State-independent uncertainty relations and entanglement detection

The uncertainty relation is one of the key ingredients of quantum theory. Despite the great efforts devoted to this subject, most of the variance-based uncertainty relations are state-dependent and suffering from the triviality problem of zero lower bounds. Here we develop a method to get uncertainty relations with state-independent lower bounds. The method works by exploring the eigenvalues of a Hermitian matrix composed by Bloch vectors of incompatible observables and is applicable for both pure and mixed states and for arbitrary number of N-dimensional observables. The uncertainty relation for the incompatible observables can be explained by geometric relations related to the parallel postulate and the inequalities in Horn’s conjecture on Hermitian matrix sum. Practical entanglement criteria are also presented based on the derived uncertainty relations.     

Uncertainty under quantum measures and quantum memory

The uncertainty principle restricts potential information one gains about physical properties of the measured particle. However, if the particle is prepared in entanglement with a quantum memory, the corresponding entropic uncertainty relation will vary. Based on the knowledge of correlations between the measured particle and quantum memory, we have investigated the entropic uncertainty relations for two and multiple measurements and generalized the lower bounds on the sum of Shannon entropies without quantum side information to those that allow quantum memory. In particular, we have obtained generalization of Kaniewski–Tomamichel–Wehner’s bound for effective measures and majorization bounds for noneffective measures to allow quantum side information. Furthermore, we have derived several strong bounds for the entropic uncertainty relations in the presence of quantum memory for two and multiple measurements. Finally, potential applications of our results to entanglement witnesses are discussed via the entropic uncertainty relation in the absence of quantum memory.     

Probing entropic uncertainty relations under a two-atom system coupled with structured bosonic reservoirs

The uncertainty principle imposes constraints on an observer’s ability to make precision measurements for two incompatible observables; thus, uncertainty relations play a key role in quantum precision measurement in the field of quantum information science. Here, our aim is to examine non-Markovian effects on quantum-memory-assisted entropic uncertainty relations in a system consisting of two atoms coupled with structured bosonic reservoirs. Explicitly, we explore the dynamics of the uncertainty relations via entropic measures in non-Markovian regimes when two atomic qubits independently interact with their own infinite degree-of-freedom bosonic reservoir. We show that measurement uncertainty vibrates with periodically increasing amplitude with growing non-Markovianity of the observed system and ultimately saturates toward a fixed value at a long time limit. It is worth noting that there are several appealing conclusions raised by us: First, the uncertainty’s lower bound does not entirely depend on the quantum correlations within the two-qubit system, being affected by an interplay between the quantum discord and the minimal von Neumann conditional entropy \(\mathcal{S}_\mathrm{ce}\). Second, the dynamic characteristic of the measurement uncertainty is considerably distinctive with regard to Markovian and non-Markovian regimes, respectively. Third, the measurement uncertainty is closely correlated with the Bell non-locality \({\mathcal{B}}\). Moreover, we claim that the entropic uncertainty relation could be a promising tool with which to probe entanglement in current architecture.     

Variance-based uncertainty relations for incompatible observables

Quantum Information Processing - We formulate uncertainty relations for arbitrary finite number of incompatible observables. Based on the sum of variances of the observables, both Heisenberg-type...     

Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis’ entropies

Uncertainty relations for more than two observables have found use in quantum information, though commonly known relations pertain to a pair of observables. We present novel uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Tsallis $\alpha $ -entropies. For all real $\alpha \in (0;1]$ and integer $\alpha \ge 2$ , lower bounds on the sum of three $\alpha $ -entropies are obtained. These bounds are tight in the sense that they are always reached with certain pure states. The necessary and sufficient condition for equality is that the qubit state is an eigenstate of one of the Pauli observables. Using concavity with respect to the parameter $\alpha $ , we derive approximate lower bounds for non-integer $\alpha \in (1;+\infty )$ . In the case of pure states, the developed method also allows to obtain upper bounds on the entropic sum for real $\alpha \in (0;1]$ and integer $\alpha \ge 2$ . For applied purposes, entropic bounds are often used with averaging over the individual entropies. Combining the obtained bounds leads to a band, in which the rescaled average $\alpha $ -entropy ranges in the pure-state case. A width of this band is essentially dependent on $\alpha $ . It can be interpreted as an evidence for sensitivity in quantifying the complementarity.     

