## Biography

I am an Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) fellow and Associate Professor with the Centre for Quantum Software and Information at the School of Computer Science.

I received a Master of Science degree from the Department of Electrical Engineering and Information Technology at ETH Zurich, and a doctorate in Theoretical Physics also from ETH Zurich. Before joining UTS, I have worked at the University of Sydney as a University of Sydney Postdoctoral Fellow and as a Research Fellow and Senior Research Fellow at the Centre for Quantum Technologies at the National University of Singapore.

## Professional

I am a senior member of the IEEE Information Theory Society.

### Links

#### Can supervise: YES

#### Research Interests

My research interests lie in the intersection of information theory, cryptography and quantum mechanics. My main focus is on the mathematical foundations of quantum information theory, for example the study of entropy and other information measures, as well as theoretical questions that arise in quantum communication and cryptography when the available resources are limited.

#### Publications

Tomamichel, M 2016, *Quantum Information Processing with Finite Resources - Mathematical Foundations*, Springer, Cham, Switzerland.View/Download from: Publisher's site

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One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answer such questions. A strong emphasis is given to information measures that are essential for the study of devices of finite size, including R\'enyi entropies and smooth entropies. The presentation is self-contained and includes rigorous and concise proofs of the most important properties of these measures. The first chapters will introduce the formalism of quantum mechanics, with particular emphasis on norms and metrics for quantum states. This is necessary to explore quantum generalizations of R\'enyi divergence and conditional entropy, information measures that lie at the core of information theory. The smooth entropy framework is discussed next and provides a natural means to lift many arguments from information theory to the quantum setting. Finally selected applications of the theory to statistics and cryptography are discussed.

Fang, K, Wang, X, Tomamichel, M & Berta, M 2020, 'Quantum Channel Simulation and the Channel's Smooth Max-Information', *IEEE Transactions on Information Theory*, vol. 66, no. 4, pp. 2129-2140.View/Download from: Publisher's site

McKinlay, A & Tomamichel, M 2020, 'Decomposition rules for quantum Rényi mutual information with an application to information exclusion relations', *Journal of Mathematical Physics*, vol. 61, no. 7.View/Download from: Publisher's site

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© 2020 Author(s). We prove decomposition rules for quantum Rényi mutual information, generalizing the relation I(A: B) = H(A) - H(A|B) to inequalities between Rényi mutual information and Rényi entropy of different orders. The proof uses Beigi's generalization of Reisz-Thorin interpolation to operator norms [J. Math. Phys. 54(12), 122202 (2013)] and a variation of the argument developed by Dupuis [J. Math. Phys. 56, 022203 (2015)], which was used to show chain rules for conditional Rényi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for Rényi mutual information, generalizing the original relation by Hall [Phys. Rev. Lett. 74, 3307 (1995)].

Youssry, A, Chapman, RJ, Peruzzo, A, Ferrie, C & Tomamichel, M 2020, 'Modeling and control of a reconfigurable photonic circuit using deep learning', *Quantum Science and Technology*, vol. 5, no. 2, pp. 025001-025001.View/Download from: Publisher's site

Cheng, HC, Hsieh, MH & Tomamichel, M 2019, 'Quantum Sphere-Packing Bounds with Polynomial Prefactors', *IEEE Transactions on Information Theory*, vol. 65, no. 5, pp. 2872-2898.View/Download from: Publisher's site

#### View description

© 1963-2012 IEEE. We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a finite blocklength sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of $o(\log n / n)$ , indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

Anshu, A, Berta, M, Jain, R & Tomamichel, M 2019, 'A minimax approach to one-shot entropy inequalities', *Journal of Mathematical Physics*, vol. 60, no. 12, pp. 122201-122201.View/Download from: Publisher's site

Buscemi, F, Sutter, D & Tomamichel, M 2019, 'An information-theoretic treatment of quantum dichotomies', *Quantum*, vol. 3.View/Download from: Publisher's site

#### View description

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

Chubb, CT, Tomamichel, M & Korzekwa, K 2019, 'Moderate deviation analysis of majorization-based resource interconversion', *Physical Review A*, vol. 99, no. 3.View/Download from: Publisher's site

#### View description

© 2019 American Physical Society. We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorization relation, e.g., the resource theories of entanglement and coherence (for pure-state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a nontrivial trade-off between the rate rn at which n copies of a resource state ρ can be transformed into nrn copies of another resource state σ, and the error level ϵn of the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rn approaches its optimum in the asymptotic limit of unbounded resources (n→∞), while the error ϵn vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behavior which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.

Fang, K, Wang, X, Tomamichel, M & Duan, R 2019, 'Non-Asymptotic Entanglement Distillation', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 65, no. 10, pp. 6454-6465.View/Download from: Publisher's site

Korzekwa, K, Chubb, CT & Tomamichel, M 2019, 'Avoiding Irreversibility: Engineering Resonant Conversions of Quantum Resources', *PHYSICAL REVIEW LETTERS*, vol. 122, no. 11.View/Download from: Publisher's site

Mohamed, A, Ferrie, C & Tomamichel, M 2019, 'Efficient online quantum state estimation using a matrix-exponentiated gradient method', *New Journal of Physics*, vol. 21.View/Download from: Publisher's site

#### View description

In this paper, we explore an efficient online algorithm for quantum state

estimation based on a matrix-exponentiated gradient method previously used in

the context of machine learning. The state update is governed by a learning

rate that determines how much weight is given to the new measurement results

obtained in each step. We show convergence of the running state estimate in

probability to the true state for both noiseless and noisy measurements. We

find that in the latter case the learning rate has to be chosen adaptively and

decreasing to guarantee convergence beyond the noise threshold. As a practical

alternative we then propose to use running averages of the measurement

statistics and a constant learning rate to overcome the noise problem. The

proposed algorithm is numerically compared with batch maximum-likelihood and

least-squares estimators. The results show a superior performance of the new

algorithm in terms of accuracy and runtime complexity.

Quadeer, M, Tomamichel, M & Ferrie, C 2019, 'Minimax quantum state estimation under Bregman divergence', *Quantum*, vol. 3.View/Download from: Publisher's site

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We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.

Taranto, P, Pollock, FA, Milz, S, Tomamichel, M & Modi, K 2019, 'Quantum Markov Order', *Physical Review Letters*, vol. 122, no. 14.View/Download from: Publisher's site

#### View description

We formally extend the notion of Markov order to open quantum processes by

accounting for the instruments used to probe the system of interest at

different times. Our description recovers the classical Markov order property

in the appropriate limit: when the stochastic process is classical and the

instruments are non-invasive, i.e., restricted to orthogonal projective

measurements. We then prove that there do not exist non-Markovian quantum

processes that have finite Markov order with respect to all possible

instruments; the same process exhibits distinct memory effects with respect to

different probing instruments. This naturally leads to a relaxed definition of

quantum Markov order with respect to specified sequences of instruments. The

memory effects captured by different choices of instruments vary dramatically,

providing a rich landscape for future exploration.

