⊗ P XP^{⊥}) (Γ + Ex (γ ⊗ 𝟙))\})^{\frac{1}{2}} .

Note that tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ} = tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Γ}, which was already estimated in (54). Together with (50) we obtain

tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ} ≤ tr_{1} {(P XP ⊗ P^{⊥}XP^{⊥}) Γ}
≤ 4tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

The two exchange terms have to be treated separately. Using P^{⊥}, P ≤ 𝟙, we find

tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Ex (γ ⊗ 𝟙)} = tr_{1} {P^{⊥}XP γP XP^{⊥}}
= tr_{1} \{P^{⊥}X\sqrt{γ}P \sqrt{γ}XP^{⊥}\}
≤ tr_{1} {Xγ} ,

and

tr_{2} {(P^{⊥}XP ⊗ P XP^{⊥}) Ex (γ ⊗ 𝟙)} = tr_{1} {P XP^{⊥}γP^{⊥}XP }
= tr_{1} {XP XXP^{⊥}γP^{⊥}X}
≤ tr_{1} {XP X} tr_{1} {XP^{⊥}γP^{⊥}X}
≤ 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} ,

where cyclic permutation in the argument of the trace is used together with X = X^{2}, (P^{⊥})^{2} = P^{⊥} and P^{2} = P to expand the argument of the trace. Then one can use XP X ≥ 0 and XP^{⊥}γP^{⊥}X ≥ 0 to estimate by XP X ≤ 𝟙 tr_{1} {XP X}. Afterwards, (46) can be used. Merging the results, we arrive at the assertion.

4.4 Estimation of the Main Part

In this section it is shown that the remaining terms of (47),

T_{MP} = tr_{2} {(P XP ⊗ P XP) Γ} + 4 ℜ𝔢 {tr_{2} {(P XP^{⊥} ⊗ P XP ) Γ}}
+ 4 tr_{2} {(P XP^{⊥} ⊗ P^{⊥}XP ) Γ} ,

are large enough to cover the HF-part tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)} in (43). Due to this, we call this terms the Main Part. As mentioned, the Main Part was extended by an additional term. This extension allows for the following observation.

Lemma 4.18. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Then

T_{MP} = tr_{2} {(P X (P + 2P^{⊥}) ⊗ (P + 2P^{⊥}) XP ) Γ} .  (57)

Proof. Expanding the parentheses on the right side leads to the assertion by using tr_{2} {(A ⊗ B) Γ} = tr_{2} {(B ⊗ A) Γ}.

For A := (P + 2P^{⊥}) XP we have T_{MP} = tr_{2} {(A^{∗} ⊗ A) Γ}. This provides the use of the G-Condition.

Theorem 4.19. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G-Condition. Then

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)} ≥ − 22 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} . (58)

Proof. The proof is split into two parts. In the first part, the trace of the HFpart is calculated. In the second part, the Main Part is estimated by applying the G-Condition with A := (P + 2P^{⊥}) XP.

a) As in (44), the trace of the HF part can be written as

tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)} = (tr_{1} {Xγ})^{2} − tr_{1} {XγXγ} .

b) Owing to (57), the G-Condition can be applied directly on the Main Part:

T_{MP} = tr_{2} {(P X (P + 2P^{⊥}) ⊗ (P + 2P^{⊥}) XP ) Γ}
≥ |tr_{1} {P X (P + 2P^{⊥}) γ}|^{2}

− tr_{1} {P X (P + 2P^{⊥}) γ (P + 2P^{⊥}) XP } .

Due to cyclical permutation, [γ, P] = [γ, P^{⊥}] = 0 and P^{⊥}P = P P^{⊥} = 0, some traces vanish. The result is

T_{MP} ≥ |tr_{1} {P Xγ}|^{2} − tr_{1} {XP XP γ} − 4 tr_{1} {XP XP^{⊥}γ}
≥ (tr_{1} {P Xγ})^{2} − tr_{1} {P XγX} − 4 tr_{1} {XP } tr_{1} {XP^{⊥}γ} .

In tr_{1} {XP XP γ}, [P, γ] = 0 and P ≤ 𝟙 is used to write tr_{1} \{XP XP γ\} = tr_{1} \{P X\sqrt{γ}P \sqrt{γ}XP \} ≤ tr_{1} \{P XγX\}. In the last trace, XP ≤ 𝟙 tr_{1} {XP } is used. This is possible since tr_{1} \{XP XP^{⊥}γ\} = tr_{1} \{XP XXP^{⊥}γX\} and XP^{⊥}γX = |XP^{⊥}\sqrt{γ}|^{2} ≥ 0 together with XP X ≥ 0.

Before adding up the estimates, we note that

tr_{1} \{P^{⊥}XγXγ\} = tr_{1} \{\sqrt{γ}X\sqrt{γ}P^{⊥}\sqrt{γ}X\sqrt{γ}\} ≥ 0

and

(tr_{1} {Xγ})^{2} − (tr_{1} {P Xγ})^{2}
= (tr_{1} {Xγ} + tr_{1} {P Xγ} )(tr_{1} {Xγ} − tr_{1} {P Xγ} )
= (tr_{1} {Xγ} + tr_{1} {P Xγ} )tr_{1} {P^{⊥}Xγ} .  (59)

Furthermore, one has tr_{1} {XP } tr_{1} {XP^{⊥}γ} ≤ 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})}. These results can now be applied together with a) and b) to the left side of (58):

