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Physical Review A

Reversibility-unitality-disturbance tradeoff in quantum channels

Yuan Sun, Nan Li

Published Jan 3, 2024• 4 citations
Publisher

Abstract

In this work, we employ the squared Hilbert-Schmidt norm of the Gram matrix for a quantum channel as a reversibility quantifier of this channel, which is shown to be complementary to the entropy of this channel, and derive a complementary relation between the reversibility of a quantum channel and its complementary channel. For a natural unitality measure of a quantum channel, we show that it is equivalent to the entropy of the corresponding complementary channel. By quantifying the disturbance of a quantum channel as the decrease of correlations in a maximally entangled state locally passing through this channel, we eventually establish a reversibility-unitality-disturbance triality relation. To illustrate and compare these quantities, we further evaluate them for some prototypical channels associated with some special quantum information processing tasks and computations, such as the quantum teleportation channel, DQC1 channel, Mach-Zehnder interferometry channel, dephrasure channel and so on, including both unital and nonunital cases.

DOI
10.1103/physreva.109.012407

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In this work, we employ the squared Hilbert-Schmidt norm of the Gram matrix for a quantum channel as a reversibility quantifier of this channel, which is shown to be complementary to the entropy of this channel, and derive a complementary relation between the reversibility of a quantum channel and its complementary channel. For a natural unitality measure of a quantum channel, we show that it is equivalent to the entropy of the corresponding complementary channel. By quantifying the disturbance of a quantum channel as the decrease of correlations in a maximally entangled state locally passing through this channel, we eventually establish a reversibility-unitality-disturbance triality relation. To illustrate and compare these quantities, we further evaluate them for some prototypical channels associated with some special quantum information processing tasks and computations, such as the quantum teleportation channel, DQC1 channel, Mach-Zehnder interferometry channel, dephrasure channel and so on, including both unital and nonunital cases.
Quantum channels are indispensable instruments for transmitting, extracting, and processing information. According to the spirit of the Bohr complementarity principle and the Heisenberg uncertainty principle, one expects that there are intrinsic tradeoff relations between state disturbance and information gain for any channel, which indeed have been widely studied and characterized from various angles. In this work, we investigate this issue from the perspective of information conservation. More specifically, we divide the information associated with a channel into three categories: transmitted information (quantified by operation fidelity), disturbance (quantified by the Hilbert-Schmidt norm), and extracted information (quantified by the increase of Tsallis-2 entropy), which are all derived naturally from a channel with direct physical motivations. We reveal basic properties of these three information-theoretic quantities and establish a triality relation between them. As applications, we apply these quantities to shed insights into the Mach-Zehnder interferometry and several prototypical channels.
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.
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.
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.
Many important properties of quantum channels are quantified by means of entropic functionals. Characteristics of such a kind are closely related to different representations of a quantum channel. In the Jamiołkowski–Choi representation, the given quantum channel is described by the so-called dynamical matrix. Entropies of the rescaled dynamical matrix known as map entropies describe a degree of introduced decoherence. Within the so-called natural representation, the quantum channel is formally posed by another matrix, obtained as the reshuffling of the dynamical matrix. The corresponding entropies characterize an amount, with which the receiver a priori knows the channel output. As was previously shown, the map and receiver entropies are mutually complementary characteristics. Indeed, there exists a non-trivial lower bound on their sum. First, we extend the concept of receiver entropy to the family of unified entropies. Developing the previous results, we further derive non-trivial lower bounds on the sum of the map and receiver ( q , s )-entropies. The derivation is essentially based on some inequalities with the Schatten norms and anti-norms.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.