Extracting Many-Body Quantum Resources within One-Body Reduced Density Matrix Functional Theory

Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a novel functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matricial values can universally be determined by the one-body reduced density matrix (1-RDM), avoiding thus the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, becoming thus the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information.

Introduction.-Theconcept of quantum correlations is transversal for many areas of quantum physics ranging from condensed matter [1,2] and quantum chemistry [3,4] to high-energy physics [5,6].Among different measures of quantum correlations, the quantum Fisher information (QFI) [7] is a key quantity that not only gives an operational meaning for multipartite entanglement for quantum-enhanced metrology of spins [8], bosons [9,10] and fermions [11], but can also be used to probe quantum criticality [12,13], nonlocality [14][15][16], and quantum geometry [17].This widespread range of applications makes QFI a fundamentally important concept in quantum physics [18][19][20][21].However, due to the problem of finding an optimal measurement which yields the QFI, its determination in quantum many-body systems remains an important theoretical and technological challenge [22,23].
Experimentally, the value of the QFI was extracted, in the form of a lower bound, and used to prove entanglement of non-Gaussian many-body states [24] and paircorrelated states of twin matter waves [25].For thermal states, the QFI was directly related to dynamic susceptibilities [13] and to the quench dynamics in linear response [11].With the former method, the QFI was measured and used to quantify entanglement in a spin-1/2 Heisenberg antiferromagnetic chain [26] and for nitrogenvacancy center in diamonds [27].Recently, a method to determine the optimal use of a given entangled state for quantum sensing was proposed based on the QFI matrix (QFIM) [28] showing its usefulness to determine optimal quantum technology protocols.Despite this experimental progress, extracting quantum correlations in manybody systems, and, thus, quantifying their quantum re-sources through QFI, is still hampered by the Hilbert space's exponential growth, rendering the computation a formidably demanding task [29,30].
A strategy to alleviate the cost of computing global quantities of quantum many-body systems is to estimate them from local measurements.For instance, artificial neural networks can be trained to learn the entanglement entropy or the two-point density correlations of interacting fermions from local correlations [31][32][33][34][35].The intuition behind this is that some physical properties of quantum systems (usually the ones of ground states) can unambiguously be determined by certain reduced quantities.Many rigorous theorems establish the existence of one-to-one maps between ground states |ψ⟩ and appropriately chosen sets of reduced quantities {ω µ }, such as the particle density or the reduced density matrix, justifying thus the functional notation |ψ[{ω µ }]⟩ [36][37][38][39][40].As a consequence, all observables of the system's ground state are also implicitly functionals of those reduced quantities.Yet, while this is true, almost all research in functional theories focuses on developing energy functionals [41][42][43][44][45][46][47][48][49][50][51].Questions about multipartite quantum correlations or nonlocality in the systems these functionals describe are usually neither addressed nor even posed [52].
Here we initiate and develop a functional-theoretical framework for the QFI.We will show that for ground states of identical particles the QFIM can be determined by the one-body reduced density matrix (1-RDM) γ, obtained by tracing out N −1 particles of the N -body quantum state, avoiding thus the pre-computation of wave functions that expand into exponentially large Hilbert spaces.We will unveil two surprising links between the one-body reduced density matrix functional theory (1-RDMFT) and the QFI functional theory introduced in this work: (i) QFI functionals correspond to the derivatives of the 1-RDM functionals with respect to the cou-pling strengths, revealing the ability of 1-RDMFT to capture itself quantum correlations, and (ii) the energy functional of the 1-RDM can be fully reconstructed from the functionals of the QFI.
The paper is structured as follows: First, we present a general framework for the functional theory of the 1-RDM and recap the concept of QFIM.Next, we present the main ingredients of our novel functional theory of QFI, showcasing our approach for a Bose-Hubbard model.We finish with conclusions.Three additional appendices contain additional technical details.
