Entropic Uncertainty Relations Research Articles

Entropic uncertainty relations and entanglement detection from quantum designs

Abstract Uncertainty relations and quantum entanglement are pivotal concepts in quantum theory. Beyond their fundamental significance in shaping our understanding of the quantum world, they also underpin crucial applications in quantum information theory. In this article, we investigate entropic uncertainty relations and entanglement detection with an emphasis on quantum measurements with design structures. On the one hand, we derive improved Rényi entropic uncertainty relations for design-structured measurements, exploiting the property that the sum of powered (e.g., squared) probabilities of obtaining different measurement outcomes is now invariant under unitary transformations of the measured system and can be easily computed. On the other hand, the above property essentially imposes a state-independent upper bound, which is achieved at all pure states, on one’s ability to predict local outcomes when performing a set of design-structured measurements on quantum systems. Realizing this, we also obtain criteria for detecting multi-partite entanglement with design-structured measurements.

Entropic uncertainty relations and quantum coherence in the two-dimensional XXZ spin model with Dzyaloshinskii–Moriya interaction

Quantum renormalization group (QRG) is a tractable method for studying the criticalities of one-dimensional (1D) and two-dimensional (2D) many-body systems. By employing the QRG method, we first derive the effective Hamiltonian and QRG equations of a 2D XXZ model with Dzyaloshinskii–Moriya (DM) interaction analytically. The linear-entropy-based uncertainty, the quantum discord (QD), and the multipartite quantum coherence based on the square root of the quantum Jensen–Shannon divergence of the 2D XXZ model are then studied as the indicators of quantum phase transitions (QPTs). The nonanalytic and scaling behaviors of the uncertainty, QD and quantum coherence are also analyzed through numerical calculations. Moreover, we investigate the effect of the easy-axis anisotropy parameter and DM interaction on the QPT. We find that the uncertainty, QD, and quantum coherence can all be utilized to detect QPTs. Our findings could shed new light on the observable of the QPT of the many-body system with the uncertainty and quantum coherence, and enrich the application of QRG method to Heisenberg spin models.

Entropic distinguishability of quantum fields in phase space

We present a general way of quantifying the entropic uncertainty of quantum field configurations in phase space in terms of entropic distinguishability with respect to the vacuum. Our approach is based on the functional Husimi Q-distribution and a suitably chosen relative entropy, which we show to be non-trivially bounded from above by the uncertainty principle. The resulting relative entropic uncertainty relation is as general as the concept of coherent states and thus holds for quantum fields of bosonic and fermionic type. Its simple form enables diverse applications, among which we present a complete characterization of the uncertainty surplus of arbitrary states in terms of the total particle number for a scalar field and the fermionic description of the Ising model. Moreover, we provide a quantitative interpretation of the role of the uncertainty principle for quantum phase transitions.

Entropic uncertainty relations in Schwarzschild space-time

The uncertainty principle is deemed as one of cornerstones in quantum mechanics, and exploring its lower limit of uncertainty will be helpful to understand the principle's nature. In this study, we propose a generalized entropic uncertainty relation for arbitrary multiple-observable in multipartite system, and further derive a tighter lower bound by considering Holevo quality and mutual information. Importantly, we detailedly discuss the proposed uncertainty relations and quantum coherence in the context of Schwarzschild space-time. It is interesting to find that Hawking radiation will damage the coherence of the physically accessible region and increase the uncertainty. Furthermore, we argue that the properties of the uncertainty in Schwarzschild space-time can be explained from the systems' purity and the information redistribution of the different regions. Therefore, it is believed that our findings provide the generalized entropic uncertainty relations in multipartite systems, which may facilitate us deeper understanding of quantumness and information paradox of the black holes.

Proving the Security of Mediated Semi‐Quantum Key Distribution Using Entropic Uncertainty Relation

AbstractIn recent years, mediated semi‐quantum key distribution (MSQKD) has become a hot topic in quantum cryptography. In this study, the original MSQKD protocol is revisited and a new scheme for proving security based on information theory is developed. At first, a new bound on the key rate of the protocol is derived using an entropic uncertainty relation, thus proving the unconditional security of the protocol. In addition, in the asymptotic scenario, a higher noise tolerance that improves the previous results is found. The legitimate communicating parties have to abort the protocol when they observe the error rate is larger than the noise tolerance. Furthermore, the security of a single‐state MSQKD protocol and a single‐state semi‐quantum key distribution (SQKD) protocol is proven using a similar scheme.

