This paper considers the comparison of noisy channels from the viewpoint of statistical decision theory. Various orderings are discussed, all formalizing the idea that one channel is “better” than another for information transmission. The main result is an equivalence relation that is proved for classical channels, quantum channels with classical encoding, and quantum channels with quantum encoding.
This paper considers the comparison of noisy channels from the viewpoint of statistical decision theory. Various orderings are discussed, all formalizing the idea that one channel is “better” than another for information transmission. The main result is an equivalence relation that is proved for classical channels, quantum channels with classical encoding, and quantum channels with quantum encoding.
Two partial orderings among communication channels, namely “being degradable into” and “being less noisy than,” are reconsidered in the light of recent results about statistical comparisons of quantum channels. Though our analysis covers at once both classical and quantum channels, we also provide a separate treatment of classical noisy channels and show how in this case an alternative self-contained proof can be constructed, with its own particular merits with respect to the general result.
Reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones. This is dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one. The Quantum Reverse Shannon Theorem is extensively studied by Bennett and co-authors in [IEEE Trans. Inf. Theory, 2014]. They present two distinct theorems, each tailored to classical and quantum channel simulations respectively, explaining the fact that these theorems remain incomparable due to the fundamentally different nature of correlations they address. The authors leave as an open question the challenge of formulating a unified theorem that could encompass the principles of both and unify them. We unify these two theorems into a single, comprehensive theorem, extending it to the most general case by considering correlations with a general mixed-state reference system. Furthermore, we unify feedback and non-feedback theorems by simulating a general side information system at the encoder side.
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.
The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.
The data processing inequality is the most basic requirement for any meaningful measure of information. It essentially states that distinguishability measures between states decrease if we apply a quantum channel and is the centerpiece of many results in information theory. Moreover, it justifies the operational interpretation of most entropic quantities. In this work, we revisit the notion of contraction coefficients of quantum channels, which provide sharper and specialized versions of the data processing inequality. A concept closely related to data processing is partial orders on quantum channels. First, we discuss several quantum extensions of the well-known less noisy ordering and relate them to contraction coefficients. We further define approximate versions of the partial orders and show how they can give strengthened and conceptually simple proofs of several results on approximating capacities. Moreover, we investigate the relation to other partial orders in the literature and their properties, particularly with regard to tensorization. We then examine the relation between contraction coefficients with other properties of quantum channels such as hypercontractivity. Next, we extend the framework of contraction coefficients to general f-divergences and prove several structural results. Finally, we consider two important classes of quantum channels, namely Weyl-covariant and bosonic Gaussian channels. For those, we determine new contraction coefficients and relations for various partial orders.
Founded by Shannon, information theory deals with the fundamental principles of communication. The two most important questions answered by this theory are how much we can compress a given data source and how much data we can transmit in a given communication channel. Information theory is essentially statistically minded. Data sources are modeled as random processes, and transmission channels are also modeled in probabilistic terms. The theory does not deal with the content of information – it deals with the frequency at which symbols (letters, figures, etc.) appear or are processed but not their meaning. A statistical model is not the only option. Non-statistical theories also exist (e.g., Kolmogorov complexity). However, in this section and throughout this book, we will only use the statistical tools. Information theory is of central importance in quantum cryptography. It may be used to model the transmission of the key elements from Alice to Bob. Note that what may happen on the quantum channel is better described using quantum information theory – see Chapter 4. Yet, the key elements chosen by Alice and those obtained by Bob after measurement are classical values so, for instance, the transmission errors can accurately be modeled using classical information theory. Reconciliation, in particular, requires classical information-theoretic techniques. Source coding Source coding is the first problem that information theory addresses. Assume that a source emits symbols x i from an alphabet χ and that it can be modeled by the random variable X on χ. For instance, the source can be the temperature measured by some meteorological station at regular intervals or the traffic on a network connection.