arXiv:1807.05299v2 [math-ph] 9 Apr 2019 where tity vlaino 1 s oee,patclyipsil u t due impossible practically space however, state is, (1) of evaluation rbe a erdcddatclyb elcn h states the replacing by drastically reduced be can problem problem representability Abstract steBnc pc fbuddoeaoson operators bounded of space Banach the is otn aki h rudsaeenergy state ground the is task portant k ftesse,where system, the of n h Hamiltonian the and ibr pc fteessesi ie ytefrinFc sp Fock fermion the by given is s systems More these of space approximation). non- Hilbert as (Born-Oppenheimer modeled systems usually are fermion molecules chemistry, problems quantum Representability In Motivation: 1.1 Introduction 1 bd prtron operator -body rhgnlzto ffermion of Orthogonalization r τ oyosrals rprudrtnigo hsprojectio the this to of related understanding proper intimately A observables. body iesoa,frinFc pc a edfie steiaeun image the as ont geometry defined Hilbert-Schmidt the be in can projection orthogonal space Fock fermion dimensional, k etit oa rhnra ai ftesaeof space the of ope basis space orthonormal Fock an of space space to ortho the Hilbert restricts of an basis one-particle orthonormal Given an finite-dimensional construct the chemistry. in quantum basis computational in problem h the , . ste(ope)Hletsaeo igeeeto (e.g. electron single a of space Hilbert (complex) the is h reduced The S k . bd reduction -body ah okrRuh Robert Rauch, Volker Bach, F B ⊆ S k H e hsclqatt hs optto sa im- an is computation whose quantity physical key A . pril est arxo est arxo finite- on matrix density a of matrix density -particle n representability and suulyatobd prtro,mr eeal,a generally, more or, operator two-body a usually is ( F of ersnaiiyproblem representability ) E ′ pi 0 2019 10, April 0 τ sasial e fsae on states of set suitable a is ( htol noe h xetto ausof values expectation the encodes only that , H Abstract ) inf = . 1 ϕ ∈S shsbe ieyosre,this observed, widely been has As ϕ ( H F (1) ) k and k Bd operators -Body ogsadn open long-standing a , bd prtr o all for operators -body B ( F h atsz fthe of size vast the o ) h pc of space the o ′ eexplicitly we , B stherefore is n t ul direct A dual. its τ eaiitcmany- relativistic aoswhich rators h ( S ∈ F ace = eicly the pecifically, ,where ), L e the der normal F 2 yaquan- a by ( R = 3 k ) F - ⊗ B f C ( ( F h 2 ), ), ) k-body operators in the state τ. More precisely, denote by k( ) ( ) the ′ O F ⊆ B F subspace of k-body operators on and let τ ( ) , then rτ can be defined as the restriction τ ( F)′. In other∈ words,B F if i : ( ) ( ) |Ok(F) ∈ Ok F k Ok F → B F denotes the inclusion map then the mapping τ rτ is given by the dual ′ ′ ′ 7→ map ik : ( ) k( ) , which we call the k-body reduction map. Now, if H ( B) thenF →τ(H O) =F (i′ τ)(H) for all τ ( )′ and (1) can be rewritten as ∈ Ok F k ∈B F H H H H E0( ) = inf τ( ) = inf rτ ( ) = inf′ r( ), (2) τ∈S τ∈S r∈ik(S) thus the evaluation of (1) is, in principle, simplified, because the infimum has to ′ be taken over the much smaller set ik( ). To explicitly compute the right hand side of (2) however, one has to find anS efficient parametrization of the set i′ ( ). k S The representability problem for (and k N0) amounts to characterize the ′ S ∈ image ik( ) of representable functionals on k( ) in a computationally efficient way. S O F
Traditional representability problems The general framework of repre- sentability problems as discussed here is usually invisible in the pertinent litera- ture, because in concrete applications is almost always chosen to be (a subset S ′ of) the set of density matrices on and k( ) is identified with a suitable subspace of ( ). Moreover, in applicationsF O ofF physics or chemistry the by far most importantB F case is k = 2, as the Hamiltonian usually is a two-body opera- ′ tor. In this case the two-body reduction ik(ρ) of an N-particle density matrix can be identified with the (customary) 2-RDM, which is a bounded operator on 2 h. V Erdahl’s representability framework In this paper, only the case dim h < is considered, which is sufficient for many important applications. For ex- ∞ample, in quantum chemistry one commonly starts by choosing a finite subset of L2(R3) C2 of spin orbitals and then considers their span h. In the finite- dimensional⊗ case, the reduced k-body reduction of a density matrix ρ can be introduced as the image πk(ρ) under the orthogonal projection onto k( ) [see 8], O F π : 2( ) ( ) 2( ). (3) k L F → Ok F ⊆ L F As it turns out, in the finite-dimensional case πk is an equivalent description of ′ the map ik introduced above. The reason for this is that in the finite-dimensional case ( )= 2( ), where 2( ) denotes the Hilbert space of Hilbert-Schmidt B F L F L F ′ 2 ′ operators on , and we may identify ( ) ∼= ( ) and k( ) ∼= k( ) via the Riesz isomorphisms.F Under these identifications,B F L F the k-bodyO F reductionO F map ′ ∗ ∗ ik is given by the adjoint ik of ik and πk = ikik. This geometric interpretation of the representability problem is visualized in Fig. 1. Note that Erdahl’s representability framework breaks down in the infinite-dimensional case, because then k-body operators are generally not Hilbert-Schmidt anymore.
2 1.2 Related work The idea of replacing density matrices by their reduced density matrices to sim- plify the evaluation of (1) can be traced back to Husimi [10]. First extensive analyses were carried out in the 1950’s and 1960’s and lead, e. g., to the solu- tion of the representability problem for one-body reduced density matrices of N-particle density matrices [5, 9, 21] and the development of (still very inaccu- rate) lower bound methods based on representability conditions. In 1978 Erdahl introduced a new class of representability conditions [8], which were found to significantly increase the accuracy of lower bound methods [4]. In 2005 the rep- resentability problem for the one-body reduced density matrices of pure states was solved by Klyachko [11] based on results from quantum information theory. In 2012 Mazziotti established a hierarchy of representability conditions provid- ing a formal solution of the representability problem for the two-body RDMs of N-particle density matrices [15]. However, the general representability problem has been found to be computationally intractible [15], even on a quantum com- puter [12]. Computational advances [13] enabled a range of recent applications [17, 18, 16]. Representability methods have also proved useful in Hartree-Fock theory [2]. For a more detailed overview on the history of representability prob- lems, we refer to [14] and [6].