Wang, X, Fang, K & Tomamichel, M 2019, 'On converse bounds for classical communication over quantum channels', *IEEE Transactions on Information Theory*, vol. 65, no. 7, pp. 4609-4619.View/Download from: Publisher's site

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© 1963-2012 IEEE. We explore several new converse bounds for classical communication over quantum channels in both the one-shot and asymptotic regimes. First, we show that the Matthews-Wehner meta-converse bound for entanglement-Assisted classical communication can be achieved by activated, no-signaling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new efficiently computable meta-converse on the amount of classical information unassisted codes can transmit over a single use of a quantum channel. As applications, we provide a finite resource analysis of classical communication over quantum erasure channels, including the second-order and moderate deviation asymptotics. Third, we explore the asymptotic analogue of our new meta-converse, the \Upsilon-information of the channel. We show that its regularization is an upper bound on the classical capacity, which is generally tighter than the entanglement-Assisted capacity and other known efficiently computable strong converse bounds. For covariant channels, we show that the \Upsilon-information is a strong converse bound.

Berta, M, Scholz, VB & Tomamichel, M 2018, 'Rényi Divergences as Weighted Non-commutative Vector-Valued Lp-Spaces', *Annales Henri Poincaré*, vol. 19, no. 6, pp. 1843-1867.View/Download from: Publisher's site

#### View description

© 2018, Springer International Publishing AG, part of Springer Nature. We show that Araki and Masuda's weighted non-commutative vector-valued Lp-spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter α=p2. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in α. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases α→{12,1,∞} leading to minus the logarithm of Uhlmann's fidelity, Umegaki's relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz–Thorin theorem for Araki–Masuda Lp-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.

Chapman, RJ, Karim, A, Huang, Z, Flammia, ST, Tomamichel, M & Peruzzo, A 2018, 'Beating the classical limits of information transmission using a quantum decoder', *Physical Review A*, vol. 97, no. 1.View/Download from: Publisher's site

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© 2018 American Physical Society. Encoding schemes and error-correcting codes are widely used in information technology to improve the reliability of data transmission over real-world communication channels. Quantum information protocols can further enhance the performance in data transmission by encoding a message in quantum states; however, most proposals to date have focused on the regime of a large number of uses of the noisy channel, which is unfeasible with current quantum technology. We experimentally demonstrate quantum enhanced communication over an amplitude damping noisy channel with only two uses of the channel per bit and a single entangling gate at the decoder. By simulating the channel using a photonic interferometric setup, we experimentally increase the reliability of transmitting a data bit by greater than 20% for a certain damping range over classically sending the message twice. We show how our methodology can be extended to larger systems by simulating the transmission of a single bit with up to eight uses of the channel and a two-bit message with three uses of the channel, predicting a quantum enhancement in all cases.

Pfister, C, Rol, MA, Mantri, A, Tomamichel, M & Wehner, S 2018, 'Capacity estimation and verification of quantum channels with arbitrarily correlated errors.', *Nature communications*, vol. 9, no. 1, pp. 27-27.View/Download from: Publisher's site

#### View description

The central figure of merit for quantum memories and quantum communication devices is their capacity to store and transmit quantum information. Here, we present a protocol that estimates a lower bound on a channel's quantum capacity, even when there are arbitrarily correlated errors. One application of these protocols is to test the performance of quantum repeaters for transmitting quantum information. Our protocol is easy to implement and comes in two versions. The first estimates the one-shot quantum capacity by preparing and measuring in two different bases, where all involved qubits are used as test qubits. The second verifies on-the-fly that a channel's one-shot quantum capacity exceeds a minimal tolerated value while storing or communicating data. We discuss the performance using simple examples, such as the dephasing channel for which our method is asymptotically optimal. Finally, we apply our method to a superconducting qubit in experiment.

Tomamichel, M & Hayashi, M 2018, 'Operational interpretation of Rényi information measures via composite hypothesis testing against product and markov distributions', *IEEE Transactions on Information Theory*, vol. 64, no. 2, pp. 1064-1082.View/Download from: Publisher's site

#### View description

© 1963-2012 IEEE. We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case, we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold), and the second order asymptotics (at small deviations from the threshold). We apply our results to find the operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by the variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the variations of Rényi conditional mutual information. In either case, the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, this paper also strengthens the view that different definitions of Rényi mutual information, conditional entropy, and conditional mutual information are adequate depending on the context in which the measures are used.

Cheng, HC, Hsieh, MH & Tomamichel, M 2017, 'Exponential decay of matrix Φ-entropies on Markov semigroups with applications to dynamical evolutions of quantum ensembles', *Journal of Mathematical Physics*, vol. 58, no. 9.View/Download from: Publisher's site

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In this work, we extend the theory of quantum Markov processes on a single quantum state to a broader theory that covers Markovian evolution of an ensemble of quantum states, which generalizes Lindblad's formulation of quantum dynamical semigroups. Our results establish the equivalence between an exponential decrease of the matrix Φ-entropies and the Φ-Sobolev inequalities, which allows us to characterize the dynamical evolution of a quantum ensemble to its equilibrium. In particular, we study the convergence rates of two special semigroups, namely, the depolarizing channel and the phase-damping channel. In the former, since there exists a unique equilibrium state, we show that the matrix Φ-entropy of the resulting quantum ensemble decays exponentially as time goes on. Consequently, we obtain a stronger notion of monotonicity of the Holevo quantity-the Holevo quantity of the quantum ensemble decays exponentially in time and the convergence rate is determined by the modified log-Sobolev inequalities. However, in the latter, the matrix Φ-entropy of the quantum ensemble that undergoes the phase-damping Markovian evolution generally will not decay exponentially. There is no classical analogy for these different equilibrium situations. Finally, we also study a statistical mixing of Markov semigroups on matrix-valued functions. We can explicitly calculate the convergence rate of a Markovian jump process defined on Boolean hypercubes and provide upper bounds to the mixing time. Published by AIP Publishing.

Berta, M, Fawzi, O & Tomamichel, M 2017, 'On variational expressions for quantum relative entropies', *Letters in Mathematical Physics*, vol. 107, no. 12, pp. 2239-2265.View/Download from: Publisher's site

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© 2017, Springer Science+Business Media B.V. Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback–Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow general positive operator-valued measures. Second, we extend the result to Rényi relative entropies and show that for non-commuting states the sandwiched Rényi relative entropy is strictly larger than the measured Rényi relative entropy for α∈(12,∞) and strictly smaller for α∈[0,12). The latter statement provides counterexamples for the data processing inequality of the sandwiched Rényi relative entropy for α<12. Our main tool is a new variational expression for the measured Rényi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.

