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.
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.
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.
It is important to distinguish between classical information and quantum information in quantum information theory. In this paper, we first extend the concept of metric-adjusted correlation measure and some related measures to non-Hermitian operators, and establish several relations between the metric-adjusted skew information with different operator monotone functions. By employing operator monotone functions, we next introduce three uncertainty matrices generated by channels: the total uncertainty matrix, the classical uncertainty matrix, and the quantum uncertainty matrix. We establish a decomposition of the total uncertainty matrix into classical and quantum parts and further investigate their basic properties. As applications, we employ uncertainty matrices to quantify the decoherence caused by the action of quantum channels on quantum states, and calculate the uncertainty matrices of some typical channels to reveal certain intrinsic features of the corresponding channels. Moreover, we establish several uncertainty relations that improve the traditional Heisenberg uncertainty relations involving variance.
Quantum uncertainty lies at the heart of quantum mechanics, fundamentally distinguishing it from classical physics. Traditionally quantified through variance-based measures, these approaches often rely on the expectation values of observables, providing a macroscopic statistical view rather than an intrinsic characterization of the wavefunction's inherent indeterminacy. This paper introduces a novel commutator-induced metric designed to directly quantify quantum uncertainty within the wavefunction itself, independent of specific measurement outcomes or a pair of conjugate variables. By leveraging the fundamental non-commutativity of quantum operators, this metric provides a more fundamental and intrinsic measure of the delocalization and inherent indeterminacy encoded in the quantum state. We define this metric based on the expectation value of an operator derived from the self-commutator of the wavefunction's defining operators, mapping the non-commutativity inherent in the Hilbert space onto a scalar value that reflects the degree of quantum uncertainty. This approach offers a new perspective on interpreting the uncertainty principle, providing a generalized framework that extends beyond canonical conjugate pairs and potentially offering insights into the structure of quantum information. The paper details the theoretical construction of this metric, explores its mathematical properties, and discusses its potential applications in quantum information theory, quantum measurement, and the foundational understanding of quantum mechanics.
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.
Coherence of a quantum state (with respect to a fixed incoherent basis) is usually quantified by distancelike quantities, among which a particularly convenient and intuitive quantifier of coherence is based on the Hilbert-Schmidt distance between the quantum state and its dephased counterpart. This quantifier has a simple structure and many nice properties, although it is not a coherence monotone in the conventional resource-theoretic framework of coherence. Here we reveal its information-theoretic significance by showing that it coincides with uncertainty, as quantified by the variance of the state in the incoherent basis. The key point here is to regard the state as an observable, and to regard the incoherent basis as an ensemble of states rather than as measurement operators. Our result provides a natural decomposition of coherence into components along the incoherent basis. Furthermore, in terms of the Tsallis 2-entropy, which is also a measure of uncertainty, we provide two alternative interpretations of coherence: as increase of uncertainty caused by decoherence and as the conditional Tsallis 2-entropy in the context of purification. An intrinsic relation between the maximal coherence and the Brukner-Zeilinger invariant information is also established. These identifications of coherence with increase of uncertainty lead us to interpret coherence as a manifestation of quantum uncertainty, which may have implications for both quantum foundations and applications.