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DOI: 10.1039/D4FD00059E
(Paper)
Faraday Discuss., 2024, Advance Article

Damiano Aliverti-Piuri^{ab},
Kaustav Chatterjee^{ab},
Lexin Ding^{ab},
Ke Liao^{ab},
Julia Liebert^{ab} and
Christian Schilling*^{ab}
^{a}Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany. E-mail: c.schilling@physik.uni-muenchen.de
^{b}Munich Center for Quantum Science and Technology (MCQST), Schellingstrasse 4, 80799 München, Germany

Received
13th March 2024
, Accepted 9th April 2024

First published on 12th April 2024

It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts, such as entanglement and correlation, into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on ‘electron correlation’, leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions, while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.

At the same time, the concepts of correlation and complexity lie also at the heart of quantum information theory (QIT). With distinguishable parties as subsystems, concise and operationally meaningful characterization of various correlation types has long been established, the most famous one being entanglement.^{65–68} Together with the geometric picture of quantum states,^{69} different correlation types can be elegantly unified under the same theory.^{70} Such characterization of correlation is mathematically rigorous, thus offering precise assessment of the correlation in quantum systems. Crucially, it is also operationally meaningful, in that QIT quantifies correlation and entanglement as the exact amount of available resources in quantum systems for distinct information processing tasks, such as quantum teleportation^{71,72} and superdense coding.^{72,73} As we transition into the second quantum revolution, the practical aspects of quantum technologies also emerge as major challenges that cannot be tackled by the field of QIT alone. In particular, physical realization of qubits using molecular systems, and storing and manipulating quantum information therein, are all on-going interdisciplinary endeavors.

We illustrate and summarize the resulting interplay between QIT and QChem based on their different expertise and research activities in Fig. 1. Given the distinct strengths of both fields and their needs and long-term goals, the two fields can complement each other and form a powerful synergy. On one hand, QIT offers precise characterization of various aspects of electron correlation, which could simplify descriptions of correlated many-electron wave functions, refine our understanding of static and dynamic correlation, and even inspire new (classical or quantum) approaches towards the electron correlation problem.^{41,74–76} In order to achieve any of these, it is essential to adapt the correlation and entanglement theory from QIT, which was designed specifically for distinguishable systems, to systems of indistinguishable electrons. This is indeed a nontrivial task, as major theoretical considerations are involved regarding fermionic antisymmetry, superselection rules^{77,78} or the N-representability problem,^{79} amongst others. On the other hand, expertise from QChem is absolutely essential for the development of effective and novel quantum registers based on atoms and molecules, and for exploiting the resourcefulness of the entanglement therein.

It is therefore of great importance that the two communities join forces and open up a communication line for active discussions, which is exactly the purpose of this work. Our article focuses more on how quantum chemistry can benefit from QIT, and we structure it as follows. In Section II, we revisit the key concepts of the geometry of quantum states, as well as the entanglement and correlation in systems of distinguishable particles as studied in QIT. In Section III and IV, we explain how one can adapt these concepts to the setting of indistinguishable fermions, in both the orbital and particle picture, respectively. In Section V, we demonstrate several applications of using fermionic entanglement and correlation as tools for simplifying the structure of molecular ground states.

(1) |

(2) |

Equipped with a basic notion of quantum states, we can introduce expectation values of observables as 〈O〉_{ρ} = Tr[Oρ]. It is instructive to interpret this as a complex linear functional,

(3) |

The Hilbert space of such bipartite systems is given as the tensor product

(4) |

(5) |

We first focus on one of the two subsystems only, say, Alice's one, and discuss in the next section the interplay between both subsystems. Resorting to the abstract and more elegant notion of quantum states (3), one identifies the joint system with the algebra of operators and Alice's subsystem with its subalgebra

(6) |

Indeed, the latter is closed under taking linear combinations, products and the adjoint. From a more fundamental point of view, it has actually been established by Zanardi^{83} that general subalgebras are precisely those mathematical objects that define subsystems.

