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Prof. Dr. Otfried Gühne

Professor

Room: B-109

Phone: +49 271 740 3707

Lectures

see the teaching page

Preprints

Publications

2020

Fabian Bernards and Otfried Gühne
Generalizing Optimal Bell Inequalities
Phys. Rev. Lett. 125, 200401 (2020), arXiv:2005.08687

Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particle inequality by Froissart [Il Nuovo Cimento B64, 241 (1981)], also known as I3322 inequality, to three particles. For the simplest of these inequalities, we study their quantum mechanical properties and demonstrate that they are relevant, in the sense that they detect nonlocality of quantum states, for which all two-setting inequalities fail to do so.

H. Chau Nguyen and Otfried Gühne
Some Quantum Measurements with Three Outcomes Can Reveal Nonclassicality where All Two-Outcome Measurements Fail to Do So
Phys. Rev. Lett. 125, 230402 (2020), arXiv:2001.03514

Measurements serve as the intermediate communication layer between the quantum world and our classical perception. So, the question which measurements efficiently extract information from quantum systems is of central interest. Using quantum steering as a nonclassical phenomenon, we show that there are instances, where the results of all two-outcome measurements can be explained in a classical manner, while the results of some three-outcome measurements cannot. This points at the important role of the number of outcomes in revealing the nonclassicality hidden in a quantum system. Moreover, our methods allow to improve the understanding of quantum correlations by delivering novel criteria for quantum steering and improved ways to construct local hidden variable models.

Cornelia Spee, Costantino Budroni and Otfried Gühne
Simulating extremal temporal correlations
New J. Phys. 22, 103037 (2020), arXiv:2004.14854

The correlations arising from sequential measurements on a single quantum system form a polytope. This is defined by the arrow-of-time (AoT) constraints, meaning that future choices of measurement settings cannot influence past outcomes. We discuss the resources needed to simulate the extreme points of the AoT polytope, where resources are quantified in terms of the minimal dimension, or "internal memory" of the physical system. First, we analyze the equivalence classes of the extreme points under symmetries. Second, we characterize the minimal dimension necessary to obtain a given extreme point of the AoT polytope, including a lower scaling bound in the asymptotic limit of long sequences. Finally, we present a general method to derive dimension-sensitive temporal inequalities for longer sequences, based on inequalities for shorter ones, and investigate their robustness to imperfections.

Andreas Ketterer, Nikolai Wyderka and Otfried Gühne
Entanglement characterization using quantum designs
Quantum 4, 325 (2020), arXiv:2004.08402

We present in detail a statistical approach for the reference-frame-independent detection and characterization of multipartite entanglement based on moments of randomly measured correlation functions. We start by discussing how the corresponding moments can be evaluated with designs, linking methods from group and entanglement theory. Then, we illustrate the strengths of the presented framework with a focus on the multipartite scenario. We discuss a condition for characterizing genuine multipartite entanglement for three qubits, and we prove criteria that allow for a discrimination of $W$-type entanglement for an arbitrary number of qubits.

Zhen-Peng Xu, Jing-Ling Chen and Otfried Gühne
Proof of the Peres conjecture for contextuality
Phys. Rev. Lett. 124, 230401 (2020), arXiv:2001.07656

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics cannot be reconciled with classical models that are noncontextual for ideal measurements. The first explicit derivation by Kochen and Specker was rather complex, but considerable simplifications have been achieved thereafter. We propose a systematic approach to find minimal Hardy-type and Greenberger-Horne-Zeilinger-type (GHZ-type) proofs of the Kochen-Specker theorem, these are characterized by the fact that the predictions of classical models are opposite to the predictions of quantum mechanics. Based on our results, we show that the Kochen-Specker set with 18 vectors from Cabello et al. [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] is the minimal set for any dimension, verifying a longstanding conjecture by Peres. Our results allow to identify minimal contextuality scenarios and to study their usefulness for information processing.

Andreas Ketterer and Otfried Gühne
Entropic uncertainty relations from quantum designs
Phys. Rev. Research se, rch 2, 023130 (2020), arXiv:1911.07533

In the course of the last decades entropic uncertainty relations have attracted much attention not only due to their fundamental role as manifestation of non-classicality of quantum mechanics, but also as major tools for applications of quantum information theory. Amongst the latter are protocols for the detection of quantum correlations or for the secure distribution of secret keys. In this work we show how to derive entropic uncertainty relations for sets of measurements whose effects form quantum designs. The key property of quantum designs is their indistinguishability from truly random quantum processes as long as one is concerned with moments up to some finite order. Exploiting this characteristic enables us to evaluate polynomial functions of measurement probabilities which leads to lower bounds on sums of generalized entropies. As an application we use the derived uncertainty relations to investigate the incompatibility of sets of binary observables.

H. Chau Nguyen and Otfried Gühne
Quantum steering of Bell-diagonal states with generalized measurements
Phys. Rev. A 10, , 042125 (2020), arXiv:1909.03963

The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states characterised by the correlation matrix $T$ that a local hidden state model exists if and only if $R_T= 2 \pi N_T |\det (T)| \ge 1$, where $N_T$ is defined by an integral over the Bloch sphere. For generalized measurements described by positive operator valued measures we construct a model if $R_T \ge 6/5$. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly-symmetric Werner states.

Nikolai Wyderka and Otfried Gühne
Characterizing quantum states via sector lengths
J. Phys. A: Math. Theor. 53, 345302 (2020), arXiv:1905.06928

Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different $k$, the amount of $k$-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the $n$-representability problem. For the case of two- and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.

Roope Uola, Ana C. S. Costa, H. Chau Nguyen and Otfried Gühne
Quantum Steering
Rev. Mod. Phys , , 15001 (2020), arXiv:1903.06663

Quantum correlations between two parties are essential for the argument of Einstein, Podolsky, and Rosen in favour of the incompleteness of quantum mechanics. Schr\"odinger noted that an essential point is the fact that one party can influence the wave function of the other party by performing suitable measurements. He called this phenomenon quantum steering and studied its properties, but only in the last years this kind of quantum correlation attracted significant interest in quantum information theory. In this paper the theory of quantum steering is reviewed. First, the basic concepts of steering and local hidden state models are presented and their relation to entanglement and Bell nonlocality is explained. Then various criteria for characterizing steerability and structural results on the phenomenon are described. A detailed discussion is given on the connections between steering and incompatibility of quantum measurements. Finally, applications of steering in quantum information processing and further related topics are reviewed.

Cornelia Spee, Hendrik Siebeneich, Timm Florian Gloger, Peter Kaufmann, Michael Johanning, Matthias Kleinmann, Christof Wunderlich and Otfried Gühne
Genuine temporal correlations can certify the quantum dimension
New J. Phys. 22, 023028 (2020), arXiv:1811.12259

Temporal correlations in quantum mechanics are the origin of several non-classical phenomena, but they depend on the dimension of the underlying quantum system. This allows one to use such correlations for the certification of a minimal Hilbert space dimension. Here we provide a theoretical proposal and an experimental implementation of a device-independent dimension test, using temporal correlations observed on a single trapped $^{171}$Yb$^+$ ion. Our test goes beyond the prepare-and-measure scheme of previous approaches, demonstrating the advantage of genuine temporal correlations.

Yue Dai, Yuli Dong, Zhenyu Xu, Wenlong You, Chengjie Zhang and Otfried Gühne
Experimentally accessible lower bounds for genuine multipartite entanglement and coherence measures
Phys. Rev. A pl, ed 13, 054022 (2020), arXiv:1810.07492

Experimentally quantifying entanglement and coherence are extremely important for quantum resource theory. However, because the quantum state tomography requires exponentially growing measurements with the number of qubits, it is hard to quantify entanglement and coherence based on the full information of the experimentally realized multipartite states. Fortunately, other methods have been found to directly measure the fidelity of experimental states without quantum state tomography. Here we present a fidelity-based method to derive experimentally accessible lower bounds for measures of genuine multipartite entanglement and coherence. On the one hand, the method works for genuine multipartite entanglement measures including the convex-roof extended negativity, the concurrence, the G-concurrence, and the geometric measure for genuine multipartite entanglement. On the other hand, the method also delivers observable lower bounds for the convex roof of the $l_{1}$-norm of coherence, the geometric measure of coherence, and the coherence of formation. Furthermore, all the lower bounds are based on the fidelity between the chosen pure state and the target state, and we obtain the lower bounds of several real experimental states as examples of our results.

