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# 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.
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, Quantum Information Processing with Finite Resources - Mathematical Foundations, Springer International Publishing.
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', Proceedings IEEE International Symposium on Information Theory (ISIT 2016), IEEE, Barcelona, pp. 760-764.
Tomamichel, M. & Hayashi, M. 2016, 'Operational interpretation of R enyi conditional mutual information via composite hypothesis testing against Markov distributions', Proceedings IEEE International Symposium on Information Theory (ISIT 2016), IEEE, Barcelona, pp. 585-589.
Berta, M., Fawzi, O. & Tomamichel, M. 2016, 'Exploiting variational formulas for quantum relative entropy', Proceedings IEEE International Symposium on Information Theory (ISIT 2016), IEEE, Barcelona, pp. 2844-2848.
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.
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, Honolulu, USA, pp. 1469-1473.
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.
Tan, V.Y.F. & Tomamichel, M. 2014, 'The third-order term in the normal approximation for the AWGN channel', IEEE International Symposium on Information Theory - Proceedings, pp. 2077-2081.
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-Verd&uacute; (2010) and matches the upper bound proved by the same authors. &copy; 2014 IEEE.
Tomamichel, M. & Tan, V.Y.F. 2014, 'Second order refinements for the classical capacity of quantum channels with separable input states', IEEE International Symposium on Information Theory - Proceedings, pp. 141-145.
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. &copy; 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, pp. 731-735.
Recently a new quantum generalization of the R&eacute;nyi divergence and the corresponding conditional R&eacute;nyi entropies was proposed. Here we report on a surprising relation between conditional R&eacute;nyi entropies based on this new generalization and conditional R&eacute;nyi entropies based on the quantum relative R&eacute;nyi entropy that was used in previous literature. This generalizes the well-known duality relation H(AB)+H(AC) = 0 for tripartite pure states to R&eacute;nyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional R&eacute;nyi entropies. &copy; 2014 IEEE.
Tomamichel, M. & Tan, V.Y.F. 2013, 'A tight upper bound for the third-order asymptotics of discrete memoryless channels', IEEE International Symposium on Information Theory - Proceedings, pp. 1536-1540.
This paper shows that the logarithm of the -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). &copy; 2013 IEEE.
Tomamichel, M. & Tan, V.Y.F. 2013, 'epsilon-Capacity and Strong Converse for Channels with General State', 2013 IEEE INFORMATION THEORY WORKSHOP (ITW), IEEE Information Theory Workshop (ITW), IEEE, Seville, SPAIN.
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), pp. 609-625.
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. &copy; 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.
Witzigmann, B., Steiger, S., Tomamichel, M., Veprek, R. & Schwarz, U.T. 2007, 'Optical gain in 407nm and 470nm InGaN/GaN heterostructures: Signature of quantum-dot states', Proceedings of SPIE - The International Society for Optical Engineering.
In this contribution, a detailed analysis of optical gain in InGaN/GaN quantum structures with Indium content of 10% and 20% is presented. Experimental data are obtained from Hakki-Paoli characterization of edge-emitting Fabry-Perot lasers. A gain model that includes many-particle effects on a microscopic level, as well as combined quantum-well and quantum-dot density of states, is used to explain the experimental findings. Inhomogeneous broadening arising from local Indium clusters is included via a statistical fluctuation of the electronic density of states. Excellent agreement is obtained for the characteristic gain spectra from structures emitting at 405nm (10% In content) and 470nm (20% In content), and a systematic analysis of the microscopic physics shows signature of quantum-dot states.

## Journal articles

Coles, P.J., Berta, M., Tomamichel, M. & Wehner, S. 2017, 'Entropic uncertainty relations and their applications', Reviews of Modern Physics, vol. 89, no. 1.
&copy; 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.
Wilde, M.M., 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.
&copy; 1963-2012 IEEE. This paper establishes several 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, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and the receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.
Sutter, D., Berta, M. & Tomamichel, M. 2017, 'Multivariate Trace Inequalities', Communications in Mathematical Physics, vol. 352, no. 1, pp. 37-58.
Tomamichel, M., Wilde, M.M. & Winter, A. 2016, 'Strong Converse Rates for Quantum Communication', IEEE Transactions on Information Theory, vol. PP, no. 99.
&copy; 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.
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.