1.3 Goal and main results
The goal of the present work is to shed more light on the projection πk in the finite-dimensional case. As a result, we explicitly diagonalize the orthogonal projections πk simultaneously for all k N0. More specifically, we prove the following.1 ∈
Theorem 1 (Main Theorem) Let dimC h = n < and ϕ ,...,ϕ be an ∞ 1 n . orthonormal basis of h. For I = i1 < ... < ij 1,...,n define cI = . ∗ { }⊆{ } c(ϕij ) c(ϕi1 ) and nI = cI cI , where c(ϕ) denotes the usual fermion annihila- tion operator.· · · Then the following is found 1. An orthonormal basis B of 2( ) is given by the elements L F 1 |A| ∗ ( 2) nAcI cJ , (4) √2n−|I∪˙ J| − AX⊆L where I,J,L run over all mutually disjoint subsets of 1,...,n . { } 2. For any k N , B ( ) is an orthonormal basis of ( ). ✷ ∈ 0 ∩ Ok F Ok F Orthogonal decompositions of 2( ) as implied by Theorem 1 have already been introduced, e. g., in [8, Sec. 8],L whereF an orthogonal decomposition ( )= B F n,m Λ(n,m) is used to derive new classes of representability conditions. The spaces Λ(n,m) are generated by elements of the form (69), see Sec. 5. The L 1See Fig. 1 for a geometric interpretation of this result and its relation to the representabil- ity problem.
3 πk P1
B
( ) Ok F
Figure 1: Geometric interpretation of the representability problem for density matrices in finite dimensions: the mapping of density matrices ρ 1 to its k- ∈P 2 body reduction as orthogonal projection πk onto the subspace k( ) ( ) of k-body operators. The representability problem amounts toO findF an⊆ efficient L F characterization of the image πk( 1) within k( ). The orthonormal basis B given in Theorem 1 is adapted toP this situationO asF it restricts to an orthonormal basis B ( ) of ( ) for every k N . ∩ Ok F Ok F ∈ 0 orthonormal basis elements given in Theorem 1, however, have the additional property of being normal ordered, which can be used to express πk(ρ) in terms of the customary reduced density matrices, as in the following example. Corollary 2 Let ρ be a particle number-preserving density matrix, γ (h) ∗ ∈ B its 1-RDM and dΓ(γ)= i,j γjici cj the (differential) second quantization of γ. Then 2nπ (ρP) = (n + 1) 2 tr γ 2Nˆ +4dΓ(γ), (5) 1 − { }− Nˆ ∗ where = i ci ci denotes the particle number operator. ✷
A similar formulaP for π2(ρ) exists, but is much more complicated.
1.4 Overview of the paper In Sec. 2, we introduce the necessary terminology and notation of fermion many- particle systems and general density matrix theory, as well as, some features specific to the finite-dimensional setting. In Sec. 3, we compute the Hilbert- Schmidt scalar product of specific monomials in creation and annihilation oper- ators (Proposition 11). In Sec. 4 we prove Theorem 1 in two steps, as follows.
1. The orthonormal basis B of 2( ) is constructed in Theorem 14. L F 2. In Theorem 16 we show that B ( ) is a basis of ( ) for all k N . ∩Ok F Ok F ∈ 0 R In many cases one also considers the space k ( ) of selfadjoint k-body opera- tors. We generalize the above results in TheoremO F 19, where we apply a suitable
4 unitary transformation U on 2( ) and show that the orthonormal basis U(B) 2 L F R N of ( ) restricts to an orthonormal basis of k ( ) for all k 0. Finally, in Sec.L 5F we present an alternative approach for constructingO F an orthonormal∈ basis of 2( ) with properties as in Theorem 1, which was first communicated to us byL GossetF 2 and turned out to be already present in [8].
1.5 Motivating application We illustrate the virtue of having orthonormal bases of the space of operators ex- plicitly available on the following example: Consider a fermionic many-particle system with finite-dimensional one-particle Hilbert space h, a two-body Hamil- tonian of the form 1 H = t c∗c + V c∗c∗c c , (6) ij i j 2 ij;kl i j l k i,j X i,j,k,lX . where Vij;kl = ϕi ϕj V (ϕk ϕl) is a matrix element of a repulsive two- body potential Vh ⊗0. Let| be⊗ an orthonormali basis of 2( ). Then for any . ≥ B 1 2 L F we have PA = θ∈A θ θ θ∈B θ θ = L (F) and, under suitable A⊆Bpositivity requirements on the| ih potential|≤ V ,| weih obtain| P P 1 . H t c∗c + V c∗c∗P c c = H . (7) ≥ ij i j 2 ij;kl i j A l k A i,j X i,j,k,lX
Thus E0(HA) is a lower bound, which are usually more difficult to derive than upper bounds, for the ground-state energy E0(H) of the original quantum system. In many situations, after a suitable choice of an orbital basis ϕ1,...,ϕn of h, the orthonormal basis B given by Theorem 1 and a suitable choice of A⊂B leads to a nontrivial lower bound E0(HA) of E0(H).
2 Foundations
Throughout this work, h denotes the one-particle Hilbert space, i.e., a separable complex Hilbert space. We consider only the finite-dimensional case here and . assume n = dimC h < throughout the paper. ∞ 2.1 General notions In this subsection, we will recall some relevant notions from general density matrix theory of fermion many-particle systems that are also valid when dim h = . ∞ [email protected]
5 Hilbert spaces If not stated otherwise, all Hilbert spaces are assumed to be complex. For a Hilbert space , the inner product between elements ϕ, ψ H ∈ H is denoted by ϕ ψ H and is assumed to be anti-linear in the first and linear in the second component.h | i When there is no risk of confusion, we will freely omit the subscript of the inner product. By ( ) we denote the C*-algebra of linear boundedH operators on . B H H Hilbert-Schmidt operators The space of Hilbert-Schmidt operators on a Hilbert space is denoted by 2( ) and is a Hilbert space with respect to the H . L ∗H 2 inner product a b 2 = tr a b . Furthermore, ( ) is endowed with a h | iL (H) { } L F natural real structure (i.e., a complex conjugate involution) given by the Her- mitian adjoint.
Fermion Fock space For a Hilbert space h, the associated fermion Fock . k space = (h) is the completion of the Grassmann algebra h = k≥0 h with respectF F to the inner product defined by V L V k . det ( ϕi ψj )i,j=1 if k = l, ϕ1 ϕk ψ1 ψl = h | i (8) h ∧···∧ | ∧···∧ i (0 otherwise.
