# Priv.-Doz. Dr. Matthias Kleinmann

LecturerRoom: B-006

Phone: +49 271 740 3799

ORCID: orcid.org/0000-0002-5782-804X

## Lectures

see the teaching page

## Preprints

See also arxiv

Carlos de Gois, Matthias Kleinmann*User-friendly confidence regions for quantum state tomography*

arXiv:2308.01851

Quantum state tomography is the standard technique for reconstructing a quantum state from experimental data. Given finite statistics, experimental data cannot give perfect information about the quantum state. A common way to express this limited knowledge is by providing confidence regions in state space. Though plenty of confidence regions have been previously proposed, they are often too loose to use for large systems or difficult to apply to nonstandard measurement schemes. Starting from a vector Bernstein inequality, we consider concentration bounds for random vectors following multinomial distributions and analyse optimal strategies to distribute a fixed budget of samples across them. Interpreting this as a tomography experiment leads to two confidence regions, one of which performs comparably well to the best regions in the literature. The regions describe an ellipsoid in the state space and have the appeal of being efficient in the required number of samples as well as being easily applicable to any measurement scheme.

Adán Cabello, Marco Túlio Quintino, Matthias Kleinmann*Logical possibilities for physics after MIP*=RE*

arXiv:2307.02920

MIP*=RE implies that C_{qa} (the closure of the set of tensor product correlations) and C_{qc} (the set of commuting correlations) can be separated by a hyperplane (i.e., a Bell-like inequality) and that there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. We point out that there are four logically possible universes after this result. Each possibility is interesting because it reveals either limitations in accepted physical theories or opportunities to test crucial aspects of nature. We list some open problems that may help us to design a road map to learn in which of these universes we are.

Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann*Certifying activation of quantum correlations with finite data*

arXiv:2305.03748

Quantum theory allows for different classes of correlations, such as entanglement, steerability or Bell-nonlocality. Experimental demonstrations of the preparation of quantum states within specific classes and their subsequent interconversion have been carried out; however, rigorous statements on the statistical significance are not available. Behind this are two difficulties: the lack of a method to derive a suitable confidence region from the measured data and an efficient technique to classify the quantum correlations for every state in the confidence region. In this work, we show how both of these problems can be addressed. Specifically, we introduce a confidence polytope in the form of a hyperoctahedron and provide a computationally efficient method to verify whether a quantum state admits a local hidden state model, thus being unsteerable and, consequently, Bell-local. We illustrate how our methods can be used to analyse the activation of quantum correlations by local filtering, specifically for Bell-nonlocality and quantum steerability.

Pascal Höhn, Zhen-Peng Xu, Matthias Kleinmann*Systematic construction of quantum contextuality scenarios with rank advantage*

arXiv:2207.11183

A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.

Massimiliano Smania, Matthias Kleinmann, Adán Cabello, Mohamed Bourennane*Avoiding apparent signaling in Bell tests for quantitative applications*

arXiv:1801.05739

Bell tests have become a powerful tool for quantifying security, randomness, entanglement, and many other properties, as well as for investigating fundamental physical limits. In all these cases, the specific experimental value of the Bell parameter is important as it leads to a quantitative conclusion. However, most experimental implementations aiming for high values of the Bell parameter suffer from the defect of showing signaling. This signaling can be attributed to systematic errors occurring due to weaknesses in the experimental designs. Here we point out the importance, for quantitative applications, to identify and address this problem. We present a set of experiments with polarization-entangled photons in which we point out common sources of systematic errors and demonstrate approaches to avoid them. This allows us to establish a reliable estimate for the Bell parameter.

## Publications

Jessica O. de Almeida, Matthias Kleinmann, Gael Sentís*Comparison of confidence regions for quantum state tomography*

New J. Phys. *25*,
113018
(2023),
arXiv:2303.07136

The quantum state associated to an unknown experimental preparation procedure can be determined by performing quantum state tomography. If the statistical uncertainty in the data dominates over other experimental errors, then a tomographic reconstruction procedure must express this uncertainty. A rigorous way to accomplish this is via statistical confidence regions (CRs) in state space. Naturally, the size of this region decreases when increasing the number of samples, but it also depends critically on the construction method of the region. We compare recent methods for constructing CRs as well as a reference method based on a Gaussian approximation. For the comparison, we propose an operational measure with the finding, that there is a significant difference between methods, but which method is preferable can depend on the details of the state preparation scenario.

