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arXiv:quant-ph/0612171

On the measurement probability of quantum phases

Thomas Schürmann

Published Dec 20, 2006• 0 citations
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Abstract

We consider the probability by which quantum phase measurements of a given precision can be done successfully. The least upper bound of this probability is derived and the associated optimal state vectors are determined. The probability bound represents an unique and continuous transition between macroscopic and microscopic measurement precisions.

DOI
10.48550/arxiv.quant-ph/0612171

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We consider the probability by which quantum phase measurements of a given precision can be done successfully. The least upper bound of this probability is derived and the associated optimal state vectors are determined. The probability bound represents an unique and continuous transition between macroscopic and microscopic measurement precisions.
We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let PPP be the conditional probability to measure the phase alpha\\alphaalpha with precision Deltaalpha\\Delta\\alphaDeltaalpha, given a previous measurement of kkk photons with precision Deltak\\Delta kDeltak. Two upper bounds of the probability are derived. For arbitrary given precisions, these bounds refer to the variation of kkk, alpha\\alphaalpha, and the state vector psi\\psipsi in Hilbert space. The first (weaker) bound is given by the inequality PleqxiP\\leq\\xiPleqxi, with xi=fracDeltaalpha(Deltak+1)2pi\\xi=\\frac{\\Delta\\alpha (\\Delta k+1)}{2\\pi}xi=fracDeltaalpha(Deltak+1)2pi. It is nontrivial for measurements with xi<1\\xi<1xi<1. As our main result the least upper bound of PPP is determined. We obtain an analytical representation of this bound in the asymptotic limit Deltaktoinfty\\Delta k\\to\\inftyDeltaktoinfty and Deltaalphato0\\Delta\\alpha\\to 0Deltaalphato0 such that xi>0\\xi>0xi>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/pi1/pi1/pi.
When measuring the phase of quantum states of light, the optimal single-shot measurement implements a projection on the unphysical phase states. If we want to improve the precision further we need to accept a reduced probability of success, either by implementing a probabilistic measurement or by probabilistically manipulating the measured quantum state by means of noiseless amplification. We analyze the limits of this approach by finding the optimal probabilistic measurement that, for a given rate of success, maximizes the precision with which the phase can be measured.
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.