Decoherence effect on quantum-memory-assisted entropic uncertainty relations

Uncertainty principle significantly provides a bound to predict precision of measurement with regard to any two incompatible observables, and thereby plays a nontrivial role in quantum precision measurement. In this work, we observe the dynamical features of the quantum-memory-assisted entropic uncertainty relations (EUR) for a pair of incompatible measurements in an open system characterized by local generalized amplitude damping (GAD) noises. Herein, we derive the dynamical evolution of the entropic uncertainty with respect to the measurement affecting by the canonical GAD noises when particle A is initially entangled with quantum memory B. Specifically, we examine the dynamics of EUR in the frame of three realistic scenarios: one case is that particle A is affected by environmental noise (GAD) while particle B as quantum memory is free from any noises, another case is that particle B is affected by the external noise while particle A is not, and the last case is that both of the particles suffer from the noises. By analytical methods, it turns out that the uncertainty is not full dependent of quantum correlation evolution of the composite system consisting of A and B, but the minimal conditional entropy of the measured subsystem. Furthermore, we present a possible physical interpretation for the behavior of the uncertainty evolution by means of the mixedness of the observed system; we argue that the uncertainty might be dramatically correlated with the systematic mixedness. Furthermore, we put forward a simple and effective strategy to reduce the measuring uncertainty of interest upon quantum partially collapsed measurement. Therefore, our explorations might offer an insight into the dynamics of the entropic uncertainty relation in a realistic system, and be of importance to quantum precision measurement during quantum information processing.     

Entropic uncertainty relations in the Heisenberg <Emphasis Type="Italic">XXZ</Emphasis> model and its controlling via filtering operations

The uncertainty principle is recognized as an elementary ingredient of quantum theory and sets up a significant bound to predict outcome of measurement for a couple of incompatible observables. In this work, we develop dynamical features of quantum memory-assisted entropic uncertainty relations (QMA-EUR) in a two-qubit Heisenberg XXZ spin chain with an inhomogeneous magnetic field. We specifically derive the dynamical evolutions of the entropic uncertainty with respect to the measurement in the Heisenberg XXZ model when spin A is initially correlated with quantum memory B. It has been found that the larger coupling strength \( J \) of the ferromagnetism (\( J < 0 \)) and the anti-ferromagnetism (\( J > 0 \)) chains can effectively degrade the measuring uncertainty. Besides, it turns out that the higher temperature can induce the inflation of the uncertainty because the thermal entanglement becomes relatively weak in this scenario, and there exists a distinct dynamical behavior of the uncertainty when an inhomogeneous magnetic field emerges. With the growing magnetic field \( \left| B \right| \), the variation of the entropic uncertainty will be non-monotonic. Meanwhile, we compare several different optimized bounds existing with the initial bound proposed by Berta et al. and consequently conclude Adabi et al.’s result is optimal. Moreover, we also investigate the mixedness of the system of interest, dramatically associated with the uncertainty. Remarkably, we put forward a possible physical interpretation to explain the evolutionary phenomenon of the uncertainty. Finally, we take advantage of a local filtering operation to steer the magnitude of the uncertainty. Therefore, our explorations may shed light on the entropic uncertainty under the Heisenberg XXZ model and hence be of importance to quantum precision measurement over solid state-based quantum information processing.     

Robust identification and fault diagnosis based on uncertain multiple input–multiple output linear parameter varying parity equations and zonotopes

We present a robust fault diagnosis method for uncertain multiple input–multiple output (MIMO) linear parameter varying (LPV) parity equations. The fault detection methodology is based on checking whether measurements are inside the prediction bounds provided by the uncertain MIMO LPV parity equations. The proposed approach takes into account existing couplings between the different measured outputs. Modelling and prediction uncertainty bounds are computed using zonotopes. Also proposed is an identification algorithm that estimates model parameters and their uncertainty such that all measured data free of faults will be inside the predicted bounds. The fault isolation and estimation algorithm is based on the use of residual fault sensitivity. Finally, two case studies (one based on a water distribution network and the other on a four-tank system) illustrate the effectiveness of the proposed approach.     

