We present a complete statistical analysis of quantum optical measurement schemes based on photodetection. Statistical distributions of quantum observables determined from a finite number of experimental runs are characterized with the help of the generating function, which we derive using the exact statistical description of raw experimental outcomes. We use the developed formalism to point out that the statistical uncertainty results in substantial limitations of the determined information on the quantum state: though a family of observables characterizing the quantum state can be safely evaluated from experimental data, its further use to obtain the expectation value of some operators generates exploding statistical errors. These issues are discussed using the example of phase-insensitive measurements of a single light mode. We study reconstruction of the photon number distribution from photon counting and random phase homodyne detection. We show that utilization of the reconstructed distribution to evaluate a simple well-behaved observable, namely the parity operator, encounters difficulties due to accumulation of statistical errors. As the parity operator yields the Wigner function at the phase space origin, this example also demonstrates that transformation between various experimentally determined representations of the quantum state is a quite delicate matter.
We investigate the effect of imperfections in realistic detectors upon the problem of quantum state and parameter estimation by continuous monitoring of an open quantum system. Specifically, we have reexamined the system of a two-level atom with an unknown Rabi frequency introduced by Gambetta and Wiseman [Phys. Rev. A 64, 042105 (2001)]. We consider only direct photodetection and use the realistic quantum trajectory theory reported by Warszawski, Wiseman, and Mabuchi [Phys. Rev. A 65, 023802 (2002)]. The most significant effect comes from a finite bandwidth, corresponding to an uncertainty in…
Measuring time means counting the occurrence of periodic phenomena. Over the past centuries a major effort was put to make stable and precise oscillators to be used as clock regulators. Here we consider a different class of clocks based on stochastic clicking processes. We provide a rigorous statistical framework to study the performances of such devices and apply our results to a single coherently driven two-level atom under photodetection as an extreme example of non-periodic clock. Quantum Jump MonteCarlo simulations and photon counting waiting time distribution will provide independent che…
In this thesis we present a direct scheme for measuring quasidistribution functions of light. This scheme, based on photon counting, is derived from a simple relation linking the Wigner function with photon statistics. We develop a full multimode theory of the scheme, and show that the principle of the measurement can be straightforwardly generalized to the detection of the multimode radiation. We also discuss practical aspects of the scheme, including statistical error, non-unit detection efficiency, and imperfect interference visibility. Next, we report experimental realization of the scheme…
We analyse nonperturbatively signal transmission patterns in Green's functions of interacting quantum fields. Quantum field theory is re-formulated in terms of the nonlinear quantum-statistical response of the field. This formulation applies equally to interacting relativistic fields and nonrelativistic models. Of crucial importance is that all causality properties to be expected of a response formulation indeed hold. Being by construction equivalent to Schwinger's closed-time-loop formalism, this formulation is also shown to be related naturally to both Kubo's linear response and Glauber's ma…
Exploiting the geometric nature of statistical divergences, we devise a way to define associated induced uncertainty measures for discrete and finite probability distributions. We also report new uncertainty measures and discuss their properties. Further, we apply a similar technique to measure the uncertainty in the preparation of a quantum state.
We show within a statistical model of quantization reported in the previous work based on Hamilton-Jacobi theory with a random constraint that the statistics of fluctuations of the actual trajectories around the classical trajectories in velocity and position spaces satisfy a reciprocal uncertainty relation. The relation is objective (observation independent) and implies the standard quantum mechanical uncertainty relation.
For a simple set of observables we can express, in terms of transition probabilities alone, the Heisenberg Uncertainty Relations, so that they are proven to be not only necessary, but sufficient too, in order for the given observables to admit a quantum model. Furthermore distinguished characterizations of strictly complex and real quantum models, with some ancillary results, are presented and discussed.
It is well known that the problem of divergence in the physical interpretation of quantum mechanics originating from the uncertainty principle has not yet been resolved. Attempting to clear the constraints and confusion of this situation for the further development of quantum technology, this article traces the original derivation and the physical meaning of the Heisenberg uncertainty principle, analyzes the Einstein photon-box thought-experiment, and studies the limits of the relationships under different action scenarios. By analyzing the statistical distribution of quantum mechanical quanti…
Precise measurement of frequency sensitivity coefficients (FSCs) of some physical effects contributing to uncertainty is an important work in type-B uncertainty (u_B) evaluation of atomic frequency standards. We proposed a method based on statistical correlation of noise to measure FSCs and evaluate u_B for atomic frequency standards, and giving a full statistical expression of FSC as K_I=COV_A(y,x_I)/(sigma(x_I))^2 , where y and x_I are fractional frequency of atomic frequency standards and noise independent variable of Ith physical effects, respectively. The confidence value and the effect o…
A non stationary state in the one-dimensional infinite square well formed by a combination of the ground state and the first excited one is considered. The statistical complexity and the Fisher-Shannon entropy in position and momentum are calculated with time for this system. These measures are compared with the Heisenberg uncertainty relation, ΔxΔp. It is observed that the extreme values of ΔxΔp coincide in time with extreme values of the other two statistical magnitudes.