Focusing on operators rather than on vectors in the Hilbert space has an immediate advantage when one defines reduced states of subsystems. For our setting, a given state can be ‘reduced’ to a state by restricting the action of ω to Alice's subalgebra, see eqn (6), i.e., to operators taking the form :

(7) |

As stated in Section II A, one can then univocally associate to ω_{A} a density operator by requiring that the equality ω_{A}(O_{A}) = Tr[ρ_{A}O_{A}] is valid . If ρ_{AB} is the density operator corresponding to ω_{AB}, the defining equality (7) then corresponds to

(8) |

The reduced density operator ρ_{A} turns out to be the familiar partial trace (over the complementary subsystem B) of ρ_{AB}, i.e., ρ_{A} = Tr_{B}[ρ_{AB}].

In analogy to Alice’s subsystem, Bob's subsystem B is associated with the subalgebra of operators , where . An important remark is that any two ‘local’ operators and commute. This property of commutativity has been identified^{83} as the defining property of a general notion of independent subsystems: correlations between subsystems are understood as correlations between the outcomes of joint but independent measurements on subsystems A and B. The independence corresponds, on the mathematical level, to precisely the commutativity of the ‘local’ algebras of observables.

While correlations are conceptually rooted in the concept of (local) measurements and operators, they are encoded in terms of properties of the system's quantum states. Specifically, different types of correlations are related to different types of states (uncorrelated states, entangled states, etc.), while the amount of correlations can be assessed using correlation measures, i.e., functions M(ρ) of the density operator ρ.

(9) |

If for some pair of O_{A}, O_{B} the above function vanishes, it can still be non-zero for some other pair . This observation suggests that a state ρ_{AB} is uncorrelated if and only if its correlation function vanishes for any pair of local observables. The states for which this holds true are exactly those of the form ρ_{AB} = ρ_{A} ⊗ ρ_{B}, the so-called product states, forming the set

(10) |

These states are precisely those that Alice and Bob can prepare through local operations. If Alice and Bob are in addition allowed to communicate classically, i.e., if we refer to the scheme of ‘local operations and classical communication’ (LOCC),^{65,84} they can create probabilistic mixtures of such uncorrelated states. These are the so-called ‘separable’ or ‘unentangled’ states, which form the set

(11) |

In fact, is the convex hull of the set , with the extremal points given by the uncorrelated pure states |ψ〉_{A}〈ψ|_{A} ⊗ |ϕ〉_{B}〈ϕ|_{B}. In the following, we skip the subscript AB of ρ_{AB} whenever it is clear from the context that we refer to the joint quantum state.

Based on the definition of uncorrelated and unentangled states one can now introduce measures of correlation and entanglement by quantifying, e.g., through the quantum relative entropy^{66,72,85}

S(ρ‖σ) := Tr[ρ(log_{2}ρ − log_{2}σ)],
| (12) |

(13) |

The geometrical nature of the measures in (13), as well as the different state manifolds, are graphically illustrated in Fig. 2.

One remarkable fact is that the minimization underlying I in (13) can be performed explicitly. Given a state ρ_{AB}, the minimizer is the product state ρ_{A} ⊗ ρ_{B}.^{70} This yields

(14) |

Another important aspect of the definition of I is that it universally bounds from above^{89–91} the correlation function of eqn (9) according to

(15) |

The bound (15) quantitatively confirms our intuition: whenever a state is close to , then for any choice of local observables O_{A}, O_{B} the correlation function is small. In particular, when the correlation I is zero, then for all possible local operators the correlation function vanishes. A large value of correlation I is generally regarded unfavorable from a computation viewpoint, as such a state would require a larger amount of computational resources for preparing, storing and manipulating it. To the contrary, from the viewpoint of quantum information, such states are resourceful (see Section II E) and, thus, can be used to realize quantum information processing tasks, e.g., in quantum communication or quantum cryptography.^{92–94} Unlike for I, there does not exist a closed form for the (quantum relative entropy of) entanglement E (13). Nonetheless, it can be calculated for certain states that possess a large number of symmetries.^{95} In particular, for pure states |ψ〉 in there exists a well-known closed form^{66}

(16) |

For general states, one of the crucial properties of E is that it does not increase under LOCC^{96} operations. This means when the parties A, B are restricted to LOCC, they can only degrade the entanglement content of their state. This relates nicely to the idea that entanglement is a resource that is useful for quantum information processing tasks.

