We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision dk, given a previously successful position measurement q with precision dq. Several upper bounds for the probability P are determined. For arbitrary, but given precision dq, dk, these bounds refer to the variation of q, k and the state vector of the particle. A weak bound is given by the inequality P <= dkdq/h, where h is Planck's quantum of action. It is non-trivial for all measurements with dkdq < h. A sharper bound is obtained by applying the Hilbert-Schmidt-norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.
We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision dk, given a previously successful position measurement q with precision dq. Several upper bounds for the probability P are determined. For arbitrary, but given precision dq, dk, these bounds refer to the variation of q, k and the state vector of the particle. A weak bound is given by the inequality P <= dkdq/h, where h is Planck's quantum of action. It is non-trivial for all measurements with dkdq < h. A sharper bound is obtained by applying the Hilbert-Schmidt-norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.
We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with accuracy ∆k, given a previously successful position measurement q with accuracy ∆q. Several upper bounds for the probability P are determined. For arbitrary, but given accuracies ∆q, ∆k, these bounds refer to the variation of q, k, the state vector ψ of the particle as well as to an infinite set of measuring partitions. A simple bound is given by the inequality P ≤ ∆k∆q h, where h is Planck’s quantum of action. It is nontrivial for all measurements with ∆k∆q < h. The second bound is obtained by considering the Hilbert-Schmidt norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.
The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.
We prove an uncertainty relation, which imposes a bound on any joint measurement of position and momentum. It is of the form (\Delta P)(\Delta Q)\geq C\hbar, where the `uncertainties' quantify the difference between the marginals of the joint measurement and the corresponding ideal observable. Applied to an approximate position measurement followed by a momentum measurement, the uncertainties become the precision \Delta Q of the position measurement, and the perturbation \Delta P of the conjugate variable introduced by such a measurement. We also determine the best constant C, which is attained for a unique phase space covariant measurement.
We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let P be the conditional probability to measure the phase alpha with precision Deltaalpha, given a previous measurement of k photons with precision Deltak. Two upper bounds of the probability are derived. For arbitrary given precisions, these bounds refer to the variation of k, alpha, and the state vector psi in Hilbert space. The first (weaker) bound is given by the inequality Pleqxi, with xi=fracDeltaalpha(Deltak+1)2pi. It is nontrivial for measurements with xi<1. As our main result the least upper bound of P is determined. We obtain an analytical representation of this bound in the asymptotic limit Deltaktoinfty and Deltaalphato0 such that xi>0 is fixed. Finally, we present a rigorous prove that the well-known 'Heisenberg limit' in precision phase measurement can never be attained with measurement probabilities greater than 1/pi.
Standard-deviation and entropic formulations of uncertainty principle capture the spread of the probability distribution but say little about the probability itself contained in a small region. We introduce the confidence uncertainty Δcx(θx) as the minimal Lebesgue measure of the support set in which the particle is found with probability at least θx, and the companion interval confidence uncertainty ΔIx(θx) which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For θx+θp≤1 both confidence uncertainties can be made arbitrarily small simultaneously, so that no nontrivial product bound holds; in particular, position and momentum can be jointly localised with probability at least~50%. (ii) For θx+θp>1 a lower bound holds: combining Lenard's projection inequality with the Donoho--Stark operator-norm bound we obtain ΔcxΔcp≥2πℏ(θxθp−(1−θx)(1−θp))2, and for the interval version we obtain the sharp implicit Landau--Pollak bound ΔIxΔIp≥4ℏλ0−1((θxθp−(1−θx)(1−θp))2), where λ0(c) is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of λ0(c), provide closed-form small-c and large-c asymptotics, compute the optimal Slepian-superposition states that saturate the interval bound, and compare the resulting product against the variance Heisenberg--Kennard, the Białynicki-Birula--Mycielski entropic, and the Donoho--Stark concentration bounds. The unified picture provides a complete phase diagram on (θx,θp)∈[0,1]2.