Chubb, CT, Tan, VYF & Tomamichel, M 2017, 'Moderate Deviation Analysis for Classical Communication over Quantum Channels', *Communications in Mathematical Physics*, vol. 355, no. 3, pp. 1283-1315.View/Download from: Publisher's site

#### View description

© 2017, Springer-Verlag GmbH Germany. We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altŭg and Wagner as well as Polyanskiy and Verdú. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

Coles, PJ, Berta, M, Tomamichel, M & Wehner, S 2017, 'Entropic uncertainty relations and their applications', *Reviews of Modern Physics*, vol. 89, no. 1, pp. 1-58.View/Download from: Publisher's site

#### View description

© 2017 American Physical Society. Heisenberg's uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg's idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography.

Hiai, F, König, R & Tomamichel, M 2017, 'Generalized Log-Majorization and Multivariate Trace Inequalities', *Annales Henri Poincaré*, vol. 18, no. 7, pp. 2499-2521.View/Download from: Publisher's site

#### View description

© 2017, Springer International Publishing. We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5) for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regard to logarithmic integral averages of vectors of singular values.

Sutter, D, Berta, M & Tomamichel, M 2017, 'Multivariate Trace Inequalities', *Communications in Mathematical Physics*, vol. 352, no. 1, pp. 37-58.View/Download from: Publisher's site

Tomamichel, M & Leverrier, A 2017, 'A largely self-contained and complete security proof for quantum key distribution', *Quantum*, vol. 1, pp. 14-52.View/Download from: Publisher's site

#### View description

In this work we present a security analysis for quantum key distribution, establishing a rigorous tradeoff between various protocol and security parameters for a class of entanglement-based and prepare-and-measure protocols. The goal of this paper is twofold: 1) to review and clarify the stateof-the-art security analysis based on entropic uncertainty relations, and 2) to provide an accessible resource for researchers interested in a security analysis of quantum cryptographic protocols that takes into account finite resource effects. For this purpose we collect and clarify several arguments spread in the literature on the subject with the goal of making this treatment largely self-contained.

More precisely, we focus on a class of prepare-and-measure protocols based on the Bennett-Brassard (BB84) protocol as well as a class of entanglement-based protocols similar to the Bennett-Brassard-Mermin (BBM92) protocol. We carefully formalize the different steps in these protocols, including randomization, measurement, parameter estimation, error correction and privacy amplification, allowing us to be mathematically precise throughout the security analysis. We start from an operational definition of what it means for a quantum key distribution protocol to be secure and derive simple conditions that serve as sufficient condition for secrecy and correctness. We then derive and eventually discuss tradeoff relations between the block length of the classical computation, the noise tolerance, the secret key length and the security parameters for our protocols. Our results significantly improve upon previously reported tradeoffs.

Tomamichel, M, Martinez-Mateo, J, Pacher, C & Elkouss, D 2017, 'Fundamental finite key limits for one-way information reconciliation in quantum key distribution', *Quantum Information Processing*, vol. 16, no. 11.View/Download from: Publisher's site

#### View description

© 2017, Springer Science+Business Media, LLC. The security of quantum key distribution protocols is guaranteed by the laws of quantum mechanics. However, a precise analysis of the security properties requires tools from both classical cryptography and information theory. Here, we employ recent results in non-asymptotic classical information theory to show that one-way information reconciliation imposes fundamental limitations on the amount of secret key that can be extracted in the finite key regime. In particular, we find that an often used approximation for the information leakage during information reconciliation is not generally valid. We propose an improved approximation that takes into account finite key effects and numerically test it against codes for two probability distributions, that we call binary–binary and binary–Gaussian, that typically appear in quantum key distribution protocols.

Tomamichel, M, Wilde, MM & Winter, A 2017, 'Strong Converse Rates for Quantum Communication', *IEEE Transactions on Information Theory*, vol. 63, no. 1, pp. 715-727.View/Download from: Publisher's site

#### View description

© 2016 IEEE. We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. As an application of this result, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.

Wilde, MM, Tomamichel, M & Berta, M 2017, 'Converse Bounds for Private Communication Over Quantum Channels', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 63, no. 3, pp. 1792-1817.View/Download from: Publisher's site

Wilde, MM, Tomamichel, M, Lloyd, S & Berta, M 2017, 'Gaussian Hypothesis Testing and Quantum Illumination', *Physical Review Letters*, vol. 119, no. 12.View/Download from: Publisher's site

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© 2017 American Physical Society. Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.

Berta, M & Tomamichel, M 2016, 'The Fidelity of Recovery Is Multiplicative', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 62, no. 4, pp. 1758-1763.View/Download from: Publisher's site

Datta, N, Tomamichel, M & Wilde, MM 2016, 'On the second-order asymptotics for entanglement-assisted communication', *QUANTUM INFORMATION PROCESSING*, vol. 15, no. 6, pp. 2569-2591.View/Download from: Publisher's site

Hayashi, M & Tomamichel, M 2016, 'Correlation detection and an operational interpretation of the Renyi mutual information', *JOURNAL OF MATHEMATICAL PHYSICS*, vol. 57, no. 10.View/Download from: Publisher's site

Pfister, C, Kaniewski, J, Tomamichel, M, Mantri, A, Schmucker, R, McMahon, N, Milburn, G & Wehner, S 2016, 'A universal test for gravitational decoherence.', *Nature Communications*, vol. 7, pp. 13022-13022.View/Download from: Publisher's site

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Quantum mechanics and the theory of gravity are presently not compatible. A particular question is whether gravity causes decoherence. Several models for gravitational decoherence have been proposed, not all of which can be described quantum mechanically. Since quantum mechanics may need to be modified, one may question the use of quantum mechanics as a calculational tool to draw conclusions from the data of experiments concerning gravity. Here we propose a general method to estimate gravitational decoherence in an experiment that allows us to draw conclusions in any physical theory where the no-signalling principle holds, even if quantum mechanics needs to be modified. As an example, we propose a concrete experiment using optomechanics. Our work raises the interesting question whether other properties of nature could similarly be established from experimental observations alone-that is, without already having a rather well-formed theory of nature to make sense of experimental data.

Sutter, D, Tomamichel, M & Harrow, AW 2016, 'Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map', *IEEE Transactions on Information Theory*, vol. 62, no. 5, pp. 2907-2913.View/Download from: Publisher's site

#### View description

The quantum relative entropy between two states

satisfies a monotonicity property meaning that applying the same

quantum channel to both states can never increase their relative

entropy. It is known that this inequality is only tight when there

is a recovery map that exactly reverses the effects of the quantum

channel on both states. In this paper, we strengthen this inequality

by showing that the difference of relative entropies is bounded

below by the measured relative entropy between the first state

and a recovered state from its processed version. The recovery

map is a convex combination of rotated Petz recovery maps and

perfectly reverses the quantum channel on the second state. As a

special case, we reproduce recent lower bounds on the conditional

mutual information, such as the one proved by Fawzi and Renner.

