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.
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.
Abstract Two quantum channels are called compatible if they can be obtained as marginals from a single broadcasting channel; otherwise they are incompatible. We derive a characterization of the compatibility relation in terms of concatenation and conjugation, and we show that all pairs of sufficiently noisy quantum channels are compatible. The complement relation of incompatibility can be seen as a unifying aspect for several important quantum features, such as impossibility of universal broadcasting and unavoidable measurement disturbance. We show that the concepts of entanglement breaking channel and antidegradable channel can be completely characterized in terms compatibility.
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.
Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.
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.
Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the notion of parallel compatibility and show that these two notions are inequivalent. We then argue that the notion of traditional compatibility is conceptually incomplete and prove that, while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not capture channel compatibility. Hence, we propose parallel compatibility as the conceptually complete definition of the compatibility of quantum instruments.
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.