Entanglement is not the only form of correlation that exists. Separable states can possess yet another type of correlation that is useful in quantum information protocols. Such correlations are called quantum correlations beyond entanglement. To be more specific, we first define the set of classical states,

(17) |

Given the set , we again use the quantum relative entropy to quantify the ‘distance’ of a given state ρ to that set. This leads to the following definition of quantum correlation:

(18) |

Similar to E(ρ), there is no closed formula for Q(ρ) in general. The quantity Q is sometimes regarded as symmetric discord or geometric discord and plays an important role for the realization of tasks such as quantum state merging^{99} or quantum key generation.^{100} Moreover, quantum correlation can be converted to entanglement, E, via distinct activation protocols.^{101}

Note that any state of the form ρ_{A} ⊗ ρ_{B} can be written as , which is of the form (17). Furthermore, it is clear by definition (17) that classical states are in particular separable. This leads to the following inclusion hierarchy

(19) |

I(ρ) ≥ Q(ρ) ≥ E(ρ). | (20) |

The hierarchy of the various sets of states and the geometric notion of the measures is presented in Fig. 2. Given the definition of quantum correlation contained in a state ρ, one can also quantify the amount of classical correlation contained in a state ρ. For this, one first finds the closest classical state , which fulfills Q(ρ) = S(ρ‖χ_{ρ}). Then the amount of classical correlation in ρ is defined as C(ρ) = I(χ_{ρ}), i.e., the correlation contained in the state after all quantum correlation has been extracted.^{70}

The above geometric ideas straightforwardly extend^{70} to systems composed of N > 2 distinguishable subsystems. Here the set of uncorrelated states contains density operators of form ρ = ⊗^{N}_{i=1}ρ_{i}. The fully separable states are given by convex combination of uncorrelated states and classical states are given by , where {|〉 := ⊗^{N}_{i = 1}|k_{i}〉} is any product basis for . As a generalization of eqn (14), the corresponding correlation I can be explicitly evaluated and follows as

(21) |

(22) |

To discuss a second important protocol, we consider Alice and Bob being connected via a quantum communication channel. Alice holds a single electron that she can send to Bob in order to transmit information. Without additional resources, the best she can do is to encode the classical bit (0 or 1) onto the spin of the electron and pass it to Bob via the channel, who then measures the spin along the z-axis to determine the value of the classical bit. In this way, Alice can send one classical bit of information to Bob. However, if Alice and Bob share in addition an entangled state |ϕ^{+}〉_{AB}, then Alice can communicate two bits of information. To see this, notice that the set of states coincides with BELL up to phase factors and thus forms an orthonormal basis for (here ). This means Alice can choose to apply one of the four unitary operators σ_{i} to her system and send her electron afterwards to Bob. Bob then performs a joint measurement on both electrons in the Bell basis to determine which of the four operators was applied by Alice. Accordingly, Alice could encode two classical bits using the four σ_{i} operators and Bob can decode by performing a Bell measurement. This protocol is called superdense coding.^{72,73}

If we denote a unit of entanglement (|ϕ^{+}〉) with [qq], a quantum channel that transmits single qubits with [q → q] and a classical channel that transmits one bit with [c → c], then the above protocols can concisely be summarized as:

(23) |

In these so-called resource inequalities,^{102} the sign ≥ emphasizes that the left hand side is at least as resourceful as the right hand side. To conclude, these two remarkable quantum protocols univocally demonstrate the necessity for a concise and operationally meaningful quantification of entanglement and various other correlation types.