Martin Plávala, Matthias Kleinmann*Generalized dynamical theories in phase space and the hydrogen atom*

Phys. Rev. A *108*,
052212
(2023),
arXiv:2212.12267

We show that the phase-space formulation of general probabilistic theories can be extended to include a generalized time evolution and that it can describe a nonquantum hydrogenlike system which is stable, has discrete energy levels, and includes the Zeeman effect. This allows us to study dynamical effects such as excitations of the hydrogenlike system by a resonant laser and Rutherford scattering. Our construction demonstrates that classical theory and quantum theory can be seen as specific choices of general probabilistic theory in phase space and that other probabilistic theories also lead to measurable predictions.

Michael Gaida, Matthias Kleinmann*Seven definitions of bipartite bound entanglement*

J. Phys. A: Math. Theor. *56*,
385302
(2023),
arXiv:2212.11015

An entangled state is bound entangled if one cannot combine any number of copies of the state to a maximally entangled state by using only local operations and classical communication. If one formalizes this notion of bound entanglement, one arrives immediately at four different definitions. In addition, at least three more definitions are commonly used in the literature, in particular in the very first paper on bound entanglement. Here we review critical distillation protocols and we examine how different results from quantum information theory interact in order to prove that all seven definitions are eventually equivalent. Our self-contained analysis unifies and extends previous results scattered in the literature and reveals details of the structure of bound entanglement.

Giuseppe Vitagliano, Matteo Fadel, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, Géza Tóth*Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles*

Quantum *7*,
914
(2023),
arXiv:2104.05663

We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.

Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan-Åke Larsson*Kochen-Specker contextuality*

Rev. Mod. Phys. *94*,
045007
(2022),
arXiv:2102.13036

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed.

Martin Plávala, Matthias Kleinmann*Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator*

Phys. Rev. Lett. *128*,
040405
(2022),
arXiv:2101.08323

We show how to construct general probabilistic theories that contain an energy observable dependent on position and momentum. The construction is in accordance with classical and quantum theory and allows for physical predictions, such as the probability distribution for position, momentum, and energy. We demonstrate the construction by formulating a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with a nonpositive Wigner function as well as a state that has tunneling properties. The toy model demonstrates that operational theories can be a viable alternative approach for formulating physical theories.

Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann*Minimal scheme for certifying three-outcome qubit measurements in the prepare-and-measure scenario*

Phys. Rev. A *104*,
062431
(2021),
arXiv:2105.09925

The number of outcomes is a defining property of a quantum measurement, in particular, if the measurement cannot be decomposed into simpler measurements with fewer outcomes. Importantly, the number of outcomes of a quantum measurement can be irreducibly higher than the dimension of the system. The certification of this property is possible in a semi-device-independent way either based on a Bell-like scenario or by utilizing the simpler prepare-and-measure scenario. Here we show that in the latter scenario the minimal scheme for certifying an irreducible three-outcome qubit measurement requires three state preparations and only two measurements and we provide experimentally feasible examples for this minimal certification scheme. We also discuss the dimension assumption characteristic of the semi-device-independent approach and to what extent it can be mitigated.

Zhen-Peng Xu, Xiao-Dong Yu, Matthias Kleinmann*State-independent quantum contextuality with projectors of nonunit rank*

New J. Phys. *23*,
043025
(2021),
arXiv:2011.04048

Virtually all of the analysis of quantum contextuality is restricted to the case where events are represented by rank-one projectors. This restriction is arbitrary and not motivated by physical considerations. We show here that loosening the rank constraint opens a new realm of quantum contextuality and we demonstrate that state-independent contextuality (SIC) can even require projectors of nonunit rank. This enables the possibility of SIC with less than 13 projectors, which is the established minimum for the case of rank one. We prove that for any rank, at least 9 projectors are required. Furthermore, in an exhaustive numerical search we find that 13 projectors are also minimal for the cases where all projectors are uniformly of rank two or uniformly of rank three.

