Efficient system identification and characterization for quantum many-body systems

The analytical, experimental, and even numerical investigation of quantum many-body systems is pushed to its limits rather rapidly as the dimension of the Hilbert space scales exponentially with the number of subsystems. As a result, tasks such as the exact diagonalization of general quantum many-body Hamiltonians and the analysis of quantum experiments are limited to small system sizes. In this thesis, we address two issues related to the exponential scaling of the Hilbert space dimension: (i) The approximation of the ground state energy of many-body systems with a lower bound and (ii) efficient strategies for quantum state tomography experiments. While upper bounds on the ground state energy of quantum many-body systems are readily available using well-established techniques such as density matrix renormalization group methods, lower bounds are hard to come by. Here, we formulate this problem without an explicit representation of the ground state but rather minimize the energy over the set of reduced density matrices by imposing partial N -representability conditions. The so-obtained lower bounds now allow us to rigorously benchmark conventional techniques that estimate the ground state energy and hence – together with the upper bounds obtained by DMRG methods – bound the ground state energy of many-body systems from above and below. Quantum state tomography addresses the problem of learning the quantum state of a wellcontrolled physical system by inverting the measurements performed on identically prepared realizations of the state in the laboratory. If a large number of particles is involved, state tomography suffers from the exponential scaling of both the number of measurements that are required to uniquely determine the quantum state and the post-processing resources that are needed to invert the measurements performed on the system. We discuss how these limitations can be overcome for a large class of physically relevant quantum states and states that are of interest for quantum information processing, namely, ground states of gapped local Hamiltonians, thermal states of local Hamiltonians, the classes of GHZ-type states and W-type states, and the permutationally invariant states of indistinguishable particles. These states are similar in that their parameterization scales algebraically in the number of subsystems. Some of them are matrix product operators (matrix product states if they are pure) of low bond dimension, or very well approximated by them. Others have a non-zero support only in the symmetric subspace of the whole Hilbert space. We show that a vast majority of these states is uniquely specified if only partial information about the full quantum state is available. For instance, we discuss states on one-dimensional lattices which are uniquely determined by local information, that is, by measuring the reduced density matrices to all blocks of contiguous sites. Furthermore, the n-particle reduced density matrix of permutationally invariant states of N > n indistinguishable particles allows to determine, e.g., coherent and spin-squeezed states – at least approximately. Further, we discuss and introduce several reconstruction algorithms that are scalable in the system size and hence allow for an efficient inversion of these measurements. Extensive numerical simulations and the application to real experimental data demonstrate that these schemes enable quantum state tomography to scale up to very large system sizes.