Non-Asymptotic Lower Bounds for the Data Complexity of Statistical Attacks on Symmetric Cryptosystems

A method is proposed for obtaining the lower bounds of data complexity of statistical attacks on block or stream ciphers. The method is based on the Fano inequality and, unlike the available methods, doesn’t use any asymptotic relations, approximate formulas or heuristic assumptions about the considered cipher. For a lot of known types of attacks, the obtained data complexity bounds have the classical form. For other types of attacks, these bounds allow us to introduce reasonable parameters that characterize the security of symmetric cryptosystems against these attacks.     

Macroscopic Entanglement and Phase Transitions

This paper summarises the results of our research on macroscopic entanglement in spin systems and free Bosonic gases. We explain how entanglement can be observed using entanglement witnesses which are themselves constructed within the framework of thermodynamics and thus macroscopic observables. These thermodynamical entanglement witnesses result in bounds on macroscopic parameters of the system, such as the temperature, the energy or the susceptibility, below which entanglement must be present. The derived bounds indicate a relationship between the occurrence of entanglement and the establishment of order, possibly resulting in phase transition phenomena. We give a short overview over the concepts developed in condensed matter physics to capture the characteristics of phase transitions in particular in terms of order and correlation functions. Finally we want to ask and speculate whether entanglement could be a generalised order concept by itself, relevant in (quantum induced) phase transitions such as BEC, and that taking this view may help us to understand the underlying process of high-T superconductivity. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

Two lower bounds on the covariance for nonlinear estimation problems

Two Convariance lower bounds for nonlinear state estimation problems are presented. These bounds are based upon the Cramer-Rao bound for treating nuisance parameters and they can be applied to filtering, smoothing, and prediction problems. The tightness of these bounds are examined using a nonlinear system where the recursive equation for covariance computation can be obtained. These results are also compared with the bound of Bobrovsky and Zakai.     

Controlling quantum memory-assisted entropic uncertainty in non-Markovian environments

Quantum memory-assisted entropic uncertainty relation (QMA EUR) addresses that the lower bound of Maassen and Uffink’s entropic uncertainty relation (without quantum memory) can be broken. In this paper, we investigated the dynamical features of QMA EUR in the Markovian and non-Markovian dissipative environments. It is found that dynamical process of QMA EUR is oscillation in non-Markovian environment, and the strong interaction is favorable for suppressing the amount of entropic uncertainty. Furthermore, we presented two schemes by means of prior weak measurement and posterior weak measurement reversal to control the amount of entropic uncertainty of Pauli observables in dissipative environments. The numerical results show that the prior weak measurement can effectively reduce the wave peak values of the QMA-EUA dynamic process in non-Markovian environment for long periods of time, but it is ineffectual on the wave minima of dynamic process. However, the posterior weak measurement reversal has an opposite effects on the dynamic process. Moreover, the success probability entirely depends on the quantum measurement strength. We hope that our proposal could be verified experimentally and might possibly have future applications in quantum information processing.     

Intrinsic constraints in space-time filtering: a new approach torepresenting uncertainty in low-level vision

Describes how, in the process of extracting the optical flow through space-time filtering, one has to consider the constraints associated with the motion uncertainty, as well as the spatial and temporal sampling rates of the sequence of images. The motion uncertainty satisfies the Cramer-Rao (CR) inequality, which is shown to be a function of the filter parameters. On the other hand, the spatial and temporal sampling rates have lower bounds, which depend on the motion uncertainty, the maximum support in the frequency domain, and the optical flow. These lower bounds on the sampling rates and on the motion uncertainty are constraints that constitute an intrinsic part of the computational structure of space-time filtering. The author shows that if he uses these constraints simultaneously, the filter parameters cannot be arbitrarily determined but instead have to satisfy consistency constraints. By using explicit representations of uncertainties in extracting visual attributes, one can constrain the range of values assumed by the filter parameters     

Obtaining accurate confidence regions for the estimated zeros andpoles in system identification problems

In this note, a method is proposed to generate improved uncertainty bounds on the poles and zeros of an identified system. The uncertainty ellipsoids are replaced by dedicated uncertainty regions with a given probability level. The method can be applied to Markov estimators in the time or frequency domain     