Giulio Chiribella, Adán Cabello, Matthias Kleinmann, Markus P. Müller*General Bayesian theories and the emergence of the exclusivity principle*

Phys. Rev. Res. *2*,
042001(R)
(2020),
arXiv:1901.11412

We address the problem of reconstructing quantum theory from the perspective of an agent who makes bets about the outcomes of possible experiments. We build a general Bayesian framework that can be used to organize the agent's beliefs and update them when new information becomes available. Our framework includes as special cases classical and quantum probability theory, as well as other forms of probabilistic reasoning that may arise in future physical theories. Building on this framework, we develop a notion of an ideal experiment, which in quantum theory coincides with the notion of projective measurement. We then prove that, in every general Bayesian theory, ideal experiments must satisfy the exclusivity principle, a property of projective measurements that plays a central role in the characterization of quantum correlations. Our result suggests that the set of quantum correlations may be completely characterized in terms of Bayesian consistency conditions.

Jonathan Steinberg, H. Chau Nguyen, Matthias Kleinmann*Quaternionic quantum theory admits universal dynamics only for two-level systems*

J. Phys. A: Math. Theor. *53*,
375304
(2020),
arXiv:2001.05482

We revisit the formulation of quantum mechanics over the quaternions and investigate the dynamical structure within this framework. Similar to standard complex quantum mechanics, time evolution is then mediated by a unitary operator which can be written as the exponential of the generator of time shifts. By imposing physical assumptions on the correspondence between the energy observable and the generator of time shifts, we prove that quaternionic quantum theory admits a time evolution only for systems with a quaternionic dimension of at most two. Applying the same strategy to standard complex quantum theory, we reproduce that the correspondence dictated by the Schrödinger equation is the only possible choice, up to a shift of the global phase.

Fabian Pokorny, Chi Zhang, Gerard Higgins, Adán Cabello, Matthias Kleinmann, Markus Hennrich*Tracking the Dynamics of an Ideal Quantum Measurement*

Phys. Rev. Lett. *124*,
080401
(2020),
arXiv:1903.10398

The existence of ideal quantum measurements is one of the fundamental predictions of quantum mechanics. In theory, an ideal measurement projects a quantum state onto the eigenbasis of the measurement observable, while preserving coherences between eigenstates that have the same eigenvalue. The question arises whether there are processes in nature that correspond to such ideal quantum measurements and how such processes are dynamically implemented in nature. Here we address this question and present experimental results monitoring the dynamics of a naturally occurring measurement process: the coupling of a trapped ion qutrit to the photon environment. By taking tomographic snapshots during the detection process, we show that the process develops in agreement with the model of an ideal quantum measurement with an average fidelity of 94%.

Cornelia Spee, Hendrik Siebeneich, Timm Florian Gloger, Peter Kaufmann, Michael Johanning, Matthias Kleinmann, Christof Wunderlich, 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 ¹⁷¹Yb⁺ ion. Our test goes beyond the prepare-and-measure scheme of previous approaches, demonstrating the advantage of genuine temporal correlations.

Costantino Budroni, Gabriel Fagundes, Matthias Kleinmann*Memory cost of temporal correlations*

New J. Phys. *21*,
093018
(2019),
arXiv:1902.06517

A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the temporal correlations achievable with different theories, namely, classical, quantum, and GPTs beyond quantum mechanics. Already for the simplest nontrivial scenario, we derive inequalities able to distinguish temporal correlations where the underlying system is classical, quantum, or more general.

Fabian Bernards, Matthias Kleinmann, Otfried Gühne, 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 et al., npj Quant. Inf. 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.

Gael Sentís, Johannes N. Greiner, Jiangwei Shang, Jens Siewert, Matthias Kleinmann*Bound entangled states fit for robust experimental verification*

Quantum *2*,
113
(2018),
arXiv:1804.07562

Preparing and certifying bound entangled states in the laboratory is an intrinsically hard task, due to both the fact that they typically form narrow regions in state space, and that a certificate requires a tomographic reconstruction of the density matrix. Indeed, the previous experiments that have reported the preparation of a bound entangled state relied on such tomographic reconstruction techniques. However, the reliability of these results crucially depends on the extra assumption of an unbiased reconstruction. We propose an alternative method for certifying the bound entangled character of a quantum state that leads to a rigorous claim within a desired statistical significance, while bypassing a full reconstruction of the state. The method is comprised by a search for bound entangled states that are robust for experimental verification, and a hypothesis test tailored for the detection of bound entanglement that is naturally equipped with a measure of statistical significance. We apply our method to families of states of 3×3 and 4×4 systems, and find that the experimental certification of bound entangled states is well within reach.