(24) |

We also introduce the Fock space

(25) |

{f_{i},f^{†}_{j}} = δ_{ij}, {f_{i},f_{j}} = {f^{†}_{i},f^{†}_{j}} = 0,
| (26) |

With the given reference basis {|i〉}, we can build an ‘occupation number’ basis for the Fock space , composed of 2^{d} Slater determinant state vectors

(27) |

(28) |

We remark that this mapping depends on the underlying reference basis of , or, at a more abstract level, on the choice of a subspace of . Also, notice that some sign ambiguities in eqn (28) must be understood and resolved by means of the parity superselection rule (as explained in Section III C).

We now discuss in more detail some instances of this formalism in the case of electron systems, where the spin degree of freedom comes into play as a factor in the one-particle space,

(29) |

Particularly relevant partitions of the reference basis of the d spin–orbitals |ϕ_{i},σ〉 are the following ones:

• Finest partition: each of the d spin–orbitals |ϕ_{i},σ〉 constitutes a subsystem.

• Finest orbital partition: for i = 1, …, d/2 each pair {|ϕ_{i},↑〉,|ϕ_{i},↓〉} constitutes a subsystem, namely the one of the i-th spatial orbital.

• 1 vs. rest: one (spin-)orbital |ϕ_{i}〉 (|ϕ_{i},σ〉) defines a two-mode (one-mode) subsystem; all other (spin-)orbitals form the second subsystem.

• 1 vs. 1 vs. rest: two (spin-)orbitals are identified as two subsystems; all other (spin-)orbitals form the third subsystem.

• Closed (doubly occupied) vs. active vs. virtual (empty) orbitals: this general tripartition underlies the idea of complete active spaces.

In particular, for valence bond theory the partition of choice is the one where two (orthonormalized) atomic orbitals i and j are singled out as subsystems, and all other orbitals constitute a third subsystem to be discarded.^{103} For |〉 = |n_{1↑},n_{1↓}, …, n_{d/2},_{↑},n_{d/2,↓}〉, eqn (28) is then adapted according to

(30) |

This mapping associates to each of the two orbitals of interest a ‘small’ Fock space spanned by just four state vectors characterized by spin occupancies: |0〉_{i/j}, |↑〉_{i/j}, |↓〉_{i/j}, and |↑↓〉_{i/j}. The idea of discarding orbitals |ϕ_{k}〉 with k ≠ i, j corresponds to a partial trace over all such orbitals, whose result is the reduced state ρ_{ij} of the two orbitals.

The two-orbital reduced state ρ_{ij} describes a bipartite system made up of two distinguishable subsystems, namely orbital |ϕ_{i}〉 and orbital |ϕ_{j}〉. By evaluating the correlation measures introduced in Section II D on ρ_{ij}, one can quantify the correlations between the two orbitals. Such correlations in turn describe how bonded the orbitals |ϕ_{i}〉 and |ϕ_{j}〉 are. As an example, we consider a pure state ρ_{ij} = |Ψ〉〈Ψ|, where |ϕ_{i}〉 and |ϕ_{j}〉 host two electrons with opposite spin in the ‘bonding’ orbital . Then, using eqn (30), we can identify |Ψ〉 with an element of the tensor product of the two Fock spaces of orbitals |ϕ_{i}〉 and |ϕ_{j}〉 mentioned above. We obtain

From a quantum information theoretical perspective, one notices that in this representation, |Ψ〉 looks like a maximally entangled state. Using eqn (16) and ignoring for the moment the role of superselection rules, the entanglement between the two orbitals evaluates to E(|Ψ〉) = log_{2}(4) = 2.^{103}

