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Matthias Kleinmann

Matthias Kleinmann Lecturer

Room: B-110

Phone: +49 271 740 3799

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


See also arxiv

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

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, highly entangled states containing thousands of particles have been generated, outnumbering any other physical system by orders of magnitude. 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. Since 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 and Matthias Kleinmann
Memory cost for simulating all quantum correlations of 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 and 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 and Geza 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.

Matthias Kleinmann, Tamás Vértesi and 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 and Geza Toth
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 and Adan 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 and 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 and 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.

Adan Cabello, Matthias Kleinmann and 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 and 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 L\"uders'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, Adan Cabello, Matthias Kleinmann and 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 and 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 and 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 and 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 and Albert H. Werner
Typical local measurements in generalised 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 and Rainer Blatt
Certifying experimental 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.

E. Amselem, M. Bourennane, C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, J. -Å. Larsson and 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.

Otfried Gühne and Matthias Kleinmann
Auf den Kontext kommt es an
Physik Journal 02, 25 (2013)

Was hat die Frage „Können wir alles wissen?“ mit der Quantenmechanik zu tun?

Sönke Niekamp, Matthias Kleinmann and 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-Ake Larsson, Otfried Gühne and Adan 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.

M. Kleinmann, H. Kampermann and D. Bruß
Asymptotically perfect discrimination in the LOCC 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 and 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 and Adán Cabello
Violating noncontextual realism through sequential measurements
AIP Conference Proceedings 1327, 1, 401 (2011)

M. Kleinmann, H. Kampermann and D. 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.

M. Kleinmann, H. Kampermann and D. Bruss
Unambiguous state discrimination: 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 and 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.

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

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 and 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.

R. Hübener, M. Kleinmann, T. -C. Wei, C. González-Guillén and O. Gühne
The 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.

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

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 and 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.

M. Kleinmann, H. Kampermann, Ph. Raynal and D. Bruss
Commutator Relations Reveal Solvable Structures in Unambiguous State Discrimination
J. Phys. A: Math. Theor. 40, F871-F878 (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 and 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.

M. Kleinmann, H. Kampermann, T. Meyer and D. 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 and 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.

M. Kleinmann, H. Kampermann and D. Bruss
On the 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.