Quantum enhanced estimation of a multi-dimensional field

We present a framework for the quantum enhanced estimation of multiple parameters corresponding to non-commuting unitary generators. Our formalism provides a recipe for the simultaneous estimation of all three components of a magnetic field. We propose a probe state that surpasses the precision of estimating the three components individually and discuss measurements that come close to attaining the quantum limit. Our study also reveals that too much quantum entanglement may be detrimental to attaining the Heisenberg scaling in quantum metrology.Comment: 9 pages, 1 figur

Multi-parameter Quantum Metrology

The simultaneous quantum estimation of multiple parameters can provide a better precision than estimating them individually. This is an effect that is impossible classically. We review the rich background of multi-parameter quantum metrology, some of the main results in the field and its recent advances. We close by highlighting future challenges and open questions

Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion

Phase diffusion invariably accompanies all phase estimation strategies – quantum or classical. A precise esti- mation of the former can often provide valuable understanding of the physics of the phase generating phenom- ena itself. We theoretically examine the performance of fixed-particle number probe states in the simultaneous estimation of phase and collective phase diffusion. We derive analytical quantum limits associated with the si- multaneous local estimation of phase and phase diffusion within the quantum Crame ́r-Rao bound framework in the regimes of large and small phase diffusive noise. The former is for a general fixed-particle number state and the latter for Holland Burnett states, for which we show quantum-enhanced estimation of phase as well as phase diffusion. We next investigate the simultaneous attainability of these quantum limits using projective measure- ments acting on a single copy of the state in terms of a trade-off relation. In particular, we are interested how this trade-off varies as a function of the dimension of the state. We derive an analytical bound for this trade-off in the large phase diffusion regime for a particular form of the measurement, and show that the maximum of 2, set by the quantum Crame ́r-Rao bound, is attainable. Further, we show numerical evidence that as diffusion approaches zero, the optimal trade-off relation approaches 1 for Holland-Burnett states. These numerical results are valid in the small particle number regime and suggest that the trade-off for estimating one parameter with quantum-limited precision leads to a complete lack of precision for the other parameter as the diffusion strength approaches zero. Finally, we provide numerical results showing behaviour of the trade-off for a general value of phase diffusion when using Holland-Burnett probe states

Lower Bounds for Ground States of Condensed Matter Systems

Standard variational methods tend to obtain upper bounds on the ground state energy of quantum many-body systems. Here we study a complementary method that determines lower bounds on the ground state energy in a systematic fashion, scales polynomially in the system size and gives direct access to correlation functions. This is achieved by relaxing the positivity constraint on the density matrix and replacing it by positivity constraints on moment matrices, thus yielding a semi-definite programme. Further, the number of free parameters in the optimization problem can be reduced dramatically under the assumption of translational invariance. A novel numerical approach, principally a combination of a projected gradient algorithm with Dykstra's algorithm, for solving the optimization problem in a memory-efficient manner is presented and a proof of convergence for this iterative method is given. Numerical experiments that determine lower bounds on the ground state energies for the Ising and Heisenberg Hamiltonians confirm that the approach can be applied to large systems, especially under the assumption of translational invariance.Comment: 16 pages, 4 figures, replaced with published versio

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