Quantum channels often cause information loss of a quantum system, and it is desirable to quantify such loss. Alternatively, one may ask how much information can be preserved by a quantum channel. In this work we introduce a quantifier for information loss of quantum channels via uncertainty of the corresponding Choi states determined by the channel-state duality, reveal its basic properties, derive its ranges for various channels, and identify the extremal channels (which depend crucially on the relation between the input and output system dimensions) with the minimal and maximal information loss. Furthermore, we evaluate this quantifier for some prototypical quantum channels and their corresponding complementary channels. We further compare information loss and irreversibility by illuminating their similarities and differences. These results shed light on channel structures from an information-theoretic perspective.
Quantum channels often cause information loss of a quantum system, and it is desirable to quantify such loss. Alternatively, one may ask how much information can be preserved by a quantum channel. In this work we introduce a quantifier for information loss of quantum channels via uncertainty of the corresponding Choi states determined by the channel-state duality, reveal its basic properties, derive its ranges for various channels, and identify the extremal channels (which depend crucially on the relation between the input and output system dimensions) with the minimal and maximal information loss. Furthermore, we evaluate this quantifier for some prototypical quantum channels and their corresponding complementary channels. We further compare information loss and irreversibility by illuminating their similarities and differences. These results shed light on channel structures from an information-theoretic perspective.
In contrast to unitary evolutions, which are reversible, generic quantum processes (operations and quantum channels) are often irreversible. However, the degree of irreversibility is different for different channels, and it is desirable to have a quantitative characterization of irreversibility. In this paper, by exploiting the channel–state duality implemented by the Jamiołkowski–Choi isomorphism, we quantify the irreversibility of channels via entropy of the Jamiołkowski–Choi states of the corresponding channels and compare it with the notions of entanglement fidelity and entropy exchange. General properties of a reasonable measure of irreversibility are discussed from an intuitive perspective, and entropic measures of irreversibility are introduced. Several relations between irreversibility, entanglement fidelity, the degree of nonunitality, and decorrelating power are established. Some measures of irreversibility for a variety of prototypical channels are evaluated explicitly, revealing some information-theoretic aspects of the structure of channels from the perspective of irreversibility.
Reversible quantum channels play a fundamental role in quantum dynamics and quantum information processing. A quantum channel is reversible if there exists another quantum channel acting as its left inverse. Due to their intrinsic significance and wide applications, it is desirable to characterize reversible quantum channels from diverse perspectives. In this work, we study reversible quantum channels on finite-dimensional Hilbert spaces, with particular emphasis on the case of different input and output dimensions. We systematically present twenty-one equivalent characterizations of reversible quantum channels from algebraic, geometrical, and information-theoretical perspectives. Among these characterizations, some are well known, while others, implicit in the literature or formulated in other contexts, are clarified here; the Choi-state characterization is derived in this work. Specifically, we prove that the Choi states of reversible quantum channels admit three equivalent forms: the spectral, direct-sum, and tensor-product representations. These twenty-one equivalent characterizations establish a comprehensive framework for reversible quantum channels, provide diverse insights into the structural and information-theoretic properties of quantum channels, and facilitate applications of reversibility in quantum information processing such as quantum error correction, quantum teleportation, and quantum thermodynamics.
States and channels are fundamental and instrumental ingredients of quantum mechanics. Their interplay not only encodes information about states but also reflects uncertainties of channels. In order to quantify intrinsic uncertainties generated by channels, we exploit the action of a channel on an orthonormal basis in the space of observables from three different perspectives. The first concerns the uncertainty generated by a channel via noncommutativity between the Kraus operators of the channel and an orthonormal basis of observables, which can be interpreted as a kind of quantifier of the total uncertainty generated by a channel. The second concerns the uncertainty in terms of the Tsallis- 2 entropy of the Jamiołkowski–Choi state associated with the channel via the channel–state duality, which can be interpreted as a quantifier of the classical uncertainty generated by a channel. The third concerns the uncertainty of a channel as the deviation from the identity channel in terms of the Hilbert–Schmidt distance, which can be interpreted as a kind of quantifier of the quantum uncertainty generated by a channel. We reveal basic properties of these quantifiers of uncertainties and establish a relation between them. We identify channels producing the minimal/maximal uncertainties for these three quantifiers. Finally, we explicitly evaluate these uncertainty quantifiers for various important channels, use them to gain insights into the channels from an information-theoretic perspective, and comparatively study the quantifiers.
Informationally complete (IC) measurements are a useful class of measurements, as their outcome statistics uniquely determine an unknown quantum state. Hence, they are important for certain tasks such as quantum state tomography, quantum process tomography, etc. In this work, we study the quantification of informational completeness for arbitrary quantum measurements by introducing and characterizing a faithful measure for it. We explicitly evaluate the informational completeness of qubit symmetric informationally complete (SIC) measurements and show that it is an upper bound for all qubit minimal informationally complete measurements. Furthermore, by introducing a faithful measure, we try to quantify and characterize the ability of an arbitrary quantum channel to preserve the informational completeness of any IC measurement when the channel acts on it in the Heisenberg picture. We call this measure informational completeness-preservability (IC preservability) of quantum channels. After studying its properties, we establish its relation to another quantity, namely, the absolute output coherence of a quantum channel, which quantifies the minimum amount of coherence (w.r.t. an arbitrary incoherent basis) that can always be obtained from the output of that channel. Finally, we illustrate our results with an example from light-matter interaction scenarios. Thus, in this work, not only do we try to provide a quantitative framework for studying both the informational completeness of quantum measurements and the ability of quantum channels to preserve it, but we also try to offer key insight into the conceptual relation between informational completeness and quantum coherence.
Abstract Both irreversibility and incompatibility are important features of quantum channels. Irreversibility of quantum channels characterizes the fundamental limitation in reconstructing information from their outputs. This concept has traditionally been studied for individual channels in a global sense. In this work we generalize the conventional definition of reversible channels to relatively reversible channels, where a quantum channel is reversible relative to another one; thus it is in a relative sense. Incompatibility refers to the impossibility of simultaneously implementing certain pairs of quantum channels, and lies at the very heart of quantum theory. By leveraging the concept of complementary channels, we obtain a direct connection between relative reversibility and channel compatibility. We further propose a quantifier of channel irreversibility in terms of incompatibility between the complementary channels and the identity channel. To illustrate and compare the quantifier of irreversibility with some other quantifiers in the literature, we evaluate them for some prototypical channels. Our results provide insights into the interplay between irreversibility and incompatibility, which may have potential applications in quantum error correction and the resource theory of incompatibility.
Quantitative analysis of discontinuity of information characteristics of quantum states and channels is presented. Estimates for discontinuity jump (loss) of the von Neumann entropy for a given converging sequence of states are obtained. It is shown, in particular, that for any sequence the loss of entropy is upper bounded by the loss of mean energy (with the coefficient characterizing the Hamiltonian of a system). Then we prove that discontinuity jumps of basic measures of classical and quantum correlations in composite quantum systems are upper bounded by the loss of one of the marginal entropies (with a corresponding coefficient). Quantitative discontinuity analysis of the output entropy of a quantum operation and of basic information characteristics of a quantum channel considered as functions of a pair (channel, input state) is presented.