Hubbard-like Hamiltonian.-Webegin with the Hamiltonian: Ĥ = ĥ + Ŵ . ( The one-body part ĥ = − ⟨ij⟩ t ij b † i bj describes the tunneling, while the two-body term is of the general form The operators b † j ( bj ) create (annihilate) a particle on site j.For a standard Hubbard model with on-site interaction, the couplings are nonzero only when all the indices are equal V jjjj = U .We keep the general matrix elements to take into account non-standard Hubbard models accounting, for example, for dipolar couplings [53].For notational convenience and in order to relate the functional formalism to quantum information concepts, we define site-dependent angular-momentum Hermitian operators, i.e., Ĵij , where σ α , with α = 0, x, y, z, are Pauli matrices.The two-body operator (2) can be rewritten, up to a one-body operator that can be incorporated into h, as where {A, B} = AB + BA and u ijkl αβ are real coupling strengths subject to additional constraints stemming from the symmetry of the operators { Ĵij α , Ĵkl β }.Our central result is that (3) leads to a universal functional F[γ; u], with u = {u ij αβ }, that serves as a generating functional of the QFI.We first introduce both concepts (F and QFI) and then show this connection.
One-body reduced-density-matrix functional theory.-Givena N -body density matrix ρ, we define the 1-RDM as Here, (α, i, j) is the collective index of the vector γ.
Although this representation of γ is not standard [54], it will be convenient to develop our functional formulation of QFI.The ground-state problem for a many-body Hamiltonian of the form in Eq. ( 1) can be solved without resorting to wave functions by proving the existence of a 1-RDM-functional F[γ] [39], which describes the twobody interactions in terms of γ: for any choice of h the ground-state 1-RDM follows from the minimization of where the first term depends linearly on γ.Since the functional F[γ] depends only on the fixed interaction W (not on the one-particle Hamiltonians) it is called universal [55].The constrained-search approach [56] indicates a route for the calculation of F[γ] by minimizing the expectation value of the interacting energy over all states |ψ⟩ that lead to the same γ.Symbolically, Explicit calculations of this functional have been carried out for bosons [57][58][59][60] and fermions [49,61,62].The minimizers |ψ[γ]⟩ = argmin ψ→γ ⟨ψ|W |ψ⟩ do not only correspond to ground states but to the entire set of (representable) 1-RDMs [63].
Crucially, for W from Eq. ( 3), the functional not only depends on the 1-RDM but also on the set of coupling constants.This fact seems to be missed in the literature, and below we show how the QFI can be extracted from F when the latter is considered as a generating functional.To this end, we introduce the following general notation: Quantum Fisher information.-Considera transformation of the density matrix ρ, describing the state of the quantum system, using a unitary operator Û (ϕ).The resulting state becomes ρ(ϕ) = Û (ϕ)ρ Û † (ϕ).The distance between these two density matrices (and the response of the quantum state to perturbations) can be quantified by the QFI.Specifically, for the response ∂ρ(ϕ)/∂ϕ a ≡ (ρ La + La ρ)/2, where La is called the Symmetric Logarithmic Derivative and ϕ a is the a-th parameter of the vector ϕ, the QFIM is M ab = Tr[ρ La Lb ].Since large QFI implies high sensitivity, originally the QFI was introduced in the context of quantum metrology and sensing via the quantum Cramér-Rao lower bound, which quantifies the ultimate precision limit in estimation protocols [64][65][66].
We parametrize the unitary with the angular-momentum operators, i.e., Û (ϕ) = exp(i α ij ϕ ij α Ĵij α ).For pure states, which are considered in this work, ρ = |ψ⟩⟨ψ|, the QFIM becomes the covariance matrix An impediment to the calculation of this matrix is the prior knowledge of the quantum state, which involves computations of such complexity that hampers broad applicability.This problem, we will now see, can be circumvented by relying upon the 1-RDMs and the universal functional F. We note that the QFI is an entanglement measure [9,10]: for a quantum system of N bosons or spins, the state exhibits at least (m + 1)-particle entanglement [67] if the single-parameter QFI surpasses the quantum limit, e.g., , where s is an integer part of N/m, and m = 1, 2, 3, . . .quantifies entanglement depth; the vector n is normalized, α n 2 α = 1.For m = 1, the right-hand side, equal to N , is the standard quantum limit [22].Similar inequality was derived for fermions [11].If the inequality is violated, the entanglement structure is not revealed.
