Entropic Uncertainty Relations Research Articles

Complementary relation between quantum entanglement and entropic uncertainty

Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.

Lower and upper bounds for unilateral coherence and applying them to the entropic uncertainty relations

In this paper, a method is provided for obtaining the entropic uncertainty relations in the presence of a quantum memory by using quantum coherence. In the method, firstly, one can use the quantum relative entropy of quantum coherence to obtain the uncertainty relations. Secondly, these relations are applied to obtain the entropic uncertainty relations in the presence of a quantum memory. In comparison with other methods this approach is much simpler. Also, for a given state, the upper bounds on the sum of the relative entropies of unilateral coherences are provided, and it is shown which one is tighter. In addition, using the upper bound obtained for unilateral coherence, the nontrivial upper bound on the sum of the entropies for different observables is derived in the presence of a quantum memory.

Tightening the tripartite quantum-memory-assisted entropic uncertainty relation

The uncertainty principle determines the distinction between the classical and quantum worlds. This principle states that it is not possible to measure two incompatible observables with the desired accuracy simultaneously. In quantum information theory, Shannon entropy has been used as an appropriate measure to express the uncertainty relation. According to the applications of entropic uncertainty relation, studying and trying to improve the bound of this relation is of great importance. Uncertainty bound can be altered by considering an extra quantum system as the quantum memory $B$ which is correlated with the measured quantum system $A$. One can extend the bipartite quantum memory assisted entropic uncertainty relation to tripartite quantum memory assisted entropic uncertainty relation in which the memory is split into two parts. In this work, we obtain a lower bound for the tripartite quantum memory assisted entropic uncertainty relation. Our lower bound has two additional terms compared to the lower bound in [Phys. Rev. Lett. 103, 020402 (2009)] which depending on the conditional von Neumann entropy, the Holevo quantity and mutual information. It is shown that the bound obtained in this work is more tighter than other bounds. In addition, using our lower bound, a lower bound for the quantum secret key rate has been obtained. The lower bound is also used to obtain the states for which the strong subadditivity inequality and Koashi-Winter inequality is satisfied with equality.

Entropic uncertainty relation of a qubit–qutrit Heisenberg spin model and its steering

We investigate the quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in a Heisenberg XYZ mixed-spin (1/2, 1) model. Coupling strength, Dzyaloshinskii–Moriya (DM) interaction and inhomogeneous magnetic field, respectively, contributing to QMA-EUR by a thermal entanglement in the hybrid-spin model are studied in detail. Furthermore, we compare the uncertainty of the bipartite hybrid model with those of qubit–qubit and qutrit–qutrit systems. Meanwhile, the effects of local PT-symmetric operation and weak measurement on the steering of entropic uncertainty are analyzed. We find that the local PT-symmetric operation can reduce the entropic uncertainty, and the entropic uncertainty can also be decreased by weak measurement reversal.

Multiple uncertainty relation for accelerated quantum information

The uncertainty principle, first introduced by Heisenberg in inertial frames, clearly distinguishes quantum theories from classical mechanics. In non-inertial frames, its information-theoretic expressions, namely entropic uncertainty relations, have been extensively studied through delocalized quantum fields, and localization of the quantum fields were discussed as well. However, infeasibility of measurements applied on a delocalized quantum field due to the finite size of measurement apparatuses is left unexplained. Therefore, physical clarification of a quantum protocol revealing entropic uncertainty relations still needs investigation. Building on advances in quantum field theories and theoretical developments in entropic uncertainty relations, we demonstrate a relativistic protocol of an uncertainty game in the presence of localized fermionic quantum fields inside cavities. Moreover, a novel lower bound for entropic uncertainty relations with multiple quantum memories is given in terms of the Holevo quantity, which implies how acceleration affects uncertainty relations.

