# Dr Marco Tomamichel

### Biography

I am an Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) fellow and Senior Lecturer with the Centre for Quantum Software and Information at the University of Technology Sydney. 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 the University of Sydney as a University of Sydney Postdoctoral Fellow, I have worked as a Research Fellow and Senior Research Fellow at the Centre for Quantum Technologies in Singapore.

### Professional

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

## Links

**Senior Lecturer,**School of Software

**Core Member,**Centre for Quantum Software and Information

M.Sc. in Electrical Engineering and Information Technology, Ph.D. in Theoretical Physics

Senior Member, Institute of Electrical and Electronics Engineers

**Phone**

+61 2 9514 1829

**ORCID**

### 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.

**Can supervise:**Yes

## Books

Tomamichel, M. 2016,

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*Quantum Information Processing with Finite Resources - Mathematical Foundations*, Springer International Publishing.View/Download from: UTS OPUS or Publisher's site

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.

## Conferences

Sutter, D., Tomamichel, M. & Harrow, A.W. 2016, 'Strengthened monotonicity of relative entropy via pinched Petz recovery map',

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*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 R enyi conditional mutual information via composite hypothesis testing against Markov distributions',

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*Proceedings IEEE International Symposium on Information Theory (ISIT 2016)*, IEEE, Barcelona, pp. 585-589.View/Download from: Publisher's site

Berta, M., Fawzi, O. & Tomamichel, M. 2016, 'Exploiting variational formulas for quantum relative entropy',

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*Proceedings IEEE International Symposium on Information Theory (ISIT 2016)*, IEEE, Barcelona, pp. 2844-2848.View/Download from: Publisher's site

Hayashi, M. & Tomamichel, M. 2015, 'Correlation detection and an operational interpretation of the Rényi mutual information',

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*Proceedings IEEE International Symposium on Information Theory (ISIT 2015)*, IEEE, Hong Kong, China, pp. 1447-1451.View/Download from: Publisher's site

Tomamichel, M., Martinez-Mateo, J., Pacher, C. & Elkouss, D. 2014, 'Fundamental Finite Key Limits for Information Reconciliation in Quantum Key Distribution',

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*Proceedings of the IEEE International Symposium on Information Theory (ISIT 2014)*, IEEE, Honolulu, USA, pp. 1469-1473.View/Download from: UTS OPUS or Publisher's site

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., Renner, R., Schaffner, C. & Smith, A. 2010, 'Leftover Hashing against quantum side information',

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*Proceedings of the IEEE Symposium on Information Theory (ISIT 2010)*, IEEE, Austin, USA, pp. 2703-2707.View/Download from: Publisher's site

## Journal articles

Tomamichel, M., Wilde, M.M. & Winter, A. 2017, 'Strong Converse Rates for Quantum Communication',

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*IEEE Transactions on Information Theory*, vol. 63, no. 1, pp. 715-727.View/Download from: Publisher's site

Coles, P.J., Berta, M., Tomamichel, M. & Wehner, S. 2017, 'Entropic uncertainty relations and their applications',

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*Reviews of Modern Physics*, vol. 89, no. 1.View/Download from: UTS OPUS or Publisher's site

Wilde, M.M., Tomamichel, M. & Berta, M. 2017, 'Converse bounds for private communication over quantum channels',

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*IEEE Transactions on Information Theory*, pp. 1-1.View/Download from: Publisher's site

Sutter, D., Berta, M. & Tomamichel, M. 2017, 'Multivariate Trace Inequalities',

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*Communications in Mathematical Physics*, vol. 352, no. 1, pp. 37-58.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',

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*Nature Communications*, vol. 7, pp. 13022-13022.View/Download from: Publisher's site

Hayashi, M. & Tomamichel, M. 2016, 'Correlation detection and an operational interpretation of the Rényi mutual information',

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*Journal of Mathematical Physics*, vol. 57, no. 10, pp. 102201-102201.View/Download from: UTS OPUS or Publisher's site

Datta, N., Tomamichel, M. & Wilde, M.M. 2016, 'On the second-order asymptotics for entanglement-assisted communication',

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*Quantum Information Processing*, vol. 15, no. 6, pp. 2569-2591.View/Download from: Publisher's site

Sutter, D., Tomamichel, M. & Harrow, A.W. 2016, 'Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map',

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*IEEE Transactions on Information Theory*, vol. 62, no. 5, pp. 2907-2913.View/Download from: UTS OPUS or Publisher's site

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, J.M. 2016, 'Quantum coding with finite resources',