Our proof only relies on the elementary properties of pinching

maps and the operator logarithm

Tomamichel, M, Berta, M & Renes, JM 2016, 'Quantum coding with finite resources', *NATURE COMMUNICATIONS*, vol. 7.View/Download from: Publisher's site

Lin, MS & Tomamichel, M 2015, 'Investigating Properties of a Family of Quantum Renyi Divergences', *Quantum Information Processing*, vol. 14, no. 4, pp. 1501-1512.View/Download from: Publisher's site

#### View description

Audenaert and Datta recently introduced a two-parameter family of relative

R\'{e}nyi entropies, known as the $\alpha$-$z$-relative R\'{e}nyi entropies.

The definition of the $\alpha$-$z$-relative R\'{e}nyi entropy unifies all

previously proposed definitions of the quantum R\'{e}nyi divergence of order

$\alpha$ under a common framework. Here we will prove that the

$\alpha$-$z$-relative R\'{e}nyi entropies are a proper generalization of the

quantum relative entropy by computing the limit of the $\alpha$-$z$ divergence

as $\alpha$ approaches one and $z$ is an arbitrary function of $\alpha$. We

also show that certain operationally relevant families of R\'enyi divergences

are differentiable at $\alpha = 1$. Finally, our analysis reveals that the

derivative at $\alpha = 1$ evaluates to half the relative entropy variance, a

quantity that has attained operational significance in second-order quantum

hypothesis testing.

Lunghi, T, Kaniewski, J, Bussieres, F, Houlmann, R, Tomamichel, M, Wehner, S & Zbinden, H 2015, 'Practical Relativistic Bit Commitment', *PHYSICAL REVIEW LETTERS*, vol. 115, no. 3.View/Download from: Publisher's site

Tan, VYF & Tomamichel, M 2015, 'The Third-Order Term in the Normal Approximation for the AWGN Channel', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 61, no. 5, pp. 2430-2438.View/Download from: Publisher's site

Tomamichel, M & Tan, VYF 2015, 'Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels', *COMMUNICATIONS IN MATHEMATICAL PHYSICS*, vol. 338, no. 1, pp. 103-137.View/Download from: Publisher's site

Dupuis, F, Szehr, O & Tomamichel, M 2014, 'Decoupling Approach to Classical Data Transmission Over Quantum Channels', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 60, no. 3, pp. 1562-1572.View/Download from: Publisher's site

Furrer, F, Berta, M, Tomamichel, M, Scholz, VB & Christandl, M 2014, 'Position-momentum uncertainty relations in the presence of quantum memory', *JOURNAL OF MATHEMATICAL PHYSICS*, vol. 55, no. 12.View/Download from: Publisher's site

Furrer, F, Franz, T, Berta, M, Leverrier, A, Scholz, VB, Tomamichel, M & Werner, RF 2014, 'Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security Against Coherent Attacks (vol 109, 100502, 2012)', *PHYSICAL REVIEW LETTERS*, vol. 112, no. 1.View/Download from: Publisher's site

Kaniewski, J, Tomamichel, M & Wehner, S 2014, 'Entropic uncertainty from effective anticommutators', *PHYSICAL REVIEW A*, vol. 90, no. 1.View/Download from: Publisher's site

Tomamichel, M & Tan, VYF 2014, 'Second-Order Coding Rates for Channels With State', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 60, no. 8, pp. 4427-4448.View/Download from: Publisher's site

Tomamichel, M, Berta, M & Hayashi, M 2014, 'Relating different quantum generalizations of the conditional Renyi entropy', *JOURNAL OF MATHEMATICAL PHYSICS*, vol. 55, no. 8.View/Download from: Publisher's site

Kaniewski, J, Tomamichel, M, Haenggi, E & Wehner, S 2013, 'Secure Bit Commitment From Relativistic Constraints', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 59, no. 7, pp. 4687-4699.View/Download from: Publisher's site

Lim, CCW, Portmann, C, Tomamichel, M, Renner, R & Gisin, N 2013, 'Device-Independent Quantum Key Distribution with Local Bell Test', *Physical Review X*, vol. 3, pp. 1-11.View/Download from: Publisher's site

#### View description

Device-independent quantum key distribution (DIQKD) in its current design requires a violation of a

Bell's inequality between two parties, Alice and Bob, who are connected by a quantum channel. However,

in reality, quantum channels are lossy and current DIQKD protocols are thus vulnerable to attacks

exploiting the detection loophole of the Bell test. Here, we propose a novel approach to DIQKD that

overcomes this limitation. In particular, we propose a protocol where the Bell test is performed entirely on

two casually independent devices situated in Alice's laboratory. As a result, the detection loophole caused

by the losses in the channel is avoided

Lunghi, T, Kaniewski, J, Bussieres, F, Houlmann, R, Tomamichel, M, Kent, A, Gisin, N, Wehner, S & Zbinden, H 2013, 'Experimental Bit Commitment Based on Quantum Communication and Special Relativity', *PHYSICAL REVIEW LETTERS*, vol. 111, no. 18.View/Download from: Publisher's site

Mueller-Lennert, M, Dupuis, F, Szehr, O, Fehr, S & Tomamichel, M 2013, 'On quantum Renyi entropies: A new generalization and some properties', *JOURNAL OF MATHEMATICAL PHYSICS*, vol. 54, no. 12.View/Download from: Publisher's site

Szehr, O, Dupuis, F, Tomamichel, M & Renner, R 2013, 'Decoupling with unitary approximate two-designs', *NEW JOURNAL OF PHYSICS*, vol. 15.View/Download from: Publisher's site

Tomamichel, M & Haenggi, E 2013, 'The link between entropic uncertainty and nonlocality', *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, vol. 46, no. 5.View/Download from: Publisher's site

Tomamichel, M & Hayashi, M 2013, 'A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 59, no. 11, pp. 7693-7710.View/Download from: Publisher's site

Tomamichel, M & Tan, VYF 2013, 'A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 59, no. 11, pp. 7041-7051.View/Download from: Publisher's site

Tomamichel, M, Fehr, S, Kaniewski, J & Wehner, S 2013, 'A monogamy-of-entanglement game with applications to device-independent quantum cryptography', *NEW JOURNAL OF PHYSICS*, vol. 15.View/Download from: Publisher's site

Vitanov, A, Dupuis, F, Tomamichel, M & Renner, R 2013, 'Chain Rules for Smooth Min- and Max-Entropies', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 59, no. 5, pp. 2603-2612.View/Download from: Publisher's site

Furrer, F, Franz, T, Berta, M, Leverrier, A, Scholz, VB, Tomamichel, M & Werner, RF 2012, 'Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks', *PHYSICAL REVIEW LETTERS*, vol. 109, no. 10.View/Download from: Publisher's site

Tomamichel, M, Lim, CCW, Gisin, N & Renner, R 2012, 'Tight finite-key analysis for quantum cryptography', *Nature Communications*, vol. 3, pp. 1-6.View/Download from: Publisher's site

#### View description

Despite enormous theoretical and experimental progress in quantum cryptography, the security of most current implementations of quantum key distribution is still not rigorously established. One significant problem is that the security of the final key strongly depends on the number, M, of signals exchanged between the legitimate parties. Yet, existing security proofs are often only valid asymptotically, for unrealistically large values of M. Another challenge is that most security proofs are very sensitive to small differences between the physical devices used by the protocol and the theoretical model used to describe them. Here we show that these gaps between theory and experiment can be simultaneously overcome by using a recently developed proof technique based on the uncertainty relation for smooth entropies.