2019

Fabian Bernards, Matthias Kleinmann, Otfried Gühne and Mauro Paternostro
Daemonic ergotropy: generalised measurements and multipartite settings
Entropy 21, 771 (2019), arXiv:1907.01970

Recently, the concept of daemonic ergotropy has been introduced to quantify the maximum energy that can be obtained from a quantum system through an ancilla-assisted work extraction protocol based on information gain via projective measurements [G. Francica {\it et al.}, npj Quant. Inf. {\bf 3}, 12 (2018)]. We prove that quantum correlations are not advantageous over classical correlations if projective measurements are considered. We go beyond the limitations of the original definition to include generalised measurements and provide an example in which this allows for a higher daemonic ergotropy. Moreover, we propose a see-saw algorithm to find a measurement that attains the maximum work extraction. Finally, we provide a multipartite generalisation of daemonic ergotropy that pinpoints the influence of multipartite quantum correlations, and study it for multipartite entangled and classical~states.

Xiao-Dong Yu, Jiangwei Shang and Otfried Gühne
Optimal verification of general bipartite pure states
npj Quantum Information 5, 112 (2019), arXiv:1901.09856

The efficient and reliable verification of quantum states plays a crucial role in various quantum information processing tasks. We consider the task of verifying entangled states using one-way and two-way classical communication and completely characterize the optimal strategies via convex optimization. We solve these optimization problems using both analytical and numerical methods, and the optimal strategies can be constructed for any bipartite pure state. Compared with the nonadaptive approach, our adaptive strategies significantly improve the efficiency of quantum state verification. Moreover, these strategies are experimentally feasible, as only few local projective measurements are required.

Christina Ritz, Cornelia Spee and Otfried Gühne
Characterizing multipartite entanglement classes via higher-dimensional embeddings
J. Phys. A: Math. Theor. 52, 335302 (2019), arXiv:1901.08847

Witness operators are a central tool to detect entanglement or to distinguish among the different entanglement classes of multiparticle systems, which can be defined using stochastic local operations and classical communication (SLOCC). We show a one-to-one correspondence between general SLOCC witnesses and a class of entanglement witnesses in an extended Hilbert space. This relation can be used to derive SLOCC witnesses from criteria for full separability of quantum states; moreover, given SLOCC witness can be viewed as entanglement witnesses. As applications of this relation we discuss the calculation of overlaps between different SLOCC classes, and the SLOCC classification in $2\times 3\times 3$-dimensional systems.

Timo Simnacher, Nikolai Wyderka, René Schwonnek and Otfried Gühne
Entanglement detection with scrambled data
Phys. Rev. A 99, 062339 (2019), arXiv:1901.07946

In the usual entanglement detection scenario the possible measurements and the corresponding data are assumed to be fully characterized. We consider the situation where the measurements are known, but the data is scrambled, meaning the assignment of the probabilities to the measurement outcomes is unknown. We investigate in detail the two-qubit scenario with local measurements in two mutually unbiased bases. First, we discuss the use of entropies to detect entanglement from scrambled data, showing that Tsallis- and R\'enyi entropies can detect entanglement in our scenario, while the Shannon entropy cannot. Then, we introduce and discuss scrambling-invariant families of entanglement witnesses. Finally, we show that the set of non-detectable states in our scenario is non-convex and therefore in general hard to characterize.

Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu and Otfried Gühne
Quantifying quantum resources with conic programming
Phys. Rev. Lett. 122, 130404 (2019), arXiv:1812.09216

Resource theories can be used to formalize the quantification and manipulation of resources in quantum information processing such as entanglement, asymmetry and coherence of quantum states, and incompatibility of quantum measurements. Given a certain state or measurement, one can ask whether there is a task in which it performs better than any resourceless state or measurement. Using conic programming, we prove that any general robustness measure (with respect to a convex set of free states or measurements) can be seen as a quantifier of such outperformance in some discrimination task. We apply the technique to various examples, e.g. joint measurability, POVMs simulable by projective measurements, and state assemblages preparable with a given Schmidt number.

Timo Simnacher, Nikolai Wyderka, Cornelia Spee, Xiao-Dong Yu and Otfried Gühne
Certifying quantum memories with coherence
Phys. Rev. A 99, 062319 (2019), arXiv:1809.03403

Quantum memories are an important building block for quantum information processing. Ideally, these memories preserve the quantum properties of the input. We present general criteria for measures to evaluate the quality of quantum memories. Then, we introduce a quality measure based on coherence satisfying these criteria, which we characterize in detail for the qubit case. The measure can be estimated from sparse experimental data and may be generalized to characterize other building blocks, such as quantum gates and teleportation schemes.

H. Chau Nguyen, Huy-Viet Nguyen and Otfried Gühne
The geometry of the Einstein--Podolsky--Rosen correlations
Phys. Rev. Lett. 122, 240401 (2019), arXiv:1808.09349

Correlations between distant particles are central to many puzzles and paradoxes of quantum mechanics and, at the same time, underpin various applications like quantum cryptography and metrology. Originally in 1935, Einstein, Podolsky and Rosen (EPR) used these correlations to argue against the completeness of quantum mechanics. To formalise their argument, Schr\"odinger subsequently introduced the notion of quantum steering. Still, the question which quantum states can be used for the EPR argument and which not remained open. Here we show that quantum steering can be viewed as an inclusion problem in convex geometry. For the case of two spin-$\frac{1}{2}$ particles, this approach completely characterises the set of states leading to the EPR argument and consequently to a full description of the quantum correlations that can be used for steering. Our results find applications in various protocols in quantum information processing, and moreover they are linked to quantum mechanical phenomena such as uncertainty relations and the question which observables in quantum mechanics are jointly measurable.

Xiao-Dong Yu and Otfried Gühne
Detecting coherence via spectrum estimation
Phys. Rev. A 99, 062310 (2019), arXiv:1808.08884

Coherence is a basic phenomenon in quantum mechanics and considered to be an essential resource in quantum information processing. Although the quantification of coherence has attracted a lot of interest, the lack of efficient methods to measure the coherence in experiments limits the applications. We address this problem by introducing an experiment-friendly method for coherence and spectrum estimation. This method is based on the theory of majorization and can not only be used to prove the presence of coherence, but also result in a rather precise lower bound of the amount of coherence. As an illustration, we show how to characterize the freezing phenomenon of coherence with only two local measurements for any $N$-qubit quantum systems. Our approach also has other applications in quantum information processing, such as the characterization of distillability and entanglement transformations.

Mariami Gachechiladze, Otfried Gühne and Akimasa Miyake
Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states
Phys. Rev. A 99, 052304 (2019), arXiv:1805.12093

While the circuit model of quantum computation defines its logical depth or "computational time" in terms of temporal gate sequences, the measurement-based model could allow totally different temporal ordering and parallelization of logical gates. By developing techniques to analyze Pauli measurements on multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ) gates, we introduce a deterministic scheme of universal measurement-based computation. In contrast to the cluster-state scheme, where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. A logarithmic-depth implementation of an N-times Controlled-Z gate illustrates a novel trade-off between space and time complexity.