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*Nature Communications*, vol. 7, pp. 11419-11419.View/Download from: UTS OPUS or Publisher's site

Berta, M. & Tomamichel, M. 2016, 'The Fidelity of Recovery Is Multiplicative',

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*IEEE Transactions on Information Theory*, vol. 62, no. 4, pp. 1758-1763.View/Download from: UTS OPUS or 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',

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*Nature Communications*, vol. 7, pp. 13022-13022.View/Download from: UTS OPUS or Publisher's site

Tan, V.Y.F. & Tomamichel, M. 2015, 'The Third-Order Term in the Normal Approximation for the AWGN Channel',

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*IEEE Transactions on Information Theory*, vol. 61, no. 5, pp. 2430-2438.View/Download from: Publisher's site

Lunghi, T., Kaniewski, J., Bussières, F., Houlmann, R., Tomamichel, M., Wehner, S. & Zbinden, H. 2015, 'Practical Relativistic Bit Commitment',

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*Physical Review Letters*, vol. 115, no. 3.View/Download from: Publisher's site

Tomamichel, M. & Tan, V.Y.F. 2015, 'Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels',

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*Communications in Mathematical Physics*, vol. 338, no. 1, pp. 103-137.View/Download from: Publisher's site

Lin, M.S. & Tomamichel, M. 2015, 'Investigating Properties of a Family of Quantum Renyi Divergences',

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*Quantum Information Processing*, vol. 14, no. 4, pp. 1501-1512.View/Download from: UTS OPUS or Publisher's site

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.

Furrer, F., Berta, M., Tomamichel, M., Scholz, V.B. & Christandl, M. 2014, 'Position-momentum uncertainty relations in the presence of quantum memory',

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*Journal of Mathematical Physics*, vol. 55, no. 12, pp. 122205-122205.View/Download from: Publisher's site

Kaniewski, J., Tomamichel, M. & Wehner, S. 2014, 'Entropic uncertainty from effective anticommutators',

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*Physical Review A*, vol. 90, no. 1.View/Download from: Publisher's site

Furrer, F., Franz, T., Berta, M., Leverrier, A., Scholz, V.B., Tomamichel, M. & Werner, R.F. 2014, 'Erratum: Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security Against Coherent Attacks [Phys. Rev. Lett. 109, 100502 (2012)]',

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*Physical Review Letters*, vol. 112, no. 1.View/Download from: Publisher's site

Dupuis, F., Szehr, O. & Tomamichel, M. 2014, 'A Decoupling Approach to Classical Data Transmission Over Quantum Channels',

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*IEEE Transactions on Information Theory*, vol. 60, no. 3, pp. 1562-1572.View/Download from: Publisher's site

Tomamichel, M., Berta, M. & Hayashi, M. 2014, 'Relating different quantum generalizations of the conditional Rényi entropy',

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*Journal of Mathematical Physics*, vol. 55, no. 8, pp. 082206-082206.View/Download from: Publisher's site

Tomamichel, M. & Tan, V.Y.F. 2014, 'Second-Order Coding Rates for Channels With State',

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*IEEE Transactions on Information Theory*, vol. 60, no. 8, pp. 4427-4448.View/Download from: Publisher's site

Lim, C.C.W., Portmann, C., Tomamichel, M., Renner, R. & Gisin, N. 2013, 'Device-Independent Quantum Key Distribution with Local Bell Test',

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*Physical Review X*, vol. 3, no. 3.View/Download from: Publisher's site

Szehr, O., Dupuis, F., Tomamichel, M. & Renner, R. 2013, 'Decoupling with unitary approximate two-designs',

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*New Journal of Physics*, vol. 15, no. 5, pp. 053022-053022.View/Download from: Publisher's site

Kaniewski, J., Tomamichel, M., Hanggi, E. & Wehner, S. 2013, 'Secure Bit Commitment From Relativistic Constraints',

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*IEEE Transactions on Information Theory*, vol. 59, no. 7, pp. 4687-4699.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',

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*New Journal of Physics*, vol. 15, no. 10, pp. 103002-103002.View/Download from: Publisher's site

Lunghi, T., Kaniewski, J., Bussières, F., Houlmann, R., Tomamichel, M., Kent, A., Gisin, N., Wehner, S. & Zbinden, H. 2013, 'Experimental Bit Commitment Based on Quantum Communication and Special Relativity',

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*Physical Review Letters*, vol. 111, no. 18.View/Download from: Publisher's site