Dynamical characteristic of entropic uncertainty relation in the long-range Ising model with an arbitrary magnetic field

In this work, we explore the quantum-memory-assisted entropic uncertainty relation (QMA-EUR) and the entanglement concurrence for the long-range Ising (LRI) model in the presence of an arbitrary magnetic field. The long-range interaction is considered as type of the Calogero–Moser interaction that is inversely proportional to the square of the distance between the spins. It is shown that the measurement’s uncertainty and the entanglement concurrence are extremely sensitive to the effects of the distance between spins, the magnitude, and the direction of the external magnetic field. The evaluation of the QMA-EUR is fully related to the entanglement concurrence in case of the LRI model. At larger distances between the spins, there is inflation in the measuring uncertainty due to the fragile entanglement between them. The evolutions of the entanglement and the uncertainty show only similar dynamical characteristics in antiferromagnetic and ferromagnetic frames when the magnetic field is perpendicular to the z-axis.

Sequences of lower bounds for entropic uncertainty relations from bistochastic maps

Given two orthonormal bases and , the basic form of the entropic uncertainty principle is stated in terms of the sum of the Shannon entropies of the probabilities of measuring and onto a given quantum state. State independent lower bounds for this sum encapsulate the degree of incompatibility of the observables diagonal in the and bases, and are usually derived by extracting as much information as possible from the unitary operator U connecting the two bases. Here we show a strategy to derive sequences of lower bounds based on alternating sequences of measurements onto and . The problem can be mapped into the multiple application of bistochastic processes that can be described by the powers of the unistochastic matrices directly derivable from U. By means of several examples we study the applicability of the method. The results obtained show that the strategy can allow for an advantage both in the pure state and in the mixed state scenario. The sequence of lower bounds is obtained with resources which are polynomial in the dimension of the underlying Hilbert space, and it is thus suitable for studying high dimensional cases.

Entropic uncertainty lower bound for a two-qubit system coupled to a spin chain with Dzyaloshinskii–Moriya interaction

The uncertainty principle is one of the key concepts in quantum theory. This principle states that it is not possible to measure two incompatible observables simultaneously and accurately. In quantum information theory, the uncertainty principle is formulated using the concept of entropy. In this work we consider the entropic uncertainty relation in the presence of quantum memory. We study the dynamics of entropic uncertainty bound for a two-qubit quantum system coupled to a spin chain with Dzyaloshinskii–Moriya interaction and we investigate the effect of environmental parameters on the entropic uncertainty bound. Notably, our results reveal that there exist some environmental parameters which can be changed to suppress the entropic uncertainty bound.

Entropic uncertainty relation in Garfinkle-Horowitz-Strominger dilation black hole

Heisenberg's uncertainty principle is a fundamental element in quantum mechanics. It sets a bound on our ability to predict the measurement outcomes of two incompatible observables simultaneously. In quantum information theory, the uncertainty principle can be expressed using entropic measures. The entropic uncertainty relation can be improved by considering an additional particle as a memory particle. The presence of quantum correlation between the memory particle and the measured particle reduces the uncertainty. In a curved space-time, the presence of the Hawking radiation can reduce quantum correlation. Therefore, concerning the relationship between the quantum correlation and entropic uncertainty lower bound, we expect that the Hawking radiation increases the entropic uncertainty lower bound. In this work, we investigate the entropic uncertainty relation in Garfinkle-Horowitz-Strominger (GHS) dilation black hole. We consider a model in which the memory particle is located near the event horizon outside the black hole, while the measured particle is free falling. To study the proposed model, we will consider examples with Dirac fields. We also explore the effect of the Hawking radiation on the quantum secret key rate.

Einstein-Podolsky-Rosen steering and nonlocality in quantum dot systems

We examine the relationship among the quantum steering, Bell nonlocality and entanglement of formation in a quantum dot system taking into account Coulomb-blocked and electron-electron interaction. We find that the steerability of the quantum dot system state is dependent on the main model parameters and the degradation of the quantum steering can be reduced by a suitable choice of these parameters. We show that the quantum dot system states with Bell nonlocality can violate the entropy uncertainty relation steering inequality, and that not every entangled quantum state of the dot system is steerable with violation of the Bell nonlocality. Moreover, we find that there is an optimal value of the temperature for which the quantum steering, nonlocality and entanglement are maximal. Finally, we display the monotonic dependence of these classes of quantum correlations on the main physical parameters. • Quantum Physics. • Quantum dot system. • Electron-electron interaction and Coulomb-blocked systems. • Quantum steering. • Bell nonlocality and entanglement of formation.