Hayashi, M. & Tomamichel, M. 2016, 'Correlation detection and an operational interpretation of the Rényi mutual information', Journal of Mathematical Physics, vol. 57, no. 10.
A variety of new measures of quantum R&eacute;nyi mutual information and quantum R&eacute;nyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the R&eacute;nyi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternative hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a "useless" channel whose output is decoupled from the environment. Similarly, we establish an operational interpretation of R&eacute;nyi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of R&eacute;nyi mutual information and R&eacute;nyi conditional entropy.
Datta, N., Tomamichel, M. & Wilde, M.M. 2016, 'On the second-order asymptotics for entanglement-assisted communication', Quantum Information Processing, vol. 15, no. 6, pp. 2569-2591.
&copy; 2016, Springer Science+Business Media New York. The entanglement-assisted classical 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 classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical 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 thus completely characterize the convergence toward the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.
Sutter, D., Tomamichel, M. & Harrow, A.W. 2016, 'Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map', IEEE Transactions on Information Theory, vol. 62, no. 5, pp. 2907-2913.
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.', Nat Commun, vol. 7, p. 11419.
The quantum capacity of a memoryless channel determines the maximal rate at which we can communicate reliably over asymptotically many uses of the channel. Here we illustrate that this asymptotic characterization is insufficient in practical scenarios where decoherence severely limits our ability to manipulate large quantum systems in the encoder and decoder. In practical settings, we should instead focus on the optimal trade-off between three parameters: the rate of the code, the size of the quantum devices at the encoder and decoder, and the fidelity of the transmission. We find approximate and exact characterizations of this trade-off for various channels of interest, including dephasing, depolarizing and erasure channels. In each case, the trade-off is parameterized by the capacity and a second channel parameter, the quantum channel dispersion. In the process, we develop several bounds that are valid for general quantum channels and can be computed for small instances.
Berta, M. & Tomamichel, M. 2016, 'The fidelity of recovery is multiplicative', IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 1758-1763.
&copy; 1963-2012 IEEE. Fawzi and Renner recently established a lower bound on the conditional quantum mutual information (CQMI) of tripartite quantum states $ABC$ in terms of the fidelity of recovery (FoR), i.e., the maximal fidelity of the state $ABC$ with a state reconstructed from its marginal $BC$ by acting only on the $C$ system. The FoR measures quantum correlations by the local recoverability of global states and has many properties similar to the CQMI. Here, we generalize the FoR and show that the resulting measure is multiplicative by utilizing semi-definite programming duality. This allows us to simplify an operational proof by Brand&atilde;o et al. of the above-mentioned lower bound that is based on quantum state redistribution. In particular, in contrast to the previous approaches, our proof does not rely on de Finetti reductions.
Pfister, C., Kaniewski, J., Tomamichel, M., Mantri, A., Schmucker, R., McMahon, N., Milburn, G. & Wehner, S. 2016, 'A universal test for gravitational decoherence.', Nat Commun, vol. 7, p. 13022.
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.
Tan, V.Y.F. & 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.
&copy; 1963-2012 IEEE. 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 (1/2) log n + O(1). This improves on the lower bound by Polyanskiys-Poor-Verd&uacute; (2010) and matches the upper bound proved by the same authors.
Lunghi, T., Kaniewski, J., Bussières, F., Houlmann, R., Tomamichel, M., Wehner, S. & Zbinden, H. 2015, 'Practical Relativistic Bit Commitment', Physical Review Letters, vol. 115, no. 3.
&copy; 2015 au. &copy; 2015 American Physical Society. American Physical Society. Bit commitment is a fundamental cryptographic primitive in which Alice wishes to commit a secret bit to Bob. Perfectly secure bit commitment between two mistrustful parties is impossible through an asynchronous exchange of quantum information. Perfect security is, however, possible when Alice and Bob each split into several agents exchanging classical information at times and locations suitably chosen to satisfy specific relativistic constraints. In this Letter we first revisit a previously proposed scheme [C. Cr&eacute;peau et al., Lect. Notes Comput. Sci. 7073, 407 (2011)] that realizes bit commitment using only classical communication. We prove that the protocol is secure against quantum adversaries for a duration limited by the light-speed communication time between the locations of the agents. We then propose a novel multiround scheme based on finite-field arithmetic that extends the commitment time beyond this limit, and we prove its security against classical attacks. Finally, we present an implementation of these protocols using dedicated hardware and we demonstrate a 2 ms-long bit commitment over a distance of 131 km. By positioning the agents on antipodal points on the surface of Earth, the commitment time could possibly be extended to 212 ms.