2018

Ana C. S. Costa, Roope Uola and Otfried Gühne
Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems
Entropy 20, 763 (2018), arXiv:1808.01198

The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and R\'enyi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

Christopher Eltschka, Felix Huber, Otfried Gühne and Jens Siewert
Exponentially many entanglement and correlation constraints for multipartite quantum states
Phys. Rev. A 98, 052317 (2018), arXiv:1807.09165

We present a family of correlations constraints that apply to all multipartite quantum systems of finite dimension. The size of this family is exponential in the number of subsystems. We obtain these relations by defining and investigating the generalized state inversion map. This map provides a systematic way to generate local unitary invariants of degree two in the state and is directly linked to the shadow inequalities proved by Rains [IEEE Trans. Inf. Theory 46, 54 (2000)]. The constraints are stated in terms of linear inequalities for the linear entropies of the subsystems. For pure quantum states they turn into monogamy relations that constrain the distribution of bipartite entanglement among the subsystems of the global state.

Mariami Gachechiladze, Nikolai Wyderka and Otfried Gühne
The structure of ultrafine entanglement witnesses
J. Phys. A: Math. Theor. 51, 365307 (2018), arXiv:1805.06404

An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification of entanglement. Second, we present necessary and sufficient conditions that two product observables are able to detect entanglement. Finally, we explain some fallacies in the original construction of ultrafine entanglement witnesses [F. Shahandeh et al., Phys. Rev. Lett. 118, 110502 (2017)].

Jiangwei Shang, Ali Asadian, Huangjun Zhu and Otfried Gühne
Enhanced entanglement criterion via symmetric informationally complete measurements
Phys. Rev. A 98, 022309 (2018), arXiv:1805.03955

We show that a special type of measurements, called symmetric informationally complete positive operator-valued measures (SIC POVMs), provide a stronger entanglement detection criterion than the computable cross-norm or realignment criterion based on local orthogonal observables. As an illustration, we demonstrate the enhanced entanglement detection power in simple systems of qubit and qutrit pairs. This observation highlights the significance of SIC POVMs for entanglement detection.

2017

Nikolai Wyderka, Felix Huber and Otfried Gühne
Almost all four-particle pure states are determined by their two-body marginals
Phys. Rev. A 96, 010102 (2017), arXiv:1703.10950

We show that generic pure states (states drawn according to the Haar measure) of four particles of equal internal dimension are uniquely determined among all other pure states by their two-body marginals. In fact, certain subsets of three of the two-body marginals suffice for the characterization. We also discuss generalizations of the statement to pure states of more particles, showing that these are almost always determined among pure states by three of their $(n-2)$-body marginals. Finally, we present special families of symmetric pure four-particle states that share the same two-body marginals and are therefore undetermined. These are four-qubit Dicke states in superposition with generalized GHZ states.

Frank E. S. Steinhoff, Christina Ritz, Nikolai Miklin and Otfried Gühne
Qudit Hypergraph States
Phys. Rev. A 95, 052340 (2017), arXiv:1612.06418

We generalize the class of hypergraph states to multipartite systems of qudits, by means of constructions based on the d-dimensional Pauli group and its normalizer. For simple hypergraphs, the different equivalence classes under local operations are shown to be governed by a greatest common divisor hierarchy. Moreover, the special cases of three qutrits and three ququarts is analysed in detail.

Nikoloz Tsimakuridze and Otfried Gühne
Graph states and local unitary transformations beyond local Clifford operations
J. Phys. A: Math. Theor. 50, 195302 (2017), arXiv:1611.06938

Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation.

Felix Huber, Otfried Gühne and Jens Siewert
Absolutely maximally entangled states of seven qubits do not exist
Phys. Rev. Lett. 118, 200502 (2017), arXiv:1608.06228

Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure $((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.

M. Gachechiladze, N. Tsimakuridze, O. Gühne
Graphical description of unitary transformations on hypergraph states
J. Phys. A: Math. Theor. 50, 19LT01 (2017), arXiv:1612.01447

Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hypergraph states.

Felix Huber, Christopher Eltschka, Jens Siewert, Otfried Gühne
Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity
J. Phys. A: Math. Theor. 51, 1175301 (2017), arXiv:1708.06298

A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a 2×3×3×3 system that shows maximal entanglement across every bipartition.

I. Apellaniz, M. Kleinmann, O. Gühne, G. Toth
Optimal witnessing of the quantum Fisher information with few measurements
Phys. Rev. A 95, 032330 (2017), arXiv: 1511.05203

We show how to verify the metrological usefulness of quantum states based on the expectation values of an arbitrarily chosen set of observables. In particular, we estimate the quantum Fisher information as a figure of merit of metrological usefulness. Our approach gives a tight lower bound on the quantum Fisher information for the given incomplete information. We apply our method to the results of various multiparticle quantum states prepared in experiments with photons and trapped ions, as well as to spin-squeezed states and Dicke states realized in cold gases. Our approach can be used for detecting and quantifying metrologically useful entanglement in very large systems, based on a few operator expectation values. We also gain new insights into the difference between metrological useful multipartite entanglement and entanglement in general.

M. Gachechiladze, O. Gühne
Completing the proof of "Generic quantum nonlocality"
Phys. Lett. A 381, 1281 (2017), arXiv: 1607.02948

In a paper by Popescu and Rohrlich [Phys. Lett. A 166, 293 (1992)] a proof has been presented showing that any pure entangled multiparticle quantum state violates some Bell inequality. We point out a gap in this proof, but we also give a construction to close this gap. It turns out that with some extra effort all the results from the aforementioned publication can be proven. Our construction shows how two-particle entanglement can be generated via performing local projections on a multiparticle state.

O. Gühne, M. Kleinmann, T. Moroder
Analysing multiparticle quantum states
R. Bertlmann, A. Zeilinger (Eds.), Quantum [Un]speakables II, Springer , p. 345 (2017), arXiv:1506.06976

The analysis of multiparticle quantum states is a central problem in quantum information processing. This task poses several challenges for experimenters and theoreticians. We give an overview over current problems and possible solutions concerning systematic errors of quantum devices, the reconstruction of quantum states, and the analysis of correlations and complexity in multiparticle density matrices.

2016

S. Wölk, O. Gühne
Characterizing the width of entanglement
New J. Phys. 18, 123024 (2016), arXiv:1507.07226

The size of controllable quantum systems has grown in recent times. Therefore, the spatial degree of freedom becomes more and more important in experimental quantum systems. However, the investigation of entanglement in many-body systems mainly concentrated on the number of entangled particles and ignored the spatial degree of freedom so far. As a consequence, a general concept together with experimentally realizable criteria has been missing to describe the spatial distribution of entanglement. We close this gap by introducing the concept of entanglement width as measure of the spatial distribution of entanglement in many-body systems. We develop criteria to detect the width of entanglement based solely on global observables. As a result, our entanglement criteria can be applied easily to many-body systems since single-particle addressing is not necessary.

S. Altenburg, S. Wölk, G. Toth, O. Gühne
Optimized parameter estimation in the presence of collective phase noise
Phys. Rev. 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.

A. Neven, P. Mathonet, O. Gühne, T. Bastin
Quantum fidelity of symmetric multipartite states
Phys. Rev. A 94, 052332 (2016), arXiv: 1606.02675

For two symmetric quantum states one may be interested in maximizing the overlap under local operations applied to one of them. The question arises whether the maximal overlap can be obtained by applying the same local operation to each party. We show that for two symmetric multiqubit states and local unitary transformations this is the case; the maximal overlap can be reached by applying the same unitary matrix everywhere. For local invertible operations (SLOCC equivalence), however, we present counterexamples, demonstrating that considering the same operation everywhere is not enough.

A. Cabello, M. Gu, O. Gühne, J.-Å. Larsson, K. Wiesner
Thermodynamical cost of some interpretations of quantum theory
Phys. Rev. A 94, 052127 (2016), arXiv: 1509.03641

The interpretation of quantum theory is one of the longest-standing debates in physics. Type I interpretations see quantum probabilities as determined by intrinsic properties of the observed system. Type II see them as relational experiences between an observer and the system. It is usually believed that a decision between these two options cannot be made simply on purely physical grounds but requires an act of metaphysical judgment. Here we show that, under some assumptions, the problem is decidable using thermodynamics. We prove that type I interpretations are incompatible with the following assumptions: (i) The choice of which measurement is performed can be made randomly and independently of the system under observation, (ii) the system has limited memory, and (iii) Landauer's erasure principle holds.