. 0 The neutral element 1 C = h of the wedge product on is also called ∈ ⊂F F the (Fock) vacuum and denoted by ΩF . V CAR Associated with , there are natural linear, respectively anti-linear, maps c∗,c : h ( ) calledF the creation- and annihilation operators which are . . defined for f →Bh andF ω by c(ϕ) = [c∗(ϕ)]∗ and c∗(f)ω = f ω, respectively. They satisfy the∈ canonical∈F anti-commutation relations (CAR) ∧
c∗(ϕ),c∗(ψ) = c(ϕ),c(ψ) =0, c∗(ϕ),c(ψ) = ϕ ψ , ϕ, ψ h, (9) { } { } { } h | i ∀ ∈ ∗ and c(ϕ)ΩF = 0 for all ϕ h. The mappings c ,c : h ( ) induce a representation of the (abstract)∈ CAR algebra generated by h→[see B 3,F Sec. 5.2.2], called the Fock representation.
. Density matrices We denote by = 1 ( ) 2( ) the cone of positive, P L+ F ⊆ L F trace-class operators on . Elements ρ from the convex subset 1 which are normalized in the senseF that tr ρ = 1 are called density matricesP ⊆ P on . { } F Elements of 1 uniquely represent the normal states on the C*-algebra ( ) [see 1, TheoremP 2.7]. B F
2.2 Finite-dimensional features We conclude this section by summarizing some more specific notions, which (partly) depend on the finite-dimensionality of h.
6 Generalized creation- and annihilation operators By the CAR, we may extend c,c∗ to linear, respectively anti-linear, maps c∗, c : ( ) via F→B F . . c∗(ω)η = ω η, c(ω) = [c∗(ω)]∗ . (10) ∧
Note that the definition of c is such that c(ϕ1 ϕk) = c(ϕk) c(ϕ1), ∗ ∧···∧ · · · for all ϕ1,...ϕk h. We call c , c the generalized creation- and annihilation operators3. Note∈ that the CAR (9) do not hold for c∗ and c, when ϕ, ψ h are replaced by general ω, η . ∈ ∈F Polynomials in Creation- and Annihilation-Operators We are partic- ularly interested in operators on , which are “polynomials in creation- and annihilation” operators, i.e., elementsF in the complex -subalgebra ( ) generated by c∗(ϕ) ϕ h . In the finite-dimensional∗ case, = A⊆B( ) [seeF 3, Theorem 5.2.5]{ and we| have∈ } a natural linear map A B F
Θ : ¯ ω η¯ c∗(ω)c(η) , (11) F ⊗ F ∋ ⊗ 7→ ∈ A where ¯ denotes the conjugate Hilbert space of [see 7, Sec. 1.2]. In fact, by the WickF Theorem, Θ is surjective and thereforeF an isomorphism, as the vector spaces involved are all finite-dimensional. k-Body Operators Let k N0. We call a sum of operators of the form ∗ ∈ c (ω)c(η) with ω r, η s and r + s = 2k a k-particle operator. More generally, a sum of∈l-particle F ∈ operatorsF with l k is called a k-body operator, ≤ and we denote the space of k-body operators by k( ). We also consider the R R O F -subspace k ( ) k( ) of selfadjoint (or real) elements of k( ), which are called k-bodyO F observables⊆ O F . O F Remark 3 (On the Terminology of k-Body Operators) There are differ- ent conventions regarding the notion of a k-body operator. Especially in the physics literature this terminology usually refers to what we call a k-particle operator. For example, a typical Hamiltonian in second quantization is given by (6). In the physical literature, this operator would then often be considered as a sum of a one- and two-body operator, whereas in our convention (6) is a sum of a one- and two-particle operator and therefore a two-body operator. ✷
The Hilbert-Schmidt geometry Since in the finite-dimensional case we have 2( ) = ( ), the mappings Θ, c∗ and c introduced above are in fact mappingsL F betweenB F (finite-dimensional) complex Hilbert spaces. In particular, ¯ 2 using the natural isomorphism ∼= ( ) the map Θ defined in (11) gives rise to a linear automorphism F⊗ F L F
α : 2( ) ω η c∗(ω)c(η) 2( ). (12) L F ∋ | ih | 7→ ∈ L F 3This terminology is also used, e.g, in [19].
7 3 Trace Formulas
The goal of this section is to prove Proposition 11, which provides a formula for the Hilbert-Schmidt inner product a b L2(F) between certain monomials a,b in creation and annhiliation operators.h | Ouri approach is to evaluate
∗ ∗ a b 2 = tr a b = ϕ a bϕ (13) h | iL (F) { } h I | I iF XI for a suitable basis (ϕI )I of (Proposition 7). The main work then is to characterize the set M of thoseF I with non-vanishing contributions in (13) (Proposition 8).
3.1 Basic notation Set-theory For a set X, we denote by X N 0, the number of elements | | ∈ ∪{ ∞} in X and by P (X) the system of all subsets of X. Given sets A1,...,AΛ P (X), we write A ˙ ˙ A for their union A A when we want∈ 1 ∪ ··· ∪ Λ 1 ∪···∪ Λ to indicate or require the A1,...,AΛ to be mutually disjoint, i.e., Aα Aβ = for all 1 α<β Λ. Given a proposition p (e.g., a set-theoretic relation∩ like∅ x A B≤) we write≤ ∈ ∩ . 1 if p is true, 1(p) = (14) (0 otherwise.
In the case where p is of the form a = b, we also write δa,b for 1(p) (the Kronecker Delta).
