## Abstracts

Koenraad Audenaert:Quantum Multi-Hypothesis Testing

In this talk I will consider one of the most fundamental problems in quantum information theory, namely to find the optimal quantum measurement that distinguishes between any of the quantum states in a given set of $K$ states. For $K=2$, this problem can be solved analytically, whereas for larger $K$ the problem has to be solved numerically and the only analytical results are bounds on the error probability. In my talk I will consider a variety of such bounds and I will focus on those bounds that are off by at most a constant (independent of the dimension of the Hilbert space in which these states live). The goal is to use these bounds to prove that the asymptotic error rate for $K>2$ is given by the smallest pairwise quantum Chernoff information, thereby extending previous results for the $K=2$ case, and I will present recent progress on this problem by myself and others.

Cristina Butucea: Rank-penalized estimation for the tomography of a quantum system

We consider estimation of a joint quantum state of n qubits. We suppose tomography has been performed on each qubit independently, on different copies of the same state. We propose a least squares-type of estimator penalized by the rank of the resulting matrix. The final estimator is transformed into a proper quantum state. A study on synthetical data will show how the method performs before we apply it to real data.

Daniel Burgarth: Quantum System Identification with limited resouces

The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. Prior knowledge on some elements of the black box helps the system identification. We present an example in which a Bell measurement is more efficient to identify the system. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.

Matthias Christandl: Reliable Quantum State Tomography

Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state, and therefore have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing the confidence regions is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science. This is joint work with Renato Renner.

Marcus Cramer: Scalable reconstruction of pure and mixed quantum many-body states

I will show how one can do exponentially better than direct state tomography for a wide range of quantum states, in particular those that are well approximated by matrix product states or operators (e.g., states like the W state, the GHZ state, cluster states, and ground and thermal states of local Hamiltonians). The scheme requires only a linear number of local measurements and classical post-processing that is polynomial in the system size. If the state in the laboratory is close to a pure state, the accuracy of the reconstructed state can be rigorously certified without any assumptions.

Jens Eisert: Learning much from little

Richard Gill: Sparsity and state and detector tomography

Teiko Heinosaari: Minimal informationally complete observables and sequential measurements

I will demonstrate that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, and that it is possible to implement a joint observable as a sequential measurement. This gives a very natural way to implement a minimal informationally complete observable. An interesting fact is related to the joint observable with minimal noise in marginals, in which case the joint observable is unique. If d is odd, then this observable is informationally complete. But if d is even, then the joint observable is not informationally complete and one has to allow more noise in order to obtain informational completeness.

In the scenario without any prior knowledge, the minimal number of measurement outcomes (POVM elements) is d^2. A prior knowledge on input states typically decreases the minimal number of the required measurement outcomes, and I will discuss some methods how this minimal number can be deduced.

Theodore Kypraios: Rank based model selection for quantum state tomography

Thomas Monz: Experimental state and process reconstruction

Daniel Oi: Maximum Likelihood and Bayesian Signal Analysis for Quantum System Identification

Determining the dynamics and response of a quantum system is vital for engineering quantum information processing devices. With limited initial resources for state preparation, coherent control and measurement, it is challenging to extract system Hamiltonian or open system dynamics. The problem of inverting signals to obtain these parameters can be greatly simplified with judicious model selection. This demonstrates the importance of incorporating as much as possible prior knowledge or the underlying physics of the device. Here, we show how Baysian signal analysis and Maximum likelihood techniques can be used in such situations.

Phillipp Schindler: Model testing of tomographic data

Kristan Temme: Quantum Chi-Squared and Goodness of Fit Testing

The density matrix in quantum mechanics parametrizes the statistical properties of the system under observation, just like a classical probability distribution does for classical systems. The expectation value of observables cannot be measured directly, it can only be approximated by applying classical statistical methods to the frequencies by which certain measurement outcomes (clicks) are obtained. In this paper, we make a detailed study of the statistical fluctuations obtained during an experiment in which a hypothesis is tested, i.e. the hypothesis that a certain setup produces a given quantum state. Although the classical and quantum problem are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. Just as in the case of classical hypothesis testing, the confidence in quantum

hypothesis testing scales exponentially in the number of copies. In this paper, we will argue 1) that the physically relevant data of quantum experiments is only contained in the frequencies of the measurement outcomes, and that the statistical fluctuations of the experiment are essential, so that the correct formulation of the conclusions of a quantum experiment should be given in terms of hypothesis tests, 2) that the (classical) $\chi^2$ test for distinguishing two quantum states gives rise to the quantum $\chi^2$

divergence when optimized over the measurement basis, 3) present a max-min characterization for the optimal measurement basis for quantum goodness of fit testing, find the quantum measurement which leads both to the maximal divergence rate and determine the associated divergence rates.