Xiao-Min Hu, Bi-Heng Liu, Yu Guo, Guo-Yong Xiang, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Matthias Kleinmann, Tamás Vértesi, Adán Cabello*Observation of Stronger-than-Binary Correlations with Entangled Photonic Qutrits*

Phys. Rev. Lett. *120*,
180402
(2018),
arXiv:1712.06557

We present the first experimental confirmation of the quantum-mechanical prediction of stronger-than-binary correlations. These are correlations that cannot be explained under the assumption that the occurrence of a particular outcome of an n≥3-outcome measurement is due to a two-step process in which, in the first step, some classical mechanism precludes n−2 of the outcomes and, in the second step, a binary measurement generates the outcome. Our experiment uses pairs of photonic qutrits distributed between two laboratories, where randomly chosen three-outcome measurements are performed. We report a violation by 9.3 standard deviations of the optimal inequality for nonsignaling binary correlations.

Karsten Lange, Jan Peise, Bernd Lücke, Ilka Kruse, Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Géza Tóth, Carsten Klempt*Entanglement between two spatially separated atomic modes*

Science *360*,
416
(2018),
arXiv:1708.02480

Modern quantum technologies in the fields of quantum computing, quantum simulation, and quantum metrology require the creation and control of large ensembles of entangled particles. In ultracold ensembles of neutral atoms, nonclassical states have been generated with mutual entanglement among thousands of particles. The entanglement generation relies on the fundamental particle-exchange symmetry in ensembles of identical particles, which lacks the standard notion of entanglement between clearly definable subsystems. Here, we present the generation of entanglement between two spatially separated clouds by splitting an ensemble of ultracold identical particles prepared in a twin Fock state. Because the clouds can be addressed individually, our experiments open a path to exploit the available entangled states of indistinguishable particles for quantum information applications.

Gabriel Fagundes, Matthias Kleinmann*Memory cost for simulating all quantum correlations from the Peres–Mermin scenario*

J. Phys. A: Math. Theor. *50*,
325302
(2017),
arXiv:1611.07515

Sequences of compatible quantum measurements can be contextual and any simulation with a classical model accounting for the quantum observations, needs to use some internal memory. In Ref. [New J. Phys. 13, 113011 (2011)], it was shown that simulating the sequences of compatible observables from the Peres-Mermin scenario requires at least three different memory states in order to be not in contradiction with the deterministic predictions of quantum theory. We extend this analysis to the probabilistic quantum predictions and ask how much memory is required to simulate the quantum correlations generated by any quantum state. We find, that even in this comprehensive approach only three internal states are required for the perfect simulation of the quantum correlations in the Peres-Mermin scenario.

Otfried Gühne, Matthias Kleinmann, Tobias Moroder*Analysing Multiparticle Quantum States*

R. Bertlmann, A. Zeilinger (Eds.), Quantum [Un]speakables II ,
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.

Iagoba Apellaniz, Matthias Kleinmann, Otfried Gühne, Géza Tóth*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.

Matthias Kleinmann, Tamás Vértesi, Adán Cabello*Proposed experiment to test fundamentally binary theories*

Phys. Rev. A *96*,
032104
(2017),
arXiv:1611.05761

Fundamentally binary theories are nonsignaling theories in which measurements of many outcomes are constructed by selecting from binary measurements. They constitute a sensible alternative to quantum theory and have never been directly falsified by any experiment. Here we show that fundamentally binary theories are experimentally testable with current technology. For that, we identify a feasible Bell-type experiment on pairs of entangled qutrits. In addition, we prove that, for any n, quantum n-ary correlations are not fundamentally (n-1)-ary. For that, we introduce a family of inequalities that hold for fundamentally (n-1)-ary theories but are violated by quantum n-ary correlations.

Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, Géza Tóth*Entanglement and extreme spin squeezing of unpolarized states*

New J. Phys. *19*,
013027
(2017),
arXiv:1605.07202

We present an optimal set of criteria detecting the depth of entanglement in macroscopic systems of general spin-j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems composed of a very large number of particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spin-squeezing parameters for Dicke states. As a by-product, we obtain also an analytic lower bound to the condition derived in [A.S. Sorensen and K. Molmer, Phys. Rev. Lett. 86, 4431 (2001)]. We also extend the results to systems with fluctuating number of particles.