Functional theory of QFIM.-The central quantity of the framework presented above is the 1-RDM.Noteworthy, it already contains information about quantum correlations, via the correlation entropy S[γ] = −Tr[γ ln γ] [31, [68][69][70].Surprisingly, 1-RDMFT is mainly focused on the goal of computing the ground-state energy and some associated observables [54,[71][72][73][74][75][76], but this powerful formalism has not been used to scrutiny multipartite entanglement or nonlocality.Indeed, the map γ → |ψ[γ]⟩ can be used for the calculation of functionals of QFIM: by using Eq. ( 7) one can view QFIM as explicitly, universal functionals dependent on the 1-RDM: Let's emphasize that this functional is defined in a domain whose degrees of freedom do not scale with the number of particles.We now explain how M ijkl αβ can be directly obtained from the universal functional F.
Generation of QFI and reconstruction of F.-The universal functional is a function of the couplings u, cf.Eq. (6).By applying the Hellmann-Feynman theorem to where γ is kept fixed in the derivative and we indicated that M depends on u.We note that one can substitute here F instead of E, but the latter is accessible experimentally.This is our central result, which shows that, once the universal functional is determined as a function of the coupling strengths, it provides access to quantum resources quantified by the QFI without referring to the quantum state |ψ[γ]⟩.Furthermore, the average of Eq. ( 3) with respect to |ψ[γ]⟩ yields the relation Hence, the QFIM, together with its content about quantum resources, enters into the universal functional.These results show that the knowledge of the QFIM allows for full reconstruction of the universal 1-RDM functional.Two-well Bose-Hubbard model.-Two-wellHubbard models played a historical role as analytical tests for density functional theory in its ground-state [78], timedependent [79], and excited-state ensemble [80] versions in order to unveil analytical properties of the functionals [57,62,81,82].It was already used in the context of bosonic Josephson Junctions to obtain quantum resources in terms of spin squeezing and QFI [24,83].An advantage of the bosonic model is that, while it can be filled with an arbitrary number of particles, the functionals can be visualized as 3D graphs.The Hamiltonian is: where nj = b † j bj is the particle-number operator in the site j = r (right) or l (left); the on-site interactions in Eq. ( 2) are V rrrr = V llll = u and the rest are zero.The term Ŵ = u j∈{l,r} nj (n j − 1) is the relevant quantity of what follows.As only two sites are considered we drop the site indices: with α γ 2 α ⩽ N 2 /4.This means that all 1-RDMs lie inside the Bloch sphere of radius N/2.
The universal functional contains all the information to reconstruct one of the diagonals of the QFIM.This is a consequence of the relation between Ŵ and Ĵz : where N = nl + nr .Hence, since |ψ⟩ has a fixed number of particles, we obtain: which is a special case of Eq. ( 9) with a single coupling strength.It is instructive to write the expression of the functional for N = 2.With real wave functions (γ y = 0), In Fig. 1 the exact functionals for M xx , M yy , M zz and M xz are presented for U > 0. Since the amplitudes of the wavefunctions are real and { Ĵx , Ĵy } and { Ĵy , Ĵz } are skew symmetrical operators, M xy = M yz = 0, everywhere.With a gray disk, we mark the value of the standard quantum limit and thus the values M αα > N = 2 signal entanglement.We observe that for M yy , all the states are entangled apart from the surface of the Bloch sphere, describing spin coherent states, at which the quantum limit is not surpassed also for α = x, z, i.e., M αα ⩽ N .We notice that the above results are valid for repulsive interaction, and, therefore, not all quantum states can become a minimizer.For instance, the NOON state (|2, 0⟩ + |0, 2⟩)/ √ 2 shows up in the functionals with attractive interactions U < 0. They are sketched in Fig. 2. While some features are similar, they differ greatly from the functionals in Fig. 1: for the attractive case, the value of the QFI of most of the ground states lies above the  16)).The white line marks the standard quantum limit M zz = N .quantum limit, although there are states within the Bloch sphere which do not exhibit entanglement.