(31) |

(32) |

However, for A and B to qualify as valid subsystems, operators pertaining to modes in A should commute with those pertaining to modes in B, as per our discussion in Section II C. This is clearly not the case, given that the fermionic creation and annihilation operators associated with the modes in A anticommute with those relative to modes in B (see eqn (26)). This issue is overcome by imposing the so-called fermionic parity superselection rule (P-SSR),^{77,78} a fundamental rule of nature whose violation would actually make superluminal signalling (i.e., communication faster than the speed of light) possible.^{104,105} On the level of states, P-SSR ‘forbids’ coherent superpositions of even and odd fermion-number states. On the level of local operators,^{106–108} it dictates that physical local observables on modes in A/B must always commute with the local parity operator , where Π^{(A)}_{τ} is the projector onto the subspace of local particle number parity τ ∈ {even, odd}. Imposing such a commutation rule between the local observables and local parity operator selects out observables on A commuting with those on B. This leads to a proper description of A and B as subsystems. Operationally, such a rule has a drastic effect on the accessible entanglement and correlation contained in a state ρ_{AB}, as now the physically relevant state is given by the superselected version^{106,107} ρ^{P}_{AB}:

(33) |

This implies that the various measures of correlations M = I, E, Q discussed in Section II D must be replaced by superselected versions of them, denoted by M^{P}, where

M^{P}(ρ_{AB}) := M(ρ^{P}_{AB})
| (34) |

(35) |

This subset, however, is not closed under multiplication. For instance, the product of f^{†}_{i}f_{i} and f^{†}_{j}f_{j} is the two-particle operator f^{†}_{i}f_{i}f^{†}_{j}f_{j} = −f^{†}_{i}f^{†}_{j}f_{i}f_{j}. Therefore, this subset does not qualify as a proper physical subsystem.

The embedding of the N-fermion antisymmetric space into the larger space seemingly allows one to recover a tensor product structure. Yet, this approach is dubious and could easily lead to incorrect conclusions. This can be illustrated by considering two (spin-polarized) electrons that have never interacted and that occupy two orbitals |ϕ_{1}〉, |ϕ_{2}〉 localized in far-away regions. In fact, the corresponding Slater determinant state of the two electrons can be written as

(36) |

At first sight, this looks like an entangled state, in striking contrast to the fact that the two electrons have never interacted. Yet, this is merely an artefact of the misleading embedding into the Hilbert space of N = 2 distinguishable particles: the underlying algebra of observables is still the one of fermionic particles, rather than the one of distinguishable particles. Therefore these ‘exchange correlations’ are purely mathematical and do not exist in reality.

(37) |

(38) |

The outer summation in eqn (38) runs over all possible occupation vectors and the ‘partition function’ ensuring normalization reads . States of the form given by eqn (38) are also sometimes referred to as number-conserving fermionic Gaussian states.^{116}

A fundamental property of free states, which in mathematical physics often serves as their actual definition, is that they satisfy a so-called generalized Wick theorem: every correlation function of a free state splits into a product of two-point correlation functions, which only involves the one-particle reduced density matrix (1RDM).^{111,113,117,118}

The 1RDM of any state ρ on the Fock space follows via

(39) |

(40) |

(41) |

In general, 1RDMs do not have a unique preimage in the set and thus also do not in the set of all states on the Fock space . This changes considerably if we restrict ourselves to the free states, since they are in a one-to-one correspondence with 1RDMs:^{113,115} for a free state Γ (38), the corresponding 1RDM reads

(42) |

(43) |

We illustrate this one-to-one relation between free states and their 1RDMs in Fig. 3. For different particle numbers N, N′, the free states mapping to 1RDMs in or lie on hyperplanes of fixed average particle number 〈〉_{Γ} = N,N′.