Tomamichel, M & Renner, R 2011, 'Uncertainty Relation for Smooth Entropies', *PHYSICAL REVIEW LETTERS*, vol. 106, no. 11.View/Download from: Publisher's site

Tomamichel, M, Schaffner, C, Smith, A & Renner, R 2011, 'Leftover Hashing Against Quantum Side Information', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 57, no. 8, pp. 5524-5535.View/Download from: Publisher's site

Winkler, S, Tomamichel, M, Hengl, S & Renner, R 2011, 'Impossibility of Growing Quantum Bit Commitments', *PHYSICAL REVIEW LETTERS*, vol. 107, no. 9.View/Download from: Publisher's site

Tomamichel, M, Colbeck, R & Renner, R 2010, 'Duality Between Smooth Min- and Max-Entropies', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 56, no. 9, pp. 4674-4681.View/Download from: Publisher's site

Tomamichel, M, Colbeck, R & Renner, R 2009, 'A Fully Quantum Asymptotic Equipartition Property', *IEEE TRANSACTIONS ON INFORMATION THEORY*, vol. 55, no. 12, pp. 5840-5847.View/Download from: Publisher's site

Witzigmann, B, Tomamichel, M, Steiger, S, Veprek, RG, Kojima, K & Schwarz, UT 2008, 'Analysis of gain and luminescence in violet and blue GaInN-GaN quantum wells', *IEEE JOURNAL OF QUANTUM ELECTRONICS*, vol. 44, no. 1-2, pp. 144-149.View/Download from: Publisher's site

Gour, G & Tomamichel, M, 'Entropy and relative entropy from information-theoretic principles'.

#### View description

We introduce an axiomatic approach to entropies and relative entropies that

relies only on minimal information-theoretic axioms, namely monotonicity under

mixing and data-processing as well as additivity for product distributions. We

find that these axioms induce sufficient structure to establish continuity in

the interior of the probability simplex and meaningful upper and lower bounds,

e.g., we find that every relative entropy must lie between the R\'enyi

divergences of order $0$ and $\infty$. We further show simple conditions for

positive definiteness of such relative entropies and a characterisation in term

of a variant of relative trumping. Our main result is a one-to-one

correspondence between entropies and relative entropies.

Gour, G & Tomamichel, M, 'Optimal Extensions of Resource Measures and their Applications'.

#### View description

We develop a framework to extend resource measures from one domain to a

larger one. We find that all extensions of resource measures are bounded

between two quantities that we call the maximal and minimal extensions. As an

application to the framework, we show that any relative entropy (i.e. an

additive function on pairs of quantum states that satisfies the data processing

inequality) must be bounded by the min and max relative entropies. Along the

way we use the framework to prove optimality properties of the generalized

trace distance, the generalized fidelity, and the purified distance. As an

application of the framework to entanglement theory we introduce a new

technique to extend pure state entanglement measures to mixed bipartite states.

Korzekwa, K, Puchała, Z, Tomamichel, M & Życzkowski, K, 'Encoding classical information into quantum resources'.

#### View description

We introduce and analyse the problem of encoding classical information into

different resources of a quantum state. More precisely, we consider a general

class of communication scenarios characterised by encoding operations that

commute with a unique resource destroying map and leave free states invariant.

Our motivating example is given by encoding information into coherences of a

quantum system with respect to a fixed basis (with unitaries diagonal in that

basis as encodings and the decoherence channel as a resource destroying map),

but the generality of the framework allows us to explore applications ranging

from super-dense coding to thermodynamics. For any state, we find that the

number of messages that can be encoded into it using such operations in a

one-shot scenario is upper-bounded in terms of the information spectrum

relative entropy between the given state and its version with erased resources.

Furthermore, if the resource destroying map is a twirling channel over some

unitary group, we find matching one-shot lower-bounds as well. In the

asymptotic setting where we encode into many copies of the resource state, our

bounds yield an operational interpretation of resource monotones such as the

relative entropy of coherence and its corresponding relative entropy variance.

Pirandola, S, Andersen, UL, Banchi, L, Berta, M, Bunandar, D, Colbeck, R, Englund, D, Gehring, T, Lupo, C, Ottaviani, C, Pereira, J, Razavi, M, Shaari, JS, Tomamichel, M, Usenko, VC, Vallone, G, Villoresi, P & Wallden, P, 'Advances in Quantum Cryptography'.

#### View description

Quantum cryptography is arguably the fastest growing area in quantum

information science. Novel theoretical protocols are designed on a regular

basis, security proofs are constantly improving, and experiments are gradually

moving from proof-of-principle lab demonstrations to in-field implementations

and technological prototypes. In this review, we provide both a general

introduction and a state of the art description of the recent advances in the

field, both theoretically and experimentally. We start by reviewing protocols

of quantum key distribution based on discrete variable systems. Next we

consider aspects of device independence, satellite challenges, and high rate

protocols based on continuous variable systems. We will then discuss the

ultimate limits of point-to-point private communications and how quantum

repeaters and networks may overcome these restrictions. Finally, we will

discuss some aspects of quantum cryptography beyond standard quantum key

distribution, including quantum data locking and quantum digital signatures.

Bravyi, S, Gosset, D, Koenig, R & Tomamichel, M 2019, 'Quantum Advantage with Noisy Shallow Circuits in 3D', *Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society Press, Baltimore, MD, USA, USA, pp. 995-999.View/Download from: Publisher's site

#### View description

© 2019 IEEE. Prior work has shown that there exists a relation problem which can be solved with certainty by a constant-depth quantum circuit composed of geometrically local gates in two dimensions, but cannot be solved with high probability by any classical constant depth circuit composed of bounded fan-in gates. Here we provide two extensions of this result. Firstly, we show that a separation in computational power persists even when the constant-depth quantum circuit is restricted to geometrically local gates in one dimension. The corresponding quantum algorithm is the simplest we know of which achieves a quantum advantage of this type. Our second, main result, is that a separation persists even if the shallow quantum circuit is corrupted by noise. We construct a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value. On the other hand, the problem cannot be solved with high probability by a noise-free classical circuit of constant depth. A key component of the proof is a quantum error-correcting code which admits constant-depth logical Clifford gates and single-shot logical state preparation. We show that the surface code meets these criteria.

