Uncertainty relation for successive measurements

It is noted that the Heisenberg uncertainty relations set a lower bound on the product of variances of two observablesA, B when they are separately measured on two distinct, but identically prepared ensembles. A new uncertainty relation is derived for the product of the variances of the two observablesA, B when they are measured sequentially on a single ensemble of systems. It is shown that the two uncertainty relations differ significantly wheneverA andB are not compatible.     

Uncertainty relation of successive measurements based on Wigner–Yanase skew information

Wigner-Yanase skew information could quantify the quantum uncertainty of the observables that are not commuting with a conserved quantity.We present the uncertainty principle for two successive projective measurements in terms of Wigner-Yanase skew information based on a single quantum system.It could capture the incompatibility of the observables,i.e.the lower bound can be nontrivial for the observables that are incompatible with the state of the quanaim system.Furthermore,the lower bound is also constrained by the quantum Fisher information.In addition,we find the complementarity relation between the uncertainties of the observable which operated on the quantum state and the other observable that performed on the post-measured quantum state and the uncertainties formed by the non-degenerate quantum observables performed on the quantum state,respectively.     

Complementarity in atomic (finite-level quantum) systems: an information-theoretic approach

We develop an information theoretic interpretation of the number-phase complementarity in atomic systems, where phase is treated as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as an upper bound on a sum of knowledge of these two observables for the case of two-level systems. A tighter bound characterizing the uncertainty relation is obtained numerically in terms of a weighted knowledge sum involving these variables. We point out that complementarity in these systems departs from mutual unbiasededness in two significant ways: first, the maximum knowledge of a POVM variable is less than log (dimension) bits; second, surprisingly, for higher dimensional systems, the unbiasedness may not be mutual but unidirectional in that phase remains unbiased with respect to number states, but not vice versa. Finally, we study the effect of non-dissipative and dissipative noise on these complementary variables for a single-qubit system.     

Weak convergence in Hilbert space and weak uncertainty relations in quantum theory

A suitable weak topology is considered on the Hilbert phase space of a quantum-mechanical system. It is then shown that if two bounded observables of the system have no common eigenvector, the sum of their variances in any state is always greater than some positive constant. Consequences of this result on some observables of simple physical systems are examined. First of all, the case of the position and momentum of the elementary particle in one dimension is studied and a comparation with Heisenberg's indeterminacy principle is carried out. Then, the case of angular variables is also examined, with special emphasis on spin 1/2. An experiment with neutrons is finally suggested and analysed with the help of the theory developed.     

Optimal observables and phase-space ambiguities

Optimal observables are known to lead to minimal statistical errors on parameters for a given normalised event distribution of a physics reaction. Thereby all statistical correlations are taken into account. Therefore, on the one hand they are a useful tool to extract values on a set of parameters from measured data. On the other hand one can calculate the minimal constraints on these parameters achievable by any data-analysis method for the specific reaction. In case the final states can be reconstructed without ambiguities optimal observables have a particularly simple form. We give explicit formulae for the optimal observables for generic reactions in case of ambiguities in the reconstruction of the final state and for general parameterisation of the final-state phase space.Received: 12 January 2005, Published online: 7 March 2005     

Tighter Bound of Entropic Uncertainty under the Unruh Effect

The uncertainty principle limits the ability to simultaneously predict measurement outcomes for two non-commuting observables of a quantum particle. However, the uncertainty can be violated by considering a particle as a quantum memory correlated with the primary particle. By modeling an Unruh–Dewitt detector coupled to a massless scalar field, it is explored how the Unruh effect affects the entropic uncertainty and the tighter lower bound for a pair of entangled detectors is probed when one of them is accelerated. It is found that Unruh thermal noise really gives rise to an increase of entropic uncertainty for the given conditions since the correlation between quantum memory and the measured system is decreased. It is shown that the bound of the entropic uncertainty relations, in the presence of memory, can be formulated by introducing the Holevo quantity and mutual information. It is also noticed that Adabi's lower bound is tighter than that of Berta, and just the optimal bound under the Unruh effect. Moreover, it is shown that Berta's lower bound is unrelated to the choice of complementary observables, while the optimal Adabi's lower bound is dependent on the measurement choice. It is worth mentioning that the investigations may offer a better understanding of the entropic uncertainty in a relativistic motion.     

A Lower Bound on Concurrence

We derive an analytical lower bound on the concurrence for bipartite quantum systems with an improved computable cross norm or realignment criterion and an improved positive partial transpose criterion respectively. Furthermore we demonstrate that our bound is better than that obtained from the local uncertainty relations criterion with optimal local orthogonal observables which is known as one of the best estimations of concurrence.     

A new physical characterisation of classical systems in quantum mechanics

A physical characterisation of classical systems in quantum mechanics is given in terms of the set of ensembles in contrast to the well-known characterisations concerning the effects or observables: A quantum mechanical system is classical if and only if each two decompositions of every ensemble are compatible.     

Optimal entropic uncertainty relation in two-dimensional Hilbert space

The exact lower bound on the sum of the information entropies is obtained for arbitrary pairs of observables in two-dimensional Hilbert space. The result coincides with that given by Garrett and Gull for the particular case of real transformation matrices and state vectors. A weaker analytical bound is also obtained.     