Equipped with the definition of free states, one defines the so-called nonfreeness^{115,119}

(44) |

(45) |

To make the connection between orbital correlations and nonfreeness/particle correlation precise, we now present a remarkable result that has not been properly acknowledged yet in quantum chemistry, despite its potential far-reaching implications. For this, we first consider the total orbital correlation in a state obeying the parity superselection rule (P-SSR, as per III C), i.e., we refer to the finest partition into single modes (associated with operators a^{(†)}_{i}) of the one-particle Hilbert space. This means we quantify correlation with respect to d subsystems given by the individual modes; see Section III B. Based on eqn (21), it means that we need to compute the entropy S(ρ_{i}) of every single mode reduced state ρ_{i}, obtained by tracing out all modes except the i-th one, that is ρ_{i} = Tr_{ic}[ρ]. Given that our total state ρ is parity superselected, the reduced states take the form ρ_{i} = (1 − p_{i})|0〉〈0| + p_{i}|1〉〈1|. Accordingly, the total orbital correlation follows as

(46) |

(47) |

The proof of this equality uses tools from majorization theory^{121,122} to show that the minimising basis is simply the one of the natural modes/spin–orbitals. In words, relation (47) means nothing else than that the particle correlation measured through the nonfreeness is identical to the total orbital correlation minimized over all bases . Accordingly, particle correlation corresponds to the minimal, thus intrinsic, complexity of many-electron wave functions, while the orbital correlations quantify their complexity relative to a fixed basis. Hence, the relation (47) explains to what extent orbital optimization can reduce the computational complexity of an N-electron quantum state in quantum chemistry.

Solving the ground-state problem for large systems is cursed by the exponential scaling of the dimension of the N-fermion Hilbert space (24) with the system size. Most numerical methods to calculate the ground-state energy are based on the Rayleigh–Ritz variational principle, which corresponds to a minimization of the expectation value 〈Ψ|H|Ψ〉 over all . This does not exploit, however, that most physical Hamiltonians include only pair-wise interactions. Based on this observation, Coulson formulated, in the closing speech at the 1959 Boulder conference in Colorado, the vision to replace the N-fermion wave function with the two-particle reduced density matrix (2RDM)

(48) |

(49) |

The minimization over N-fermion states in eqn (49) can be relaxed to states on the Fock space by introducing a chemical potential that fixes the total particle number N. For general states on with indefinite particle number, their 2RDMs D can be defined in a similar fashion as in eqn (39) by introducing a map μ^{(2)},

(50) |

Moreover, the ground-state energy can be approximated by restricting the variational energy minimization to a submanifold of states. A quite crude but well-known example thereof is Hartree–Fock (HF) theory, where the minimization is restricted to the manifold of Slater determinants. Since the 1RDMs of Slater determinants are idempotent, γ^{2} = γ, their 2RDMs are given by

(51) |

Thus, the HF energy E_{HF} for a Hamiltonian H = h + W, where W denotes the two-body interaction, follows from minimizing the HF energy functional over all idempotent 1RDMs (we now skip the superscript (2) in the Hamiltonian denoting its restriction to the two-particle level). In his seminal work, Lieb showed that for positive semi-definite interactions W ≥ 0 this minimization can be relaxed to the set of all ensemble N-representable 1RDMs .^{135} By exploiting the one-to-one relation between 1RDMs and free states, it further follows that a relaxation from Slater determinant states on to free states on the Fock space does not alter the outcome of the energy minimization for W ≥ 0.^{113} This result exploits the remarkable fact that the 2RDM of any free state is given by precisely D_{HF}, that is

(52) |

Thus, despite D_{HF}(γ) being not ensemble N-representable for γ^{2} ≠ γ, it is indeed representable to a free state on the Fock space for all . Moreover, the corresponding free state is simply given by the free state (43) mapping to the 1RDM γ. In fact, the so-called Lieb variational principle^{133,135,136} and resulting HF functional F_{HF}(γ) := Tr_{2}[WD_{HF}(γ)] provided the foundation for more sophisticated functional approximations in one-particle reduced density matrix functional theory.^{28,134,137–140}

