Generalized Landauer Bound for Information Processing: Proof and Applications

A generalized form of Landauer’s bound on the dissipative cost of classical information processing in quantum-mechanical systems is proved using a new approach. This approach sidesteps some prominent objections to standard proofs of Landauer’s bound—broadly interpreted here as a nonzero lower bound on the amount of energy that is irreversibly transferred from a physical system to its environment for each bit of information that is lost from the system—while establishing a far more general result. Specializations of our generalized Landauer bound for ideal and non-ideal information processing operations, including but not limited to the simplified forms for erasure and logical operations most familiar from the literature, are presented and discussed. These bounds, taken together, enable reconsideration of the links between logical reversibility, physical reversibility, and conditioning of operations in contexts that include but are far more general than the thermodynamic model systems that are most widely invoked in discussions of Landauer’s Principle. Because of the strategy used to prove the generalized bounds and these specializations, this work may help to illuminate and resolve some longstanding controversies related to dissipation in computation.     

Information erasure in quantum systems

The physical cost of information erasure is considered within a new approach that regards erasure as loss of correlation between the state of an erasable quantum system and that of an enduring “referent” system holding classical information. A physical model of information erasure built on this referential picture is described in detail, and lower bounds on entropic and energetic costs are obtained from quantum dynamics and entropic inequalities alone.     

Landauer’s limit and the physicality of information

Landauer’s lower bound on the dissipative cost of information erasure is revisited within a new physical conception of information. The notion of strong physical information is introduced, and the new conception of physical information – observer-local referential (OLR) information – is defined, shown to be strongly physical, and related to other measures that arise in physical information contexts. A generalization of Landauer’s limit is then obtained for OLR information from quantum dynamics and entropic inequalities alone. Specializations of this bound are compared and contrasted to similar bounds under conditions for which they coincide, and important distinctions between seemingly identical bounds expressed in terms of various information measures are discussed. The controversial distinction between Landauer erasure of known and unknown data – and the alleged difference between their respective erasure costs – is then explored via OLR information. This physically grounds and clarifies distinctions between known and unknown data and between unconditional and conditional erasure operations, enables a straightforward physical accounting of associated lower bounds on erasure costs, and illustrates the advantages of OLR information for resolution of controversies related to the dissipative cost of information erasure. Applications of OLR information to determination of irreversibility induced dissipation bounds in more complex computing scenarios are briefly discussed.     

Quantum-Memory-Assisted Entropic Uncertainty Relation and Quantum Coherence in Structured Reservoir

The uncertainty principle is regarded as one of basics in quantum mechanics, which sets up a strict lower bound to quantify the prediction on the outcome concerning a set of incompatible measurements. In this paper, we investigate the dynamic behaviors of quantum-memory-assisted entropic uncertainty relation (EUR), and quantum coherence in structured reservoir. The results shown that the EUR is smallest in the vanishing limit of noise regardless of the forms of the initial sate we considered, while the coherence keeps the maximal value. During the time-evolution process, the uncertainty bound is lifted and the coherence damps monotonously. Subsequently, the EUR converges to an asymptotic nonzero constant in the long-time limit, yet the coherence asymptotically decays to zero. Moreover, the initial state purity plays a deterministic role in the initial amounts of EUR and coherence, i.e. the larger purity the less EUR and larger coherence. As an application, we employ the EUR to witness the coherence, and prove that the corresponding witnessing efficiencies are only depended on the version of coherence, while are insensitive to the reservoir.

     

Quantum Coherences and Classical Inhomogeneities as Equivalent Thermodynamics Resources

Quantum energy coherences represent a thermodynamic resource, which can be exploited to extract energy from a thermal reservoir and deliver that energy as work. We argue that there exists a closely analogous classical thermodynamic resource, namely, energy-shell inhomogeneities in the phase space distribution of a system’s initial state. We compare the amount of work that can be obtained from quantum coherences with the amount that can be obtained from classical inhomogeneities, and find them to be equal in the semiclassical limit. We thus conclude that coherences do not provide a unique thermodynamic advantage of quantum systems over classical systems, in situations where a well-defined semiclassical correspondence exists.     