Tomamichel, M. & Tan, V.Y.F. 2015, 'Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels', Communications in Mathematical Physics, vol. 338, no. 1, pp. 103-137.
&copy; 2015, Springer-Verlag Berlin Heidelberg. We study non-asymptotic fundamental limits for transmitting classical information over memoryless quantum channels, i.e. we investigate the amount of classical information that can be transmitted when a quantum channel is used a finite number of times and a fixed, non-vanishing average error is permissible. In this work we consider the classical capacity of quantum channels that are image-additive, including all classical to quantum channels, as well as the product state capacity of arbitrary quantum channels. In both cases we show that the non-asymptotic fundamental limit admits a second-order approximation that illustrates the speed at which the rate of optimal codes converges to the Holevo capacity as the blocklength tends to infinity. The behavior is governed by a new channel parameter, called channel dispersion, for which we provide a geometrical interpretation.
Lin, M.S. & Tomamichel, M. 2015, 'Investigating Properties of a Family of Quantum Renyi Divergences', Quantum Information Processing, vol. 14, no. 4, pp. 1501-1512.
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', Journal of Mathematical Physics, vol. 55, no. 12.
&copy; 2014 AIP Publishing LLC. A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recen tly, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg's original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.
Kaniewski, J., Tomamichel, M. & Wehner, S. 2014, 'Entropic uncertainty from effective anticommutators', Physical Review A - Atomic, Molecular, and Optical Physics, vol. 90, no. 1.
We investigate entropic uncertainty relations for two or more binary measurements, for example, spin-12 or polarization measurements. We argue that the effective anticommutators of these measurements, i.e., the anticommutators evaluated on the state prior to measuring, are an expedient measure of measurement incompatibility. Based on the knowledge of pairwise effective anticommutators we derive a class of entropic uncertainty relations in terms of conditional R&eacute;nyi entropies. Our uncertainty relations are formulated in terms of effective measures of incompatibility, which can be certified in a device-independent fashion. Consequently, we discuss potential applications of our findings to device-independent quantum cryptography. Moreover, to investigate the tightness of our analysis we consider the simplest (and very well studied) scenario of two measurements on a qubit. We find that our results outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)PRLTAO0031-900710.1103/PhysRevLett.60.1103] and provide an analytical expression for the minimum uncertainty which also outperforms some recent bounds based on majorization. &copy; 2014 American Physical Society.
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 (Physical Review Letters (2012) 109 (100502))', Physical Review Letters, vol. 112, no. 1.
Dupuis, F., Szehr, O. & Tomamichel, M. 2014, 'A decoupling approach to classical data transmission over quantum channels', IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1562-1572.
Most coding theorems in quantum Shannon theory can be proven using the decoupling technique. To send data through a channel, one guarantees that the environment gets no information about it. Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method, sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a dequantizing theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher- Westmoreland theorem to the one-shot scenario, where a quantum channel can be applied only once. &copy; 1963-2012 IEEE.
Tomamichel, M., Berta, M. & Hayashi, M. 2014, 'Relating different quantum generalizations of the conditional Rényi entropy', Journal of Mathematical Physics, vol. 55, no. 8.
&copy; 2014 AIP Publishing LLC. Recently a new quantum generalization of the R&eacute;nyi divergence and the corresponding conditional R&eacute;nyi entropies was proposed. Here, we report on a surprising relation between conditional R&eacute;nyi entropies based on this new generalization and conditional R&eacute;nyi entropies based on the quantum relative R&eacute;nyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to R&eacute;nyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional R&eacute;nyi entropies and derive a new entropic uncertainty relation.
Tomamichel, M. & Tan, V.Y.F. 2014, 'Second-order coding rates for channels with state', IEEE Transactions on Information Theory, vol. 60, no. 8, pp. 4427-4448.
We study the performance limits of state-dependent discrete memoryless channels with a discrete state available at both the encoder and the decoder. We establish the -capacity as well as necessary and sufficient conditions for the strong converse property for such channels when the sequence of channel states is not necessarily stationary, memoryless, or ergodic. We then seek a finer characterization of these capacities in terms of second-order coding rates. The general results are supplemented by several examples including independent identically distributed and Markov states and mixed channels. &copy; 2014 IEEE.