Tomamichel, M. & Hayashi, M. 2013, 'A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks',

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*IEEE Transactions on Information Theory*, vol. 59, no. 11, pp. 7693-7710.View/Download from: Publisher's site

Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S. & Tomamichel, M. 2013, 'On quantum Rényi entropies: A new generalization and some properties',

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*Journal of Mathematical Physics*, vol. 54, no. 12, pp. 122203-122203.View/Download from: Publisher's site

Vitanov, A., Dupuis, F., Tomamichel, M. & Renner, R. 2013, 'Chain Rules for Smooth Min- and Max-Entropies',

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*IEEE Transactions on Information Theory*, vol. 59, no. 5, pp. 2603-2612.View/Download from: Publisher's site

Tomamichel, M. & Hänggi, E. 2013, 'The link between entropic uncertainty and nonlocality',

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*Journal of Physics A: Mathematical and Theoretical*, vol. 46, no. 5, pp. 055301-055301.View/Download from: Publisher's site

Tomamichel, M. & Tan, V.Y.F. 2013, 'A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels',

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*IEEE Transactions on Information Theory*, vol. 59, no. 11, pp. 7041-7051.View/Download from: Publisher's site

Tomamichel, M., Lim, C.C.W., Gisin, N. & Renner, R. 2012, 'Tight finite-key analysis for quantum cryptography',

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*Nature Communications*, vol. 3, pp. 1-6.View/Download from: UTS OPUS or Publisher's site

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.

Furrer, F., Franz, T., Berta, M., Leverrier, A., Scholz, V.B., Tomamichel, M. & Werner, R.F. 2012, 'Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks',

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*Physical Review Letters*, vol. 109, no. 10.View/Download from: Publisher's site

Winkler, S., Tomamichel, M., Hengl, S. & Renner, R. 2011, 'Impossibility of Growing Quantum Bit Commitments',

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*Physical Review Letters*, vol. 107, no. 9.View/Download from: Publisher's site

Tomamichel, M., Schaffner, C., Smith, A. & Renner, R. 2011, 'Leftover Hashing Against Quantum Side Information',

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*IEEE Transactions on Information Theory*, vol. 57, no. 8, pp. 5524-5535.View/Download from: Publisher's site

Tomamichel, M. & Renner, R. 2011, 'Uncertainty Relation for Smooth Entropies',

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*Physical Review Letters*, vol. 106, no. 11.View/Download from: Publisher's site

Tomamichel, M., Colbeck, R. & Renner, R. 2010, 'Duality Between Smooth Min- and Max-Entropies',

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*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',

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*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, R.G., Kojima, K. & Schwarz, U.T. 2008, 'Analysis of Gain and Luminescence in Violet and Blue GaInN–GaN Quantum Wells',

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*IEEE Journal of Quantum Electronics*, vol. 44, no. 2, pp. 144-149.View/Download from: Publisher's site

## Other

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

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.

Tomamichel, M. & Leverrier, A., 'A largely self-contained and complete security proof for quantum key distribution'.

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
state-of-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. & Hayashi, M., 'Operational Interpretation of Renyi Information Measures via Composite Hypothesis Testing Against Product and Markov Distributions'.

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 various Renyi 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
variations of Renyi mutual information. In case the alternative hypothesis
consists of tripartite distributions satisfying the Markov property, we find
that the optimal exponents are determined by variations of Renyi conditional
mutual information. In either case the relevant notion of Renyi mutual
information depends on the precise choice of the alternative hypothesis. As
such, our work also strengthens the view that different definitions of Renyi
mutual information, conditional entropy and conditional mutual information are
adequate depending on the context in which the measures are used.

Berta, M., Fawzi, O. & Tomamichel, M., 'On Variational Expressions for Quantum Relative Entropies'.

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
valued measures. To do this, we prove that maximizing the Kullback-Leibler
divergence over general positive valued measures results in the measured
relative entropy. Second, we extend the result to R\'enyi relative entropies
and show that for non-commuting states the sandwiched R\'enyi relative entropy
is strictly larger than the measured R\'enyi relative entropy for $\alpha \in
(\frac12, \infty)$, and strictly smaller for $\alpha \in [0,\frac12)$. The
latter statement provides counterexamples for the data-processing inequality of
the sandwiched R\'enyi relative entropy for $\alpha < \frac12$. Our main tool
is a new variational expression for the measured R\'enyi relative entropy,
which we further exploit to show that certain lower bounds on quantum
conditional mutual information are superadditive.