Controlling of the Entropic Uncertainty in Open Quantum System

We investigate the dynamics of quantum-memory-assisted entropic uncertainty relations under two typical categories of noise: phase damping channel and depolarizing channel in detail. It shows that, owing to the dissipation, the entropic uncertainty monotonically increases and tends to a steady-state value with the increase of the decoherence in phase damping channel, and can always keep its lower bound during the evolution when the initial state is the maximum entangled state. The larger correlated dephasing rate is favorable for suppressing the amount of entropic uncertainty. In contrast, under the depolarizing channel with memory, the entropic uncertainty always fails to reach its lower bound. Besides, the entropic uncertainty and its lower bound firstly increase with time, then turn down and tend to a steady-state value. The larger correlated decay rate has no benefit to improve the accuracy of quantum measurement. Our investigations might offer an insight into the dynamics of the measurement uncertainty under decoherence, and be important to quantum precision measurement in open systems.

     

Entropic Uncertainty for Two Coupled Dipole Spins Using Quantum Memory under the Dzyaloshinskii–Moriya Interaction

In the thermodynamic equilibrium of dipolar-coupled spin systems under the influence of a Dzyaloshinskii–Moriya (D–M) interaction along the z-axis, the current study explores the quantum-memory-assisted entropic uncertainty relation (QMA-EUR), entropy mixedness and the concurrence two-spin entanglement. Quantum entanglement is reduced at increased temperature values, but inflation uncertainty and mixedness are enhanced. The considered quantum effects are stabilized to their stationary values at high temperatures. The two-spin entanglement is entirely repressed if the D–M interaction is disregarded, and the entropic uncertainty and entropy mixedness reach their maximum values for equal coupling rates. Rather than the concurrence, the entropy mixedness can be a proper indicator of the nature of the entropic uncertainty. The effect of model parameters (D–M coupling and dipole–dipole spin) on the quantum dynamic effects in thermal environment temperature is explored. The results reveal that the model parameters cause significant variations in the predicted QMA-EUR.     

Hidden Dissipation and Irreversibility in Maxwell’s Demon

Maxwell’s demon is an entity in a 150-year-old thought experiment that paradoxically appears to violate the second law of thermodynamics by reducing entropy without doing work. It has increasingly practical implications as advances in nanomachinery produce devices that push the thermodynamic limits imposed by the second law. A well-known explanation claiming that information erasure restores second law compliance fails to resolve the paradox because it assumes the second law a priori, and does not predict irreversibility. Instead, a purely mechanical resolution that does not require information theory is presented. The transport fluxes of mass, momentum, and energy involved in the demon’s operation are analyzed and show that they imply “hidden” external work and dissipation. Computing the dissipation leads to a new lower bound on entropy production by the demon. It is strictly positive in all nontrivial cases, providing a more stringent limit than the second law and implying intrinsic thermodynamic irreversibility. The thermodynamic irreversibility is linked with mechanical irreversibility resulting from the spatial asymmetry of the demon’s speed selection criteria, indicating one mechanism by which macroscopic irreversibility may emerge from microscopic dynamics.     

Tighter Bound of Entropic Uncertainty under the Unruh Effect

The uncertainty principle limits the ability to simultaneously predict measurement outcomes for two non-commuting observables of a quantum particle. However, the uncertainty can be violated by considering a particle as a quantum memory correlated with the primary particle. By modeling an Unruh–Dewitt detector coupled to a massless scalar field, it is explored how the Unruh effect affects the entropic uncertainty and the tighter lower bound for a pair of entangled detectors is probed when one of them is accelerated. It is found that Unruh thermal noise really gives rise to an increase of entropic uncertainty for the given conditions since the correlation between quantum memory and the measured system is decreased. It is shown that the bound of the entropic uncertainty relations, in the presence of memory, can be formulated by introducing the Holevo quantity and mutual information. It is also noticed that Adabi's lower bound is tighter than that of Berta, and just the optimal bound under the Unruh effect. Moreover, it is shown that Berta's lower bound is unrelated to the choice of complementary observables, while the optimal Adabi's lower bound is dependent on the measurement choice. It is worth mentioning that the investigations may offer a better understanding of the entropic uncertainty in a relativistic motion.     