Matthias Kleinmann, Adán Cabello*Quantum Correlations Are Stronger Than All Nonsignaling Correlations Produced by n-Outcome Measurements*

Phys. Rev. Lett. *117*,
150401
(2016),
arXiv:1505.04179

We show that, for any n, there are m-outcome quantum correlations, with m>n, which are stronger than any nonsignaling correlation produced from selecting among n-outcome measurements. As a consequence, for any n, there are m-outcome quantum measurements that cannot be constructed by selecting locally from the set of n-outcome measurements. This is a property of the set of measurements in quantum theory that is not mandatory for general probabilistic theories. We also show that this prediction can be tested through high-precision Bell-type experiments and identify past experiments providing evidence that some of these strong correlations exist in nature. Finally, we provide a modified version of quantum theory restricted to having at most n-outcome quantum measurements.

Adán Cabello, Matthias Kleinmann, José R. Portillo*Quantum state-independent contextuality requires 13 rays*

J. Phys. A: Math. Theor. *49*,
38LT01
(2016),
arXiv:1606.01848

We show that, regardless of the dimension of the Hilbert space, there exists no set of rays revealing state-independent contextuality with less than 13 rays. This implies that the set proposed by Yu and Oh in dimension three [Phys. Rev. Lett. 108, 030402 (2012)] is actually the minimal set in quantum theory. This contrasts with the case of Kochen-Specker sets, where the smallest set occurs in dimension four.

Esteban S. Gómez, Santiago Gómez, Pablo González, Gustavo Cañas, Johanna F. Barra, Aldo Delgado, Guilherme B. Xavier, Adán Cabello, Matthias Kleinmann, Tamás Vértesi, Gustavo Lima*Device-Independent Certification of a Nonprojective Qubit Measurement*

Phys. Rev. Lett. *117*,
260401
(2016),
arXiv:1604.01417

Quantum measurements on a two-level system can have more than two independent outcomes, and in this case, the measurement cannot be projective. Measurements of this general type are essential to an operational approach to quantum theory, but so far, the nonprojective character of a measurement can only be verified experimentally by already assuming a specific quantum model of parts of the experimental setup. Here, we overcome this restriction by using a device-independent approach. In an experiment on pairs of polarization-entangled photonic qubits we violate by more than 8 standard deviations a Bell-like correlation inequality that is valid for all sets of two-outcome measurements in any dimension. We combine this with a device-independent verification that the system is best described by two qubits, which therefore constitutes the first device-independent certification of a nonprojective quantum measurement.

Adán Cabello, Matthias Kleinmann, Costantino Budroni*Necessary and Sufficient Condition for Quantum State-Independent Contextuality*

Phys. Rev. Lett. *114*,
250402
(2015),
arXiv:1501.03432

We solve the problem of whether a set of quantum tests reveals state-independent contextuality and use this result to identify the simplest set of the minimal dimension. We also show that identifying state-independent contextuality graphs [R. Ramanathan and P. Horodecki, Phys. Rev. Lett. 112, 040404 (2014)] is not sufficient for revealing state-independent contextuality.

Christian Schwemmer, Lukas Knips, Daniel Richart, Tobias Moroder, Matthias Kleinmann, Otfried Gühne, Harald Weinfurter*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 for state reconstruction, we observe a systematic underestimation of the fidelity and an overestimation of entanglement. A solution for this problem can be achieved by a linear evaluation of the data yielding reliable and computational simple bounds including error bars.

Matthias Kleinmann*Sequences of projective measurements in generalized probabilistic models*

J. Phys. A: Math. Theor. *47*,
455304
(2014),
arXiv:1402.3583

We define a simple rule that allows to describe sequences of projective measurements for a broad class of generalized probabilistic models. This class embraces quantum mechanics and classical probability theory, but, for example, also the hypothetical Popescu-Rohrlich box. For quantum mechanics, the definition yields the established Lüders's rule, which is the standard rule how to update the quantum state after a measurement. In the general case it can be seen as the least disturbing or most coherent way to perform sequential measurements. As example we show that Spekkens's toy model is an instance of our definition. We also demonstrate the possibility of strong post-quantum correlations as well as the existence of triple-slit correlations for certain non-quantum toy models.