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)}

≥ − ((tr_{1} {Xγ})^{2} − (tr_{1} {P Xγ})^{2}) + tr_{1} {XγXγ} − tr_{1} {P XγX}

− 16 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

At this point we use (59), split tr_{1} {XγXγ} into tr_{1} {P XγXγ} + tr_{1} {P^{⊥}XγXγ}, and rearrange:

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)}
≥ − (tr_{1} {Xγ} + tr_{1} {P Xγ}) tr_{1} {P^{⊥}Xγ}
− tr_{1} {P XγX (𝟙 − γ)} + tr_{1} {P^{⊥}XγXγ}
− 16 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

Then, with P ≤ 𝟙, P^{⊥} ≤ 2 (1 − γ), and tr_{1} {P^{⊥}XγXγ} ≥ 0, one obtains

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)}
≥ − 4 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} − tr_{1} {P XγX (𝟙 − γ)}
− 16 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

We continue with the inequality XγX ≤ 𝟙tr_{1} {XγX}. This is allowed because tr_{1} {P XγX (𝟙 − γ)} = tr_{1} {XγXX (𝟙 − γ) P X} and X (𝟙 − γ) P X = X\sqrt{𝟙 − γ}P \sqrt{𝟙 − γ}X = |X\sqrt{𝟙 − γ}P |^{2} ≥ 0 together with XγX ≥ 0:

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)}

≥ − 20 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} − tr_{1} {XγX} tr_{1} {X (𝟙 − γ) P X}

≥ − 20 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} − 2 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})}
= − 22 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

The last inequality follows from P ≤ 2γ.

Finally, the proof of Theorem 4.1 is completed by the estimation of T_{R} in Theorem 4.15, T_{MET} in Theorem 4.17 and T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)} in Theorem 4.19. In each of this theorems, the G-Condition was used to generate bounds. The P-Condition was only applied to provide the use of the Cauchy– Schwarz inequality. In the end, it is remarkable that the Q-Condition is not needed for the proof of the correlation estimate.

5 Summary

We have obtained several results in the last section, which were merged in the main theorem, Theorem 4.1:

tr_{2} {(X ⊗ X) (Γ − (1 − Ex) (γ ⊗ γ))} ≥ − tr_{1} {Xγ} ,

T_{R} ≥ − 16 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} ,

T_{MET} ≥ − 2 tr_{1} \{Xγ\} [8 tr_{1} \{X (γ − γ^{2})\} (1 + 4 tr_{1} \{X (γ − γ^{2})\} )]^{\frac{1}{2}} ,

T_{MP} − tr_{2} {(X ⊗ X) (𝟙 − Ex) (γ ⊗ γ)} ≥ − 22 tr_{1} {Xγ} tr_{1} {X (γ − γ^{2})} .

Denoting b := tr_{1} \{Xγ\} and a := \sqrt{tr_{1} \{X (γ − γ^{2})\}}, one can rewrite the estimates for tr_{2} {(X ⊗ X) Γ^{(T)}} = tr_{2} {(X ⊗ X) (Γ − (1 − Ex) (γ ⊗ γ))} as follows:

tr_{2} \{(X ⊗ X) Γ^{(T)}\} ≥ − b min \{1; a (38a + 2\sqrt{8 + 32a^{2}})\} .  (60)

A suitable choice of a ≤ b in (60) leads to the following correlation estimation.

Theorem 5.1. Let X, and P and P^{⊥} be as defined in (42) and (45), respectively. Assume that (γ, Γ) is admissible and fulfills the G- and P-Conditions. Then

tr_{2} \{(X ⊗ X) Γ^{(T)}\} ≥ − tr_{1} \{Xγ\} min \{1; 10\sqrt{tr_{1} \{X (γ − γ^{2})\}}\} .

Documenta Mathematica 17 (2012) 451–481

Proof. The minimum in (60) is a (38a + 2\sqrt{8 + 32a^{2}}) for 0 < a ≤ \frac{1}{\sqrt{94}} and, thus, \frac{1}{a} ≥ (38a + 2\sqrt{8 + 32a^{2}}). Since (38a + 2\sqrt{8 + 32a^{2}}) is monotonously increasing in a, we find

(38a + 2\sqrt{8 + 32a^{2}}) ≤ \sqrt{94} < 10,  (61)

which implies the assertion.

Remark 5.2. In section 4.1, we have split the eigenvalues of γ in eigenvalues which are larger than \frac{1}{2} and lower or equal \frac{1}{2}. In fact, this split turns out to be almost optimal and (61) cannot be sharpened by another choice of P and P^{⊥}. Up to the constant (61), Theorem 5.1 is exactly the result which was already obtained in [1]. The difference of the constants comes, on the one hand, from a different arrangement of the terms of tr_{2} {(X ⊗ X) Γ^{(T)}} and, on the other hand, from the fact that in [1] also the Q-Condition was used, which can be seen implicitly in estimate (68) in [1]. With the result of Theorem 5.1 we can immediately perform the integration in the Feffermann–de la Llave identity according to [1] which leads to an estimate of tr_{2} {V (Γ − (1 − Ex)(γ ⊗ γ))}.

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Volker Bach v.bach@tu-bs.de Hans Konrad Knörr h-k.knörr@tu-bs.de

Edmund Menge e.menge@tu-bs.de

Institut für Analysis und Algebra TU Braunschweig Pockelsstraße 14, 38118 Braunschweig 482

Documenta Mathematica 17 (2012)