M. Paraschiv, S. Wölk, T. Mannel, O. Gühne
Generalized Effective Operator Formalism for Decaying Systems
Phys. Rev. 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.

G. Sentís, C. Eltschka, O. Gühne, M. Huber, J. Siewert
Quantifying entanglement of maximal dimension in bipartite mixed states
Phys. Rev. Lett. 117, 190502 (2016), arXiv: 1605.09783

The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the maximum possible Schmidt number of an arbitrary mixed state. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.

F. Huber, O. Gühne
Characterizing ground and thermal states of few-body Hamiltonians
Phys. Rev. Lett. 117, 010403 (2016), arXiv: 1601.01630

The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two- and, more generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.

L.E. Buchholz, T. Moroder, O. Gühne
Evaluating the geometric measure of multiparticle entanglement
Ann. Phys. (Berlin) 528, 278 (2016), arXiv:1412.7471

We present an analytical approach to evaluate the geometric measure of multiparticle entanglement for mixed quantum states. Our method allows the computation of this measure for a family of multiparticle states with a certain symmetry and delivers lower bounds on the measure for general states. It works for an arbitrary number of particles, for arbitrary classes of multiparticle entanglement, and can also be used to determine other entanglement measures.

T. Moroder, O. Gittsovich, M. Huber, R. Uola, O. Gühne
Steering maps and their application to dimension-bounded steering
Phys. Rev. Lett. 116, 090403 (2016), arXiv:1412.2623

The existence of quantum correlations that allow one party to steer the quantum state of another party is a counterintuitive quantum effect that has been described already at the beginning of the past century. Steering occurs if entanglement can be proven although the description of the measurements on one party is not known, while the other side is characterized. We introduce the concept of steering maps that allow to unlock the sophisticated techniques developed in regular entanglement detection to be used for certifying steerability. As an application we show that this allows to go even beyond the canonical steering scenario, enabling a generalized dimension-bounded steering where one only assumes the Hilbert space dimension on the characterized side, but no description of the measurements. Surprisingly this does not weaken the detection strength of very symmetric scenarios that have recently been carried out in experiments.

M. Gachechiladze, C. Budroni, O. Gühne
Extreme violation of local realism in quantum hypergraph states
Phys. Rev. Lett. 116, 070401 (2016), arXiv:1507.03570

Hypergraph states form a family of multiparticle quantum states that generalizes the well-known concept of Greenberger-Horne-Zeilinger states, cluster states, and more broadly graph states. We study the nonlocal properties of quantum hypergraph states. We demonstrate that the correlations in hypergraph states can be used to derive various types of nonlocality proofs, including Hardy-type arguments, Bell inequalities for genuine multiparticle nonlocality, and an exponentially increasing violation of local realism. Our results suggest that certain classes of hypergraph states are novel resources for quantum metrology and measurement-based quantum computation.

N. Miklin, T. Moroder, O. Gühne
Multiparticle entanglement as an emergent phenomenon
Phys. Rev. A 93, 020104(R) (2016), arXiv:1506.05766

The question whether global entanglement of a multiparticle quantum system can be inferred from local properties is of great relevance for the theory of quantum correlations as well as for experimental implementations. We present a method to systematically find quantum states, for which the two- or three-body marginals do not contain any entanglement, nevertheless, the knowledge of these reduced states is sufficient to prove genuine multiparticle entanglement of the global state. With this, we show that the emergence of global entanglement from separable local quantum states occurs frequently and for an arbitrary number of particles. We discuss various extensions of the phenomenon and present examples where global entanglement can be proven from marginals, even if entanglement cannot be localized in the marginals with measurements on the other parties.

O. Gühne
Keine Ausreden mehr!
Physik Journal 15(2), 18 (2016)

Drei Experimente schließen durch Verletzung der Bellschen Ungleichungen lokal-realistische Modelle aus.

2015

R. Uola, C. Budroni, O. Gühne, J.-P. Pellonpää
One-to-one mapping between steering and joint measurability problems
Phys. Rev. Lett. 115, 230402 (2015), arXiv:1507.08633

Quantum steering refers to the possibility for Alice to remotely steer Bob's state by performing local measurements on her half of a bipartite system. Two necessary ingredients for steering are entanglement and incompatibility of Alice's measurements. In particular, it has been recently proven that for the case of pure states of maximal Schmidt rank the problem of steerability for Bob's assemblage is equivalent to the problem of joint measurability for Alice observables. We show that such an equivalence holds in general, namely, the steerability of any assemblage can always be formulated as a joint measurability problem, and vice versa. We use this connection to introduce steering inequalities from joint measurability criteria and develop quantifiers for the incompatibility of measurements.

C. Lancien, O. Gühne, R. Sengupta, M. Huber
Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes
J. Phys. A: Math. Theor. 48, 505302 (2015), arXiv:1504.01029

While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive analytic upper-bounds on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split exponentially outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.

C. Budroni, G. Vitagliano, G. Colangelo, R.J. Sewell, O. Gühne, G. Toth, M. Mitchell
Quantum non-demolition measurement enables macroscopic Leggett-Garg tests
Phys. Rev. Lett. 115, 200403 (2015), arXiv:1503.08433

We show how a test of macroscopic realism based on Leggett-Garg inequalities (LGIs) can be performed in a macroscopic system. Using a continuous-variable approach, we consider quantum non-demolition (QND) measurements applied to atomic ensembles undergoing magnetically-driven coherent oscillation. We identify measurement schemes requiring only Gaussian states as inputs and giving a significant LGI violation with realistic experimental parameters and imperfections. The predicted violation is shown to be due to true quantum effects rather than to a classical invasivity of the measurement. Using QND measurements to tighten the "clumsiness loophole" forces the stubborn macrorealist to re-create quantum back action in his or her account of measurement.

A. Asadian, C. Budroni, F. E. S. Steinhoff, P. Rabl, O. Gühne
Contextuality in phase space
Phys. Rev. Lett. 114, 250403 (2015), arXiv:1502.05799

We present a general framework for contextuality tests in phase space using displacement operators. First, we derive a general condition that a single-mode displacement operator should fulfill in order to construct Peres-Mermin square and similar scenarios. This approach offers a straightforward scheme for experimental implementations of the tests via modular variable measurements. In addition to the continuous variable case, our condition can be also applied to finite-dimensional systems in discrete phase space, using Heisenberg-Weyl operators. This approach, therefore, offers a unified picture of contextuality with a geometric flavor.

G. Toth, T. Moroder, O. Gühne
Evaluating convex roof entanglement measures
Phys. Rev. Lett. 114, 160501 (2015), arXiv:1409.3806

We show an efficient method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the quantum state for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples.

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, O. Gühne
Systematic errors in current quantum state tomography tools
Phys. Rev. Lett. 114, 080403 (2015), arXiv:1310.8465

Common tools for obtaining physical density matrices in experimental quantum state tomography are shown here to cause systematic errors. For example, using maximum likelihood or least squares optimization to obtain physical estimates for the quantum state, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. Such strongly biased estimates can be avoided using linear evaluation of the data or by linearizing measurement operators yielding reliable and computational simple error bounds.

O. Gittsovich, T. Moroder, A. Asadian, O. Gühne, P. Rabl
Non-classicality tests and entanglement witnesses for macroscopic mechanical superposition states
Phys. Rev. A 91, 022114 (2015), arXiv:1412.2167

We describe a set of measurement protocols for performing non-classicality tests and the verification of entangled superposition states of macroscopic continuous variable systems, such as nanomechanical resonators. Following earlier works, we first consider a setup where a two-level system is used to indirectly probe the motion of the mechanical system via Ramsey measurements and discuss the application of this methods for detecting non-classical mechanical states. We then show that the generalization of this techniques to multiple resonator modes allows the conditioned preparation and the detection of entangled mechanical superposition states. The proposed measurement protocols can be implemented in various qubit-resonator systems that are currently under experimental investigation and find applications in future tests of quantum mechanics at a macroscopic scale.