Maximally Entangled Squeezed Spin Coherent States and Entropic Uncertainty Relation

Maximally Entangled Squeezed Spin Coherent States and Entropic Uncertainty Relation

The entropic uncertainty preservation in non-Markovian environment via classical driving fields

Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. In quantum information theory, the preferred mathematical quantity to express the entropic uncertainty relation is the Shannon’s entropy. In this work, we investigate the effect of a classical driving field on entropic uncertainty lower bound (EULB). In this regard, we consider a particle as the quantum memory correlated with a measured particle, where during the interaction with the environment, the quantum memory is driven by the external classical field. Our result reveals that in the presence of the classical driving fields, the EULB is reduced and consequently, the measurement accuracy is increased.

Evaluating extractable work of quantum batteries via entropic uncertainty relations.

In this study, we investigate the effectiveness of entropic uncertainty relations (EURs) in discerning the energy variation in quantum batteries (QBs) modelled by battery-charger field in the presence of bosonic and fermionic reservoirs. Our results suggest that the extractable works (exergy and ergotropy) have versatile characteristics in different scenarios, resulting in a complex relationship between tightness and extractable work. It is worth noting that the tightness of the lower bound of entropic uncertainty can be a good indicator for energy conversion efficiency in charging QBs. Furthermore, we disclose how the EUR including uncertainty and lower bound contributes to energy conversion efficiency in the QB system. It is believed that these findings will be beneficial for better understanding the role of quantum uncertainty in evaluating quantum battery performance.

Tight Bounds from Multiple‐Observable Entropic Uncertainty Relations

AbstractThe additivity properties for both bipartite and multipartite systems are investigated by using entropic uncertainty relations (EUR) defined in terms of the joint Shannon entropy of probabilities of local measurement outcomes. In particular, state‐independent and state‐dependent entropic inequalities are introduced. Interestingly, the violation of these inequalities is strictly connected with the presence of quantum correlations. It is shown that the additivity of EUR holds only for EUR that involve two observables, while this is not the case for inequalities that consider more than two observables or the addition of the von Neumann entropy of a subsystem. They are applied to bipartite systems and to several classes of states of a three‐qubit system.

Entropic uncertainty relations for measurements assigned to a projective two-design

The current study aims to examine uncertainty relations for quantum measurements assigned to a projective two-design. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are important cases of such measurements. To characterize the amount of uncertainty, we use the Tsallis and Rényi entropies as well as the probabilities of separate outcomes. The obtained results are based on an estimation of the index of coincidence. They improve some uncertainty relations given in the literature.

Thermal entanglement versus quantum-memory-assisted entropic uncertainty relation in a two-qubit Heisenberg system with Herring–Flicker coupling under Dzyaloshinsky–Moriya interaction

Thermal entanglement versus quantum-memory-assisted entropic uncertainty relation in a two-qubit Heisenberg system with Herring–Flicker coupling under Dzyaloshinsky–Moriya interaction

The influence of a magnetic field on Shannon entropy and thermal properties in graphene

We conducted an investigation into the impact of an external magnetic field on graphene under two proposed conditions: a constant magnetic field (ω), and a field characterized by exponential decay (μ). The resulting eigenvalue solutions were utilized to analyze the Shannon entropy of low-lying states and thermodynamic properties. Regarding Shannon entropy, we observed that both conditions led to increased fermion localization in position space and delocalization in momentum space. Furthermore, we calculated the BBM entropic uncertainty relation and confirmed its validity in one dimension. As for the thermodynamic properties, we found that augmenting the influence of the magnetic fields altered their respective behaviors.

Coherence versus quantum-memory-assisted entropic uncertainty relation of double quantum dots with Rashba spin–orbit interaction

Coherence versus quantum-memory-assisted entropic uncertainty relation of double quantum dots with Rashba spin–orbit interaction

Does conditional entropy squeezing indicate normalized entropic uncertainty relation steering?