Otfried Gühne, Costantino Budroni, Adán Cabello, Matthias Kleinmann, Jan-Åke 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.

Costantino Budroni, Tobias Moroder, Matthias Kleinmann, Otfried 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.

Jochen Szangolies, Matthias Kleinmann, Otfried 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 is not well defined. 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.

Sönke Niekamp, Tobias Galla, Matthias Kleinmann, Otfried 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.

Matthias Kleinmann, Tobias J. Osborne, Volkher B. Scholz, Albert H. Werner*Typical Local Measurements in Generalized Probabilistic Theories: Emergence of Quantum Bipartite Correlations*

Phys. Rev. Lett. *110*,
040403
(2013),
arXiv:1205.3358

What singles out quantum mechanics as the fundamental theory of Nature? Here we study local measurements in generalised probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalisation of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.

Tobias Moroder, Matthias Kleinmann, Philipp Schindler, Thomas Monz, Otfried Gühne, Rainer 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.

Elias Amselem, Mohamed Bourennane, Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan-Åke Larsson, Marcin 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.

Otfried Gühne, Matthias Kleinmann*Auf den Kontext kommt es an*

Physik Journal *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.

Sönke Niekamp, Matthias Kleinmann, Otfried Gühne*Entropic uncertainty relations and the stabilizer formalism*

J. Math. Phys. *53*,
012202
(2012),
arXiv:1103.2316

Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen-Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.

Matthias Kleinmann, Costantino Budroni, Jan-Åke Larsson, Otfried Gühne, Adán 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.

Jan-Åke Larsson, Matthias Kleinmann, Constantino Budroni, Otfried Gühne, Adán Cabello*Maximal violation of state-independent contextuality inequalities*

AIP Conference Proceedings *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.

Matthias Kleinmann, Hermann Kampermann, Dagmar Bruß*Asymptotically perfect discrimination in the local-operation-and-classical-communication paradigm*

Phys. Rev. A *84*,
042326
(2011),
arXiv:1105.5132

We revisit the problem of discriminating orthogonal quantum states within the local quantum operation and classical communication (LOCC) paradigm. Our particular focus is on the asymptotic situation where the parties have infinite resources and the protocol may become arbitrarily long. Our main result is a necessary condition for perfect asymptotic LOCC discrimination. As an application, we prove that for complete product bases, unlimited resources are of no advantage. On the other hand, we identify an example, for which it still remains undecided whether unlimited resources are superior.

Matthias Kleinmann, Otfried Gühne, José R. Portillo, Jan-Åke Larsson, Adán 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 in order to characterize the difference between quantum and classical physics. For a simulation of the phenomenon of state-independent quantum contextuality, we show that the minimal 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.

Jan-Åke Larsson, Matthias Kleinmann, Otfried Gühne, Adán Cabello*Violating noncontextual realism through sequential measurements*

AIP Conference Proceedings *1327*,
401
(2011)

The question of whether noncontextual hidden variable models can give the quantum‐mechanical predictions has been under discussion for a long time. The question originates in the sixties, perhaps best known from the 1967 paper by Kochen and Specker where it was shown that no noncontextual hidden‐variable model can give these predictions. Recently, the question has gained interest from experimentalists trying to test this in the laboratory. The experimental setups in question have used a sequential setup rather than the alternative, joint (simultaneous) measurement. There has been discussion in the community whether this is appropriate. This brief paper argues that sequential measurement is not only the correct choice, but the best possible.

Matthias Kleinmann, Hermann Kampermann, Dagmar Bruß*Structural approach to unambiguous discrimination of two mixed quantum states*

J. Math. Phys. *51*,
032201
(2010),
arXiv:0803.1083

We analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 ("solution in 4 dimensions"). The solution is illustrated by some examples. The optimality conditions proved by Eldar et al. [Phys. Rev. A 69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of optimal unambiguous state discrimination is summarized via a general strategy.

Matthias Kleinmann, Hermann Kampermann, Dagmar Bruß*Unambiguous discrimination of mixed quantum states: Optimal solution and case study*

Phys. Rev. A *81*,
020304(R)
(2010),
arXiv:0807.3923

We present a generic study of unambiguous discrimination between two mixed quantum states. We derive operational optimality conditions and show that the optimal measurements can be classified according to their rank. In Hilbert space dimensions smaller or equal to five this leads to the complete optimal solution. We demonstrate our method with a physical example, namely the unambiguous comparison of n quantum states, and find the optimal success probability.

