Entropic Uncertainty Relations Research Articles

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.

Quantum memory and entanglement dynamics induced by interactions of two moving atoms with a coherent cavity

This study explores the dynamics of atomic entanglement of formation and entropic uncertainty relation (EUR) of two moving atoms interacting with an even coherent cavity during a two-photon interaction. We investigate the impact of the initial even coherent amplitude, the atom–photon coupling strengths, and the atomic motion coupling on the dynamics of quantum memory (QM) entropic uncertainty and entanglement. The dynamics of the entropic uncertainty and entanglement of formation are significantly altered by the atom-photon-atom interaction parameters. By increasing the initial mean photon number, the QM entropic uncertainty and the atomic entanglement can be improved. Bob’s capacity to limit his uncertainty regarding Alice’s measurement results depends on the initial even coherent state, the atom–photon couplings, and the atomic motion parameter. The atomic location couplings produce QM entropic uncertainty and entanglement with regular oscillatory behavior. The system parameters control the sudden death-birth entanglement phenomenon, which is improved by increasing the atomic location coupling.

Coherent States of the $$p$$-Adic Heisenberg Group and Entropic Uncertainty Relations

Coherent States of the $$p$$-Adic Heisenberg Group and Entropic Uncertainty Relations

The relationship between the degree of violation of Bell-CHSH inequality and measurement uncertainty in two classes of spin squeezing mechanisms

In solid-state systems, spin squeezing states have acquired much interest in quantum information processing since they are not only essential from a basic physics standpoint but also have many applications, notably reducing noise in interferometry. This paper addresses the study of quantum nonlocality as examined by the violation of the Bell and Clauser–Horne–Shimony–Holt (CHSH) inequality in two classes of spin squeezing states and how this relates to the quantum-memory-assisted entropic uncertainty relation (QMA–EUR). The spin squeezing states in the two classes are systematically generated by two twisting nonlinear interactions. These separately describe two spin-[Formula: see text] interaction mechanisms that undergo one-axis twisting (OAT) and two-axis countertwisting (TAC). We investigate each spin squeezing mechanism in a thermal bath ([Formula: see text]) with a transverse magnetic field. Our main results indicate that the degree of violation of the Bell-CHSH inequality and the entropic uncertainty relation (EUR) is typically anticorrelated in both mechanisms. We demonstrate that the spin squeezing states in the OAT mechanism violate the Bell-CHSH inequality and thus exhibit Bell nonlocality only at low temperatures ([Formula: see text]), whereas in the TAC mechanism, increasing the strength of the spin squeezing interaction leads to a violation of the Bell-CHSH inequality even at finite temperatures. We have also shown that there is always an expansion of the EUR and a decrease in the degree of violation of the Bell-CHSH inequality in the high-temperature regime. In addition, the transverse magnetic field exhibits different behavior under the considered temperature regimes in the two classes of spin squeezing states. Other results will also be discussed.

Sequential sharing of two-qudit entanglement based on the entropic uncertainty relation

Sequential sharing of two-qudit entanglement based on the entropic uncertainty relation

Entropic uncertainty relations with quantum memory in a multipartite scenario

Entropic uncertainty relations with quantum memory in a multipartite scenario

Measurement-Device-Independent Quantum Key Distribution

The security of the measurement-device-independent quantum key distribution protocol through untrusted nodes has been directly proven using fundamental entropy uncertainty relations. Reasons for the amazing coincidence of the expressions for the secrete key length in the BB84 and MDI protocols are presented.

Uncertainty relations from graph theory

Quantum measurements are inherently probabilistic and quantum theory often forbids precisely predicting the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. For arbitrary observables, we obtain an upper bound on the sum of the square of their expectation values. We furthermore show how this bound behaves when the observables are imprecisely calibrated. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria, and entanglement witnesses.

Entropic uncertainty relations and the fisher information for the Generalized radial Yukawa potential

The Heisenberg uncertainty relation as well as the Fisher information are presented analytically and numerically for the Generalized radial Yukawa potential. The probability density for the ground and first excited state has been analyzed via the Fisher information for this potential model. Some numerical results are obtained. From the numerical results obtained, we observed that, for n = 0, 1, the position-space Fisher information Ir increases with increasing potential parameter a, while the momentum-space Fisher information Iρ initially increases, and later decreases with increasing potential parameter a. The Fisher-information-based uncertainty relation and the Heisenberg uncertainty relation have been verified to hold for this atomic model. In addition, we observed a squeezed phenomenon in some of the results in position r and momentum ρ for the ground and first excited states.