We illustrate the relation between free states and their 2RDMs and 1RDMs in Fig. 5. The set depicts the intersection of the set of free states with the hyperplane of constant average particle number 〈〉_{Γ} = N, as explained in Section IV B. This is meaningful, since also on the two- and one-particle level we refer to a fixed particle number in Fig. 5. The set of all states on the Fock space is illustrated in gray. In contrast to in Fig. 3, the intersection with a hyperplane might have the effect that not all extremal elements of are Slater determinants (red dots) anymore. We then depict the image of the set under the map μ^{(2)} on the two-fermion level with the yellow set. As explained above, this set only intersects with the set of ensemble N-representable 2RDMs at those 2RDMs whose preimage is a Slater determinant (illustrated by red dots on the right side). On the one-particle level, we compose the map μ^{(1)} introduced above with the spectral map spec(·), which maps a 1RDM to its vector of eigenvalues λ_{i}. Due to the one-to-one relation between free states and 1RDMs the image under this composed map is given by the Pauli hypercube (blue) whose vertices correspond to idempotent 1RDMs with eigenvalues λ_{i} ∈ {0, 1} (red dots).

Fig. 5 Illustration of the intersection of with a hyperplane of fixed average particle number N (see also Fig. 3) denoted by and its image under the maps μ^{(1/2)} leading to the respective 1RDMs and 2RDMs. The red dots (left) illustrate the Slater determinants. The image of (light blue, left) under μ^{(2)} (yellow set, right) only intersects with the set of ensemble N-representable 2RDMs (green) at those 2RDMs whose preimages are Slater determinants. On the one-particle level, we illustrate the Pauli hypercube (blue) of admissible natural occupation number vectors (see text for more details). |

(53) |

In the last equation of this configuration interaction (CI) expansion we collected the CI coefficients into a vector . In the natural orbital basis, |Ψ_{2e}〉 admits a compact form containing at most K Slater determinants,^{141}

(54) |

(55) |

To verify this conjecture, we sampled various one-particle bases . To be more specific, for the case K = 2 (the first row in Fig. 6), 10^{5} + 4 × 4 orthogonal matrices were sampled uniformly from the orthogonal group, which transform the natural orbitals to a target basis. For the case K = 3 (the second row in Fig. 6), 10^{6} + 6 × 6 orthogonal matrices were sampled from the orthogonal group, and 10^{6} additional ones are sampled around the identity. In Fig. 6, we present both quantities and , for various states with different levels of intrinsic multireference character modulated by the choice of parameters p_{i} in eqn (54) (i.e., the natural occupation numbers, up to a square). When only one |p_{i}|^{2} is nonzero and therefore equal to 1, |Ψ_{2e}〉 is a single Slater determinant. When various |p_{i}|^{2}'s are fractional, |Ψ_{2e}〉 always contains some multireference character in any orbital basis.

Fig. 6 Relation between total orbital correlation and Shannon entropy of the CI expansion coefficients of the two-electron quantum state (54) in 2K modes for numerous randomly sampled orbital reference bases . The horizontal and vertical red lines indicate the minima of the two quantities, given by H({|p_{i}|^{2}}) and (nonfreeness), respectively. In the first row we set K = 2, and in the second row K = 3. Different plots in the same row correspond to different choices of the parameters {|p_{i}|^{2}}^{K}_{i=1} in (54) that uniquely determine the multireference structure of the state. |

First, we observe that indeed, for each two-electron state |Ψ_{2e}〉, the two correlation quantities and are simultaneously minimized, specifically by the natural spin-orbital basis. This provides the first evidence for our conjecture (55). In particular, when only two |p_{i}|^{2}'s are nonzero (which is the general case if the number of modes is 2K = 4), the minima of the two quantities can be shown to be related by

(56) |

Second, it is clear from the plots that when reduces, both the upper and lower bounds of are also reduced. In particular, when approaches its minimum, the gap between the two bounds of closes. Third, as the |p_{i}|^{2}'s get close to each other and the intrinsic multireference character of the state thus increases, the range of the values of shrinks. This range collapses to a point when the natural occupation numbers become identical (which is evidenced by the line of blue dots sitting right on top of the vertical red line in the last column in Fig. 6). A similar effect is observed for , but much less pronounced. Finally, if we focus on the natural spin-orbitals (represented by the data point at the intersection of the two red lines), we see that the CI entropy is monotonic with the total spin–orbital correlation.