Orbital bases and induced Fock bases For the remainder of this paper, . let h be finite-dimensional, dim h = n < , and assume that ϕ ,...,ϕ is . ∞ { 1 n} a fixed orthonormal basis. Let Nn = 1,...,n and P (Nn) be the family of subsets of N . For A = a , ,a {N with }a < . ϕa1 ϕak A = , ϕA = ∧···∧ 6 ∅ (15) Ω for A = . ( F ∅ Then, by definition (8) of the inner product on , (ϕ ) N is an orthonor- F A A⊆ n mal basis of and, using Diracs Bra-ket notation, ( ϕA ϕB ) N is an or- F | ih | A,B⊆ n thonormal basis of 2( ). Applying the generalized creation and annihilation operators, we furtherL defineF for A, B N the monomials ⊆ n ∗ . ∗ . . ∗ . cA = c (ϕA), cA = c(ϕA), cA,B = cAcB, nA = cA,A. (16) 3.2 Monomials acting on the induced Fock bases To efficiently deal with the signs occurring in computations with the monomials of the form (16), we introduce for A ,...,A ,B ,...,B N the multi-sign 1 k 1 l ⊆ n A1 ... Ak . = ϕA1 ϕAk ϕB1 ϕBl . (17) B1 ... Bl h ∧···∧ | ∧···∧ i 8 The main use of these multi-signs is to account for the signs occurring when reordering products of elements of the form (15), which is made precise by the following. Lemma 4 The multi-sign (17) vanishes, unless A1 ˙ ˙ Ak = B1 ˙ ˙ Bl. However, if A ˙ ˙ A = B ˙ ˙ B , then ∪··· ∪ ∪··· ∪ 1 ∪··· ∪ k 1 ∪··· ∪ l A1 Ak · · · (ϕA1 ϕAk )= ϕB1 ϕBl . (18) B1 Bl ∧···∧ ∧···∧ · · · Proof Since the ϕ anti-commute as elements in , its clear that ϕ i F A1 ∧···∧ ϕAk = 0 whenever the Ai are not mutually disjoint (and similarly for the Bi). Therefore the right-hand side of (17) trivially vanishes unless the Ai and Bi are mutually disjoint, respectively. Now consider the case where the Ai and Bi are mutually disjoint, but their unions A respectively B are not equal, say there is a A B for some a Nn. Then ϕa ϕb = 0 for all b B, thus ϕA ϕB =0 by∈ definition\ (8) and∈ h | i ∈ h | i A1 Ak · · · = ϕA ϕB =0, (19) B1 Bl ±h | i · · · which proves the first part. For the second part, assume that A ˙ ˙ A = 1 ∪··· ∪ k B1 ˙ ˙ Bl. Then, by anti-commuting the ϕi, there is λ 1, +1 such that∪ ··· ∪ ∈ {− } . . ϕ = ϕ ϕ = λ ϕ ϕ = λ ϕ˜ (20) A1 ∧···∧ Ak · B1 ∧···∧ Bl · Using the same argument, we find thatϕ ˜ = ϕ , thus ϕ˜ 2 = 1. Consequently, ± A k k A1 Ak 2 2 · · · ϕA1 ϕAk = ϕ ϕ˜ ϕ = λ ϕ˜ ϕ˜ =ϕ ˜ B1 Bl ∧···∧ h | i k k · · · (22) = ϕ ϕ . B1 ∧···∧ Bl Lemma 5 For A, B, I N we have ⊆ n A I c∗ ϕ = 1(A I = ) ϕ (23) A I ∩ ∅ A I A∪I ∪ A I A cAϕI = 1(A I) \ ϕ . (24) ⊆ I I\A Proof ∗ If A I = then cAϕI = 0 and also the right hand side of (23) vanishes due to Lemma∩ 4.6 Otherwise,∅ if A I = then Lemma 4 implies ∩ ∅ A B c∗ ϕ = ϕ ϕ = ϕ , (25) A I A ∧ I A B A∪B ∪ 9 which completes the proof of (23). To prove (24) note that, since (ϕJ )J⊆Nn is an orthonormal basis of , we have F cAϕI = cAϕI ϕJ ϕJ . (26) N h | i JX⊆ n Unwinding the definitions and using Lemma 4, we compute I c ϕ ϕ ϕ = ϕ ϕ ϕ = h A I | J i J h I | A ∧ J i A J (28) I = 1(A I)1(J = A I) . ⊆ \ A I A \ thus (24) follows by combining (26) and (28). Remark 6 Definition (15) of the Fock space basis elements ϕA naturally gen- eralizes to the case where A is a string over the alphabet Nn. Within this gener- alized framework, the multi-sign (17) can be interpreted as the anti-symmetric Kronecker Delta (see, e.g., the “algebraic framework” in [20]). ✷ 3.3 Derivation of the trace formula Proposition 7 Let A,B,C,D N , then ⊆ n A I B I I cA,B cC,D 2 = \ (29) h | iL (F) C I D B I B D I D M IX∈ \ \ \ . where M = M(A,B,C,D) is the family of all I N such that ⊆ n 1. B D I and ∪ ⊆ 2. A ˙ (I B)= C ˙ (I D). ∪ \ ∪ \ Proof Since (ϕ ) N is an orthonormal basis of , we have I I⊆ n F ∗ ∗ ∗ ∗ cA,B cC,D = tr cBcAcC cD = cAcB ϕI cC cDϕI (30) h | i { } N h | i IX⊆ n Using Lemma 5, we compute for arbitrary I N ⊆ n I c ϕ = c∗ (c ϕ )= 1(B I) c∗ ϕ A,B I A B I ⊆ B I B A I\B \ (31) I = 1(B I)1(A (I B)= ) ϕ ϕ , ⊆ ∩ \ ∅ B I B A ∧ I\B \ and similarly for cC,DϕI , which yields A I B I I cA,BϕI cC,DϕI = 1(I M) \ . (32) h | i ∈ C I D B I B D I D \ \ \ Combining (32) with (30), the assertion follows. 10 As stated in Proposition 7, the contributing sets I Nn in (29) must sat- isfy certain set-theoretic compatibility relations with the⊆ given sets A, B, C and D. Moreover, Proposition 7 is of limited use because of the complicated signs occuring in (29). The main part of this paper therefore is to overcome these difficulties by a careful analysis of the set M of contributing subsets I N . ⊆ n Proposition 8 Let M = M(A,B,C,D) as in Proposition 7. Then the follow- ing conditions are equivalent: 1. M = , 6 ∅ 2. A ˙ (D B)= C ˙ (B D), ∪ \ ∪ \ 3. B D M, ∪ ∈ 4. A B = C D and B A = D C. \ \ \ \ In any of these cases, M = (B D) ˙ N N (A C)= . (33) { ∪ ∪ | ∩ ∪ ∅} Proof We will first show the equivalence of the conditions 1-3. The equivalence of 2 and 4 follows from a purely set-theoretic argument, see Lemma 9 below. 