2014

N. Brunner, O. Gühne, M. Huber
Editorial: Fifty years of Bell's theorem
J. Phys. A: Math. Theor. 47, 420301 (2014)

R. Uola, T. Moroder, O. Gühne
Joint measurability of generalized measurements implies classicality
Phys. Rev. Lett. 113, 160403 (2014), arXiv:1407.2224

The fact that not all measurements can be carried out simultaneously is a peculiar feature of quantum mechanics and is responsible for many key phenomena in the theory, such as complementarity or uncertainty relations. For the special case of projective measurements, quantum behavior can be characterized by the commutator but for generalized measurements it is not easy to decide whether two measurements can still be understood in classical terms or whether the already show quantum features. We prove that a set of generalized measurements which does not satisfy the notion of joint measurability is nonclassical, as it can be used for the task of quantum steering. This shows that the notion of joint measurability is, among several definitions, the proper one to characterize quantum behavior. Moreover, the equivalence allows one to derive novel steering inequalities from known results on joint measurability and new criteria for joint measurability from known results on the steerability of states.

S. Wölk, M. Huber, O. Gühne
Unified approach to entanglement criteria using the Cauchy-Schwarz and Hölder inequalities
Phys. Rev. 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.

T. Moroder, O. Gittsovich, M. Huber, O. Gühne
Steering bound entangled states: A counterexample to the stronger Peres conjecture
Phys. Rev. Lett. 113, 050404 (2014), arXiv:1405.0262

Quantum correlations are at the heart of many applications in quantum information science and, at the same time, they form the basis for discussions about genuine quantum effects and their difference to classical physics. On one hand, entanglement theory provides the tools to quantify correlations in information processing and many results have been obtained to discriminate useful entanglement, which can be distilled to a pure form, from bound entanglement, being of limited use in many applications. On the other hand, for discriminating quantum phenomena from their classical counterparts, Schr\"odinger and Bell introduced the notions of steering and local hidden variable models. We provide a method to generate systematically bound entangled quantum states which can still be used for steering and therefore to rule out local hidden state models. This sheds light on the relations between the various views on quantum correlations and disproves a widespread conjecture known as the stronger Peres conjecture. For practical applications, it implies that even the weakest form of entanglement can be certified in a semi-device independent way.

O. Gühne, M. Cuquet, F.E.S. Steinhoff, T. Moroder, M. Rossi, D. Bruß, B. Kraus, C. Macchiavello
Entanglement and nonclassical properties of hypergraph states
J. Phys. A: Math. Theor. 47, 335303 (2014), arXiv:1404.6492

Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, H. Weinfurter
Experimental Comparison of Efficient Tomography Schemes for a Six-Qubit State
Phys. Rev. Lett. 113, 040503 (2014), arXiv:1401.7526

Quantum state tomography suffers from the measurement effort increasing exponentially with the number of qubits. Here, we demonstrate permutationally invariant tomography for which, contrary to conventional tomography, all resources scale polynomially with the number of qubits both in terms of the measurement effort as well as the computational power needed to process and store the recorded data. We evaluate permutationally invariant tomography by comparing it to full tomography for six-photon states obtained from spontaneous parametric down-conversion. We show that their results are compatible within the statistical errors. For low rank states, we further optimize both schemes using compressed sensing. We demonstrate the benefits of combining permutationally invariant tomography with compressed sensing by studying the influence of the pump power on the noise present in a six-qubit symmetric Dicke state, a case where full tomography is possible only for very high pump powers.

O. Gühne, C. Budroni, A. Cabello, M. Kleinmann, J.-Å. Larsson
Bounding the quantum dimension with contextuality
Phys. Rev. A 89, 062107 (2014), arXiv:1302.2266

We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. The resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.

M. Hofmann, T. Moroder, and O. Gühne
Analytical characterization of the genuine multiparticle negativity
J. Phys. A: Math. Theor. 47, 155301 (2014), arXiv:1401.2424

The genuine multiparticle negativity is a measure of genuine multiparticle entanglement which can be computed numerically. We present several results how this entanglement measure can be characterized analytically. First, we show that with an appropriate normalization this measure can be seen as coming from a mixed convex roof construction. Based on this, we determine its value for $n$-qubit GHZ-diagonal states and four-qubit cluster-diagonal states.

M. Ali, O. Gühne
Robustness of multiparticle entanglement: specific entanglement classes and random states
J. Phys. B: At. Mol. Opt. Phys. 47, 055503 (2014), arXiv:1310.7336

We investigate the robustness of genuine multiparticle entanglement under decoherence. We consider different kinds of entangled three- and four-qubit states as well as random pure states. For amplitude damping noise, we find that the W-type states are most robust, while other states are not more robust than generic states. For phase damping noise the GHZ state is the most robust state, and for depolarizing noise several states are significantly more robust than random states.

M. Hofmann, A. Osterloh, and O. Gühne
Scaling of genuine multiparticle entanglement at a quantum phase transition
Phys. Rev. B 89, 134101 (2014), arXiv:1309.2217

We investigate the scaling and spatial distribution of genuine multiparticle entanglement in three- and four-spin reduced states of the one-dimensional XY-model at the quantum phase transition. We observe a logarithmic divergence and show that genuine three- and four-particle entanglement obeys finite-size scaling.

2013

M. Bergmann and O. Gühne
Entanglement criteria for Dicke states
J. Phys. A: Math. Theor. 46, 385304 (2013), arXiv:1305.2818

Dicke states are a family of multi-qubit quantum states with interesting entanglement properties and have been observed in many experiments. We construct entanglement witnesses for detecting genuine multiparticle entanglement in the vicinity of these states. We use the approach of PPT mixtures to derive the conditions analytically. For nearly all cases, our criteria are stronger than all conditions previously known.

O. Gühne, T. Moroder
Squeezing out more information about entanglement
Physics 6, 79 (2013)

An experimental demonstration of a novel quantum analysis tool will allow diagnosis of entanglement in a much broader set of states.

C. Budroni, T. Moroder, M. Kleinmann, O. Gühne
Bounding temporal quantum correlations
Phys. Rev. Lett. 111, 020403 (2013), arXiv:1302.6223

Sequential measurements on a single particle play an important role in fundamental tests of quantum mechanics. We provide a general method to analyze temporal quantum correlations, which allows us to compute the maximal correlations for sequential measurements in quantum mechanics. As an application, we present the full characterization of temporal correlations in the simplest Leggett-Garg scenario and in the sequential measurement scenario associated with the most fundamental proof of the Kochen-Specker theorem.

T. Moroder, J.-D. Bancal, Y.-C. Liang, M. Hofmann, O. Gühne
Device-independent entanglement quantification and related applications
Phys. Rev. Lett. 111, 030501 (2013), arXiv:1302.1336

We present a general method to quantify both bipartite and multipartite entanglement in a device-independent manner, meaning that we put a lower bound on the amount of entanglement present in a system based on observed data only but independently of any quantum description of the employed devices. Some of the bounds we obtain, such as for the Clauser-Horne-Shimony-Holt Bell inequality or the Svetlichny inequality, are shown to be tight. Besides, device-independent entanglement quantification can serve as a basis for numerous tasks. We show in particular that our method provides a rigorous way to construct dimension witnesses, gives new insights into the question whether bound entangled states can violate a Bell inequality, and can be used to construct device independent entanglement witnesses involving an arbitrary number of parties.

L. Novo, T. Moroder, O. Gühne
Genuine multiparticle entanglement of permutationally invariant states
Phys. Rev. A 88, 012305 (2013), arXiv:1302.4100

We consider the problem of characterizing genuine multiparticle entanglement for permutationally invariant states using the approach of PPT mixtures. We show that the evaluation of this necessary biseparability criterion scales polynomially with the number of particles. In practice, it can be evaluated easily up to ten qubits and improves existing criteria significantly. Finally, we show that our approach solves the problem of characterizing genuine multiparticle entanglement for permutationally invariant three-qubit states.

