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# Dr. Sabine Wölk

Postdoc

Room: B-111

Phone: +49 271 740 3799

## Preprints

Sanah Altenburg, Michał Oszmaniec, Sabine Wölk, and Otfried Gühne
Estimation of gradients in quantum metrology
arXiv: 1703.09123

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. Classically, 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 ions and therefore suffers from collective dephasing, so we concentrate on states which are insensitive to these fluctuations. States from the decoherence-free subspace (DFS) allow to measure the gradient directly, without estimating the magnetic field. We use the framework of quantum metrology to assess the maximal accuracy of the precision of gradient estimation. We find that states from the DFS achieve the highest measurement sensitivity, as quantified by the quantum Fisher information and find measurements saturating the quantum Cramér-Rao bound.

Alexander Streltsov, Hermann Kampermann, Sabine Wölk, Manuel Gessner and Dagmar Bruß
Maximal Coherence and the Resource Theory of Purity
arXiv:1612.07570

The resource theory of quantum coherence studies the off-diagonal elements of a density matrix in a distinguished basis, whereas the resource theory of purity studies all deviations from the maximally mixed state. We establish a direct connection between the two resource theories, by identifying purity as the maximal coherence, which is achievable by unitary operations. The states that saturate this maximum identify a universal family of maximally coherent mixed states. These states are optimal resources under maximally incoherent operations, and thus independent of the way coherence is quantified. For all distance-based coherence quantifiers the maximal coherence can be evaluated exactly, and is shown to coincide with the corresponding distance-based purity quantifier. We further show that purity bounds the maximal amount of entanglement and discord that can be generated by unitary operations, thus demonstrating that purity is the most elementary resource for quantum information processing. As a byproduct, our results imply that the $l_{1}$-norm of coherence can increase under maximally incoherent operations.

S. Wölk
Revealing quantum properties with simple measurements
arXiv:1611.07678

Since the beginning of quantum mechanics, many puzzling phenomena which distinguish the quantum from the classical world, have appeared such as complementarity, entanglement or contextuality. All of these phenomena are based on the existence of non-commuting observables in quantum mechanics. Furthermore, theses effects generate advantages which allow quantum technologies to surpass classical technologies. In this lecture note, we investigate two prominent examples of these phenomenons: complementarity and entanglement. We discuss some of their basic properties and introduce general methods for their experimental investigation. In this way, we find many connections between the investigation of complementarity and entanglement. One of these connections is given by the Cauchy-Schwarz inequality which helps to formulate quantitative measurement procedures to observe complementarity as well as entanglement.

Sabine Wölk, Christof Wunderlich
Quantum dynamics of trapped ions in a dynamic field gradient using dressed states
arXiv: 1606.04821

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.

## Publications

Sabine Wölk, Otfried Gühne
Characterizing the width of entanglement
New Journal of Physics 18, 123024 (2016), arXiv: 1507.07226

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.

Marius Paraschiv, Sabine Wölk, Thomas Mannel, Otfried Gühne
Generalized Effective Operator Formalism for Decaying Systems
Physical Review A 94, 042103 (2016), arXiv:1607.00797

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 171Yb or 172Yb, which may be used to simulate kaonic behavior in a quantum optical system.

Sanah Altenburg, Sabine Wölk, Geza Toth, Otfried Gühne
Optimized parameter estimation in the presence of collective phase noise
Physical Review A 94, 052306 (2016), arXiv: 1607.05160

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.

Ch. Piltz, Th. Sriarunothai, S. Ivanov, S. Wölk, Ch. Wunderlich
Versatile microwave-driven trapped ion spin system for quantum information processing
Sci. Adv. 2, e1600093 (2016), arXiv:1509.01478

A quantum simulator allows for investigating static and dynamic properties of a complex quantum system, difficult to access directly, by means of another physical system that is well understood and controlled. A universal quantum computer would be suitable for that purpose. However, other, more specialized physical systems -- already in close experimental reach -- promise groundbreaking new insight in quantum phenomena when used as quantum simulators. Here, we show how a tailored and versatile effective spin-system suitable for quantum simulations and universal quantum computation is realized using trapped atomic ions. Each single spin can be addressed individually, and, simply by the application of microwave pulses, selected spins can be decoupled from the remaining system. Furthermore, the sign of the couplings can be changed, as well as the effective strength of spin-spin coupling determined. Thus, all operations for a versatile quantum simulator are implemented. In addition, taking advantage of simultaneous coupling between three spins a coherent quantum Fourier transform -- an essential building block for many quantum algorithms -- is efficiently realized. This approach based on microwave-driven trapped ions, complementary to laser-based methods, opens a new route to overcome technical and physics challenges in the quest for a quantum simulator and quantum computer.

