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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

On various joint measurements of position and momentum observables in quantum theory

Paul Busch, PEKKA J. LAHTI

Published Apr 15, 1984• 65 citations
Publisher

Abstract

In this paper the problem of simultaneous measurability of complementary physical quantities, and the relevance of the uncertainty relations to this question, is analyzed. Formalizing carefully the crucial notations involved in the problem it is shown that the well-known apparently contradictory viewpoints on the subject matter are all true in pointing to a different aspect of one and the same fundamental fact expressed here as complementarity.

DOI
10.1103/physrevd.29.1634

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In this paper the problem of simultaneous measurability of complementary physical quantities, and the relevance of the uncertainty relations to this question, is analyzed. Formalizing carefully the crucial notations involved in the problem it is shown that the well-known apparently contradictory viewpoints on the subject matter are all true in pointing to a different aspect of one and the same fundamental fact expressed here as complementarity.
Abstract The problem of approximate joint measurement of complementary observables, like position and momentum, and the relevance of the uncertainty relations to that question were at the heart of the investigations of Paul Busch into the foundations of quantum mechanics. A good third of his published work dealt with this and closely related questions. This paper is an attempt to survey some of the steps taken in that research starting with Paul’s first papers on the subject matter in the early 1980’s and reaching its height in recent years.
Uncertainty relations and complementarity of canonically conjugate position and momentum observables in quantum theory are discussed with respect to some general coupling properties of a function and its Fourier transform. The question of joint localization of a particle on bounded position and momentum value sets and the relevance of this question to the interpretation of position-momentum uncertainty relations is surveyed. In particular, it is argued that the Heisenberg interpretation of the uncertainty relations can consistently be carried through in a natural extension of the usual Hilbert space frame of the quantum theory.
The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration shows that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.
In quantum mechanics, measurements disturb the state of the system being measured. This disturbance is largest for complementary properties (e.g. position and momentum) and hence limits the precision with which such properties can be determined simultaneously. Often, this fact is conflated with Heisenberg's uncertainty principle, which refers to an uncertainty relation between complementary properties that is intrinsic to quantum states. In this thesis, the distinction between these two fundamental characteristics of quantum mechanics is made clear. At the intersection of the two are "joint measurements", which circumvent measurement disturbance to simultaneously determine complementary properties. They have applications in quantum metrology and enable a direct measurement of quantum states. The focus of this thesis is on the latter. The thesis is structured in the following way. The first chapter serves as an introduction to joint measurements. It surveys the seminal works in the field, doing so in a chronological manner to provide some historical context. The remainder of the thesis discusses two strategies to experimentally achieve joint measurements. The first strategy is to sequentially measure the complementary properties, making these measurements weak so that they do not disrupt each other. The second strategy is to first clone the system being measured, and then measure each complementary property on a separate clone. Both strategies are experimentally demonstrated on polarized photons, but can be readily extended to other systems.
Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics.Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accepted.The uncertainty principle is shown to appear in three manifestations, in the form of uncertainty relations: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity.Whilst conceptually distinct, these three kinds of uncertainty relations are shown to be closely related formally.Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.