J. Szangolies, M. Kleinmann, and O. Gühne
Tests against noncontextual models with measurement disturbances
Phys. Rev. A 87, 050101(R) (2013), arXiv:1303.3837

The testability of the Kochen-Specker theorem is a subject of ongoing controversy. A central issue is that experimental implementations relying on sequential measurements cannot achieve perfect compatibility between the measurements and that therefore the notion of noncontextuality does not apply. We demonstrate by an explicit model that such compatibility violations may yield a violation of noncontextuality inequalities, even if we assume that the incompatibilities merely originate from context-independent noise. We show, however, that this problem can be circumvented by combining the ideas behind Leggett-Garg inequalities with those of the Kochen-Specker theorem.

T. Moroder, M. Kleinmann, P. Schindler, T. Monz, O. Gühne, and R. Blatt
Certifying Systematic Errors in Quantum Experiments
Phys. Rev. Lett. 110, 180401 (2013), arXiv:1204.3644

When experimental errors are ignored in an experiment, the subsequent analysis of its results becomes questionable. We develop tests to detect systematic errors in quantum experiments where only a finite amount of data is recorded and apply these tests to tomographic data taken in an ion trap experiment. We put particular emphasis on quantum state tomography and present three detection methods: the first two employ linear inequalities while the third is based on the generalized likelihood ratio.

S. Niekamp, T. Galla, M. Kleinmann, O. Gühne
Computing complexity measures for quantum states based on exponential families
J. Phys. A: Math. Theor. 46, 125301 (2013), arXiv:1212.6163

Given a multiparticle quantum state, one may ask whether it can be represented as a thermal state of some Hamiltonian with k-particle interactions only. The distance from the exponential family defined by these thermal states can be considered as a measure of complexity of a given state. We investigate the resulting optimization problem and show how symmetries can be exploited to simplify the task of finding the nearest thermal state in a given exponential family. We also present an algorithm for the computation of the complexity measure and consider specific examples to demonstrate its applicability.

E. Amselem, M. Bourennane, C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, J.-Å. Larsson, M. Wieśniak
Comment on "State-Independent Experimental Test of Quantum Contextuality in an Indivisible System"
Phys. Rev. Lett. 110, 078901 (2013), arXiv:1302.0617

We argue that the experiment described in the recent Letter by Zu et al. [Phys. Rev. Lett. 109, 150401 (2012); arXiv:1207.0059v1] does not allow to make conclusions about contextuality, since the measurement of the observables as well as the preparation of the state manifestly depend on the chosen context.

O. Gühne, M. Kleinmann
Auf den Kontext kommt es an
Physik Journal 12(2), 25 (2013)

Die Quantenmechanik hat viele, scheinbar paradoxe Konsequenzen. Diese Tatsache hat zu Spekulationen darüber verleitet, ob es eine übergeordnete Theorie geben könnte, die im Einklang mit der klassischen Physik ist. Neben der Bellschen Ungleichung gibt es ein weitreichendes Theorem von Ernst Specker und Simon Kochen, das es ermöglicht, „klassische Modelle“ quantenmechanischer Systeme auszuschließen. Was als Nachdenken über die logische Struktur der Quantenmechanik begann, lässt sich nun auch im Experiment beobachten.

2012

M. Kleinmann, C. Budroni, J.-A. Larsson, O. Gühne, A. Cabello
Optimal inequalities for state-independent contextuality
Phys. Rev. Lett. 109, 250402 (2012), arXiv:1204.3741

Contextuality is a natural generalization of nonlocality which does not need composite systems or spacelike separation and offers a wider spectrum of interesting phenomena. Most notably, in quantum mechanics there exist scenarios where the contextual behavior is independent of the quantum state. We show that the quest for an optimal inequality separating quantum from classical noncontextual correlations in an state-independent manner admits an exact solution, as it can be formulated as a linear program. We introduce the noncontextuality polytope as a generalization of the locality polytope, and apply our method to identify two different tight optimal inequalities for the most fundamental quantum scenario with state-independent contextuality.

Z.-H. Chen, Z.-H. Ma, O. Gühne, and S. Severini
Estimating entanglement monotones with a generalization of the Wootters formula
Phys. Rev. Lett. 109, 200503 (2012), arXiv:1207.2889

Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow to recognize novel families of multiparticle bound entangled states.

J.-A. Larsson, M. Kleinmann, C. Budroni, O. Gühne, and A. Cabello
Maximal violation of state-independent contextuality inequalities
AIP Conf. Proc. 1508, 265 (2012)

The discussion on noncontextual hidden variable models as an underlying description for the quantum-mechanical predictions started in ernest with 1967 paper by Kochen and Specker. There, it was shown that no noncontextual hidden-variable model can give these predictions. The proof used in that paper is complicated, but recently, a paper by Yu and Oh [PRL, 2012] proposes a simpler statistical proof that can also be the basis of an experimental test. Here we report on a sharper version of that statistical proof, and also explain why the algebraic upper bound to the expressions used are not reachable, even with a reasonable contextual hidden variable model. Specifically, we show that the quantum mechanical predictions reach the maximal possible value for a contextual model that keeps the expectation value of the measurement outcomes constant.

T. Moroder, P. Hyllus, G. Toth, C. Schwemmer, A. Niggebaum, S. Gaile, O. Gühne and H. Weinfurter
Permutationally invariant state reconstruction
New J. Phys. 14, 105001 (2012), arXiv:1205.4941

Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, also an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a non-linear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today's experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer.

H. Kampermann, O. Gühne, C. Wilmott, D. Bruß
An algorithm for characterizing SLOCC classes of multiparticle entanglement
Phys. Rev. A 86, 032307 (2012), arXiv:1203.5872

It is well known that the classification of pure multiparticle entangled states according to stochastic local operations leads to a natural classification of mixed states in terms of convex sets. We present a simple algorithmic procedure to prove that a quantum state lies within a given convex set. Our algorithm generalizes a recent algorithm for proving separability of quantum states [J. Barreiro {\it et al.}, Nature Phys. {\bf 6}, 943 (2010)]. We give several examples which show the wide applicability of our approach. We also propose a procedure to determine a vicinity of a given quantum state which still belongs to the considered convex set.

T. Galla and O. Gühne
Complexity measures, emergence, and multiparticle correlations
Phys. Rev. E 85, 046209 (2012), arXiv:1107.1180

We study correlation measures for complex systems. First, we investigate some recently proposed measures based on information geometry. We show that these measures can increase under local transformations as well as under discarding particles, thereby questioning their interpretation as a quantifier for complexity or correlations. We then propose a refined definition of these measures, investigate its properties and discuss its numerical evaluation. As an example, we study coupled logistic maps and study the behavior of the different measures for that case. Finally, we investigate other local effects during the coarse graining of the complex system.

S. Niekamp, M. Kleinmann, and O. Gühne
Entropic uncertainty relations and the stabilizer formalism
J. Math. Phys. 53, 012202 (2012), arXiv:1103.2316

2011

O. Gühne, B. Jungnitsch, T. Moroder, Y.S. Weinstein
Multiparticle entanglement in graph-diagonal states: Necessary and sufficient conditions for four qubits
Phys. Rev. A 84, 052319 (2011), arXiv:1107.4863

The characterization of genuine multiparticle entanglement is important for entanglement theory as well as experimental studies related to quantum information theory. Here, we completely characterize genuine multiparticle entanglement for four-qubit states diagonal in the cluster-state basis. In addition, we give a complete characterization of multiparticle entanglement for all five-qubit graph states mixed with white noise, for states diagonal in the basis corresponding to the five-qubit Y-shaped graph, and for a family of graph states with an arbitrary number of qubits.

