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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.
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.

## Journal articles

Tomamichel, M., Wilde, M.M. & Winter, A. 2017, 'Strong Converse Rates for Quantum Communication', IEEE Transactions on Information Theory, vol. 63, no. 1, pp. 715-727.
Coles, P.J., Berta, M., Tomamichel, M. & Wehner, S. 2017, 'Entropic uncertainty relations and their applications', Reviews of Modern Physics, vol. 89, no. 1.
Wilde, M.M., Tomamichel, M. & Berta, M. 2017, 'Converse bounds for private communication over quantum channels', IEEE Transactions on Information Theory, pp. 1-1.
Sutter, D., Berta, M. & Tomamichel, M. 2017, 'Multivariate Trace Inequalities', Communications in Mathematical Physics, vol. 352, no. 1, pp. 37-58.
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, pp. 102201-102201.
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.
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', Nature Communications, vol. 7, pp. 11419-11419.
Berta, M. & Tomamichel, M. 2016, 'The Fidelity of Recovery Is Multiplicative', IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 1758-1763.
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.
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.
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.
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.
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, pp. 122205-122205.
Kaniewski, J., Tomamichel, M. & Wehner, S. 2014, 'Entropic uncertainty from effective anticommutators', Physical Review A, vol. 90, no. 1.
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)]', 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.
Tomamichel, M., Berta, M. & Hayashi, M. 2014, 'Relating different quantum generalizations of the conditional Rényi entropy', Journal of Mathematical Physics, vol. 55, no. 8, pp. 082206-082206.
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.
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.
Szehr, O., Dupuis, F., Tomamichel, M. & Renner, R. 2013, 'Decoupling with unitary approximate two-designs', New Journal of Physics, vol. 15, no. 5, pp. 053022-053022.
Kaniewski, J., Tomamichel, M., Hanggi, E. & Wehner, S. 2013, 'Secure Bit Commitment From Relativistic Constraints', IEEE Transactions on Information Theory, vol. 59, no. 7, pp. 4687-4699.
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, no. 10, pp. 103002-103002.
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.
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.
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, pp. 122203-122203.
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.
Tomamichel, M. & Hänggi, E. 2013, 'The link between entropic uncertainty and nonlocality', Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 5, pp. 055301-055301.
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.
Tomamichel, M., Lim, C.C.W., Gisin, N. & Renner, R. 2012, 'Tight finite-key analysis for quantum cryptography', Nature Communications, vol. 3, pp. 634-634.
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.
Winkler, S., Tomamichel, M., Hengl, S. & Renner, R. 2011, 'Impossibility of Growing Quantum Bit Commitments', Physical Review Letters, vol. 107, no. 9.
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.
Tomamichel, M. & Renner, R. 2011, 'Uncertainty Relation for Smooth Entropies', Physical Review Letters, vol. 106, no. 11.
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.
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.
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.

## 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'.
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.