Sönke Niekamp, Matthias Kleinmann, Otfried Gühne*Discrimination strategies for inequivalent classes of multipartite entangled states*

Phys. Rev. A *82*,
022322
(2010),
arXiv:1006.1313

How can one discriminate different inequivalent classes of multiparticle entanglement experimentally? We present an approach for the discrimination of an experimentally prepared state from the equivalence class of another state. We consider two possible measures for the discrimination strength of an observable. The first measure is based on the difference of expectation values, the second on the relative entropy of the probability distributions of the measurement outcomes. The interpretation of these measures and their usefulness for experiments with limited resources are discussed. In the case of graph states, the stabilizer formalism is employed to compute these quantities and to find sets of observables that result in the most decisive discrimination.

Otfried Gühne, Matthias Kleinmann, Adán Cabello, Jan-Åke Larsson, Gerhard Kirchmair, Florian Zähringer, Rene Gerritsma, Christian F. Roos*Compatibility and noncontextuality for sequential measurements*

Phys. Rev. A *81*,
022121
(2010),
arXiv:0912.4846

A basic assumption behind the inequalities used for testing noncontextual hidden variable models is that the observables measured on the same individual system are perfectly compatible. However, compatibility is not perfect in actual experiments using sequential measurements. We discuss the resulting "compatibility loophole" and present several methods to rule out certain hidden variable models which obey a kind of extended noncontextuality. Finally, we present a detailed analysis of experimental imperfections in a recent trapped ion experiment and apply our analysis to that case.

Bastian Jungnitsch, Sönke Niekamp, Matthias Kleinmann, Otfried Gühne, He Lu, Wei-Bo Gao, Yu-Ao Chen, Zeng-Bing Chen, Jian-Wei Pan*Increasing the Statistical Significance of Entanglement Detection in Experiments*

Phys. Rev. Lett. *104*,
210401
(2010),
arXiv:0912.0645

Entanglement is often verified by a violation of an inequality like a Bell inequality or an entanglement witness. Considerable effort has been devoted to the optimization of such inequalities in order to obtain a high violation. We demonstrate theoretically and experimentally that such an optimization does not necessarily lead to a better entanglement test, if the statistical error is taken into account. Theoretically, we show for different error models that reducing the violation of an inequality can improve the significance. Experimentally, we observe this phenomenon in a four-photon experiment, testing the Mermin and Ardehali inequality for different levels of noise. Furthermore, we provide a way to develop entanglement tests with high statistical significance.

Robert Hübener, Matthias Kleinmann, Tzu-Chieh Wei, Carlos González-Guillén, Otfried Gühne*Geometric measure of entanglement for symmetric states*

Phys. Rev. A *80*,
032324
(2009),
arXiv:0905.4822

Is the closest product state to a symmetric entangled multiparticle state also symmetric? This question has appeared in the recent literature concerning the geometric measure of entanglement. First, we show that a positive answer can be derived from results concerning symmetric multilinear forms and homogeneous polynomials, implying that the closest product state can be chosen to be symmetric. We then prove the stronger result that the closest product state to any symmetric multiparticle quantum state is necessarily symmetric. Moreover, we discuss generalizations of our result and the case of translationally invariant states, which can occur in spin models.

Gerhard Kirchmair, Florian Zähringer, Rene Gerritsma, Matthias Kleinmann, Otfried Gühne, Adán Cabello, Rainer Blatt, Christian F. Roos*State-independent experimental test of quantum contextuality*

Nature *460*,
494
(2009),
arXiv:0904.1655

The question of whether quantum phenomena can be explained by classical models with hidden variables is the subject of a long lasting debate. In 1964, Bell showed that certain types of classical models cannot explain the quantum mechanical predictions for specific states of distant particles. Along this line, some types of hidden variable models have been experimentally ruled out. An intuitive feature for classical models is non-contextuality: the property that any measurement has a value which is independent of other compatible measurements being carried out at the same time. However, the results of Kochen, Specker, and Bell show that non-contextuality is in conflict with quantum mechanics. The conflict resides in the structure of the theory and is independent of the properties of special states. It has been debated whether the Kochen-Specker theorem could be experimentally tested at all. Only recently, first tests of quantum contextuality have been proposed and undertaken with photons and neutrons. Yet these tests required the generation of special quantum states and left various loopholes open. Here, using trapped ions, we experimentally demonstrate a state-independent conflict with non-contextuality. The experiment is not subject to the detection loophole and we show that, despite imperfections and possible measurement disturbances, our results cannot be explained in non-contextual terms.

