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Motivated by the Penrose–Onsager criterion for Bose–Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh–Ritz variational principle to ensemble states with spectrum w and prove a corresponding generalization of the Hohenberg–Kohn theorem: the underlying one-particle reduced density matrix determines all properties of systems of N identical particles in their w -ensemble states. Then, to circumvent the v -representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy–Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body w -ensemble N -representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli’s exclusion principle for fermions and recently discovered generalizations thereof.

The N-representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body N-representability problems that incorporate partial knowledge of the one-body RDMs (1RDMs) within an ensemble of N-fermion states with fixed weights wi. Specifically, we propose a systematic relaxation that reduces the refined problem—where full 1RDMs are fixed for certain ensemble elements—to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn’s problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble N-representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.

We initiate the recently proposed \\(\\boldsymbol{w}\\)-ensemble one-particle reduced density matrix functional theory (\\(\\boldsymbol{w}\\)-RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of \\(\\boldsymbol{w}\\)-RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional's domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories.

Motivated by the Penrose-Onsager criterion for Bose-Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh-Ritz variational principle to ensemble states with spectrum \\(\\boldsymbol{w}\\) and prove a corresponding generalization of the Hohenberg-Kohn theorem: The underlying one-particle reduced density matrix determines all properties of systems of \\(N\\) identical particles in their \\(\\boldsymbol{w}\\)-ensemble states. Then, to circumvent the \\(v\\)-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy-Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body \\(\\boldsymbol{w}\\)-ensemble \\(N\\)-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli's exclusion principle for fermions and recently discovered generalizations thereof.

Density functional theory constitutes the workhorse of modern electronic structure calculations due to its favourable computational cost despite the fact that it usually fails to describe strongly correlated systems. A particularly promising approach to overcome those difficulties is reduced density matrix functional theory (RDMFT): It abandons the complexity of the \\(N\\)-particle wave function and at the same time explicitly allows for fractional occupation numbers. It is the goal of this thesis to initiate and establish a bosonic RDMFT for both ground state and excited state energy calculations. Motivated by the Onsager and Penrose criterion which identifies RDMFT as a particularly suitable approach to describe Bose-Einstein condensates (BECs), we derive the universal functional for a homogeneous BEC in the Bogoliubov regime. Remarkably, the gradient of the universal functional is found to diverge repulsively in the regime of complete condensation. This introduces the new concept of a BEC force, which provides a universal explanation for quantum depletion since it is merely based on the geometry of quantum states. In the second part of the thesis, we propose and work out an ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour further highlights the potential of convex analysis for the development of functional theories in the future. Indeed by resorting to several concepts from convex analysis, we succeeded to provide a comprehensive foundation of \\(\\boldsymbol{w}\\)-ensemble RDMFT for bosons which is further based on a generalization of the Ritz variational principle and a constrained search formalism. In particular, we solve the emerging \\(N\\)-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states.

One-particle reduced density matrix functional theory would potentially be the ideal approach for describing Bose-Einstein condensates. It namely replaces the macroscopically complex wavefunction by the simple one-particle reduced density matrix, therefore provides direct access to the degree of condensation and still recovers quantum correlations in an exact manner. We eventually initiate and establish this novel theory by deriving the respective universal functional \\(F\\) for general homogeneous Bose-Einstein condensates with arbitrary pair interaction. Most importantly, the successful derivation necessitates a particle-number conserving modification of Bogoliubov theory and a solution of the common phase dilemma of functional theories. We then illustrate this novel approach in several bosonic systems such as homogeneous Bose gases and the Bose-Hubbard model. Remarkably, the general form of \\(F\\) reveals the existence of a universal Bose-Einstein condensation force which provides an alternative and more fundamental explanation for quantum depletion.

Excitation energy transfer in light-harvesting aggregates is highly efficient, yet whether quantum coherence plays an operational role in transport remains debated. A central challenge is that coherence is usually inferred from spectroscopic signatures, whereas transport performance is assessed through specific observables and depends on both the open system dynamics and the initial state preparation. Here we develop a resource theoretic approach that quantifies the maximum change that initial site-basis coherence can induce in a chosen readout under fixed reduced dynamics. The central quantity is the resource impact functional, which yields state independent, readout specific bounds on coherence-induced changes in signals and transport figures of merit. We apply the framework to two models. For a donor-acceptor dimer, we analyse coherence sensitivity across coupling and bath-timescale regimes and bound trapping efficiency and average transfer time in terms of the impact functional. For a multi-site chain with terminal trapping, we derive rigorous criteria that distinguish population placement from sensitivity to initial state site-basis coherence. These include upper bounds on the largest advantage over incoherent preparations, necessary delocalization requirements for achieving a prescribed improvement, and a simple pairwise sufficient condition that can be checked from local information. For quasi-local reduced dynamics, we further obtain a Lieb-Robinson-type bound that constrains when coherence prepared in a distant region can influence a localized readout at finite times. Together, these results provide operational diagnostics and rigorous bounds for benchmarking coherence effects and for identifying regimes in which they are necessarily negligible or potentially relevant in excitonic transport models.

Quantum coherence and other non-classical features are widely discussed in chemical dynamics, yet it remains difficult to quantify when such resources are operationally relevant for a given process and observable. While quantum resource theories provide a comprehensive framework for comparing free and resourceful settings, existing approaches typically rely on resource monotones or on performance bounds under free operations, and do not directly quantify the maximal influence a chosen resource can exert on a fixed chemical dynamics. Here, we introduce task specific, process level quantifiers that upper bound the largest change a quantum resource can induce in a target figure of merit. Central is a resource impact functional \\(\\mathcal{C}_M(\\Lambda)\\), defined by comparing a state with its paired resource-free counterpart under the same quantum channel \\(\\Lambda\\), which admits an operational interpretation in binary hypothesis testing. We derive variation and time bounds that constrain how rapidly a resource can modify a target signal, providing resource-aware analogues of quantum speed limits. Moreover, we show that open system dynamics can be decomposed into free and resourceful components such that only the resourceful component contributes to \\(\\mathcal{C}_M(\\Lambda)\\), thereby isolating the parts of a generator responsible for resource-induced changes in the observable. We illustrate the framework exemplary for energy transfer in a donor-acceptor dimer in two analytically solvable regimes. Our results provide a general toolbox for diagnosing and benchmarking quantum resource effects in molecular processes.

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies \\(n_i\\) of orbitals \\(\\varphi_i\\) according to \\(0 \\leq n_i \\leq 2\\). In this work, we first refine the underlying one-body \\(N\\)-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness \\(\\boldsymbol w\\) of the \\(N\\)-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope \\(\\Sigma_{N,S}(\\boldsymbol w) \\subset [0,2]^d\\). These constraints are independent of \\(M\\) and the number \\(d\\) of orbitals, while their dependence on \\(N, S\\) is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT) we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its possible natural variables and explain how symmetries could be exploited. In particular, for systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of \\(v\\)-representability is relative to the scope and choice of variable. All these fundamental concepts are then comprehensively discussed and illustrated for the Hubbard dimer and its generalization to arbitrary pair interactions \\(W\\). For this, we derive by analytical means the pure and ensemble functionals with respect to both the real- and complex-valued Hilbert space. The comparison of various functionals allows us to solve the underlying \\(v\\)-representability problems analytically and the dependence of its solution on the pair interaction is demonstrated. Intriguingly, the gradient of each universal functional is found to always diverge repulsively on the boundary of the domain. In that sense, this key finding emphasizes the universal character of the fermionic exchange force, recently discovered and proven in the context of translationally-invariant one-band lattice models.