1 2: Choose M M. By definition of M, B D M, we may write M =⇒ (B D) ˙ N so∈ that M B = (D B) ˙ N. Since∪ ⊆A (M B) = by definition∪ of M∪, also A (D B)\ A (M \B)=∪ , and similarly∩ C \(B D)=∅ . Moreover, we have A ∩N \A ⊆((D∩B))\ N)=∅A (M B)= ∩and\ similarly∅ C N = . In summary,∩ ⊆ we∩ have\ (A ∪ (D B))∩ ˙ N\ = A∅ (M B) = C ∩(M D∅) = (C (B D)) ˙ N and therefore∪ \ A ∪(D B)= C∪ (B\ D). . ∪2 3:\ By definition∪ of\ M, ∪M = B D M if and∪ only\ if A ˙∪(M \ B) = C ˙ ⇒(M D), but by construction M ∪B =∈D B and M D = B∪ D.\ ∪3 1:\ this follows trivially. \ \ \ \ Now⇒ it remains to prove (33), given the conditions 1-4 hold. Denote the right-hand side of (33) by M˜ . M M˜ : Choose some M M. Since B D M, we can write M = (B D)⊆˙ N for some N I (B∈ D) and now need∪ to⊆ show that N (A C)= . Since∪ A∪ (M B)= by⊆ definition\ ∪ of M, also A (D B) A ∩(M ∪B)= ∅, and similarly∩ C\ (B ∅D)= . Moreover, we have ∩A N\ A⊆ ((D∩ B))\ N)=∅ A (M B)= ∩ and\ similarly∅ C N = , thus N ∩(A ⊆C)=∩ . \ ∪ . ∩M˜ \ M: Let∅ M = (B D) ∩˙ N ∅ M˜ , i.e., ∩N ∪(A C)∅ = . Clearly, B D ⊆ M. Moreover, by∪ assumption∪ ∈ we have A ˙ ∩(D ∪B) = C ∅˙ (B D), thus∪ ⊆ ∪ \ ∪ \ A (M B)= A ((D B N) = (A (D B)) (A N)= . (34) ∩ \ ∩ \ ∪ ∩ \ ∪ ∩ ∅ Similarly, C (M D)= . Finally, ∩ \ ∅ A (M B)= A ((D B) N) = (A (D B)) N ∪ \ ∪ \ ∪ ∪ \ ∪ (35) = (C (B D)) N = C (M D), ∪ \ ∪ ∪ \ 11 thus M M, which completes the proof. ∈ Lemma 9 Let X be a set and A,B,C,D X. Then the following conditions are equivalent ⊆ 1. A ˙ (D B)= C ˙ (B D), ∪ \ ∪ \ 2. A B = C D and B A = D C. \ \ \ \ Proof 1 2: Let x A B. Then x A A ˙ (D B)= C ˙ (B D), thus x C. Moreover,⇒ since∈ (A\ B) D =∈A ⊆(D ∪B)=\ , we have∪ x \D, hence x ∈ C D. This shows that\ A ∩B C ∩D. Exchanging\ ∅ the roles of6∈ A, C and B,D∈ respectively,\ also C D \A ⊆B. \ Moreover, let x B \A.⊆ If x \ D then x B D C ˙ (B D) = A ˙ (D B), i.e., x A∈, contradicting\ 6∈ our assumption∈ \x ⊆B A∪. Hence,\ x D∪. Also,\ if x C then∈ x C ˙ (B D) = A ˙ (D B∈), so\ x D B, which∈ contradicts∈x B, hence∈x ∪C. This\ shows B∪ A \D C. Again,∈ by\ renaming A, B, C and D∈, we also see6∈D C B A. \ ⊆ \ 2 1: We compute \ ⊆ \ ⇒ A (D B)= A D Bc = (A B) D = (C D) D = . (37) ∩ \ ∩ ∩ \ ∩ \ ∩ ∅ Exchanging the roles of A, C and B,D, we also get C (B D)= . To show that A (D B)= C (B D), first note that ∩ \ ∅ ∪ \ ∪ \ A Dc = (A Dc B) (A Dc Bc) (B D) (A B) ∩ ∩ ∩ ∪ ∩ ∩ ⊆ \ ∪ \ (38) = (B D) (C D) C (B D) \ ∪ \ ⊆ ∪ \ and A B = A (A B) A (B A)c = A (D C)c = A (C Dc) ∩ ∩ ∩ ⊆ ∩ \ ∩ \ ∩ ∪ (39) = (A C) (A Dc) C B D, ∩ ∪ ∩ ⊆ ∪ \ where we used (38) in the last step. Consequently, we conclude (37) A A (D B)c = A (Dc B) = (A Dc) (A B) C (B D), (40) ⊆ ∩ \ ∩ ∪ ∩ ∪ ∩ ⊆ ∪ \ where we used (38) and (39) in the last step. Moreover, we have (37) D B (D B) Ac = [(D B) Ac C[ [(D B) Ac Cc] \ ⊆ \ ∩ \ ∩ ∩ ∪ \ ∩ ∩ (41) C (D Cc Ac)= C (B Ac) C B, ⊆ ∪ ∩ ∩ ∪ ∩ ⊆ ∪ and intersecting both sides of this inclusion with Bc, we obtain D B C B C. Combined with (40), this shows A (D B) C (B D) and, by\ exchanging⊆ \ ⊆ ∪ \ ⊆ ∪ \ the roles of A, C and B,D, the converse inclusion follows as well. 12 Remark 10 Lemma 9 can be further generalized by noting that the given con- ditions are also equivalent to the following (equivalent) conditions: 1. B D = A C and D B = C A, \ \ \ \ 2. B ˙ (A C)= D ˙ (C A). ✷ ∪ \ ∪ \ Proposition 11 (Trace Formula) Let K,A,B Nn and L,C,D Nn be mutually disjoint, respectively. Then ⊆ ⊆ n−|A∪B∪K∪L| n c n c 2 = δ δ 2 . (42) h K A,B | L C,DiL (F) A,C B,D · Proof Using Lemma 4 and Lemma 5, we find for any I N ⊆ n I n ϕ = c∗ (c ϕ )= 1(K I) c∗ ϕ K I K K I ⊆ K I K K I\K \ (43) I = 1(K I) ϕ ϕ = 1(K I)ϕ . ⊆ K I K K ∧ I\K ⊆ I \ Combined with Lemma 5, we therefore get for any I N ⊆ n n c ϕ = 1(K A (I B))1(B I)1(A I B = ) K A,B I ⊆ ∪ \ ⊆ ∩ \ ∅ I (44) ϕ ϕ . · B I B A ∧ I\B \ Consequently, we have with M = M(A,B,C,D) as in Proposition 8 n c ϕ n c ϕ = 1(I M)1[K A (I B)]1[L C (I D)] h K A,B I | L C,D I i ∈ ⊆ ∪ \ ⊆ ∪ \ A I B I I \ (45) · C I D B I B D I D \ \ \ Since A B = C D = by assumption, Proposition 8 implies that 1(I M)= δ δ ∩1(B I∩)1(I ∅A = ). Thus (45) equals ∈ A,C B,D ⊆ ∩ ∅ δ δ 1(B I)1(I A = )1[K L A (I B)]. (46) A,C B,D ⊆ ∩ ∅ ∪ ⊆ ∪ \ Now observe that for A = C we have L A = L C = , i.e., K L A (I B) is equivalent to K L I B, which is∩ further∩ equivalent∅ to K∪ L⊆ I∪. Hence\ (45) equals ∪ ⊆ \ ∪ ⊆ δ δ 1(I A = )1(B K L I) (47) A,C B,D ∩ ∅ ∪ ∪ ⊆ and, by summing (47) over all I N , we find ⊆ n n c n c = δ δ P[N (A B K L)] . (48) h K A,B | L C,Di A,C B,D| n \ ∪ ∪ ∪ | 13 Example 12 (Trace of the Particle Number Operator) Let dim h = n< Nˆ . n . By Lemma 5, the particle number operator = i=1 ni can be written as ∞ n n n Nˆ = k id k . Consequently, its trace is given by k . On the k=0 · Λ h P k=0 · k other hand, Proposition 11 implies tr Nˆ = n 1 n = n 2n−1. Thus we L i=1 i P proved the well-known identity { } h | i · P n n k = tr Nˆ = n 2n−1, (49) k { } · Xk=0 which also follows from differentiating (1 + x)n with respect to x and evaluating at x = 1. ✷ 4 Orthonormalization In this section, given an orthonormal basis in h, we will construct explicit