We remark that the conjectured inequality (55) can be proven analytically for any pure state of two fermions (N = 2). While a detailed proof goes beyond the scope of this article, in ref. 142 the interested reader can find the proof of a similar inequality that arises in the evaluation of the quantum correlation (eqn (18)) of any pure state of two distinguishable particles.

We present the results in Fig. 7. For both molecules, as the internuclear distance R increases from below equilibrium, the multireference character of the ground states also increases. This can be directly seen in the left column, where both the nonfreeness and the CI entropy grow with the internuclear distances. Moreover, the relation between the nonfreeness and the CI entropy is monotonic (and even almost linear for H_{2}), as the second column in Fig. 7 clearly demonstrates. This again highlights the high potential of the easily accessible nonfreeness as a universal tool for characterizing multireference wave functions, in place of the cumbersome, if not inaccessible, CI entropy .

Our numerical results confirm that some of the insights from the analytical two-electron examples indeed extend to larger systems. Both the minimized total spin–orbital correlation, i.e., the nonfreeness , and the CI entropy H({|c_{i}|^{2}}) in the natural basis are valid quantitative descriptors of the multireference character of the ground state. More importantly, the two descriptors are found to be monotonic functions of each other for both molecules. This suggests that the simple nonfreeness , which only involves the 1RDM γ, can reveal the high complexity of the wave function encoded in the CI expansion equally well as the CI entropy, which is difficult to calculate in practice.

We then demonstrated that particle correlation equals total orbital correlation minimized with respect to all orbital reference bases. Accordingly, particle correlation equals the minimal, thus intrinsic, complexity of many-electron wave functions while orbital correlation quantifies their complexity relative to a basis. From a practical point of view, the particle correlation therefore defines the correlation threshold to which orbital optimization schemes can reduce the representational complexity of the many-electron wave function. Prime examples for methods with a particularly strong dependence on the orbital reference basis are the density matrix renormalization group (DMRG)-method, as well as variational quantum eigensolvers (VQE) in quantum computing. To further strengthen the connection between the particle and orbital picture, we presented an inherent relation between free states and Hartree–Fock theory: the states that were defined as particle uncorrelated from a quantum information perspective are precisely those that underlie the construction of the pivotal Hartree–Fock functional in one-particle reduced density matrix functional theory. Accordingly, it can be expected that the measure of particle correlation might be connected to the two-electron cumulant, which is discarded in Hartree–Fock theory.

We illustrated all these concepts and analytical findings in few-electron systems. With an analytic example of two-electron states and a numerical one concerning two concrete molecules, we made two instructive observations: (i) (at least for states of two electrons) both the total spin orbital correlation and the entropy of the CI coefficients are minimized in the natural orbital basis. (ii) The particle correlation, which is the minimized total spin orbital correlation, is a good approximation (up to a factor) to the complicated CI entropy relative to the natural orbitals, which measures directly the multireference character of the wave function expansion. At the same time, these results suggested a general guiding principle for simplifying the structure of the wave function: by reducing the total (spin) orbital correlation, one effectively trims away the excessive complexity in the wave function due to a sub-optimal orbital representation.

In summary, we believe that the fermion-compatible correlation and entanglement measures presented in this work facilitate a complete characterization and successful exploitation of the quantum resourcefulness of atoms and molecules for information processing tasks. In return, the profound connection between orbital and particle correlation, as well as their role in evaluating the multireference character of wave functions, could stimulate developments of novel and more efficient approaches to the electron correlation problem.

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## Footnote |

† Note that I is also sometimes referred to as ‘total correlation’ to highlight that it contains in general both classical and quantum correlations |

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