Anshu, A, Berta, M, Jain, R & Tomamichel, M 2019, 'Second-Order Characterizations via Partial Smoothing', *IEEE International Symposium on Information Theory - Proceedings*, pp. 937-941.View/Download from: Publisher's site

#### View description

© 2019 IEEE. Smooth entropies are a tool for quantifying resource trade-offs in information theory and cryptography. However, in typical multi-partite problems some of the sub-systems are often left unchanged and this is not reflected by the standard smoothing of information measures over a ball of close states. We propose to smooth instead only over a ball of close states which also have some of the reduced states on the relevant sub-systems fixed. This partial smoothing of information measures naturally allows to give more refined characterizations of various information-theoretic problems in the one-shot setting. As a consequence, we can derive asymptotic second-order characterizations for tasks such as privacy amplification against classical side information or classical state splitting. For quantum problems like state merging the general resource trade-off is tightly characterized by partially smoothed information measures as well.

Chubb, CT, Korzekwa, K & Tomamichel, M 2019, 'Moderate deviation analysis of majorisation-based resource interconversion', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, France, pp. 3002-3006.View/Download from: Publisher's site

#### View description

© 2019 IEEE. We consider the problem of interconverting a finite amount of resources within all theories whose single-shot transformation rules are based on a majorisation relation, e.g. the resource theories of entanglement and coherence (for pure state transformations), as well as thermodynamics (for energy-incoherent transformations). When only finite resources are available we expect to see a non-trivial trade-off between the rate rn at which n copies of a resource state ρ can be transformed into nrn copies of another resource state σ, and the error level ϵn of the interconversion process, as a function of n. In this work we derive the optimal trade-off in the so-called moderate deviation regime, where the rate of interconversion rn approaches its optimum in the asymptotic limit of unbounded resources (n → ∞), while the error ϵn vanishes in the same limit. We find that the moderate deviation analysis exhibits a resonance behaviour which implies that certain pairs of resource states can be interconverted at the asymptotically optimal rate with negligible error, even in the finite n regime.

Tomamichel, M, Bravyi, S, Konig, R & Gosset, D 2019, 'Quantum advantage with noisy shallow circuits in 3D', IEEE 60th Annual Symposium on Foundations of Computer Science, Baltimore, USA.View/Download from: Publisher's site

Fang, K, Wang, X, Tomamichel, M & Berta, M 2018, 'Quantum Channel Simulation and the Channel's Smooth Max-Information', *2018 IEEE International Symposium on Information Theory (ISIT)*, International Symposium on Information Theory, IEEE, Vail, CO, USA.View/Download from: Publisher's site

#### View description

We study the general framework of quantum channel simulation, that is, the

ability of a quantum channel to simulate another one using different classes of

codes. First, we show that the minimum error of simulation and the one-shot

quantum simulation cost under no-signalling assisted codes are given by

semidefinite programs. Second, we introduce the channel's smooth

max-information, which can be seen as a one-shot generalization of the mutual

information of a quantum channel. We provide an exact operational

interpretation of the channel's smooth max-information as the one-shot quantum

simulation cost under no-signalling assisted codes. Third, we derive the

asymptotic equipartition property of the channel's smooth max-information,

i.e., it converges to the quantum mutual information of the channel in the

independent and identically distributed asymptotic limit. This implies the

quantum reverse Shannon theorem in the presence of no-signalling correlations.

Finally, we explore the simulation cost of various quantum channels.

Wang, X, Fang, K & Tomamichel, M 2018, 'On Finite Blocklength Converse Bounds for Classical Communication over Quantum Channels', *2018 IEEE International Symposium on Information Theory (ISIT)*, IEEE International Symposium on Information Theory, IEEE, Vail, CO, USA, pp. 2157-2161.View/Download from: Publisher's site

#### View description

We explore several new converse bounds for classical communication over quantum channels in the finite blocklength regime. First, we show that the Matthews-Wehner meta-converse bound for entanglement-assisted classical communication can be achieved by activated, no-signalling assisted codes, suitably generalizing a result for classical channels. Second, we derive a new meta-converse on the amount of information unassisted codes can transmit over a single use of a quantum channel. We further show that this meta-converse can be evaluated via semidefinite programming. As an application, we provide a second-order analysis of classical communication over quantum erasure channels.

Cheng, HC, Hsieh, MH & Tomamichel, M 2017, 'Sphere-packing bound for classical-quantum channels', *IEEE International Symposium on Information Theory - Proceedings*, IEEE Information Theory Workshop, IEEE, Kaohsiung, Taiwan, pp. 479-483.View/Download from: Publisher's site

#### View description

© 2017 IEEE. We study lower bounds on the optimal error probability in channel coding at rates below capacity, commonly termed sphere-packing bounds. In this work, we establish a sphere-packing bound for classical-quantum channels, which significantly improves previous prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n/n), indicating our sphere-packing bound is almost exact in the high rate regime. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

Cheng, HC, Hsieh, MH & Tomamichel, M 2017, 'Sphere-packing bound for symmetric classical-quantum channels', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Aachen, Germany, pp. 286-290.View/Download from: Publisher's site

#### View description

© 2017 IEEE. "To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award." We provide a sphere-packing lower bound for the optimal error probability in finite blocklengths when coding over a symmetric classical-quantum channel. Our result shows that the pre-factor can be significantly improved from the order of the subexponential to the polynomial, This established pre-factor is arguably optimal because it matches the best known random coding upper bound in the classical case. Our approaches rely on a sharp concentration inequality in strong large deviation theory and crucial properties of the error-exponent function.

Sutter, D, Berta, M & Tomamichel, M 2017, 'Quantum Markov chains and logarithmic trace inequalities', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Aachen, Germany, pp. 1988-1992.View/Download from: Publisher's site

#### View description

© 2017 IEEE. A Markov chain is a tripartite quantum state ρABCwhere there exists a recovery map RB→BCsuch that ρABC= RB→BC(ρAB). More generally, an approximate Markov chain ρABCis a state whose distance to the closest recovered state RB→BC(ρAB) is small. Recently it has been shown that this distance can be bounded from above by the conditional mutual information I(A: C|B)ρof the state. We improve on this connection by deriving the first bound that is tight in the commutative case and features an explicit recovery map that only depends on the reduced state pBC. The key tool in our proof is a multivariate extension of the Golden-Thompson inequality, which allows us to extend logarithmic trace inequalities from two to arbitrarily many matrices.

Tomamichel, M, Chubb, CT & Tan, V 2018, 'Moderate deviation analysis for classical communication over quantum channels', *IEEE International Symposium on Information Theory - Proceedings*, Vail, CO, USA, pp. 1544-1548.View/Download from: Publisher's site

Wilde, MM, Tomamichel, M & Berta, M 2017, 'A meta-converse for private communication over quantum channels', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Aachen, Germany, pp. 291-295.View/Download from: Publisher's site

#### View description

© 2017 IEEE. We establish a converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication.