or- thogonal bases of 2( ) which restrict to the spaces of k-body operators and k-body observables,L respectively.F 4.1 Orthonormal basis of 2( ) L F As implied by Proposition 11, the monomials (nK )K⊆Nn are not pairwise or- thogonal. Inspired by computer algebraic experiments using Gram-Schmidt or- thogonalization in low-dimensional cases, we introduce for K N the element ⊆ n . b = ( 2)|I|n 2( ). (50) K − I ∈ L F IX⊆K As we will see in Theorem 14, the bK are pairwise orthogonal and can be used to construct an orthogonal basis of 2( ). The key ingredient is the following lemma, which is essentially a consequenceL F of the binomial formula. Lemma 13 Let K,L be finite sets. Then ( 2)|I|+|J|2−|I∪J| = δ . (51) − KL IX⊆K JX⊆L . Proof Let M = K L. We compute ∩ . ( 1)|I|+|J| S = ( 2)|I|+|J|2−|I∪J| = − , − 2−|I∩J| (52) I⊆K I⊆K JX⊆L JX⊆L where we have used that I J = I + J I J . Since every I K can be | ∪ | | | .| | − | ∩ | . ⊆ written uniquely as I = I1 ˙ I2 with I1 = (I M) M and I2 = I I1 K M and (similarly for J L),∪ we find ∩ ⊆ \ ⊆ \ ⊆ |I1|+|J1| ( 1) |I2| |J2| S = − ( 1) ( 1) . (53) 2−|I1∩J1| − − I1,JX1⊆M I2⊆XK\M J2⊆XK\M 14 By the binomial formula, for any finite set X and a C we have ∈ a|Y | = (1+ a)|X|. (54) YX⊆X In particular, for a = 1 we have ( 1)|Y | = 1(X = ). Hence − Y ⊆X − ∅ ( 1)|I2| ( P1)|J2| = 1(K M = )1(L M = ) − − \ ∅ \ ∅ I2⊆XK\M J2⊆XL\M (55) = 1(K L)1(L K)= δ . ⊆ ⊆ KL Inserting (55) in (53), we find ( 1)|I|+|J| S = δKL − . (56) 2−|I∩J| I,JX⊆M To evaluate the sum in (56), instead of summing over all I, J M, we sum over . . . ⊆ all X = I J M, I3 = I X M X and J3 = J (X ˙ I3) M (X ˙ I3) and apply∩ (54)⊆ once again:\ ⊆ \ \ ∪ ⊆ \ ∪ ( 1)|I|+|J| − = 2|X| ( 1)|I3| ( 1)|J3| 2−|I∩J| − − I⊆M X⊆M I3⊆M\X J3⊆M\(X∪˙ I3) JX⊆M X X X = 2|X| ( 1)|I3|1(I = M X) − 3 \ (57) XX⊆M I3⊆XM\X = 2|X|( 1)|M\X| = ( 1)|M| ( 2)|X| − − − XX⊆M XX⊆M = ( 1)|M|( 1)|M| =1. − − Combining (56) and (57), the assertion follows. 2 Theorem 14 Let bK be defined as in (50), then an orthonormal basis of ( ) is explicitly given by L F bKcI,J 2 B = ( ) K,I,J Nn pairwise disjoint . (58) √ n−|I∪˙ J| ∈ L F ⊂ 2 Proof Let K,A,B N and L,C,D N be mutually disjoint, respectively. ⊆ n ⊆ n By definition of bK and using Proposition 11, we obtain b c b c = ( 2)|I|+|J| n c n c h K A,B | L C,Di − h I A,B | J C,Di IX⊆K JX⊆L = ( 2)|I|+|J|δ δ 2n−|(A∪˙ B)∪(I∪J)| − AC BD I⊆K J⊆L X X (59) = δ δ 2n−|A∪˙ B| ( 2)|I|+|J|2−|I∪J| AC BD − IX⊆K JX⊆L n−|A∪˙ B| = δACδBD2 δKL, 15 where we used that for A = C,B = D, I K and J L we have A B I J = A B + I J in the third step and⊆ Lemma 13⊆ (see below) in| the∪ last∪ ∪ step.| This| ∪ shows| | that∪ | (58) is an orthonormal basis of its span S. Noting that dim S = B = f : N 1, 2, 3, 4 =4n = dim 2( ), (60) | | |{ n →{ }}| L F 2 we conclude that S = ( ). L F 4.2 Orthonormal basis of k-body operators Having established B as an orthonormal basis of 2( ), we now proceed and show that B restricts to a basis of ( ) for all kL NF (Theorem 16). Ok F ∈ 0 Lemma 15 A basis of k( ) is explicitly given by . O F B = c I, J N , I + J =2l with 0 l k , (61) 0 { I,J | ⊆ n | | | | ≤ ≤ } k 2n in particular, we have dimC ( )= . Ok F l=0 2l Proof 2 Since the mapping α defined inP (12) is a linear automorphism of ( ), 2 L F the cI,J = α ( ϕI ϕJ ) with I, J Nn form a basis of ( ). An element A 2( ) of the| ih form| ⊆ L F ∈ L F A = AI,J cI,J (62) N I,JX⊆ n is a k-body operator if and only if AI,J = 0 whenever I + J is odd or I + J > 2k. In other words, (61) a basis of ( ) and | | | | | | | | Ok F k 2l k n n 2n dimC ( )= B = = , (63) Ok F | 0| i 2l i 2l i=0 Xl=0 X − Xl=0 where we used Vandermonde’s identity. Theorem 16 The orthonormal C-basis B of 2( ) given in Theorem 14 re- L F stricts to an orthonormal basis Bk of the space k( ) of k-body operators. More specifically, we have O F N . bKcI,J K,I,J n pairwise disjoint, Bk = B k( )= ⊂ . (64) ∩ O F (√2n−|I∪J| I + J +2 K =2l with 0 l k ) | | | | | | ≤ ≤ Proof Let b B, i.e., ∈ ( 2)|L| b = bK cI,J = − nLcI,J (65) √2n−|I∪J| LX⊆K for K,I,J N pairwise disjoint. Since n c = c ˙ ˙ for every L K, ⊆ n L I,J ± I∪L,J∪L ⊆ Lemma 15 implies that b k( ) if and only if I + J + 2 K = 2l for some 0 l k, which proves∈ O (64).F Finally, noting| that| | we| have| a| bijection ≤ ≤ B b c c ˙ ˙ B with inverse c b c , we conclude ∋ K I,J → I∪K,J∪K ∈ 0 I,J 7→ I∩J I\J,J\I that B = B = dim ( ) and therefore B is a basis of ( ). | k| | 0| Ok F k Ok F 16 4.3 Orthonormal basis of k-body observables The orthonormal C-basis B of 2( ) as given in Theorem 14 does not immedi- L F ately restrict to bases of k-body observables, since BC contains elements which are not self-adjoint. For example, if I N is non-empty, then ⊂ n ∗ b c = c = c∗ = b c . ∅ I,∅ I 6 I ∅ I,∅ However, BC has the special property that BC = b∗ b BC , which allows us to obtain an orthonormal basis of self-adjoint elements{ | ∈ by a suitable} unitary transformation of 2( ). The general principle of this idea is given by the following. L F Lemma 17 Let be a finite-dimensional, complex Hilbert space with real struc- ture J and B anH orthonormal C-basis with J(B) B. Then ⊆ 1. B is of the form B = (a ,...,a ,b ,b∗,...,b ,b∗) with a = a∗ 1 i k. (66) 1 k 1 1 l l i i ∀ ≤ ≤ . 