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.
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.
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 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.
Abstract A modern state-of-the-art “quantum measurement” [The term “quantum measurement” as used here implies that parameters of atomic particles are measured that emerge from a single scattering process of quantum particles.] of momentum and position of a single electron at a given time [“at a given time” means directly after the scattering process. (It should be noticed that the duration of the reaction process is typically extremely short => attoseconds).] and the precision limits for their experimental determination are discussed from an experimentalists point of view. We show—by giving examples of actually performed experiments—that in a single reaction between quantum particles at a given time only the momenta of the emitted particles but not their positions can be measured with sub-atomic resolution. This fundamental disparity between the conjugate variables of momentum and position is due to the fact that during a single-event measurement only the total momentum but not position is conserved as function of time. We highlight, that (other than prevalently perceived) Heisenberg’s “Uncertainty Relation” UR [1] does not limit the achievable resolution of momentum in a single-event measurement. Thus, Heisenberg’s statement that in a single-event measurement only either the position or the momentum (velocity) of a quantum particle can be measured with high precision contradicts a real experiment. The UR states only a correlation between the mean statistical fluctuations of a large number of repeated single-event measurements of two conjugate variables. A detailed discussion of the real measurement process and its precision with respect to momentum and position is presented.
Abstract. We consider a family of vector and operator norms which we refer to as Schmidt norms. We show that these norms have several uses in quantum information theory – they can be used to help classify k-positive linear maps (and hence entanglement witnesses), they are useful for approaching the problem of finding non-positive partial transpose bound entangled Werner states, they are related to the quantum fidelity and trace distance measures, and they are connected to the recently-defined local numerical range. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.