Patrick Skwara, Hermann Kampermann, Matthias Kleinmann, Dagmar Bruß*Entanglement witnesses and a loophole problem*

Phys. Rev. A *76*,
012312
(2007),
arXiv:0608058

We consider a possible detector-efficiency loophole in experiments that detect entanglement via the local measurement of witness operators. Here, only local properties of the detectors are known. We derive a general threshold for the detector efficiencies which guarantees that a negative expectation value of a witness is due to entanglement, rather than to erroneous detectors. This threshold depends on the local decomposition of the witness and its measured expectation value. For two-qubit witnesses we find the local operator decomposition that is optimal with respect to closing the loophole.

Matthias Kleinmann, Hermann Kampermann, Philippe Raynal, Dagmar Bruß*Commutator relations reveal solvable structures in unambiguous state discrimination*

J. Phys. A: Math. Theor. *40*,
F871
(2007),
arXiv:0705.3391

We present a criterion, based on three commutator relations, that allows to decide whether two self-adjoint matrices with non-overlapping support are simultaneously unitarily similar to quasidiagonal matrices, i.e., whether they can be simultaneously brought into a diagonal structure with 2x2-dimensional blocks. Application of this criterion to unambiguous state discrimination provides a systematic test whether the given problem is reducible to a solvable structure. As an example, we discuss unambiguous state comparison.

Matthias Kleinmann, Hermann Kampermann, Tim Meyer, Dagmar Bruß*Purifying and reversible physical processes*

Appl. Phys. B *86*,
371
(2007),
arXiv:0608053

Starting from the observation that reversible processes cannot increase the purity of any input state, we study deterministic physical processes, which map a set of states to a set of pure states. Such a process must map any state to the same pure output, if purity is demanded for the input set of all states. But otherwise, when the input set is restricted, it is possible to find non-trivial purifying processes. For the most restricted case of only two input states, we completely characterize the output of any such map. We furthermore consider maps, which combine the property of purity and reversibility on a set of states, and we derive necessary and sufficient conditions on sets, which permit such processes.

Matthias Kleinmann, Hermann Kampermann, Tim Meyer, Dagmar Bruß*Physical purification of quantum states*

Phys. Rev. A *73*,
062309
(2006),
arXiv:0509100

We introduce the concept of a physical process that purifies a mixed quantum state, taken from a set of states, and investigate the conditions under which such a purification map exists. Here, a purification of a mixed quantum state is a pure state in a higher-dimensional Hilbert space, the reduced density matrix of which is identical to the original state. We characterize all sets of mixed quantum states, for which perfect purification is possible. Surprisingly, some sets of two non-commuting states are among them. Furthermore, we investigate the possibility of performing an imperfect purification.

Tim Meyer, Hermann Kampermann, Matthias Kleinmann, Dagmar Bruß*Finite key analysis for symmetric attacks in quantum key distribution*

Phys. Rev. A *74*,
042340
(2006),
arXiv:0607141

We introduce a constructive method to calculate the achievable secret key rate for a generic class of quantum key distribution protocols, when only a finite number n of signals is given. Our approach is applicable to all scenarios in which the quantum state shared by Alice and Bob is known. In particular, we consider the six state protocol with symmetric eavesdropping attacks, and show that for a small number of signals, i.e. below the order of 10^4, the finite key rate differs significantly from the asymptotic value for n approaching infinity. However, for larger n, a good approximation of the asymptotic value is found. We also study secret key rates for protocols using higher-dimensional quantum systems.

Matthias Kleinmann, Hermann Kampermann, Dagmar Bruß*Generalization of quantum-state comparison*

Phys. Rev. A *72*,
032308
(2005),
arXiv:0503012

We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of N pure states, and find an optimal solution for some parameter range when N=3. In both cases we use the reduction method for the corresponding problem of mixed state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for N=2. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.