Quantum Thermodynamic Uncertainty Relations,Generalized Current Fluctuations and Nonequilibrium Fluctuation–Dissipation Inequalities

Thermodynamic uncertainty relations (TURs) represent one of the few broad-based and fundamental relations in our toolbox for tackling the thermodynamics of nonequilibrium systems. One form of TUR quantifies the minimal energetic cost of achieving a certain precision in determining a nonequilibrium current. In this initial stage of our research program, our goal is to provide the quantum theoretical basis of TURs using microphysics models of linear open quantum systems where it is possible to obtain exact solutions. In paper [Dong et al., Entropy 2022, 24, 870], we show how TURs are rooted in the quantum uncertainty principles and the fluctuation–dissipation inequalities (FDI) under fully nonequilibrium conditions. In this paper, we shift our attention from the quantum basis to the thermal manifests. Using a microscopic model for the bath’s spectral density in quantum Brownian motion studies, we formulate a “thermal” FDI in the quantum nonequilibrium dynamics which is valid at high temperatures. This brings the quantum TURs we derive here to the classical domain and can thus be compared with some popular forms of TURs. In the thermal-energy-dominated regimes, our FDIs provide better estimates on the uncertainty of thermodynamic quantities. Our treatment includes full back-action from the environment onto the system. As a concrete example of the generalized current, we examine the energy flux or power entering the Brownian particle and find an exact expression of the corresponding current–current correlations. In so doing, we show that the statistical properties of the bath and the causality of the system+bath interaction both enter into the TURs obeyed by the thermodynamic quantities.     

The uncertainty and quantum correlation of measurement in double quantum-dot systems

In this work, we study the entropic uncertainty and quantum discord in two double-quantum-dot (DQD) system coupled via a transmission line resonator (TLR). Explicitly, the dynamics of the systemic quantum correlation and measured uncertainty are analysed with respect to a general X-type state as the initial state. Interestingly, it is found that the different parameters, including the eigenvalue α of the coherent state, detuning amount δ, frequency ω and the coupling constant g, have subtle effects on the dynamics of the entropic uncertainty, such as the oscillation period of the uncertainty. It is clear to reveal that the quantum discord and the lower bound of the entropic uncertainty are anti-correlated when the initial state of the system is the Werner-type state, while quantum discord and the lower bound of the entropic uncertainty are not anti-correlated when the initial state of the system is the Bell-diagonal state. Thereby, we claim that the current investigation would provide an insight into the entropic uncertainty and quantum correlation in DQDs system, and are basically of importance to quantum precision measurement in practical quantum information processing.     

Quantum uncertainty relations of quantum coherence and dynamics under amplitude damping channel

In this paper, we discuss quantum uncertainty relations of quantum coherence through a different method from Ref. [52]. Some lower bounds with parameters and their minimal bounds are obtained. Moreover, we find that for two pairs of measurement bases with the same maximum overlap, quantum uncertainty relations and lower bounds with parameters are different, but the minimal bounds are the same. In addition, we discuss the dynamics of quantum uncertainty relations of quantum coherence and their lower bounds under the amplitude damping channel(ADC). We find that the ADC will change the uncertainty relations and their lower bounds, and their tendencies depend on the initial state.     

Energy Cost of Dynamical Stabilization: Stored versus Dissipated Energy

Dynamical stabilization processes (homeostasis) are ubiquitous in nature, but the needed energetic resources for their existence have not been studied systematically. Here, we undertake such a study using the famous model of Kapitza’s pendulum, which has attracted attention in the context of classical and quantum control. This model is generalized and rendered autonomous, and we show that friction and stored energy stabilize the upper (normally unstable) state of the pendulum. The upper state can be rendered asymptotically stable, yet it does not cost any constant dissipation of energy, and only a transient energy dissipation is needed. Asymptotic stability under a single perturbation does not imply stability with respect to multiple perturbations. For a range of pendulum–controller interactions, there is also a regime where constant energy dissipation is needed for stabilization. Several mechanisms are studied for the decay of dynamically stabilized states.     