Sabine Wölk, Christian Piltz, Theeraphot Sriarunothai and Christof Wunderlich
State selective detection of hyperfine qubits
Physical Review B 48, 075101 (2015), arXiv:1406.5821

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 π-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 171Yb+ where several state change are possible. To decrease the detection error for such qubits, we develope generalizations of the time-resolved method and the π-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 171Yb+. 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.

S. Wölk, M. Huber, Ot. Gühne
A unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder Inequalities
Physical Review A 90, 022315 (2014), arXiv:1405.0986

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.

J.-H. Huang, S. Wölk, S.-Y. Zhu, M. S. Zubairy
Higher order wave-particle duality
Physical Review A 87, 022107 (2013), arXiv:1212.4255

The complementarity of single-photon's particle-like and wave-like behaviors can be described by the inequality $D^2+V^2 \leq 1$, with $D$ being the path distinguishability and $V$ being the fringe visibility. In this paper, we generalize this duality relation to multi-photon case, where two new concepts, higher order distinguishability and higher order fringe visibility, are introduced to quantify the higher order particle-like and wave-like behaviors of multi-photons.

S. Wölk, W. Ge, M. S. Zubairy
Phase measurement with classical light
Physical Review A 86, 053833 (2012), arXiv:1211.2902

In this paper we investigate whether it is, in general, possible to substitute maximally path-entangled states, namely, NOON states, with classical light in Doppleron-type resonant multiphoton detection processes by studying adaptive phase measurement with classical light. We show that multiphoton detection probability using classical light coincides with that of NOON states and the multiphoton absorption rate is not hindered by the spatially unconstrained photons of the classical light in our scheme. We prove that the optimal phase variance with classical light can be achieved and scales the same as that using NOON states.

S. Wölk and W. P. Schleich
Factorization with Gauss sums:III. Algorithms with entanglement
New J. Phys. 14, 013049 (2012)

We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of Gauss sums in the probability amplitudes of two entangled states. These schemes are rather efficient provided that there exists a fast algorithm that can detect a period of a function hidden in its zeros.

S. Wölk,W. Merkel, I. Sh. Averbukh, B. Girard and W. P. Schleich
Factorization with Gauss sums: I. Mathematical background
New J. Phys. 13, 103007 (2011)

We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and scales exponentially.

W. Merkel, S. Wölk, I. Sh. Averbukh, B. Girard, G. G. Paulus and W. P. Schleich
Factorization with Gauss sums:II.Suggestions for implementations with chirped laser pulses
New J. Phys. 13, 103008 (2011)

We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series (Wölk et al 2011 New J. Phys. 13 103007): (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state and (iii) a linearly chirped onephoton transition driven by a sequence of ultrashort pulses. For each of these quantum systems, we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules on how to encode the number N to be factored in our system and how to identify the factors of N in the fluorescence signal of the excited state.

S. Wölk, C. Feiler, and W. P. Schleich
Factorization of numbers with truncated Gauss sums at rational arguments
J. Mod. Opt. 56, 2118 (2009)

Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a one-to-one relationship. As a result, the identification of factors by the maxima is unique. However, for non-integer arguments, such as rational numbers, this powerful instrument to find factors breaks down. We develop new strategies for factoring numbers using Gauss sums at rational arguments. This approach may find application in a recent suggestion to factor numbers using a light interferometer discussed in this issue (V. Tamma et al., J. Mod. Opt. in press).

## Book contributions

S. Wölk, C. Feiler, and W. P. Schleich
Quantum Mechanics Meet Number Theory
In: (2011)
url

S. Wölk and W. P. Schleich
Quantum carpets: Factorization with degeneracies
In: (2011)
url