The Heisenberg uncertainty principle sets a lower bound on the sensitivity of continuous optical measurements of force. This bound, the standard quantum limit, can only be reached when a mechanical oscillator subjected to the force is unperturbed by its environment, and when measurement imprecision from photon shot-noise is balanced against disturbance from measurement backaction. We apply an external force to the center-of-mass motion of an ultracold atom cloud in a high-finesse optical cavity. The optomechanically transduced response clearly demonstrates the trade-off between measurement imp…
Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenberg's original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [Phys. Rev. Lett. 111, 160405 (2013); Phys. Rev. A 89, 012129 (2014)]. Here we reform their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin syste…
Classical statistical average values are generally generalized to average values of quantum mechanics, it is discovered that quantum mechanics is direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, the general…
In quantum physics, all measured observables are subject to statistical uncertainties, which arise from the quantum nature as well as the experimental technique. We consider the statistical uncertainty of the so-called sampling method, in which one estimates the expectation value of a given observable by empirical means of suitable pattern functions. We show that if the observable can be written as a function of a single directly measurable operator, the variance of the estimate from the sampling method equals to the quantum mechanical one. In this sense, we say that the estimate is on the qua…
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $…
We derive a Bell-like inequality involving all correlations in local observables with uncertainty free states and show that the inequality is violated in quantum mechanics for EPR and GHZ states. If the uncertainties are allowed in local observables then the statistical predictions of hidden variable theory is well respected in quantum world. We argue that the uncertainties play a key role in understanding the non-locality issues in quantum world. Thus we can not rule out the possibility that a local, realistic hidden variable theory with statistical uncertainties in the observables might repr…
The classical and quantum evolution of a generic probability distribution is analyzed. To that end, a formalism based on the decomposition of the distribution in terms of its statistical moments is used, which makes explicit the differences between the classical and quantum dynamics. In particular, there are two different sources of quantum effects. Distributional effects, which are also present in the classical evolution of an extended distribution, are due to the fact that all moments can not be vanishing because of the Heisenberg uncertainty principle. In addition, the non-commutativity of …
We study the dynamical consequences of Maggiore's unique generalised uncertainty principle (GUP). We find that it leads naturally, and generically, to novel consequences. In the high temperature limit, there is a drastic reduction in the degrees of freedom, of the type found, for example, in strings far above the Hagedorn temperature. In view of this, the present GUP may perhaps be taken as the new version of the Heisenberg uncertainty principle, conjectured by Atick and Witten to be responsible for such reduction. Also, the present GUP leads naturally to varying speed of light and modified di…
Universally valid uncertainty relations are proven in a model independent formulation for inherent and unavoidable extra noises in arbitrary joint measurements on single systems, from which Heisenber's original uncertainty relation is proven valid for any joint measurements with statistically independent noises.
We investigate the measurement uncertainties of a triple of positive operator-valued measures (POVMs) based on statistical distance, and formulate state-independent tight uncertainty inequalities satisfied by the three measurements in terms of triple-wise joint measurability. Particularly, uncertainty inequalities for three unbiased qubit measurements are presented with analytical lower bounds which relates to the necessary and sufficient condition of the triple-wise joint measurability of the given triple. We show that the measurement uncertainties for a triple measurement are essentially dif…
We show by a statistical analysis of high-resolution scanning tunneling microscopy (STM) experiments, that the interpretation of the density of electron charge as a statistical quantity leads to a conflict with the Heisenberg uncertainty principle. Given the precision in these experiments we find that the uncertainty principle would be violated by close to two orders of magnitude, if this interpretation were correct. We are thus forced to conclude that the density of electron charge is a physically real, i.e., in principle precisely measurable quantity.
1/f noise at arbitrary low frequences is the way of existence of irreversibility in thermal motion governed by reversible laws of mechanics. This statement not once was confirmed in statistical mechanics beyond its traditional kinetical roughenings. Here we point out that in case of quantum statistical mechanics in principle it is sufficient to avoid such the roughening as the "Fermi golden rule". This means taking into account the time-energy uncertainty principle (time-frequency one in classical limit) and thus uncertainties in characteristics of real collisions and scatterings of particles …
