Characterizing the width of entanglement

We introduce the concept of entanglement width as measure of the spatial distribution of entanglement in multiparticle systems. We develop criteria to detect the width of entanglement using global observables such as energy and magnetic susceptibility. Therefore, the introduced entanglement criteria can be applied to systems where addressing of single particles is not possible. We apply the criteria to different examples such as the J1-J2- Heisenberg model and point out the difference between entanglement depth and entanglement width.Comment: 10 pages, 5 figures, major revision including new examples and criteria for entanglement widt

Quantum dynamics of trapped ions in a dynamic field gradient using dressed states

Novel ion traps that provide either a static or a dynamic magnetic gradient field allow for the use of radio frequency (rf) radiation for coupling internal and motional states of ions, which is essential for conditional quantum logic. We show that the coupling mechanism in the presence of a dynamic gradient is the same, in a dressed state basis, as in the case of a static gradient. Then, it is shown how demanding experimental requirements arising when using a dynamic gradient could be overcome. Thus, using dressed states in a dynamic gradient field could decisively reduce experimental complexity on the route towards a scalable device for quantum information science based on rf-driven trapped ions.Comment: 5 page

A unified approach to entanglement criteria using the Cauchy-Schwarz and H\"older inequalities

We present unified approach to different recent entanglement criteria. Although they were developed in different ways, we show that they are all applications of a more general principle given by the Cauchy-Schwarz inequality. We explain this general principle and show how to derive with it not only already known but also new entanglement criteria. We systematically investigate its potential and limits to detect bipartite and multipartite entanglement.Comment: 11 pages, 1 figur

Optimal distributed sensing in noisy environments

We consider distributed sensing of non-local quantities. We introduce quantum enhanced protocols to directly measure any (scalar) field with a specific spatial dependence by placing sensors at appropriate positions and preparing a spatially distributed entangled quantum state. Our scheme has optimal Heisenberg scaling and is completely unaffected by noise on other processes with different spatial dependence than the signal. We consider both Fisher and Bayesian scenarios, and design states and settings to achieve optimal scaling. We explicitly demonstrate how to measure coefficients of spatial Taylor and Fourier series, and show that our approach can offer an exponential advantage as compared to strategies that do not make use of entanglement between different sites.Comment: 7 pages (4+3), 4 figure

Estimation of gradients in quantum metrology

We develop a general theory to estimate magnetic field gradients in quantum metrology. We consider a system of $N$ particles distributed on a line whose internal degrees of freedom interact with a magnetic field. Usually gradient estimation is based on precise measurements of the magnetic field at two different locations, performed with two independent groups of particles. This approach, however, is sensitive to fluctuations of the off-set field determining the level-splitting of the particles and results in collective dephasing. In this work we use the framework of quantum metrology to assess the maximal accuracy for gradient estimation. For arbitrary positioning of particles, we identify optimal entangled and separable states allowing the estimation of gradients with the maximal accuracy, quantified by the quantum Fisher information. We also analyze the performance of states from the decoherence-free subspace (DFS), which are insensitive to the fluctuations of the magnetic offset field. We find that these states allow to measure a gradient directly, without the necessity of estimating the magnetic offset field. Moreover, we show that DFS states attain a precision for gradient estimation comparable to the optimal entangled states. Finally, for the above classes of states we find simple and feasible measurements saturating the quantum Cram\'er-Rao bound.Comment: 22 pages, 8 figures. See also the related work by I. Apellaniz et al. arXiv: 1703.09056 (2017

Generalized Effective Operator Formalism for Decaying Systems

Systems of neutral kaons can be used to observe entanglement and the violation of Bell inequalities. The decay of these particles poses some problems, however, and recently an effective formalism for treating such systems has been derived. We generalize this formalism and make it applicable to other quantum systems that can be made to behave in a similar manner. As examples, we discuss two possible implementations of the generalized formalism using trapped ions such as $^{171}$Yb or $^{172}$Yb, which may be used to simulate kaonic behavior in a quantum optical system.Comment: 11 pages, 20 figure

Optimized parameter estimation in the presence of collective phase noise

We investigate phase and frequency estimation with different measurement strategies under the effect of collective phase noise. First, we consider the standard linear estimation scheme and present an experimentally realisable optimization of the initial probe states by collective rotations. We identify the optimal rotation angle for different measurement times. Second, we show that sub-shot noise sensitivity - up to the Heisenberg limit - can be reached in presence of collective phase noise by using differential interferometry, where one part of the system is used to monitor the noise. For this, not only GHZ states but also symmetric Dicke states are suitable. We investigate the optimal splitting for a general symmetric Dicke state at both inputs and discuss possible experimental realisations of differential interferometry.Comment: 17 pages, 6 figures, v2: small revisions, final versio

State selective detection of hyperfine qubits

In order to faithfully detect the state of an individual two-state quantum system (qubit) realized using, for example, a trapped ion or atom, state selective scattering of resonance fluorescence is well established. The simplest way to read out this measurement and assign a state is the threshold method. The detection error can be decreased by using more advanced detection methods like the time-resolved method or the $\pi$-pulse detection method. These methods were introduced to qubits with a single possible state change during the measurement process. However, there exist many qubits like the hyperfine qubit of $^{171}Yb^+$ where several state change are possible. To decrease the detection error for such qubits, we develope generalizations of the time-resolved method and the $\pi$-pulse detection method for such qubits. We show the advantages of these generalized detection methods in numerical simulations and experiments using the hyperfine qubit of $^{171}Yb^+$. The generalized detection methods developed here can be implemented in an efficient way such that experimental real time state discrimination with improved fidelity is possible.Comment: 22 pages, 9 figure

Distinguishing between statistical and systematic errors in quantum process tomography

It is generally assumed that every process in quantum physics can be described mathematically by a completely positive map. However, experimentally reconstructed processes are not necessarily completely positive due to statistical or systematic errors. In this paper, we introduce a test for discriminating statistical from systematic errors which is necessary to interpret experimentally reconstructed, non-completely positive maps.Wedemonstrate the significance of the test using several examples given by experiments and simulations. In particular, we demonstrate experimentally how an initial correlation between the system to be measured and its environment leads to an experimentally reconstructed map with negative eigenvalues. These experiments are carried out using atomic 171Yb+ ions confined in a linear Paul trap, addressed and coherently manipulated by radio frequency radiation.Comment: 11 pages, 5 figure

Radio-frequency sideband cooling and sympathetic cooling of trapped ions in a static magnetic field gradient

We report a detailed investigation on near-ground state cooling of one and two trapped atomic ions. We introduce a simple sideband cooling method for confined atoms and ions, using RF radiation applied to bare ionic states in a static magnetic field gradient, and demonstrate its application to ions confined at secular trap frequencies, $\omega_z \approx 2\pi\times 117$kHz. For a single \ybplus ion, the sideband cooling cycle reduces the average phonon number, $\left\langle\,n\,\right\rangle$ from the Doppler limit to $\left\langle\,n\,\right\rangle =$ 0.30(12). This is in agreement with the theoretically estimated lowest achievable phonon number in this experiment. We extend this method of RF sideband cooling to a system of two \ybplus ions, resulting in a phonon number of $\left\langle\,n\,\right\rangle =$ 1.1(7) in the center-of-mass mode. Furthermore, we demonstrate the first realisation of sympathetic RF sideband cooling of an ion crystal consisting of two individually addressable identical isotopes of the same species.Comment: 8 pages, 7 figure