Lim, C.C.W., Portmann, C., Tomamichel, M., Renner, R. & Gisin, N. 2013, 'Device-Independent Quantum Key Distribution with Local Bell Test', Physical Review X, vol. 3, no. 3, pp. 1-11.
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
Szehr, O., Dupuis, F., Tomamichel, M. & Renner, R. 2013, 'Decoupling with unitary approximate two-designs', New Journal of Physics, vol. 15.
Consider a bipartite system, of which one subsystem, A, undergoes a physical evolution separated from the other subsystem, R. One may ask under which conditions this evolution destroys all initial correlations between the subsystems A and R, i.e. decouples the subsystems. A quantitative answer to this question is provided by decoupling theorems, which have been developed recently in the area of quantum information theory. This paper builds on preceding work, which shows that decoupling is achieved if the evolution on A consists of a typical unitary, chosen with respect to the Haar measure, followed by a process that adds sufficient decoherence. Here, we prove a generalized decoupling theorem for the case where the unitary is chosen from an approximate two-design. A main implication of this result is that decoupling is physical, in the sense that it occurs already for short sequences of random two-body interactions, which can be modeled as efficient circuits. Our decoupling result is independent of the dimension of the R system, which shows that approximate two-designs are appropriate for decoupling even if the dimension of this system is large. &copy; IOP Publishing and Deutsche Physikalische Gesellschaft.
Kaniewski, J., Tomamichel, M., Hänggi, E. & Wehner, S. 2013, 'Secure bit commitment from relativistic constraints', IEEE Transactions on Information Theory, vol. 59, no. 7, pp. 4687-4699.
We investigate two-party cryptographic protocols that are secure under assumptions motivated by physics, namely special relativity and quantum mechanics. In particular, we discuss the security of bit commitment in the so-called split models, i.e., models in which at least one of the parties is not allowed to communicate during certain phases of the protocol. We find the minimal splits that are necessary to evade the Mayers-Lo-Chau no-go argument and present protocols that achieve security in these split models. Furthermore, we introduce the notion of local versus global command, a subtle issue that arises when the split committer is required to delegate noncommunicating agents to open the commitment. We argue that classical protocols are insecure under global command in the split model we consider. On the other hand, we provide a rigorous security proof in the global command model for Kent's quantum protocol [1]. The proof employs two fundamental principles of modern physics, the no-signaling property of relativity and the uncertainty principle of quantum mechanics. &copy; 2013 IEEE.
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.
We consider a game in which two separate laboratories collaborate to prepare a quantum system and are then asked to guess the outcome of a measurement performed by a third party in a random basis on that system. Intuitively, by the uncertainty principle and the monogamy of entanglement, the probability that both players simultaneously succeed in guessing the outcome correctly is bounded. We are interested in the question of how the success probability scales when many such games are performed in parallel. We show that any strategy that maximizes the probability to win every game individually is also optimal for the parallel repetition of the game. Our result implies that the optimal guessing probability can be achieved without the use of entanglement. We explore several applications of this result. Firstly, we show that it implies security for standard BB84 quantum key distribution when the receiving party uses fully untrusted measurement devices, i.e. we show that BB84 is one-sided device independent. Secondly, we show how our result can be used to prove security of a one-round position-verification scheme. Finally, we generalize a well-known uncertainty relation for the guessing probability to quantum side information. &copy; IOP Publishing and Deutsche Physikalische Gesellschaft.
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', Physical Review Letters, vol. 111, no. 18.
Bit commitment is a fundamental cryptographic primitive in which Bob wishes to commit a secret bit to Alice. Perfectly secure bit commitment between two mistrustful parties is impossible through asynchronous exchange of quantum information. Perfect security is however possible when Alice and Bob split into several agents exchanging classical and quantum information at times and locations suitably chosen to satisfy specific relativistic constraints. Here we report on an implementation of a bit commitment protocol using quantum communication and special relativity. Our protocol is based on [A. Kent, Phys. Rev. Lett. 109, 130501 (2012)] and has the advantage that it is practically feasible with arbitrary large separations between the agents in order to maximize the commitment time. By positioning agents in Geneva and Singapore, we obtain a commitment time of 15 ms. A security analysis considering experimental imperfections and finite statistics is presented. &copy; 2013 American Physical Society.