Berta, M, Fawzi, O & Tomamichel, M 2016, 'Exploiting variational formulas for quantum relative entropy', *Proceedings IEEE International Symposium on Information Theory (ISIT 2016)*, IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 2844-2848.View/Download from: Publisher's site

#### View description

The relative entropy is the basic concept underlying various information measures like entropy, conditional entropy and mutual information. Here, we discuss how to make use of variational formulas for measured relative entropy and quantum relative entropy for understanding the additivity properties of various entropic quantities that appear in quantum information theory. In particular, we show that certain lower bounds on quantum conditional mutual information are superadditive.

Sutter, D, Tomamichel, M & Harrow, AW 2016, 'Strengthened monotonicity of relative entropy via pinched Petz recovery map', *Proceedings IEEE International Symposium on Information Theory (ISIT 2016)*, IEEE, Barcelona, pp. 760-764.View/Download from: Publisher's site

Tomamichel, M & Hayashi, M 2016, 'Operational interpretation of Rényi conditional mutual information via composite hypothesis testing against Markov distributions', *Proceedings IEEE International Symposium on Information Theory (ISIT 2016)*, IEEE International Symposium on Information Theory, IEEE, Barcelona, Spain, pp. 585-589.View/Download from: Publisher's site

#### View description

We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold) and the second-order asymptotics (at small deviations from the threshold). We apply our results to find operational interpretations of Rényi information measures. In particular, in case the alternative hypothesis consists of certain tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the Rényi conditional mutual information

Datta, N, Tomamichel, M & Wilde, MM 2015, 'Second-order coding rates for entanglement-assisted communication', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Hong Kong, China, pp. 2772-2776.View/Download from: Publisher's site

#### View description

© 2015 IEEE. The entanglement-assisted capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and characterizes the convergence towards the entanglement-assisted capacity when the number of channel uses increases. More generally, we prove that the Gaussian approximation for a second-order coding rate is achievable for all quantum channels.

Hayashi, M & Tomamichel, M 2015, 'Correlation detection and an operational interpretation of the Rényi mutual information', *Proceedings IEEE International Symposium on Information Theory (ISIT 2015)*, IEEE, Hong Kong, China, pp. 1447-1451.View/Download from: Publisher's site

Tomamichel, M, Wilde, MM & Winter, A 2015, 'Strong converse rates for quantum communication', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Hong Kong, China, pp. 2386-2390.View/Download from: Publisher's site

#### View description

© 2015 IEEE. We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any code with a rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. Our result has several applications. Most importantly, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. This for the first time conclusively settles the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.

Tan, VYF & Tomamichel, M 2014, 'The third-order term in the normal approximation for the AWGN channel', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, pp. 2077-2081.View/Download from: Publisher's site

#### View description

This paper shows that, under the average error probability formalism, the third-order term in the normal approximation for the additive white Gaussian noise channel with a maximal or equal power constraint is at least 1 over 2 log n+O(1). This improves on the lower bound by Polyanskiy-Poor-Verdú (2010) and matches the upper bound proved by the same authors. © 2014 IEEE.

Tomamichel, M & Tan, VYF 2014, 'Second order refinements for the classical capacity of quantum channels with separable input states', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Honolulu, USA, pp. 141-145.View/Download from: Publisher's site

#### View description

We study the non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of information that can be transmitted when the channel is used a finite number of times and a finite average decoding error is permissible. We show that, if we restrict the encoder to use ensembles of separable states, the non-asymptotic fundamental limit admits a Gaussian approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the number of channel uses tends to infinity. To do so, several important properties of quantum information quantities, such as the capacity-achieving output state, the divergence radius, and the channel dispersion, are generalized from their classical counterparts. Further, we exploit a close relation between classical-quantum channel coding and quantum binary hypothesis testing and rely on recent progress in the non-asymptotic characterization of quantum hypothesis testing and its Gaussian approximation. © 2014 IEEE.

Tomamichel, M, Berta, M & Hayashi, M 2014, 'A duality relation connecting different quantum generalizations of the conditional Rényi entropy', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, IEEE, Honolulu, USA, pp. 731-735.View/Download from: Publisher's site

#### View description

Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. This generalizes the well-known duality relation H(AB)+H(AC) = 0 for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies. © 2014 IEEE.

Tomamichel, M, Martinez-Mateo, J, Pacher, C & Elkouss, D 2014, 'Fundamental Finite Key Limits for Information Reconciliation in Quantum Key Distribution', *Proceedings of the IEEE International Symposium on Information Theory (ISIT 2014)*, IEEE International Symposium on Information Theory, IEEE, USA, pp. 1469-1473.View/Download from: Publisher's site

#### View description

The security of quantum key distribution protocols is guaranteed by the laws

of quantum mechanics. However, a precise analysis of the security properties

requires tools from both classical cryptography and information theory. Here,

we employ recent results in non-asymptotic classical information theory to show

that information reconciliation imposes fundamental limitations on the amount

of secret key that can be extracted in the finite key regime. In particular, we

find that an often used approximation for the information leakage during

information reconciliation is flawed and we propose an improved estimate.

Tomamichel, M & Tan, VYF 2013, 'A tight upper bound for the third-order asymptotics of discrete memoryless channels', *IEEE International Symposium on Information Theory - Proceedings*, IEEE International Symposium on Information Theory, pp. 1536-1540.View/Download from: Publisher's site

#### View description

This paper shows that the logarithm of the ε-error capacity (average error probability) for n uses of a discrete memoryless channel with positive conditional information variance at every capacity-achieving input distribution is upper bounded by the normal approximation plus a term that does not exceed 1/2 log n + O(1). © 2013 IEEE.

Tomamichel, M & Tan, VYF 2013, 'ε-Capacity and strong converse for channels with general state', *Information Theory Workshop (ITW), 2013 IEEE*, IEEE Information Theory Workshop (ITW), IEEE, Seville, Spain.View/Download from: Publisher's site

#### View description

We consider state-dependent memoryless channels with general state available at both encoder and decoder. We establish the ε-capacity and the optimistic ε-capacity. This allows us to prove a necessary and sufficient condition for the strong converse to hold. We also provide a simpler sufficient condition on the first- and second-order statistics of the state process that ensures that the strong converse holds.