M. Kleinmann, O. Gühne, J.R. Portillo, J.-A. Larsson, and A. Cabello
Memory cost of quantum contextuality
New J. Phys. 13, 113011 (2011), arXiv:1007.3650

The simulation of quantum effects requires certain classical resources, and quantifying them is an important step to characterize the difference between quantum and classical physics. For a simulation of the phenomenon of state-independent quantum contextuality, we show that the minimum amount of memory used by the simulation is the critical resource. We derive optimal simulation strategies for important cases and prove that reproducing the results of sequential measurements on a two-qubit system requires more memory than the information-carrying capacity of the system.

B. Jungnitsch, T. Moroder and O. Gühne
Entanglement Witnesses for Graph States: General Theory and Examples
Phys. Rev. A 84, 032310 (2011), arXiv:1106.1114

We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to one when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.

B. Jungnitsch, T. Moroder, and O. Gühne
Taming multiparticle entanglement
Phys. Rev. Lett. 106, 190502 (2011), arXiv:1010.6049

J.-A. Larsson, M. Kleinmann, O. Gühne, and A. Cabello
Violating noncontextual realism through sequential measurements
AIP Conf. Proc. 1327, 401 (2011)

O. Gühne
Entanglement criteria and full separability of multi-qubit quantum states
Phys. Lett. A 375, 406 (2011), arXiv:1009.3782

2010

J. T. Barreiro, P. Schindler, O. Gühne, T. Monz, M. Chwalla, C. F. Roos, M. Hennrich, and R. Blatt
Experimental multiparticle entanglement dynamics induced by decoherence
Nature Physics 6, 943 (2010), arXiv:1005.1965

W.-B. Gao, X.-C. Yao, P. Xu, H. Lu, O. Gühne, A. Cabello, C.-Y. Lu, T. Yang, Z.-B. Chen, and J.-W. Pan
Bell inequality tests of four-photon six-qubit graph states
Phys. Rev. A 82, 042334 (2010), arXiv:0906.3390

O. Gittsovich, P. Hyllus, and O. Gühne
Multiparticle covariance matrices and the impossibility of detecting graph state entanglement with two-particle correlations
Phys. Rev. A 82, 032306 (2010), arXiv:1006.1594

S. Niekamp, M. Kleinmann, and O. Gühne
Discrimination strategies for inequivalent classes of multipartite entangled states
Phys. Rev. A 82, 022322 (2010), arXiv:1006.1313

P. Hyllus, O. Gühne, and A. Smerzi
Not all pure entangled states are useful for sub shot-noise interferometry
Phys. Rev. A 82, 012337 (2010), arXiv:0912.4349

B. Jungnitsch, S. Niekamp, M. Kleinmann, O. Gühne, H. Lu, W.-B. Gao, Y.-A. Chen, Z.-B. Chen, and J.-W. Pan
Increasing the statistical significance of entanglement detection in experiments
Phys. Rev. Lett. 104, 210401 (2010), arXiv:0912.0645

T. Moroder, O. Gühne, N.J. Beaudry, M.Piani, and N. Lütkenhaus
Entanglement verification with realistic measurement devices via squashing operations
Phys. Rev. A 81, 052342 (2010), arXiv:0909.4212

O. Gittsovich and O. Gühne
Quantifying entanglement with covariance matrices
Phys. Rev. A 81, 032333 (2010), arXiv:0912.3018

O. Gühne and M. Seevinck
Separability criteria for genuine multiparticle entanglement
New J. Phys. 12, 053002 (2010), arXiv:0905.1349

O. Gühne, M. Kleinmann, A. Cabello, J.-A. Larsson, G. Kirchmair, F. Zähringer, R. Gerritsma, and C.F. Roos
Compatibility and noncontextuality for sequential measurements
Phys. Rev. A 81, 022121 (2010), arXiv:0912.4846

W.-B. Gao, C.-Y Lu, X.-C. Yao, P. Xu, O. Gühne, A. Goebel, Y.-A. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan
Experimental demonstration of a hyper-entangled ten-qubit "Schrödinger cat" state
Nature Physics 6, 331 (2010), arXiv:0809.4277

G. Tóth and O. Gühne
Separability criteria and entanglement witnesses for symmetric quantum states
Applied Phys. B 98, 617 (2010), arXiv:0908.3679

W.-B. Gao, P. Xu, X.-C. Yao, O. Gühne, A. Cabello, C.-Y. Lu, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan
Experimental Realization of a controlled-NOT gate with four-photon six-qubit cluster states
Phys. Rev. Lett. 104, 020501 (2010), arXiv:0905.2103

2009

R. Hübener, M. Kleinmann, T.-C. Wei, C. González-Guillén and O. Gühne
Geometric measure of entanglement for symmetric states
Phys. Rev. A 80, 032324 (2009), arXiv:0905.4822

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos
State-independent experimental test of quantum contextuality
Nature 460, 494 (2009), arXiv:0904.1655

F. Bodoky, O. Gühne, and M. Blaauboer
Modeling the decay of entanglement for electron spin qubits in quantum dots
J. Phys.: Condens. Matter 21, 395602 (2009), arXiv:0809.3561

G. Tóth and O. Gühne
Entanglement and permutational symmetry
Phys. Rev. Lett. 102, 170503 (2009), arXiv:0812.4453

O. Gühne and G. Tóth
Entanglement detection
Physics Reports 474, 1 (2009), arXiv:0811.2803

G. Tóth, C. Knapp, O. Gühne, and H.J. Briegel
Generalized spin squeezing criteria: Entanglement detection with collective measurements
AIP Conf. Proc. 1110, 41 (2009)

O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert
Covariance matrix criterion for separability
AIP Conf. Proc. 1110, 63 (2009)

G. Tóth, C. Knapp, O. Gühne, and H.J. Briegel
Spin squeezing and entanglement
Phys. Rev. A 79, 042334 (2009), arXiv:0806.1048

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan
Demonstrating anyonic fractional statistics with a six-qubit quantum simulator
Phys. Rev. Lett. 102, 030502 (2009), arXiv:0710.0278

2008

O. Gühne, F. Bodoky, and M. Blaauboer
Multiparticle entanglement under the influence of decoherence
Phys. Rev. A 78, 060301(R) (2008), arXiv:0805.2873

O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert
Unifying several separability conditions using the covariance matrix criterion
Phys. Rev. A 78, 052319 (2008), arXiv:0803.0757

T. Moroder, O. Gühne, and N. Lütkenhaus
Iterations of nonlinear entanglement witnesses
Phys. Rev. A 78, 032306 (2008), arXiv:0806.0855

W. Wieczorek, C. Schmidt, N. Kiesel, R. Pohlner, O. Gühne, and H. Weinfurter
Experimental observation of an entire family of four-photon entangled states
Phys. Rev. Lett. 101, 010503 (2008), arXiv:0806.1882

See also: M. Aspelmeyer and J. Eisert
Entangled families
Nature (News and Views) 455, 180 (2008)

J. Richert and O. Gühne
Low energy properties of even-legged d-dimensional quantum spin systems: a variational approach
Phys. Status Solidi B 245, 1552 (2008)

A. Cabello, O. Gühne, and D. Rodriguez
Mermin inequalities for perfect correlations
Phys. Rev. A 77, 062106 (2008), arXiv:0708.3208

O. Gühne, M. Reimpell, and R.F. Werner
Lower bounds on entanglement measures from incomplete information
Phys. Rev. A. 77, 052317 (2008), arXiv:0802.1734

A. Cabello, O. Gühne, P. Moreno and D. Rodriguez
Nonlocality for graph states
Laser Phys. 18, 335 (2008)

O. Gühne and A. Cabello
Generalized Ardehali-Bell inequalities for graph states
Phys. Rev. A 77, 032108 (2008), arXiv:0806.2769

2007

G. Tóth, C. Knapp, O. Gühne and H.J. Briegel
Optimal spin squeezing inequalities detect bound entanglement in spin models
Phys. Rev. Lett. 99, 250405 (2007), arXiv:quant-ph/0702219