量子计算与不确定性原理

对量子计算的计算潜力的高度期望源于量子力学的各种特性,如叠加原理、纠缠现象、破坏性和建设性的量子干扰。相对于经典计算,量子计算具有某些假定的优势,例如量子算法的运行速度比经典算法快;但另一方面却似乎存在影响经典算法但不影响量子算法的障碍,障碍之一是传统上归因于Werner Heisenberg的两个不确定性原理。Heisenberg最初制定的不确定性原理涉及用于测量量子系统的非量子仪器必然会对该系统造成影响。这个原理与其后来的发展有所不同,因为后来发现的不确定性所假定的是不交换可观察量在测量方面存在固有的不能精确测量的特性。在目前的技术发展状况以及当前对量子力学的形式表述与诠释的情况下,这两种不确定性皆有可能对量子计算的速度造成不良影响。近年来,针对这两种不确定性原理有了新的研究成果:1)Ozawa对Heisenberg原理提出了修改,将两种不确定性纳入其内进行并列考虑,从而可以减小Heisenberg原理的不确定性程度;2)在考虑到熵不确定性的情况下,Heisenberg不确定性可被视为Hirschmann不确定性的下界,因此除了在测量上的不确定性之外,量子计算还必须考虑来自其他如信息学的不确定性因素。     

On the robust control of robot manipulators

A simple robust nonlinear control law for n-link robot manipulators is derived using the Lyapunov-based theory of guaranteed stability of uncertain systems. The novelty of this result lies in the fact that the uncertainty bounds needed to derive the control law and to prove uniform ultimate boundedness of the tracking error depend only on the inertial parameters of the robot. In previous results of this type, the uncertainty bounds have depended not only on the inertia parameters but also on the reference trajectory and on the manipulator state vector. The presented result also removes previous assumptions regarding closeness in norm of the computed inertia matrix to the actual inertial matrix. The design used thus provides the simplest such robust design available to date     

Set-membership approach for identification of parameter and prediction uncertainty in power-law relationships: The case of sediment yield

Power laws are used to describe a large variety of natural and industrial phenomena. Consequently, they are used in a wide range of scientific research and management applications. This paper focuses on the identification of bounds on the parameter and prediction uncertainty in a power-law relation from experimental data, assuming known bounds on the error between model output and observations. The prediction uncertainty bounds can subsequently be used as constraints, for example in optimisation and scenario studies. The set-membership approach involves identification and removal of outliers, estimation of the feasible parameter set, evaluation of the feasible model-output set and tuning of the specified bounds on model-output error. As an example the procedure is applied to data of scattered sediment yield versus catchment area (Wasson, 1994). The key result is an un-falsified relationship between sediment yield and catchment area with uncertainty bounds on its parameters. The set-membership results are compared with the results from a conventional least-squares approach with first-order variance propagation, assuming a zero-mean, symmetrical error distribution.     

Estimation of uncertain models of activated sludge processes with interval observers

This paper deals with the designs of observers (software sensors) for uncertain models of wastewater treatment. We assume that the model (4 state variables) is such that the growth rate is unknown and other parameters (e.g. the organic concentration in the influent) are uncertain, i.e. we only know their upper and lower bounds. Given the measurements of the dissolved oxygen concentration, we build interval observers, giving dynamic bounds containing the variables to estimate. The width of these bounds are related to the width of the uncertainty bounds of the parameters. In some cases, we show that we can estimate exactly the unknown variables despite the uncertainties on the model.     

COMPUTATION OF BEST BOUNDS OF PROBABILITIES FROM UNCERTAIN DATA

An uncertainty reasoning method is presented in this article. The method can be used to compute from a given set of conditional probabilities the best lower bounds and the best upper bounds of those conditional probabilities that are not explicitly provided. The computation of the best upper(lower) bound of such a conditional probability relies on solution of a linear programming problem. Some reduction techniques are proposed in this article to improve the efficiency of our uncertainty reasoning method. As illustrated in Section 4.3, for many uncertainty reasoning problems in medical diagnosis, by using our reduction techniques, the best range of a conditional probability, which is specified by a lower bound and an upper bound, can be computed in polynomial time in terms of the number of basic events involved in the reasoning.