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.
We consider two fundamental tasks in quantum information theory, data compression with quantum side information, as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. These characterizations - in contrast to earlier results - enable us to derive tight second-order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree (up to logarithmic terms) with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our technique also naturally yields bounds on operational quantities for finite block lengths. &copy; 1963-2012 IEEE.
Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S. & Tomamichel, M. 2013, 'On quantum Rényi entropies: A new generalization and some properties', Journal of Mathematical Physics, vol. 54, no. 12.
The R&eacute;nyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual informationhave found many applications in information theory and beyond. Various generalizations of R&eacute;nyi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R&eacute;nyi entropies that contains the von Neumann entropy, min-entropy, collision entropythe max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relationan entropic uncertainty relation. &copy; 2013 AIP Publishing LLC.
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.
The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here, we consider the chain rule for the more general smooth min-and max-entropies, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality. However, the standard chain rule for the von Neumann entropy is retrieved asymptotically when evaluating the smooth entropies for many identical and independently distributed states. &copy; 1963-2012 IEEE.
Tomamichel, M. & Hänggi, E. 2013, 'The link between entropic uncertainty and nonlocality', Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 5.
Two of the most intriguing features of quantum physics are the uncertainty principle and the occurrence of nonlocal correlations. The uncertainty principle states that there exist pairs of incompatible measurements on quantum systems such that their outcomes cannot both be predicted. On the other hand, nonlocal correlations of measurement outcomes at different locations cannot be explained by classical physics, but appear in the presence of entanglement. Here, we show that these two fundamental quantum effects are quantitatively related. Namely, we provide an entropic uncertainty relation for the outcomes of two binary measurements, where the lower bound on the uncertainty is quantified in terms of the maximum Clauser-Horne-Shimony-Holt value that can be achieved with these measurements. We discuss applications of this uncertainty relation in quantum cryptography, in particular, to certify quantum sources using untrusted devices. &copy; 2013 IOP Publishing Ltd.
Tomamichel, M. & Tan, V.Y.F. 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.
This paper shows that the logarithm of the -error capacity (average error probability) for n uses of a discrete memoryless channel (DMC) is upper bounded by the normal approximation plus a third-order term that does not exceed 1/2log n +O(1) if the -dispersion of the channel is positive. This matches a lower bound by Y. Polyanskiy (2010) for DMCs with positive reverse dispersion. If the -dispersion vanishes, the logarithm of the -dispersion capacity is upper bounded by n times the capacity plus a constant term except for a small class of DMCs and 1/2. &copy; 1963-2012 IEEE.
Tomamichel, M., Lim, C.C.W., Gisin, N. & Renner, R. 2012, 'Tight finite-key analysis for quantum cryptography', Nature Communications, vol. 3, pp. 1-6.
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', Physical Review Letters, vol. 109, no. 10.
We provide a security analysis for continuous variable quantum key distribution protocols based on the transmission of two-mode squeezed vacuum states measured via homodyne detection. We employ a version of the entropic uncertainty relation for smooth entropies to give a lower bound on the number of secret bits which can be extracted from a finite number of runs of the protocol. This bound is valid under general coherent attacks, and gives rise to keys which are composably secure. For comparison, we also give a lower bound valid under the assumption of collective attacks. For both scenarios, we find positive key rates using experimental parameters reachable today. &copy; 2012 American Physical Society.
Winkler, S., Tomamichel, M., Hengl, S. & Renner, R. 2011, 'Impossibility of growing quantum bit commitments', Physical Review Letters, vol. 107, no. 9.
Quantum key distribution (QKD) is often, more correctly, called key growing. Given a short key as a seed, QKD enables two parties, connected by an insecure quantum channel, to generate a secret key of arbitrary length. Conversely, no key agreement is possible without access to an initial key. Here, we consider another fundamental cryptographic task, commitments. While, similar to key agreement, commitments cannot be realized from scratch, we ask whether they may be grown. That is, given the ability to commit to a fixed number of bits, is there a way to augment this to commitments to strings of arbitrary length? Using recently developed information-theoretic techniques, we answer this question in the negative. &copy; 2011 American Physical Society.
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.
The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, a strictly more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system is shown. Our result applies to almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing. &copy; 2011 IEEE.
