Abstract The possible spectra of one-particle reduced density matrices that are compatible with a pure multipartite quantum system of finite dimension form a convex polytope. We introduce a new construction of inner- and outer-bounding polytopes that constrain the polytope for the entire quantum system. The outer bound is sharp. The inner polytope stems only from doubly excited states. We find all quantum systems, where the bounds coincide giving the entire polytope. We show, that those systems are: (i) any system of two particles (ii) L qubits, (iii) three fermions on N ⩽ 7 levels, (iv) any number of bosons on any number of levels and (v) fermionic Fock space on N ⩽ 5 levels. The methods we use come from symplectic geometry and representation theory of compact Lie groups. In particular, we study the images of proper momentum maps, where our method describes momentum images for all representations that are spherical.
Abstract The possible spectra of one-particle reduced density matrices that are compatible with a pure multipartite quantum system of finite dimension form a convex polytope. We introduce a new construction of inner- and outer-bounding polytopes that constrain the polytope for the entire quantum system. The outer bound is sharp. The inner polytope stems only from doubly excited states. We find all quantum systems, where the bounds coincide giving the entire polytope. We show, that those systems are: (i) any system of two particles (ii) L qubits, (iii) three fermions on N ⩽ 7 levels, (iv) any number of bosons on any number of levels and (v) fermionic Fock space on N ⩽ 5 levels. The methods we use come from symplectic geometry and representation theory of compact Lie groups. In particular, we study the images of proper momentum maps, where our method describes momentum images for all representations that are spherical.
Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so-called Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecision, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger–Horne–Zeilinger state by stochastic local operations and classical communication, and discuss generalizations of our results to an arbitrary number of qubits.
To find whether a set of reduced density matrices comes from a common multiparty state is a hard and important problem. In this paper, we do the following. (i) We introduce a method to find out some polytopes in one-party eigenvalue space which are sufficient conditions of this problem. (ii) We point out that there are some relations between the compatible conditions and the entanglement of pure states. And we show this idea more clearly in the three-qubit case. (iii) We investigate the relations between the compatibility problem and the invariants of a matrix set under some groups. Furthermore, we show that it is one of the reasons why the compatibility problems which involve the multiparty density matrices are much more difficult than the one-party case.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.
The quantum marginal problem, concerned with the compatibility of local and global properties of a multipartite quantum system, is arguably one of the fundamental questions in quantum theory. In this article we introduce a novel family of inequalities bounding the allowed local spectra of a pure tripartite state. The number of inequalities in this family grows with the dimension of the system. Most importantly, the inequalities hold for arbitrary dimensions. For equal local dimension at most 4, they generate the complete set of eigenvalue inequalities.
Abstract 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 w i . 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.
The Pauli exclusion principle requires the spectrum of the occupation numbers of the one-electron reduced density matrix (1-RDM) to be bounded by one and zero. However, for a 1-RDM from a wave function, there exist additional conditions on the spectrum of occupation numbers, known as pure N-representability conditions or generalized Pauli conditions. For atoms and molecules, we measure through a Euclidean-distance metric the proximity of the 1-RDM spectrum to the facets of the convex set (polytope) generated by the generalized Pauli conditions. For the ground state of any spin symmetry, as long as time-reversal symmetry is considered in the definition of the polytope, we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In contrast, for excited states, we find that the 1-RDM spectrum is not pinned. Proximity of the 1-RDM to the boundary of the polytope provides a measurement and classification of electron correlation and entanglement within the quantum system. For comparison, this distance to the boundary of the generalized Pauli conditions is also compared to the distance to the polytope of the traditional Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater determinant. We explain the difference in pinning in the ground- and excited-state 1-RDMs through a connection to the N-representability conditions of the two-electron reduced density matrix.