Does conditional entropy squeezing indicate normalized entropic uncertainty relation steering?

Entropic characterization of stabilizer states and magic states

Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.

Effects of detuning on entropic uncertainty bound and quantum correlations in dissipative environments

One of the fundamental arguments in quantum information theory is the uncertainty principle. According to this principle, two incompatible observables cannot be measured with high precision at the same time. In this work, we will use the entropic uncertainty relation in the presence of quantum memory. Considering a dissipative environment, the effects of the detuning between the transition frequency of a quantum memory and the center frequency of a cavity on entropic uncertainty bound and the quantum correlation between quantum memory and measured particle will be studied. It is shown that by increasing the detuning, quantum correlation is maintained. As a result, due to the inverse relationship between the uncertainty bound and quantum correlations, the measurement results are guessed more accurately.

Stronger entropic uncertainty relations with multiple quantum memories

The uncertainty relation is the basic principle of quantum mechanics, limiting the information obtained when two arbitrary noncommutative measurements are made. In principle, the lower bound of uncertainty relation indicates the limit of uncertainty, therefore to obtain a tight and essential one is basically on demand and nontrivial. In this work, we propose a stronger tripartite entropic uncertainty relation with quantum memories by considering mutual information and Holevo quantity, being tighter than the existed ones. Besides, we generalize the tripartite scenario into a universal case, i.e., multipartite and multi-observable entropic uncertainty relation. Furthermore, the proposed tripartite EUR is well applied to security analysis of quantum key distribution (QKD) protocols and quantum randomness, attaining higher quantum secret key rates in QKD and to guarantee the desired randomness. Thus, it is believed that our investigations better shed light on the nature of the uncertainty, and are beneficial to practical quantum information processing.

Entropic uncertainty relations for multiple measurements assigned with biased weights

Entropic uncertainty relations for multiple measurements assigned with biased weights

Ascertaining the influences of auxiliary qubits on the Einstein–Podolsky–Rosen steering and its directions

Einstein–Podolsky–Rosen (EPR) steering is an example of nontrivial quantum nonlocality and characteristic in the non-classical world. The directivity (or asymmetry) is a fascinating trait of EPR steering, and it is different from other quantum nonlocalities. Here, we consider the strategy in which two atoms compose a two-qubit X state, and the two atoms are owned by Alice and Bob, respectively. The atom of Alice suffers from a reservoir, and the atom of Bob couples with a bit flip channel. The influences of auxiliary qubits on EPR steering and its directions are revealed by means of the entropy uncertainty relation. The results indicate that EPR steering declines with growing time t when adding fewer auxiliary qubits. The EPR steering behaves as damped oscillation when introducing more auxiliary qubits in the strong coupling regime. In the weak coupling regime, the EPR steering monotonously decreases as t increases when coupling auxiliary qubits. The increases in auxiliary qubits are responsible for the fact that the steerability from Alice to Bob (or from Bob to Alice) can be more effectively revealed. Notably, the introductions of more auxiliary qubits can change the situation that steerability from Alice to Bob is certain to a situation in which steerability from Bob to Alice is certain.

Particle number and interactions in the entropic uncertainty relations

The dependence of the entropic uncertainty relations on particle number and on attractive and repulsive interparticle potential strengths, is examined in the ground state of coupled harmonic oscillators. The reduced entropy sums depend on the normalization constants of the position and momentum densities, which in turn depend on particle number and interaction strength. These sums exhibit maxima as the particle number is varied, when the particles are in the presence of attractive potentials. On the other hand, no maxima are observed for systems with a repulsive potential. All reduced entropy sums increase and move away from the bound with greater interactions. Consistent behaviours are observed for both reduced one- and two-variable entropy sums. Further reductions of the N-particle densities induce that the respective entropy sums are now farther away from the corresponding bounds. The interpretation of these results is that reduction and increased interaction translates into a weakening of the strength of the uncertainty principle statement. The one- and two-variable entropy sums of these systems are shown to approximate the relation, when the intensity of the one-body harmonic potential is set at the number of particles.