2. An orthonormal R-basis of VR = v V J(v)= v is given by { ∈ | } . BR = a ,...,a , √2 (b ), √2 (b ),..., √2 (b ), √2 (b ) (67) 1 k ℜ 1 ℑ 1 ℜ l ℑ l . 1 ∗ . 1 ∗ [Here, (a) = 2 (a + a ) and (a) = 2i (a a )) denote the real- and imaginary part ofℜa, respectively] ℑ − Proof 1 Since J(B) B and J 2 = 1, J defines an action of Z/2Z on B. The set B is decomposed into⊆ the orbits of this action, which are either of length 1 or length 2 by the orbit-stabilizer Theorem. By construction, the orbits of length 1 are of the form a = a∗ and the orbits of length 2 are of the form b,b∗ , hence the desired{ form (66)} is obtained by selecting an element in each {orbit} of B. 2 Let f : V V be the C-linear map mapping B to BR. Then f is represented with respect→ to B by the unitary matrix . 1 1 i 1k U U with U = − U(2). (68) ⊕ ⊕···⊕ √2 1 i ∈ l times In particular, with|B also{z BR} is an orthonormal C-basis of V and BR = B . | | | | By construction we have BR VR, thus BR is an orthonormal R-basis of its ⊆ R-span U. Since U is an R-subspace of VR of dimension BR = B = dimC V = | | | | dimR VR, we have U = VR, i.e., BR is an orthonormal R-basis of VR. Remark 18 The ordering (66) of the basis B in Theorem 19 is not uniquely determined. However, if B is endowed with a prescribed ordering, then B can can be uniquely reordered in the form (66) by requiring a < < a and 1 · · · k b 17 Theorem 19 An orthonormal C-basis of 2( ) is explicitly given by L F b (c c ) K,I,J Nn mutually BR −n/2 N ˙ K I,J J,I ⊂ = 2 bK K n (n+1−|I±∪J|)/2 . | ⊆ ∪ ( 2 disjoint and I < J ) n o R R B restricts to an orthonormal basis of the space k ( ) of k-body observables N O RF R for every k 0. More specifically, an orthonormal -basis of k ( ) is given by ∈ O F R . R R B = B ( )= b K N and K k k ∩ Ok F { K | ⊆ n | |≤ } b (c c ) K,I,J Nn pairwise disjoint, I < J ˙ K I,J J,I ⊂ (n+1−|I±∪J|)/2 , ∪ ( 2 and I + J +2 K =2l with 0 l k ) | | | | | | ≤ ≤ where I < J is to be understood with respect to the lexicographic ordering. Proof The first statement follows immediately from Theorem 19 applied to the orthonormal C-basis B as given in Theorem 14, which has been ordered according to Remark 18 by defining b c 5 Alternative construction of an orthonormal basis In this section, we provide an alternative construction of an orthonormal basis 2 of ( ) which restricts to an orthonormal basis of k( ) in the sense of TheoremL F 16. This construction was already presentedO in [8,F Sec. 8], but the corresponding proofs were deferred to a somewhat obscure reference. Fix an orthonormal basis ϕ1,...,ϕn of the one-particle Hilbert space h and consider for j =1,..., 2n the operator ∗ . ck + ck if j =2k is even, aj = (69) i (c∗ c ) if j =2k + 1 is odd. ( k − k By definition, the aj are self-adjoint and, by the CAR (9), satisfy a ,a =2δ , a2 = 1. (70) { j k} jk j . Moreover, for a subset J = j < < j N we define a = a a . { 1 · · · l} ⊆ 2n J j1 · · · jl where a∅ = 1 by convention. The following result has been suggested to us by Gosset. We present a proof which only relies on the algebraic properties (70) of the elements aj . Theorem 20 An orthonormal C-basis of 2( ) is given by L F . B = 2−n/2a K N . (71) J | ⊆ 2n n o e 18 Moreover, B restricts to an orthonormal basis Bk of k( ) for every k N0, where O F ∈ N e . J f 2n and Bk = B k( )= aJ ⊆ . (72) ∩ O F ( J =2l with 0 l k) | | ≤ ≤ f e n Proof We will first show that aJ aK = 2 δJK for all J, K N2n. If h | i ⊆2 1 J = K = j1 < < jl then, by self-adjointness of the aj and aj = F we have { · · · } a a = tr a∗ a = tr a a a a = tr 1 =2n. (73) h J | K i { J J } { jl · · · j1 j1 · · · jl } { F } Now consider the case J = K. Without loss of generality, we may assume J K = because if i J6 K then, by (70), ∩ ∅ ∈ ∩ ∗ ∗ aJ aK 2 = tr a aK = tr a a . (74) h | iL (F) { J } ± { J\{i} K\{i}} . Moreover, by setting I = J ˙ K and noting that aJ aK = tr aI , it suffices to show that tr a = 0 for∪ all non-empty I h N | . First,i ± consider{ } the case { I } ⊆ 2n where I = l > 0 is even. Then, writing I = i1 < < il we obtain, using (70) and| | cyclicity of trace, { · · · l−1 tr aI = tr ai1 ai = ( 1) tr ai ai1 ai −1 { } { · · · l } − { l · · · l } (75) = ( 1)l−1 tr a a = tr a , − { i1 · · · il } − { I } thus tr aI = 0. On the other hand, if I is odd, then consider the natural { } | | . Nˆ . Z -grading = on induced by χ = ( 1) , i.e. = ker χ 1 . 