Toward a quantitative theory of self-generated complexity

Quantities are defined operationally which qualify as measures of complexity of patterns arising in physical situations. Their main features, distinguishing them from previously used quantities, are the following: (1) they are measuretheoretic concepts, more closely related to Shannon entropy than to computational complexity; and (2) they are observables related to ensembles of patterns, not to individual patterns. Indeed, they are essentially Shannon information needed to specify not individual patterns, but either measure-theoretic or algebraic properties of ensembles of patterns arising ina priori translationally invariant situations. Numerical estimates of these complexities are given for several examples of patterns created by maps and by cellular automata.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Strategies to measure a quantum state

We consider the problem of determining the mixed quantum state of a large but finite number of identically prepared quantum systems from data obtained in a sequence of ideal (von Neumann) measurements, each performed on an individual copy of the system. In contrast to previous approaches, we do not average over the possible unknown states but work out a “typical” probability distribution on the set of states, as implied by the experimental data. As a consequence, any measure of knowledge about the unknown state and thus any notion of “best strategy” (i.e., the choice of observables to be measured, and the number of times they are measured) depend on the unknown state. By learning from previously obtained data, the experimentalist re-adjusts the observable to be measured in the next step, eventually approaching an optimal strategy. We consider two measures of knowledge and exhibit all “best” strategies for the case of a two-dimensional Hilbert space. Finally, we discuss some features of the problem in higher dimensions and in the infinite dimensional case.     

Entropic formulation of uncertainty relations for successive measurements

An entropic formulation of uncertainty relations is obtained for the case of successive measurements. The lower bound on the overall uncertainty, that is obtained for the case of successive measurements, is shown to be larger than the recently derived Deutsch-Partovi lower bound on the overall uncertainty in the case of distinct measurements.     

Extremum uncertainty product and sum states

We consider the states with extremum products and sums of the uncertainties in non-commuting observables. These are illustrated by two specific examples of harmonic oscillator and the angular momentum states. It shows that the coherent states of the harmonic oscillator are characterized by the minimum uncertainty sum 〈(Δq)²〉 + 〈(Δp)²〉. The extremum values of the sums and products of the uncertainties of the components of the angular momentum are also obtained.     

Channel Quality-Based Optimal Status Update for Information Freshness in Internet of Things

This paper investigates the status updating policy for information freshness in Internet of things (IoT) systems, where the channel quality is fed back to the sensor at the beginning of each time slot. Based on the channel quality, we aim to strike a balance between the information freshness and the update cost by minimizing the weighted sum of the age of information (AoI) and the energy consumption. The optimal status updating problem is formulated as a Markov decision process (MDP), and the structure of the optimal updating policy is investigated. We prove that, given the channel quality, the optimal policy is of a threshold type with respect to the AoI. In particular, the sensor remains idle when the AoI is smaller than the threshold, while the sensor transmits the update packet when the AoI is greater than the threshold. Moreover, the threshold is proven to be a non-increasing function of channel state. A numerical-based algorithm for efficiently computing the optimal thresholds is proposed for a special case where the channel is quantized into two states. Simulation results show that our proposed policy performs better than two baseline policies.     

Quantum Violations of N-Qubit Svetlichny's Inequalities are Tightly Bound by the Exclusivity Principle

We first present, by using exclusivity principle, a brief proof of the complementarity principle: the sum of squared expectation values of dichotomic(±1) mutually complementary observables can not be greater than 1. Then we prove that the complementarity principle yields tight quantum bounds of violations of N-qubit Svetlichny's inequalities.This result not only demonstrates that exclusivity principle can give tight quantum bound for certain type of genuine multipartite correlations, but also illustrates the subtle relationship between quantum complementarity and quantum genuine multipartite correlations.     

Uncertainty and Certainty Relations for Successive Projective Measurements of a Qubit in Terms of Tsallis' Entropies

We study uncertainty and certainty relations for two successive measurements of two-dimensional observables.Uncertainties in successive measurement are considered within the following two scenarios.In the first scenario,the second measurement is performed on the quantum state generated after the first measurement with completely erased information.In the second scenario,the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome.Induced quantum uncertainties are characterized by means of the Tsallis entropies.For two successive projective measurement of a qubit,we obtain minimal and maximal values of related entropic measures of induced uncertainties.Some conclusions found in the second scenario are extended to arbitrary finite dimensionality.In particular,a connection with mutual unbiasedness is emphasized.     

Theory of measurement of entangled states

Problem on reconstruction of state of finite-dimension quantum information transfer channel, pure or mixed, by results of measurements of needed number of observables, is considered. It is shown that in general case it is needed to measure incompatible observables in number exceeding by one dimension of space of vectors of state. Each of incompatible observables is measured in its statistically valuable series of measurements. In special case, when one of observables is a non-demolition observable, measurement of the other observables is needed for realization of control of property of non-demolition. In case of paired channel it is shown that results of measurements of observables that do not demolish states of sub-channels are characterized by mutual distribution of probabilities while results of measurement of over-classical observables are characterized by mutual correlation only. This correlation vanishes completely in case of pure unentangled states.