Controlling of entropic uncertainty in open quantum system via proper placement of quantum register

We investigate the dynamical behaviors of quantum-memory-assisted entropic uncertainty and its lower bound in the amplitude-damping channel. The influences of different placement positions of the quantum register on the dynamics of quantum coherence, quantum entanglement, and quantum discord are analyzed in detail. The numerical simulation results show that the quantum register should be placed in the channel of the non-Markovian effect. This option is beneficial to reduce the entropic uncertainty and its lower bound. We also find that this choice does not change the evolution of the quantum coherence and quantum entanglement, but changes the dynamical process of the quantum discord of the system.These results show that quantum coherence, quantum entanglement, and quantum discord are different quantum resources with unique characteristics and properties, and quantum discord can play a key role in reducing the uncertainty of quantum systems.     

Correlations,Information Backflow,and Objectivity in a Class of Pure Dephasing Models

We critically examine the role that correlations established between a system and fragments of its environment play in characterising the ensuing dynamics. We employ a dephasing model with different initial conditions, where the state of the initial environment represents a tunable degree of freedom that qualitatively and quantitatively affects the correlation profiles, but nevertheless results in the same reduced dynamics for the system. We apply recently developed tools for the characterisation of non-Markovianity to carefully assess the role that correlations, as quantified by the (quantum) Jensen–Shannon divergence and relative entropy, as well as changes in the environmental state, play in whether the conditions for classical objectivity within the quantum Darwinism paradigm are met. We demonstrate that for precisely the same non-Markovian reduced dynamics of the system arising from different microscopic models, some exhibit quantum Darwinistic features, while others show that no meaningful notion of classical objectivity is present. Furthermore, our results highlight that the non-Markovian nature of an environment does not a priori prevent a system from redundantly proliferating relevant information, but rather it is the system’s ability to establish the requisite correlations that is the crucial factor in the manifestation of classical objectivity.     

Quantum uncertainty relations of Tsallis relative α entropy coherence based on MUBs

In this paper,we discuss quantum uncertainty relations of Tsallis relative α entropy coherence for a single qubit system based on three mutually unbiased bases.For α∈[1/2,1)U(1,2],the upper and lower bounds of sums of coherence are obtained.However,the above results cannot be verified directly for any α∈(0,1/2).Hence,we only consider the special case of α=1/n+1,where n is a positive integer,and we obtain the upper and lower bounds.By comparing the upper and lower bounds,we find that the upper bound is equal to the lower bound for the special α=1/2,and the differences between the upper and the lower bounds will increase as α increases.Furthermore,we discuss the tendency of the sum of coherence,and find that it has the same tendency with respect to the different θ or φ,which is opposite to the uncertainty relations based on the Rényi entropy and Tsallis entropy.     

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states’s hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial’s degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.     

Overwriting information: Correlations,physical costs,and environment models

In this sequel to our previous study of the entropic and energetic costs of information erasure [N.G. Anderson, Phys. Lett. A 372 (2008) 5552], we consider direct overwriting of classical information encoded in a quantum-mechanical memory system interacting with a heat bath. Lower bounds on physical costs of overwriting – in both “single-shot” and “sequential” overwriting scenarios – are obtained from globally unitary quantum dynamics and entropic inequalities alone, all within a referential approach that grounds information content in correlations between physical system states. A heterogeneous environment model, required for consistent treatment of sequential overwriting, is introduced and used to establish and relate bounds for various cases.     

Measurement Uncertainty,Purity, and Entanglement Dynamics of Maximally Entangled Two Qubits Interacting Spatially with Isolated Cavities: Intrinsic Decoherence Effect

In a system of two charge-qubits that are initially prepared in a maximally entangled Bell’s state, the dynamics of quantum memory-assisted entropic uncertainty, purity, and negative entanglement are investigated. Isolated external cavity fields are considered in two different configurations: coherent-even coherent and even coherent cavity fields. For different initial cavity configurations, the temporal evolution of the final state of qubits and cavities is solved analytically. The effects of intrinsic decoherence and detuning strength on the dynamics of bipartite entropic uncertainty, purity and entanglement are explored. Depending on the field parameters, nonclassical correlations can be preserved. Nonclassical correlations and revival aspects appear to be significantly inhibited when intrinsic decoherence increases. Nonclassical correlations stay longer and have greater revivals due to the high detuning of the two qubits and the coherence strength of the initial cavity fields. Quantum memory-assisted entropic uncertainty and entropy have similar dynamics while the negativity presents fewer revivals in contrast.