Tomamichel, M, Fehr, S, Kaniewski, J & Wehner, S 2013, 'One-sided device-independent QKD and position-based cryptography from monogamy games', *Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, International Conference on the Theory and Application of Cryptographic Techniques, Springer, Athens, Greece, pp. 609-625.View/Download from: Publisher's site

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A serious concern with quantum key distribution (QKD) schemes is that, when under attack, the quantum devices in a real-life implementation may behave differently than modeled in the security proof. This can lead to real-life attacks against provably secure QKD schemes. In this work, we show that the standard BB84 QKD scheme is one-sided device-independent. This means that security holds even if Bob's quantum device is arbitrarily malicious, as long as Alice's device behaves as it should. Thus, we can completely remove the trust into Bob's quantum device for free, without the need for changing the scheme, and without the need for hard-to-implement loophole-free violations of Bell inequality, as is required for fully (meaning two-sided) device-independent QKD. For our analysis, we introduce a new quantum game, called a monogamy-of-entanglement game, and we show a strong parallel repetition theorem for this game. This new notion is likely to be of independent interest and to find additional applications. Indeed, besides the application to QKD, we also show a direct application to position-based quantum cryptography: we give the first security proof for a one-round position-verification scheme that requires only single-qubit operations. © 2013 International Association for Cryptologic Research.

Tomamichel, M, Renner, R, Schaffner, C & Smith, A 2010, 'Leftover Hashing against quantum side information', *Proceedings of the IEEE Symposium on Information Theory (ISIT 2010)*, IEEE, Austin, USA, pp. 2703-2707.View/Download from: Publisher's site

Witzigmann, B, Steiger, S, Tomamichel, M, Veprek, R & Schwarz, UT 2007, 'Optical gain in 407nm and 470nm InGaN/GaN heterostructures: signature of quantum-dot states', *OPTOELECTRONIC MATERIALS AND DEVICES II*, Conference on Optoelectronic Materials and Devices II, SPIE-INT SOC OPTICAL ENGINEERING, Wuhan, PEOPLES R CHINA.View/Download from: Publisher's site

Memis, OG, Kong, SC, Katsnelson, A, Tomamichel, MP & Mohseni, H 2006, 'A novel avalanche free single photon detector', *2006 6th IEEE Conference on Nanotechnology, IEEE-NANO 2006*, pp. 742-745.View/Download from: Publisher's site

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We have conceived a novel single photon detector for IR wavelengths above 1 μm. The detection mechanism is based on carrier focalization and nano-injection. Preliminary measured data from unpassivated devices show a very high internal gain and low dark current at 1.55 μm at room temperature © 2006 IEEE.

Memis, OG, Kong, SC, Katsnelson, A, Tomamichel, MP & Mohseni, H, 'A Novel Avalanche Free Single Photon Detector', *2006 Sixth IEEE Conference on Nanotechnology*, 2006 Sixth IEEE Conference on Nanotechnology, IEEE.View/Download from: Publisher's site

Tomamichel, M, Bravyi, S, Gosset, D & Koenig, R 2020, 'An information-theoretic treatment of quantum dichotomies'.

Tomamichel, M, Bravyi, S, Gosset, D & Koenig, R 2020, 'Quantum advantage with noisy shallow circuits in 3D.'.

Anshu, A, Berta, M, Jain, R & Tomamichel, M 2018, 'Partially smoothed information measures'.

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Smooth entropies are a tool for quantifying resource trade-offs in (quantum)

information theory and cryptography. In typical bi- and multi-partite problems,

however, some of the sub-systems are often left unchanged and this is not

reflected by the standard smoothing of information measures over a ball of

close states. We propose to smooth instead only over a ball of close states

which also have some of the reduced states on the relevant sub-systems fixed.

This partial smoothing of information measures naturally allows to give more

refined characterizations of various information-theoretic problems in the

one-shot setting. In particular, we immediately get asymptotic second-order

characterizations for tasks such as privacy amplification against classical

side information or classical state splitting. For quantum problems like state

merging the general resource trade-off is tightly characterized by partially

smoothed information measures as well. However, for quantum systems we can so

far only give the asymptotic first-order expansion of these quantities.

Chubb, CT, Tomamichel, M & Korzekwa, K 2018, 'Beyond the thermodynamic limit: finite-size corrections to state interconversion rates'.

Huber, S, Koenig, R & Tomamichel, M 2018, 'Jointly constrained semidefinite bilinear programming with an application to Dobrushin curves'.

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We propose a branch-and-bound algorithm for minimizing a bilinear functional

of the form \[ f(X,Y) = \mathrm{tr}((X\otimes

Y)Q)+\mathrm{tr}(AX)+\mathrm{tr}(BY) , \] of pairs of Hermitian matrices

$(X,Y)$ restricted by joint semidefinite programming constraints. The

functional is parametrized by self-adjoint matrices $Q$, $A$ and $B$. This

problem generalizes that of a bilinear program, where $X$ and $Y$ belong to

polyhedra. The algorithm converges to a global optimum and yields upper and

lower bounds on its value in every step. Various problems in quantum

information theory can be expressed in this form. As an example application, we

compute Dobrushin curves of quantum channels, giving upper bounds on classical

coding with energy constraints.

Aggarwal, D, Brennen, GK, Lee, T, Santha, M & Tomamichel, M 2017, 'Quantum attacks on Bitcoin, and how to protect against them'.

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The key cryptographic protocols used to secure the internet and financial

transactions of today are all susceptible to attack by the development of a

sufficiently large quantum computer. One particular area at risk are

cryptocurrencies, a market currently worth over 150 billion USD. We investigate

the risk of Bitcoin, and other cryptocurrencies, to attacks by quantum

computers. We find that the proof-of-work used by Bitcoin is relatively

resistant to substantial speedup by quantum computers in the next 10 years,

mainly because specialized ASIC miners are extremely fast compared to the

estimated clock speed of near-term quantum computers. On the other hand, the

elliptic curve signature scheme used by Bitcoin is much more at risk, and could

be completely broken by a quantum computer as early as 2027, by the most

optimistic estimates. We analyze an alternative proof-of-work called Momentum,

based on finding collisions in a hash function, that is even more resistant to

speedup by a quantum computer. We also review the available post-quantum

signature schemes to see which one would best meet the security and efficiency

requirements of blockchain applications.

Tomamichel, M, 'A Framework for Non-Asymptotic Quantum Information Theory'.

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This thesis consolidates, improves and extends the smooth entropy framework

for non-asymptotic information theory and cryptography.

We investigate the conditional min- and max-entropy for quantum states,

generalizations of classical R\'enyi entropies. We introduce the purified

distance, a novel metric for unnormalized quantum states and use it to define

smooth entropies as optimizations of the min- and max-entropies over a ball of

close states. We explore various properties of these entropies, including

data-processing inequalities, chain rules and their classical limits. The most

important property is an entropic formulation of the asymptotic equipartition

property, which implies that the smooth entropies converge to the von Neumann

entropy in the limit of many independent copies. The smooth entropies also

satisfy duality and entropic uncertainty relations that provide limits on the

power of two different observers to predict the outcome of a measurement on a

quantum system.

Finally, we discuss three example applications of the smooth entropy

framework. We show a strong converse statement for source coding with quantum

side information, characterize randomness extraction against quantum side

information and prove information theoretic security of quantum key

distribution using an intuitive argument based on the entropic uncertainty

relation.