- Min-Hsiu Hsieh

# Quantum Shannon Theory

There are three key parameters in a communication system: (i) the coding length *n*, (i.e., the total number of channel uses); (ii) the transmission rate *R*, (i.e., the amount of information transmitted per channel use); and (iii) the error probability ε(*n*,*R*). These three parameters are interrelated. For example, both the error probability and the transmission rate are expected to be high if a large number of messages are transmitted using small error-correcting codes. Conversely, error performance would foreseeably be good if short messages were protected by large error-correcting codes; however, the transmission rate would be quite low. While such qualitative observations can quickly be made, a good benchmark for a communication system should provide a clear quantitative description of the interplay between these three parameters.

The founder of information theory, Claude Shannon, was the first to make such a quantitative statement. Define the channel capacity *C *to be the supremum rate *R *at which the error probability ε(*n*,*R*) goes to zero when *n *is infinity. Shannon showed that the transmission error could be made arbitrarily close to zero for any rate *R *below the *mutual information *of the channel if a sufficiently long code is used. This result, though elegant and powerful, only provides very rough information about a given channel and the communication system governing it. Specifically, Shannon’s result is an asymptotic statement (i.e., *n *→ ∞). It provides no information about the error probability ε(*n*,*R*) when small/moderate error correction codes (i.e. small/moderate “*n*”) are used or its relationship to the transmission rate *R*. This leads to the fundamental problem – a more precise description of system performance in the finite block-length regime, and was later extensively explored for classical channels. These outcomes significantly advanced the development of modern coding and communication theory.

I have made substantive contribution in this topic. In these endeavours, I have established refined results that work when the communication scenario only involves a finite number of channel uses. This nonasymptotic analysis has the potential to provide quantitative benchmarks for near-future quantum communication system proposals. Many of my corresponding postdoctoral research outcomes now form the backbone of six chapters in the textbook "Quantum Information Theory", Cambridge University Press, 2013. My more recent focus has been on error exponent analysis of quantum information-processing tasks, i.e., finding the trade-off relationships between key parameters in quantum communication systems. The outcomes were accepted in the Quantum Information Processing (QIP) conference in 2018. I have published 20 articles in the most prestigious information theory journal, *IEEE Transactions on Information Theory*, and now serve as an associate editor for this journal.

`Paper`

__Farzin Salek__, Anurag Anshu,**Min-Hsiu Hsieh**, Rahul Jain, Javier R. Fonollosa. One-shot Capacity bounds on the Simultaneous Transmission of Classical and Quantum Information.*IEEE Transactions on Information Theory, vol. 66, no. 4, pp. 2141-2164 (2020).*__Elton Yechao Zhu__,__Quntao Zhuang__,**Min-Hsiu Hsieh**, Peter W. Shor. Superadditivity in trade-off capacities of quantum channels.*IEEE Transactions on Information Theory, vol. 65, no. 6, pp. 3973-3989*(2019).__Hao-Chung Cheng__,**Min-Hsiu Hsieh**, Marco Tomamichel. Quantum Sphere-Packing Bounds with Polynomial Prefactors.*IEEE Transactions on Information Theory, vol. 65, no. 5, pp. 2872-2898*(2019).__Hao-Chung Cheng__,**Min-Hsiu Hsieh**. Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing.*IEEE Transactions on Information Theory,*vol. 64, no. 2, pp. 1385-1403 (2018).__Hao-Chung Cheng__,**Min-Hsiu Hsieh.**Concavity of the Auxiliary Function for Classical-Quantum Channels.*IEEE Transactions on Information Theory*, vol. 62, no. 10, pp. 5960–5965 (2016).Eric Chitambar,

**Min-Hsiu Hsieh**, Andreas Winter. The Private and Public Correlation Cost of Three Random Variables With Collaboration.*IEEE Transactions on Information Theory*, vol. 62, no. 4, pp. 2034–2043 (2016).Nilanjana Datta,

**Min-Hsiu Hsieh**, Jonathan Oppenheim. An upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution.*Journal of Mathematical Physics,*vol. 57, no. 5, p. 052203 (2016).**Min-Hsiu Hsieh**, Shun Watanabe. Channel Simulation and Coded Source Compression.*IEEE Transactions on Information Theory*, vol. 62, no.11, pp. 6609–6619 (2016).Nilanjana Datta, Milan Mosonyi,

**Min-Hsiu Hsieh**, and Fernando G. S. L. Brandao. A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels.*IEEE Transactions on Information Theory*, vol. 59, no. 12, pp. 8014–8026 (2013).Mark Wilde, Nilanjana Datta,

**Min-Hsiu Hsieh**, and Andreas Winter. Quantum rate-distortion coding with auxiliary resources.*IEEE Transactions on Information Theory*, vol. 59, no. 10, pp. 6755–6773 (2013).Nilanjana Datta,

**Min-Hsiu Hsieh**, Mark Wilde, and Andreas Winter. Quantum-to-classical rate distortion coding.*Journal of Mathematical Physics*, vol. 54, no. 4, p. 042201 (2013).Nilanjana Datta and

**Min-Hsiu Hsieh**. One-shot entanglement-assisted quantum and classical communication.*IEEE Transactions on Information Theory*, vol. 59, no. 3, pp. 1929–1939 (2013).Nilanjana Datta,

**Min-Hsiu Hsieh**, and Mark Wilde. Quantum rate distortion, reverse Shannon theorems, and source-channel separation.*IEEE Transactions on Information Theory*, vol. 59, no. 1, pp. 615–630 (2013).Mark Wilde and

**Min-Hsiu Hsieh**. Public and private resource trade-offs for a quantum channel.*Quantum Information Processing*, vol. 11, pp. 1465–1501 (2012).Mark Wilde and

**Min-Hsiu Hsieh**. The quantum dynamic capacity formula of a quantum channel.*Quantum Information Processing*, vol. 11, pp. 1431–1463 (2012).Mark Wilde, Patrick Hayden, Francesco Buscemi, and

**Min-Hsiu Hsieh**. The information-theoretic costs of simulating quantum measurements.*Journal of Physics A: Mathematical and Theoretical*, vol. 45, no. 45, p. 453001 (2012).Nilanjana Datta and

**Min-Hsiu Hsieh**. The apex of the family tree of protocols: Optimal rates and resource inequalities.*New Journal of Physics*, vol. 13, no. 9, p. 093042 (2011).**Min-Hsiu Hsieh**and Mark Wilde. Trading classical communication, quantum communication, and entanglement in quantum Shannon theory.*IEEE Transactions on Information Theory*, vol. 56, no. 9, pp. 4705–4730 (2010).**Min-Hsiu Hsieh**and Mark Wilde. Entanglement-assisted communication of classical and quantum Information.*IEEE Transactions on Information Theory*, vol. 56, no. 9, pp. 4682–4704 (2010).Nilanjana Datta and

**Min-Hsiu Hsieh**. Universal coding for transmission of private information.*Journal of Mathematical Physics*, vol. 51, no. 12, p. 122202 (2010).**Min-Hsiu Hsieh**and Mark Wilde. Public and private communication with a quantum channel and a secret key.*Physical Review A*, vol. 80, no. 2, p. 022306 (2009).**Min-Hsiu Hsieh**, Igor Devetak, and Andreas Winter. Entanglement-assisted capacity of quantum multiple-access channels.*IEEE Transactions on Information Theory*, vol. 54, no. 7, pp. 3078–3090 (2008).