Tomamichel, M. & Renner, R. 2011, 'Uncertainty relation for smooth entropies', Physical Review Letters, vol. 106, no. 11.
Uncertainty relations give upper bounds on the accuracy by which the outcomes of two incompatible measurements can be predicted. While established uncertainty relations apply to cases where the predictions are based on purely classical data (e.g., a description of the system's state before measurement), an extended relation which remains valid in the presence of quantum information has been proposed recently [Berta et al., Nature Phys.NPAHAX1745-2473 6, 659 (2010)10.1038/nphys1734] . Here, we generalize this uncertainty relation to one formulated in terms of smooth entropies. Since these entropies measure operational quantities such as extractable secret key length, our uncertainty relation is of immediate practical use. To illustrate this, we show that it directly implies security of quantum key distribution protocols. Our security claim remains valid even if the implemented measurement devices deviate arbitrarily from the theoretical model. &copy; 2011 American Physical Society.
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.
In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this paper, a recently discovered duality relation between (nonsmooth) min- and max-entropies is extended to the smooth case. More precisely, it is shown that the smooth min-entropy of a system A conditioned on a system B equals the negative of the smooth max-entropy of A conditioned on a purifying system C. This result immediately implies that certain operational quantities (such as the amount of compression and the amount of randomness that can be extracted from given data) are related. We explain how such relations have applications in cryptographic security proofs. &copy; 2010 IEEE.
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.
The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, a fully quantum generalization of this property is shown, where both the output of the experiment and side information are quantum. An explicit bound on the convergence is given, which is independent of the dimensionality of the side information. This naturally leads to a family of R&eacute;nyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case. &copy; 2009 IEEE.
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', IEEE Journal of Quantum Electronics, vol. 44, no. 2, pp. 144-149.
In this paper, gain in GaInN quantum wells with 8% and 19% indium is analyzed using a comparison of a microscopic model to experimental data. It is shown that localized valence states can explain the characteristics of the gain spectra, in particular the broadening features at the red side of the spectrum. From an analysis of experimental and simulation data, the nonradiative current component is extracted, and is shown to dominate the total current density at laser threshold operation. The increase of nonradiative current with density explains the drop in internal quantum efficiency in GaInN light-emitting diodes. &copy; 2007 IEEE.
Fang, K., Wang, X., Tomamichel, M. & Duan, R., 'Non-asymptotic entanglement distillation'.
Non-asymptotic entanglement distillation studies the trade-off between three parameters: the distillation rate, the number of independent and identically distributed prepared states, and the fidelity of the distillation. We first study the one-shot {\epsilon}-infidelity distillable entanglement under quantum operations that completely preserve positivity of the partial transpose (PPT) and characterize it as a semidefinite program (SDP). For isotropic states, it can be further simplified to a linear program. The one-shot {\epsilon}-infidelity PPT-assisted distillable entanglement can be transformed to a quantum hypothesis testing problem. Moreover, we show efficiently computable second-order upper and lower bounds for the non-asymptotic distillable entanglement with a given infidelity tolerance. Utilizing these bounds, we obtain the second order asymptotic expansions of the optimal distillation rates for pure states and some classes of mixed states. In particular, this result recovers the second-order expansion of LOCC distillable entanglement for pure states in [Datta/Leditzky, IEEE Trans. Inf. Theory 61:582, 2015]. Furthermore, we provide an algorithm for calculating the Rains bound and present direct numerical evidence (not involving any other entanglement measures, as in [Wang/Duan, arXiv:1605.00348]), showing that the Rains bound is not additive under tensor products.

## Other

Tomamichel, M. & Leverrier, A. 2017, 'A largely self-contained and complete security proof for quantum key distribution'.
Hiai, F., König, R. & Tomamichel, M. 2017, 'Generalized Log-Majorization and Multivariate Trace Inequalities'.
&copy; 2017, Springer International Publishing. We show that recent multivariate generalizations of the Araki&#8211;Lieb&#8211;Thirring inequality and the Golden&#8211;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.
Chubb, C.T., Tan, V.Y.F. & Tomamichel, M. 2017, 'Moderate Deviation Analysis for Classical Communication over Quantum Channels'.
&copy; 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 Alt&#365;g and Wagner as well as Polyanskiy and Verd&uacute;. 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.
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. & 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.
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'.
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.