Quantum Systems, Channels, Information: A Mathematical Introduction

Alexander Semenovich Holevo

Written by one of the founding fathers of Quantum Information, this book gives an accessible (albeit mathematically rigorous), self-contained introduction to quantum information theory. The central role is played by the concept of quantum channel and its entropic and information characteristics. In this revised edition, the main results have been updated to reflect the most recent developments in this very active field of research.

Teodor Strömberg

Daniel Bonior

Quantum information has the potential to revolutionize the way we store, process, transfer and acquire information [1,14,15,21,37]. In particular, quantum information offers exciting new approaches to secure communication, computation and sensing. However, in order to realize such technologies, we must first understand the effect that environmental noise has on a quantum system. This dissertation builds upon recent studies that have explored the underlying structure of quantum information and the effects of qubit channels in quantum communication protocols. This work is divided into five main chapters, with Chapter 1 being a brief introduction to quantum information. We then begin Chapter 2 by defining the error function for our qubit communication protocols. From there we explore the properties of our error functions and the topological space that they form. In Chapter 3 we consider the newly patented process Adaptive Quantum Information Processing, patent number US9838141 B2; originally outlined by Martin in [23]. We restate the adaptive scheme and exemplify its application through the Prepare and Send Protocol and Quantum Key Distribution. Applying our results from Chapter 2, we obtain an expression for the adaptability of unital channels in these two protocols and classify the channels that admit the most improvement. We dedicate Chapter 4 to the derivation of gravitational noise, and show that in certain circumstances gravity results in a channel that can be maximally improved in Adaptive QKD [3,14,16]. Lastly, we study the set of error functions through the lens of domain theory. Domain theory is a subset of mathematics that was developed in order to rigorously formalize computations. The first four chapters are all consequences of past discoveries in the mathematical structure of quantum channels. In Chapter 5 we characterize the set of error functions through domain theory, extending the mathematical foundations of quantum information. [12,18,20, 22, 23,25].

Mark M. Wilde

Mark M. Wilde

Harjas Sandhu

How do you decide which quantum channel keeps more information? There is more than one way to ask. You can compare how well two states remain distinguishable, how much information an ensemble retains, or how sensitively an output responds to a small change. These are different mathematical questions. This paper asks when they give the same ordering of quantum channels. The comparison is complete: an input may share correlations with a separate, finite quantum reference, and the reference can be as large as needed. That extra system makes the question richer than looking at the channel output alone. For finite-dimensional channels, the manuscript claims an equivalence among complete comparisons expressed through relative entropy, Holevo information, BKM and SLD Fisher information, and measured chi-squared divergence. It also studies the complementary channel—the part of a Stinespring description that goes to the environment—and claims that the information order reverses under genuine complementation. The second part asks whether these comparisons can be decided by an algorithm. The paper claims coRE-completeness for a class of strictly positive rational-complex measurements, and that no computable bound on reference size suffices for the stated SLD comparison. It also claims fixed-additive-accuracy inapproximability for a bounded entropy defect. These are claims about precisely specified mathematical decision problems, not a claim that every practical channel comparison is impossible. The companion paper, From complete channel order to capacity and one-way distillation, explores consequences for communication and distillation using the order results developed here. See the manuscript page for a further introduction and citation details.