O. Gühne, C.-Y. Lu, W.-B. Gao, and J.-W. Pan
Toolbox for entanglement detection and fidelity estimation
Phys. Rev. A 76, 030305(R) (2007), arXiv:0706.2432

O. Gühne, P. Hyllus, O. Gittsovich, and J. Eisert
Covariance matrices and the separability problem
Phys. Rev. Lett. 99, 130504 (2007), arXiv:quant-ph/0611282

T. Konrad, O. Gühne, J. Audretsch and H.J. Briegel
Parameter estimation for mixed states from a single copy
Phys. Rev. A 75, 062101 (2007), arXiv:quant-ph/0702211

O. Gühne and N. Lütkenhaus
Nonlinear entanglement witnesses, covariance matrices and the geometry of separable states
J. Phys.: Conf. Ser. 67, 012004 (2007), arXiv:quant-ph/0612108

O. Gühne and H. Häffner
Tomografie eines Quantenzustands: Verschränkung und Reinheit
Elektrotechnik und Informationstechnik 124, 131 (2007)

O. Gühne, M. Reimpell, and R.F. Werner
Estimating entanglement measures in experiments
Phys. Rev. Lett. 98, 110502 (2007), arXiv:quant-ph/0607163

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang and J.-W. Pan
Experimental entanglement of six photons in graph states
Nature Physics 3, 91 (2007), arXiv:quant-ph/0609130

2006

J. K. Korbicz, O. Gühne, M. Lewenstein, H. Häffner, C.F. Roos and R. Blatt
Generalized spin squeezing inequalities in N qubit systems: theory and experiment
Phys. Rev. A 74, 052319 (2006), arXiv:quant-ph/0601038

O. Gühne, M. Mechler, G. Tóth, and P. Adam
Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion
Phys. Rev. A 74, 010301(R) (2006), arXiv:quant-ph/0604050

O. Gühne and G. Tóth
Energy and multipartite entanglement in multidimensional and frustrated spin models
Phys. Rev. A 73, 052319 (2006), arXiv:quant-ph/0510186

O. Gühne and N. Lütkenhaus
Nonlinear entanglement witnesses
Phys. Rev. Lett. 96, 170502 (2006), arXiv:quant-ph/0512164

G. Tóth, O. Gühne, and H.J. Briegel
Two-setting Bell inequalities for graph states
Phys. Rev. A 73, 022303 (2006), arXiv:quant-ph/0510007

J. Rigas, O. Gühne, and N. Lütkenhaus
Entanglement verification for quantum key distribution systems with an underlying bipartite qubit-mode structure
Phys. Rev. A 73, 012341 (2006), arXiv:quant-ph/0510022

G. Tóth and O. Gühne
Detection of multipartite entanglement with two-body correlations
Applied Phys. B 82, 237 (2006), arXiv:quant-ph/0602068

2005

H. Häffner, W. Hänsel, C. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt
Scalable multiparticle entanglement of trapped ions
Nature 438, 643 (2005), arXiv:quant-ph/0603217

N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter
Experimental analysis of a four-qubit photon cluster state
Phys. Rev. Lett. 95, 201502 (2005), arXiv:quant-ph/0508128

G. Tóth, O. Gühne, M. Seevinck, and J. Uffink
Addendum to "Sufficient conditions for three-particle entanglement and their tests in recent experiments"
Phys. Rev. A 72, 014101 (2005), arXiv:quant-ph/0505100

P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein
Relations between entanglement witnesses and Bell inequalities
Phys. Rev. A 72, 012321 (2005), arXiv:quant-ph/0504079

O. Gühne, G. Tóth, and H.J. Briegel
Multipartite entanglement in spin chains
New J. Phys. 7, 229 (2005), arXiv:quant-ph/0502160

G. Tóth and O. Gühne
Entanglement detection in the stabilizer formalism
Phys. Rev. A 72, 022340 (2005), arXiv:quant-ph/0501020

O. Gühne, G. Tóth, P. Hyllus and H.J. Briegel
Bell inequalities for graph states
Phys. Rev. Lett. 95, 120405 (2005), arXiv:quant-ph/0410059

M. Curty, O. Gühne, M. Lewenstein and N. Lütkenhaus
Detecting quantum correlations for quantum key distribution
Proc. SPIE 5631, 9 (2005)

M. Curty, O. Gühne, M. Lewenstein and N. Lütkenhaus
Detecting two-party quantum correlations in quantum key distribution protocols
Phys. Rev. A 71, 022306 (2005), arXiv:quant-ph/0409047

G. Tóth and O. Gühne
Detecting genuine multipartite entanglement with two local measurements
Phys. Rev. Lett. 94, 060501 (2005), arXiv:quant-ph/0405165

2004

J. Eisert, P. Hyllus, O. Gühne and M. Curty
Complete hierarchies of efficient approximations to problems in entanglement theory
Phys. Rev. A 70, 062317 (2004), arXiv:quant-ph/0407135

O. Gühne and M. Lewenstein
Entropic uncertainty relations and entanglement
Phys. Rev. A 70, 022316 (2004), arXiv:quant-ph/0403219

O. Gühne and M. Lewenstein
Separability criteria from uncertainty relations
AIP Conf. Proc. 734, 230 (2004), arXiv:quant-ph/0409140

G. Tóth and O. Gühne
Two measurement settings can suffice to verify multipartite entanglement
AIP Conf. Proc. 734, 234 (2004), arXiv:quant-ph/0409132

M. Curty, O. Gühne, M. Lewenstein and N. Lütkenhaus
Quantum correlations for quantum key distribution protocols
AIP Conf. Proc. 734, 307 (2004)

M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer, H. Weinfurter, A. Cabello, O. Gühne, P. Hyllus, D.Bruß, M. Lewenstein, and A. Sanpera
Four photon polarization entanglement: tests and applications
Int. J. Quant. Inf. 2, 133 (2004)

M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera
Experimental detection of multipartite entanglement using witness operators
Phys. Rev. Lett. 92, 087902 (2004), arXiv:quant-ph/0309043

O. Gühne
Characterizing entanglement via uncertainty relations
Phys. Rev. Lett. 92, 117903 (2004), arXiv:quant-ph/0306194

2003

O. Gühne and P. Hyllus
Investigating three qubit entanglement with local measurements
Int. J. Theor. Phys. 42, 1001 (2003), arXiv:quant-ph/0301162

O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera
Experimental detection of entanglement via witness operators and local measurements
J. Mod. Opt. 50, 1079 (2003), arXiv:quant-ph/0210134

2002

O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello, and A. Sanpera
Detection of entanglement with few local measurements
Phys. Rev. A 66, 062305 (2002), arXiv:quant-ph/0205089

Book contributions

2005

M. Curty, O. Gühne, M. Lewenstein und N. Lütkenhaus
Quantum correlations as basic resource for quantum key distribution
In: G. Leuchs, T. Beth (Eds.): "Quantum information processing" (second edition), Wiley-VCH (Berlin) (2005)
url

C. Becher, J. Benhelm, D. Chek-al-kar, M. Chwalla, W. Dür, O. Gühne, H. Häffner, W. Hänsel, T. Körber, A. Kreuter, G.P.T. Lancaster, T. Monz, E.S. Phillips, U.D. Rapol, M. Riebe, C.F. Roos, C. Russo, F. Schmidt-Kaler, and R. Blatt
Entanglement of trapped ions
In: E.A. Hinds, A. Ferguson, E. Riis (Eds.) Laser Spectroscopy, p. 381-392, World Scientific (Singapore) (2005)
url

2003

K. Eckert, O. Gühne, F. Hulpke, P. Hyllus, J. Korbicz, J. Mompart, D. Bruß, M. Lewenstein, and A. Sanpera
Entanglement properties of composite quantum systems
In: G. Leuchs, T. Beth (Eds.): "Quantum information processing", Wiley-VCH (Berlin) (2003)
arXiv:quant-ph/0210107

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