Efficient methods for quantum state analysis & entanglement detection
Entanglement is a fascinating feature lying at the very heart of quantum physics. For applications and realisations of quantum information protocols, it is often required to be able to either characterise the experimentally prepared quantum state or at least provide evidence of entanglement. Quantum state tomography, the method used for obtaining full information about the quantum state, is hard to perform for larger systems. The amount of necessary measurements – scaling exponentially with the number of qubits – as well as the inherent statistical uncertainties of each single measurement lead to deviations. Prepared quantum states almost unavoidably  appear like being not positive semidefinite, a crucial property of a density matrix. In order to obtain a positive semidefinite (i.e., a physical) density matrix from the measurements, a common way is to use constrained optimisation, e.g., maximum likelihood or least squares techniques. We could show that by introducing the constraint of a physical result leads to a systematic deviation of the obtained state compared to the initial, but unknown state resulting in a bias of estimated figures of merit .
In order to assess the impact of statistical deviations to the obtained state estimate, we could derive the joint probability density function of eigenvalues in quantum state tomography . With that knowledge, one can not only survey the sample size used for state tomography. This furthermore allows to review the distribution of obtained eigenvalues in terms of systematic deviations by, for example, easily applicable tools like hypothesis testing methods .
While quantum state tomography is a helpful tool when certifying the quality of the measurement apparatus, it might not be necessary for some quantum information protocols. Depending on the scenario, other methods for entanglement detection might be sufficient and thus advantageous. In case one does not have a priori information about the quantum state, we provide a so called decision tree that proposed a sequence of subsequent measurements depending on previous measurement outcomes . With this, we could achieve a large speedup compared to a blind choice of measurements. Nonetheless, prior information about the state is highly valuable. Based on initial information, one can construct entanglement witnesses that allow not only the exclusion of full separability, but can detect genuine multipartite entanglement. We could derive and experimentally test a method based on commutation and anticommutation relations of operators that can prove genuine multipartite entanglement after solely two measurements , which in a similar way can be derived for qudits like qutrits.
At the moment, we are working on extending the toolbox of entanglement detection and state characterisation for random measurements. In this case, the different parties neither need to agree on a common reference frame nor have to be aware of their own measurement. Still, the measurement outcomes can be used to rule out separability and to identify the prepared state up to a restricted set of symmetry operations.
 W. Laskowski, C. Schwemmer, D. Richart, L. Knips, T. Paterek, H. Weinfurter, “Optimized state independent entanglement detection based on geometrical threshold criterion”, Phys. Rev. A 88, 022327 (2013).
 C. Schwemmer, L. Knips, D. Richart, T. Moroder, M. Kleinmann, O. Gühne, H. Weinfurter, “Systematic errors in current quantum state tomography tools”, Phys. Rev. Lett. 114, 080403 (2015).
 C. Schwemmer, L. Knips, M. C. Tran, A. d. Rosier, W. Laskowski, T. Paterek, H. Weinfurter, “Genuine Multipartite Entanglement without Multipartite Correlations”, Phys. Rev. Lett. 114, 180501 (2015).
 L. Knips, C. Schwemmer, N. Klein, J. Reuter, G. Tóth, H. Weinfurter, “How long does it take to obtain a physical density matrix?”, arXiv:1512.06866.
 K. Rosołek, M. Wieśniak, L. Knips, “Quadratic Entanglement Criteria for Qutrits”, arXiv:1512.07099.
 L. Knips, C. Schwemmer, N. Klein, M. Wieśniak, H. Weinfurter, “Multipartite entanglement detection with minimal effort”, Phys. Rev. Lett. 117, 210504 (2016).
 L. Vaidman, A. Ben-Israel, J. Dziewior, L. Knips, M. Weißl, J. Meinecke, C. Schwemmer, R. Ber, and H. Weinfurter, “Weak value beyond conditional expectation value of the pointer readings”, arXiv:1611.03989.
 A. Ben-Israel, L. Knips, J. Dziewior, J. Meinecke, A. Danan, H. Weinfurter, and L. Vaidman, “An Improved Experiment to Determine the `Past of a Particle‘ in the Nested Mach-Zehnder Interferometer“, Chin. Phys. Lett. 34, 020301 (2017).
 M. C. Tran, M. Zuppardo, A. d. Rosier, L. Knips, W. Laskowski, T. Paterek, and H. Weinfurter, “Genuine N-partite entanglement without N-partite correlation functions”, Phys. Rev. A 95, 062331 (2017).
September, Ringberg meeting, Ringberg castle, Germany;
September, MQC meeting (Munich)
February, ExQM seminar (Munich);
February, PhD meeting (Munich);
March, DPG spring meeting in Heidelberg, Germany;
April, MQC meeting (Munich);
May, KCIK conference in Gdansk, Poland;
May, Seminar meeting (Munich);
June, CLEO conference in Munich, Germany;
August, CQIQC conference in Toronto, Canada;
November, Bilbao (invited talk), Spain
February, DPG spring meeting in Hannover, Germany;
March, QLIPS conference in Singapore and Indonesia;
May, KCIK conference in Gdansk, Poland;
October, MPQ review meeting (Munich);
October, MQC meeting (Munich)
January, PQE Conference in Snowbird, USA;
March, PhD meeting (Munich);
March, DPG spring meeting in Mainz, Germany,
May, MQC meeting (Munich);
June, Seminar meeting (Munich);
June, ExQM seminar (Munich);
June, CLEO conference in Munich, Germany.
Wieslaw Laskowski, Marcin Wiesniak – University of Gdansk, Poland
Géza Tóth, Matthias Kleinmann – University of the Basque Country, Spain
Tomasz Paterek – Nanyang Technical University Singapore
Lev Vaidman – Tel Aviv University, Israel
Otfried Gühne – University of Siegen, Germany
Max-Planck-Institute of Quantum Optics
85748 Garching, Germany
e-mail: l u k a s . k n i p s @ m p q . m p g . d e