QFIM functional theory for BEC.-Another interesting limit of the two-well Bose-Hubbard model is the Bose-Einstein condensate (BEC) state, which has been used for quantum metrological tasks, spin squeezing and test Bell correlations [84][85][86][87].Due to the Penrose-Onsager criterion [88], BEC states lie near the border of the Bloch sphere and it is convenient to define the functionals in terms of the angles θ and φ, and the number of particles depleted from the condensate (i.e., away from the set of spin coherent states): δ = N/2 − γ (see Fig. 3).Thus, the QFIM functionals can be expressed as functions of the spherical coordinates M αβ (δ, θ, φ).In the neighborhood of the condensation point, M zz reads (see Appendix B): Here, M zz (θ) = N sin 2 (θ) represents the mean-field fluctuations which cannot surpass the standard quantum limit, i.e., M (0) zz (θ) ⩽ N , whereas the two beyond-mean field corrections scaling as δ 1/2 and δ are M zz (θ) = 8 + 2(N − 6) sin 2 (θ), respectively.The square-root scaling of the second term in Eq. ( 16) yields the diverging BEC force, that drives nonperturbatively the system away from the mean-field state [57][58][59].Since the first term does not violate the standard quantum limit, only the next orders can contribute to genuine multipartite entanglement.In Fig. 3 an example of M zz (δ, θ, φ) is shown for N = 1000 and for depletion δ = 0.1.To surpass the standard quantum limit, the mean field contribution should be large, and thus θ ≈ π/2, so γ lies close to the equator.Next, the third term in Eq. ( 16) is always positive, but is significantly overshadowed by the second term in the region cos φ > 0. However, in the region cos φ < 0 it contributes positively, and we observe enhancement of entanglement.
Conclusion.-In this paper we combined ideas from functional theories and quantum information to develop a functional approach to the QFI.In our formalism the elements of the QFI matrix are functionals of the 1-RDM, avoiding thus the exponential growth of the Hilbert space in which they are usually defined.We obtained two main results: (i) the knowledge of the QFIM allows the full reconstruction of the universal functional of the 1-RDM and (ii) QFI functionals correspond to the derivatives of the 1-RDM functional with respect to the coupling strengths, the latter being upgraded to the level of generating functional of the QFI functionals.These results show a so far unexplored ability of the 1-RDM functionals to detect genuine multipartite entanglement.Since in 1-RDM functional theory approach, we can freely adjust single-particle Hamiltonians to move in the landscape of the functionals, our work shows a novel way to extract many-body resources and to determine optimal sensing protocols [28].
We leave for future work to exploit our results in the context of quantum chemistry or condensed matter to investigate the use of strongly interacting systems for quantum metrological tasks.
In the same way, one can see that γ x = N −2n 2 sin(θ) cos(φ) and γ y = N −2n 2 sin(θ) sin(φ), which proves our statement.By noticing that nl = cos 2 (θ/2)n θ + sin 2 (θ/2)n ρ − cos (θ/2) sin (θ/2)(e iφ â † ρ âθ + e −iφ â † θ âρ ) and nr = sin Using the Hamiltonian (B4) we can compute the exact universal functional close to the full BEC state along the radius γ ρ , i.e., while keeping fixed the angles θ and φ.In fact, close to that point, the wave function reads: where β 0 and β 1 are taken real.Notice that the state (B5) gives place to a 1-RDM with β 2 1 = N/4 − γ ρ /2.At the same time, the expectation value of the particles outside the condensate is Let us take this as the important parameter δ ρ ≡ 2β 2 1 = N/2 − γ ρ .Our goal is to compute the expectation value ⟨Ψ BEC | ρ ĥ2 (θ, φ)|Ψ BEC ⟩ ρ .In order to do so we shall pre-compute the following expectation values: and Collecting (B6), (B7) and (B8) we eventually arrive at: After executing the constrained search for (U > 0), this result can be written in a much more insightful way The rest of the matricial elements of the QFIM can be computed along similar lines.

Appendix C: Derivative of the functional
To prove Eq. ( 9) we first need to show that for ground states: where we assume that the derivative keeps fixed γ.To that aim, let's first use the fact that the Hamiltonian can be written as Ĥ(u) = ĥ + Ŵ (u), where ĥ is the one-particle Hamiltonian.Thus, we have: which is what we wanted to prove.

FIG. 1 :
FIG. 1: Universal functionals of QFIM for the repulsiveBose-Hubbard model (for all U > 0) for N = 2.The limit M αβ = 2 is indicated as a disk in gray.

FIG. 2 :
FIG. 2: Universal functionals of QFIM for the attractiveBose-Hubbard model (for all U < 0) for N = 2.The limit M αβ = 2 is indicated as a disk in gray.

FIG. 3 :
FIG. 3: Representation of γ, parametrized with the angles θ and φ, inside the Bloch sphere of radius N/2.The color-code represents the value of M zz close to BEC for N = 1000 and δ = 0.1 (see Eq. (16)).The white line marks the standard quantum limit M zz = N .