2 F F+ ⊕F− F − F± { ∓ } By definition, ai is odd with respect to this grading for any i N2n, hence also a is odd when I is odd and therefore tr a = 0. We have thus∈ proved that I | | { I } a a =2nδ J, K N . (76) h J | K i JK ⊆ 2n In particular, since B =22n = dim 2( ), B is an ONB of 2( ). k L F k L F To prove (72) note that, by definition, an element a is an j-particle operator . J with j = J for any J f N2n, hence aJ is a k-bodyf operator if and only if J =2l for some| 0 | l k. By⊆ (72) and Lemma 15, | | ≤ ≤ k 2n B = = dim ( ), (77) k 2l Ok F l=0 X f thus B is an orthonormal basis of ( ). k Ok F f Remark 21 (Relation between B and B) If n> 0, the orthonormal bases B and B are different. In fact, B B = 2−n/21 , since the elements of B ∩ { F } are homogeneous with respect to the naturalf grading = k h, whereas F k≥0 thee elements a B are inhomogeneouse whenever J = . ✷ J ∈ 6 ∅ L V e 19 Acknowledgements This research is supported by the German Research Foundation (DFG Project No. 399154669). Moreover, we are grateful to D. Gosset for suggesting the alternative construc- tion presented in Sec. 5. References [1] Huzihiro Araki. Mathematical Theory of Quantum Fields. Oxford Univer- sity Press, 1999. isbn: 9780198517733. [2] Volker Bach, Hans Konrad Kn¨orr, and Edmund Menge. “Fermion cor- relation inequalities derived from G-and P-conditions”. In: Documenta Mathematica 17.14 (2012), pp. 451–481. issn: 1431-0643. [3] Ola Bratteli and Derek W. Robinson. Operator Algebras And Quantum Statistical Mechanics 2. 2nd Editio. Springer, 2002. isbn: 3-540-61443-5. [4] Eric Canc`es, Gabriel Stoltz, and Mathieu Lewin. “The electronic ground- state energy problem: A new reduced density matrix approach”. In: Jour- nal of Chemical Physics 125 (2006). doi: 10.1063/1.2222358. [5] A. J. Coleman. “Structure of fermion density matrices”. In: Reviews of Modern Physics 35.3 (1963), pp. 668–686. issn: 00346861. doi: 10.1103/RevModPhys.35.668. arXiv: 0310359v1 [arXiv:cond-mat]. [6] A. John (Albert John) Coleman and V. I. Yukalov. Reduced density ma- trices : Coulson’s challenge. Springer, 2000, p. 282. isbn: 9783540671480. [7] Jan Derezinski and Christian Gerard. Mathematics of Quantization and Quantum Fields. Cambridge: Cambridge University Press, 2013. isbn: 978- 0-511-89454-1. doi: 10.1017/CBO9780511894541. [8] Robert M. Erdahl. “Representability”. In: International Journal of Quan- tum Chemistry 13.6 (June 1978), pp. 697–718. issn: 0020-7608. doi: 10.1002/qua.560130603. [9] Claude Garrod and Jerome K Percus. “Reduction of the N-Particle Varia- tional Problem”. In: Journal of Mathematical Physics 5.12 (1964), pp. 1756– 1776. [10] Kodi Husimi. “Some Formal Properties of the Density Matrix”. In: Pro- ceedings of the Physico-Mathematical Society of Japan. 3rd Series 22.4 (1940), pp. 264–314. issn: 0370-1239. doi: 10.11429/ppmsj1919.22.4_264. [11] Alexander A Klyachko. “Quantum marginal problem and N-representability”. In: Journal of Physics: Conference Series 36 (Apr. 2006), pp. 72–86. doi: 10.1088/1742-6596/36/1/014. [12] Yi-Kai Liu, Matthias Christandl, and F. Verstraete. “Quantum Compu- tational Complexity of the N-Representability Problem: QMA Complete”. In: Physical Review Letters 98.11 (Mar. 2007), p. 110503. issn: 0031-9007. doi: 10.1103/PhysRevLett.98.110503. arXiv: 0609125 [quant-ph]. 20 [13] David Arthur Mazziotti. “Large-Scale Semidefinite Programming for Many- Electron Quantum Mechanics”. In: Physical Review Letters 106.8 (Feb. 2011), p. 083001. issn: 0031-9007. doi: 10.1103/PhysRevLett.106.083001. [14] David Arthur Mazziotti. Reduced-Density-Matrix Mechanics: With Appli- cation to Many-Electron Atoms and Molecules. Ed. by David Arthur Mazz- iotti. Wiley, 2007, p. 574. isbn: 9780471790563. doi: 10.1002/0470106603. [15] David Arthur Mazziotti. “Structure of Fermionic Density Matrices: Com- plete N-Representability Conditions”. In: Physical Review Letters 108.26 (June 2012). issn: 0031-9007. doi: 10.1103/PhysRevLett.108.263002. arXiv: 1112.5866. [16] Jason M. Montgomery and David Arthur Mazziotti. “Strong Electron Cor- relation in Nitrogenase Cofactor, FeMoco”. In: The Journal of Physical Chemistry A 122.22 (June 2018), pp. 4988–4996. issn: 1089-5639. doi: 10.1021/acs.jpca.8b00941. [17] Shiva Safaei and David Arthur Mazziotti. “Quantum signature of exciton condensation”. In: Physical Review B 98.4 (July 2018), p. 045122. issn: 2469-9950. doi: 10.1103/PhysRevB.98.045122. [18] Manas Sajjan and David Arthur Mazziotti. “Current-constrained density- matrix theory to calculate molecular conductivity with increased accu- racy”. In: Communications Chemistry 1.1 (Dec. 2018), p. 31. issn: 2399- 3669. doi: 10.1038/s42004-018-0030-2. [19] Leszek Z. Stolarczyk. “The Hodge Operator in Fermionic Fock Space”. In: Collection of Czechoslovak Chemical Communications 70.7 (2005), pp. 979– 1016. [20] Leszek Z. Stolarczyk and Hendrik J. Monkhorst. “Quasiparticle Fock- space coupled-cluster theory”. In: Molecular Physics 108.21-23 (Nov. 2010), pp. 3067–3089. issn: 0026-8976. doi: 10.1080/00268976.2010.518981. [21] Chen Ning Yang. “Concept of off-diagonal long-range order and the quan- tum phases of liquid He and of superconductors”. In: Reviews of Modern Physics 34.4 (1962), p. 694. 21