Quantum-memory-assisted entropic uncertainty relation with moving quantum memory inside a leaky cavity

Uncertainty principle has a fundamental role in quantum theory. This principle states that it is not possible to measure two incompatible observers simultaneously. The uncertainty principle is expressed logically in terms of standard deviation of the measured observables. In quantum information, it has been shown that the uncertainty principle can be expressed by means of Shannon’s entropy. Entropic uncertainty relation can be improved by considering an additional particle as the quantum memory. In the presence of quantum memory the entropic uncertainty relation is called quantum-memory-assisted entropic uncertainty relation. In this work, we will consider the case in which the quantum memory moves inside the leaky cavity. We will show that by increasing the velocity of the quantum memory, the entropic uncertainty lower bound is decreased.

Squeezing of a two-level system due to classical phase noisy laser

Based on the Heisenberg uncertainty relation and the entropy uncertainty relation, the nonclassical effects described by variance squeezing and entropy squeezing for a two-level system driven by a classical phase noisy laser are investigated. Analytical expressions of the variance squeezing and entropy squeezing subjected to the classical phase noisy laser are derived. Our results show that the nonclassical squeezed effects can be protected for a long time in both the Markovian and non-Markovian regions by controlling the ratios between the phase diffusion rate and the system-laser coupling strength. Besides, we also develop an effective strategy to enhance the nonclassical squeezed effects with the help of filtering operation. The underlying physical mechanism to enhance the squeezing is also analyzed.

Entropic uncertainty relations and the quantum-to-classical transition

Our knowledge of quantum mechanics can satisfactorily describe simple, microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.

Exploring entropic uncertainty relation and dense coding capacity in a two-qubit X-state

The measurement precision for two incompatible observables in a typical quantum system can be improved by the aid of one particle as a quantum memory. In this work, we study the entropic uncertainty relation in the presence of quantum memory (EUR-QM) and dense coding capacity (DCC) for arbitrary two-qubit X-states, and then we obtain an explicit relationship between the lower bound of the uncertainty and DCC. As an example, we examine the thermal EUR-QM as well as DCC in two kinds of two-qubit spin squeezing models (one-axis twisting model and two-axis counter-twisting model) under an external magnetic field. In the following, we relate EUR-QM to DCC and show analytically that there is an anti-correlated relation between them, and especially in the ground state. Our results show that for both the models, the entropic uncertainty and its bound can be decreased by reinforcing the spin squeezing parameters or decreasing the temperature of the system. Notably, we reveal that the valid dense coding cannot carry out in the one-axis twisting model, nevertheless, if we properly choose the Hamiltonian parameters, the two-axis counter-twisting model not only carries out the valid dense coding but also the optimal dense coding surely can be achieved. Thereby, our observations might offer new insights into quantum measurement precision and the optimal dense coding for the regimes of various solid-state systems.

Entropic uncertainty relation in neutrino oscillations

Neutrino oscillation is deemed as an interesting physical phenomenon and shows the nonclassical features made apparently by the Leggett–Garg inequality. The uncertainty principle is one of the fundamental features that distinguishes the quantum world to its classical counterpart. And the principle can be depicted in terms of entropy, which forms the so-called entropic uncertainty relations (EUR). In this work, the entropic uncertainty relations that are relevant to the neutrino-flavor states are investigated by comparing the experimental observation of neutrino oscillations to predictions. From two different neutrino sources, we analyze ensembles of reactor and accelerator neutrinos for different energies, including measurements performed by the Daya Bay collaboration using detectors at 0.5 and 1.6 km from their source, and by the MINOS collaboration using a detector with a 735km distance to the neutrino source. It is found that the entropy-based uncertainty conditions strengths exhibits non-monotonic evolutions as the energy increases. We also quantify the systemic quantumness measured by quantum correlation, and derive the intrinsic relationship between quantum correlation and EUR. Furthermore, we utilize EUR as a criterion to detect entanglement of neutrino-flavor state. Our results could illustrate the potential applications of neutrino oscillations on quantum information processing in the weak-interaction processes.