Hiai, F., Koenig, R. & Tomamichel, M., 'Generalized Log-Majorization and Multivariate Trace Inequalities'.

We show that recent multivariate generalizations of the Araki-Lieb-Thirring
inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel,
Comm. Math. Phys. (2016)] 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 regards to logarithmic integral averages
of vectors of singular values.

Cheng, H.-.C., Hsieh, M.-.H. & Tomamichel, M., 'Exponential Decay of Matrix $$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles'.

In the study of Markovian processes, one of the principal achievements is the
equivalence between the $\Phi$-Sobolev inequalities and an exponential decrease
of the $\Phi$-entropies. In this work, we develop a framework of Markov
semigroups on matrix-valued functions and generalize the above equivalence to
the exponential decay of matrix $\Phi$-entropies. This result also specializes
to spectral gap inequalities and modified logarithmic Sobolev inequalities in
the random matrix setting. To establish the main result, we define a
non-commutative generalization of the carr\'e du champ operator, and prove a de
Bruijn's identity for matrix-valued functions.
The proposed Markov semigroups acting on matrix-valued functions have
immediate applications in the characterization of the dynamical evolution of
quantum ensembles. We consider two special cases of quantum unital channels,
namely, the depolarizing channel and the phase-damping channel. In the former,
since there exists a unique equilibrium state, we show that the matrix
$\Phi$-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 $\Phi$-entropy of
the quantum ensemble that undergoes the phase-damping Markovian evolution
generally will not decay exponentially. This is because there are multiple
equilibrium states for such a channel.
Finally, we also consider examples of 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 of the mixing time on these types of examples.

Berta, M., Scholz, V.B. & Tomamichel, M., 'Rényi divergences as weighted non-commutative vector valued $L_p$-spaces'.

We show that Araki and Masuda's weighted non-commutative vector valued
$L_p$-spaces [Araki & Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)]
correspond to an algebraic generalization of the sandwiched R\'enyi divergences
with parameter $\alpha = \frac{p}{2}$. 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
$\alpha$. We thereby also give new proofs for the corresponding
finite-dimensional properties. We discuss the limiting cases $\alpha\to
\{\frac{1}{2},1,\infty\}$ 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 $L_p$-spaces and an Araki-Lieb-Thirring inequality for
states on von Neumann algebras.

Wilde, M.M., Tomamichel, M., Lloyd, S. & Berta, M., 'Gaussian hypothesis testing and quantum illumination'.

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 much
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.

Cheng, H.-.C., Hsieh, M.-.H. & Tomamichel, M., 'Sphere-Packing Bound for Symmetric Classical-Quantum Channels'.

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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. The established pre-factor is
essentially 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.

Chubb, C.T., Tan, V.Y.F. & Tomamichel, M., 'Moderate deviation analysis for classical communication over quantum channels'.

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 Altug and Wagner as well as Polyanskiy and Verdu. 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.

Pfister, C., Rol, M.A., Mantri, A., Tomamichel, M. & Wehner, S., 'Capacity estimation and verification of quantum channels with arbitrarily correlated errors'.

One of the main figures of merit for quantum memories and quantum
communication devices is their quantum capacity. It has been studied for
arbitrary kinds of quantum channels, but its practical estimation has so far
been limited to devices that implement independent and identically distributed
(i.i.d.) quantum channels, where each qubit is affected by the same noise
process. Real devices, however, typically exhibit correlated errors.
Here, we overcome this limitation by presenting protocols that estimate a
channel's one-shot quantum capacity for the case where the device acts on (an
arbitrary number of) qubits. The one-shot quantum capacity quantifies a
device's ability to store or communicate quantum information, even if there are
correlated errors across the different qubits.
We present a protocol which is easy to implement and which comes in two
versions. The first version 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 version verifies on-the-fly that a channel's
one-shot quantum capacity exceeds a minimal tolerated value while storing or
communicating data, therefore combining test qubits and data qubits in one
protocol. We discuss the performance of our method using simple examples, such
as the dephasing channel for which our method is asymptotically optimal.
Finally, we apply our method to estimate the one-shot capacity in an experiment
using a transmon qubit.

Cheng, H.-.C., Hsieh, M.-.H. & Tomamichel, M., 'Quantum Sphere-Packing Bounds with Polynomial Prefactors'.

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 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.

Chapman, R.J., Karim, A., Flammia, S.T., Tomamichel, M. & Peruzzo, A., 'Beating the Classical Limits of Information Transmission using a Quantum Decoder'.

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.

**Selected Peer-Assessed Projects**