Uncertainty Relations For Measurements 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 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.

Entropic uncertainty relations for multiple measurements assigned with biased weights

Entropic uncertainty relations for multiple measurements assigned with biased weights

Testing Heisenberg-Type Measurement Uncertainty Relations of Three Observables.

Heisenberg-type measurement uncertainty relations (MURs) of two quantum observables are essential for contemporary research in quantum foundations and quantum information science. Going beyond, here we report the first experimental study of MUR of three quantum observables. We establish rigorously MURs for triplets of unbiased qubit observables as combined approximation errors lower bounded by an incompatibility measure, inspired by the proposal of Busch et al. [Phys. Rev. A 89, 012129 (2014)PLRAAN1050-294710.1103/PhysRevA.89.012129]. We develop a convex programming protocol to numerically find the exact value of the incompatibility measure and the optimal measurements. We propose a novel implementation of the optimal joint measurements and present several experimental demonstrations with a single-photon qubit. We stress that our method is universally applicable to the study of many qubit observables. Besides, we theoretically show that MURs for joint measurement can be attained by sequential measurements in two of our explored cases. We anticipate that this work may stimulate broad interests associated with Heisenberg's uncertainty principle in the case of multiple observables, enriching our understanding of quantum mechanics and inspiring innovative applications in quantum information science.

Measurement uncertainty relation for three observables

Measurement uncertainty relation for three observables

Colloquium: Incompatible measurements in quantum information science

Some measurements in quantum mechanics disturb each other. This has puzzled physicists since the formulation of the theory, but only in recent decades has the incompatibility of measurements been analyzed in depth and detail, using the notion of joint measurability of generalized measurements. In this Colloquium joint measurability and incompatibility are reviewed from the perspective of quantum information science. The Colloquium starts by discussing the basic definitions and concepts. An overview on applications of incompatibility, such as in measurement uncertainty relations, the characterization of quantum correlations, or information processing tasks like quantum state discrimination, is then presented. Finally, emerging directions of research, such as a resource theory of incompatibility as well as other concepts to grasp the nature of measurements in quantum mechanics, are discussed.

Tripartite quantum-memory-assisted entropic uncertainty relations for multiple measurements

Quantum uncertainty relations are typically analyzed for a pair of incompatible observables, however, the concept per se naturally extends to situations of more than two observables. In this work, we obtain tripartite quantum-memory-assisted entropic uncertainty relations for multiple measurements and show that the lower bounds of these relations have three terms that depend on the complementarity of the observables, the conditional von-Neumann entropies, the Holevo quantities, and the mutual information. The saturation of these inequalities is analyzed.

Generalized uncertainty relations for multiple measurements

The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.

Optimized entropic uncertainty relations for multiple measurements

Recently, an entropic uncertainty relation for multi-measurement has been proposed by Liu et al. in [Phys. Rev. A 91, 042133 (2015)]. However, the lower bound of the relation is not always tight with respect to different measurements. Herein, we improve the lower bound of the entropic uncertainty relation for multi-measurement, termed as simply constructed bound (SCB). We verify that the SCB is tighter than Liu et al.'s result for arbitrary mutually unbiased basis measurements, which might play a fundamental and crucial role in practical quantum information processing. Moreover, we optimize the SCB by considering mutual information and the Holevo quantity, and propose an optimized SCB (OSCB). Notably, the proposed bounds are extrapolations of the behavior of two measurements to a larger collection of measurements. It is believed that our findings would shed light on entropy-based uncertainty relations in the multi-measurement scenario and will be beneficial for security analysis in quantum key distributions.

Entropic uncertainty relations in a class of generalized probabilistic theories

Entropic uncertainty relations play an important role in both fundamentals and applications of quantum theory. Although they have been well-investigated in quantum theory, little is known about entropic uncertainty in generalized probabilistic theories (GPTs). The current study explores two types of entropic uncertainty relations, preparation and measurement uncertainty relations, in a class of GPTs which can be considered generalizations of quantum theory. Not only a method for obtaining entropic preparation uncertainty relations but also an entropic measurement uncertainty relation similar to the quantum one by Buscemi et al (2014 Phys. Rev. Lett. 112 050401) are proved in those theories. These results manifest that the entropic structure in the uncertainty relations is not restricted to quantum theory and therefore is universal one. Concrete calculations of our relations in GPTs called the regular polygon theories are also demonstrated.

Experimental test of entropic noise-disturbance uncertainty relations for three-outcome qubit measurements

Information-theoretic uncertainty relations formulate the joint immeasurability of two non-commuting observables in terms of information entropies. The trade-off of the accuracy in the outcome of two successive measurements manifests in entropic noise-disturbance uncertainty relations. Recent theoretical analysis predicts that projective measurements are not optimal, with respect to the noise-disturbance trade-offs. Therefore the results in our previous letter [PRL 115, 030401 (2015)] are outperformed by general quantum measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation for three-outcome positive-operator valued measures (POVM), using neutron spin-1/2 qubits. The obtained results violate the lower bound for projective measurements as theoretically predicted.

Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation.