Entropic time–energy uncertainty relations: an algebraic approach

We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable G and the displacement r of the G-generated evolution e−irG. We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al [Phys. Rev. Lett. 122 100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gao et al [arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.

Entropic uncertainty relation and quantum phase transition in spin-1/2 Heisenberg chain

Based on the quantum-memory-assisted entropic uncertainty relation (QMA-EUR), we study the quantum phase transition (QPT) of a spin-1/2 frustrated Heisenberg chain and show that QMA entropy can be a useful tool to detect QPT. For the six-site case, we choose the reduced two-site as the detecting system and obtain the analytical result of QMA entropy and its bound. We find that the QMA entropy of the ground state is discontinuous at the first-order QPT point. Meanwhile for the excited state, the catastrophe point of QMA entropy approaches the infinite-order QPT point, which is confirmed by the numerical results in the 8-, 10- and 12-site cases. Moreover, we study the behavior of QMA entropy undergoing the dephasing process. Interestingly, two-site QMA entropy can still indicate the QPT point when the state degenerates into a mixed state. In this sense, QMA entropy is more effective than quantum correlation in reflecting the quantum criticality.

Improved tripartite uncertainty relation with quantum memory

Uncertainty principle is a striking and fundamental feature in quantum mechanics distinguishing from classical mechanics. It offers an important lower bound to predict outcomes of two arbitrary incompatible observables measured on a particle. In quantum information theory, this uncertainty principle is popularly formulized in terms of entropy. Here, we present an improvement of tripartite quantum-memory-assisted entropic uncertainty relation. The uncertainty's lower bound is derived by considering mutual information and Holevo quantity. It shows that the bound derived by this method will be tighter than the lower bound in [Phys. Rev. Lett. 103, 020402 (2009)]. Furthermore, regarding a pair of mutual unbiased bases as the incompatibility, our bound will become extremely tight for the three-qubit $\emph{X}$-state system, completely coinciding with the entropy-based uncertainty, and can restore Renes ${\emph{et al.}}$'s bound with respect to arbitrary tripartite pure states. In addition, by applying our lower bound, one can attain the tighter bound of quantum secret key rate, which is of basic importance to enhance the security of quantum key distribution protocols.

Control-based verification of multiatoms in a cavity

In this paper, we study a model of two two-level atoms interacting with a quantum field. An analytical solution is obtained which is used to study the information entropy of the system. It is shown that the nonlinear term plays a significant role in the behaviour of the minimum uncertainty (MU) compared with the concurrence (C). Our extensive study of information entropy of atoms–field interaction demonstrates that using the coupling strength between the atoms and the field as a controller parameter, one can control the dynamics of the system by increasing the lower bound of the entropic uncertainty relation or decreasing the entanglement.

Measurement Uncertainty and Its Connection to Quantum Coherence in an Inertial Unruh–DeWitt Detector

AbstractThe dynamic characteristics of measured uncertainty and quantum coherence are explored for an inertial Unruh–DeWitt detector model in an expanding de Sitter space. Using the entropic uncertainty relation, the uncertainty of interest is correlated with the evolving time t, the energy level spacing δ, and the Hubble parameter H. The investigation shows that, for short time, a strong energy level spacing and small Hubble parameter can result in a relatively small uncertainty. The evolution of quantum coherence versus the evolving time and Hubble parameter, which varies almost inversely to that of the uncertainty, is then discussed, and the relationship between uncertainty and the coherence is explicitly derived. With respect to the l1 norm of coherence, it is found that the environment for the quantum system considered possesses a strong non‐Markovian property. The dynamic behavior of coherence non‐monotonously decreases with the growth of evolving time. The dynamic features of uncertainty and coherence in the expanding space with those in flat space are also compared. Furthermore, quantum weak measurement is utilized to effectively reduce the magnitude of uncertainty, which offers realistic and important support for quantum precision measurements during the undertaking of quantum tasks.