The quantum multiparameter estimation is very different from the classical multiparameter estimation due to Heisenberg's uncertainty principle in quantum mechanics. When the optimal measurements for different parameters are incompatible, they cannot be jointly performed. We find a correspondence relationship between the inaccuracy of a measurement for estimating the unknown parameter with the measurement error in the context of measurement uncertainty relations. Taking this correspondence relationship as a bridge, we incorporate Heisenberg's uncertainty principle into quantum multiparameter estimation by giving a trade-off relation between the measurement inaccuracies for estimating different parameters. For pure quantum states, this trade-off relation is tight, so it can reveal the true quantum limits on individual estimation errors in such cases. We apply our approach to derive the trade-off between attainable errors of estimating the real and imaginary parts of a complex signal encoded in coherent states and obtain the joint measurements attaining the trade-off relation. We also show that our approach can be readily used to derive the trade-off between the errors of jointly estimating the phase shift and phase diffusion without explicitly parametrizing quantum measurements.

Experimental test of tight state-independent preparation uncertainty relations for qubits

The well-known Robertson-Schroedinger uncertainty relations miss an irreducible lower bound. This is widely attributed to the lower bound's state-dependence. Therefore, Abbott \emph{et al.} introduced a general approach to derive tight state-independent uncertainty relations for qubit measurements [Mathematics 4, 8 (2016)]. The relations are expressed in two measures of uncertainty, which are standard deviation and entropy, both functions of the expectation value. Here, we present a neutron optical test of the tight state-independent preparation uncertainty relations for non-commuting Pauli spin observables with mixed spin states. The final results, obtained in a polarimetric experiment, reproduce the theoretical predictions evidently for arbitrary initial states of variable degree of polarization.

Rényi formulation of uncertainty relations for POVMs assigned to a quantum design

Information entropies provide powerful and flexible way to express restrictions imposed by the uncertainty principle. This approach seems to be very suitable in application to problems of quantum information theory. It is typical that questions of such a kind involve measurements having one or another specific structure. The latter often allows us to improve entropic bounds that follow from uncertainty relations of sufficiently general scope. Quantum designs have found use in many issues of quantum information theory, whence uncertainty relations for related measurements are of interest. In this paper, we obtain uncertainty relations in terms of min-entropies and Rényi entropies for POVMs assigned to a quantum design. Relations of the Landau–Pollak type are addressed as well. Using examples of quantum designs in two dimensions, the obtained lower bounds are then compared with the previous ones. An impact on entropic steering inequalities is briefly discussed.

Experimental investigation of joint measurement uncertainty relations for three incompatible observables at a single-spin level.

In the light of the Busch, Lathi and Werner proposal, we explore, for the first time, the joint measurements and confirmation of uncertainty relations for three incompatible observables that reflect the original spirit proposed by Heisenberg in 1927. We first develop the error trade-off relations theoretically and then demonstrate the first experimental witness of joint measurements using a single ultracold 40Ca+ ion trapped in a harmonic potential. In addition, we report, that in contrast to the case of two observables, scarifying accuracy of any one of the three observables the rest of two can be measured with ultimate accuracy.

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.

Uncertainty and trade-offs in quantum multiparameter estimation

Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneous optimal precision is attainable in the asymptotic limit. We study trade-off relations that follow from known tight bounds in quantum multiparameter estimation. We compute trade-off curves and surfaces from Cramér–Rao type bounds which provide a compelling graphical representation of the information encoded in such bounds, and argue that bounds on simultaneously achievable precision in quantum multiparameter estimation should be regarded as measurement uncertainty relations. From the state-dependent bounds on the expected cost in parameter estimation, we derive a state-independent uncertainty relation between the parameters of a qubit system.

Entropic measurement uncertainty relations for all the infinite components of a spin vector

The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin s. For an approximate measurement of a spin vector, which gives approximate joint measurements of the spin components, we define the device information loss as the maximum loss of information per observable occurring in approximating the ideal incompatible components with the joint measurement at hand. By optimizing on the measuring device, we define the notion of minimum information loss. By using these notions, we show how to give a significant formulation of state independent MURs in the case of infinitely many target observables. The same construction works as well for finitely many observables, and we study the related MURs for two and three orthogonal spin components. The minimum information loss plays also the role of measure of incompatibility and in this respect it allows us to compare quantitatively the incompatibility of various sets of spin observables, with different number of involved components and different values of s.

Measurements of Entropic Uncertainty Relations in Neutron Optics

The emergence of the uncertainty principle has celebrated its 90th anniversary recently. For this occasion, the latest experimental results of uncertainty relations quantified in terms of Shannon entropies are presented, concentrating only on outcomes in neutron optics. The focus is on the type of measurement uncertainties that describe the inability to obtain the respective individual results from joint measurement statistics. For this purpose, the neutron spin of two non-commuting directions is analyzed. Two sub-categories of measurement uncertainty relations are considered: noise–noise and noise–disturbance uncertainty relations. In the first case, it will be shown that the lowest boundary can be obtained and the uncertainty relations be saturated by implementing a simple positive operator-valued measure (POVM). For the second category, an analysis for projective measurements is made and error correction procedures are presented.

Uncertainty equality with quantum memory and its experimental verification

As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.