Pekka Lahti

Heisenberg's uncertainty principle

Heisenbergs 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 determination of the past and the future of a physical system in quantum mechanics

The determination of the past and the future of a physical system are complementary aims of measurements. An optimal determination of the past of a system can be achieved by an informationally complete set of physical quantities. Such a set is always strongly noncommutative. An optimal determination of the future of a physical system can be obtained by a Boolean complete set of quantities. The two aims can be reconciled to a reasonable degree with using unsharp measurements.

Time observables in quantum theory

Abstract “Time” as an observable of a physical system is to be understood with reference to the evolution of some nonstationary quantity. Thus, any observable “time” is the time of occurrence of an event of a certain type, defined by the appearance of some specified value of the dynamical quantity in question. This interpretation of time observables is illustrated by means of some examples.

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Recent years have witnessed a controversy over Heisenbergs famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for {\em state-dependent} errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for {\em state-independent} errors have been proven.

Uncertainty and complementarity in axiomatic quantum mechanics

In this work an investigation of the uncertainty principle and the complementarity principle is carried through. A study of the physical content of these principles and their representation in the conventional Hilbert space formulation of quantum mechanics forms a natural starting point for this analysis. Thereafter is presented more general axiomatic framework for quantum mechanics, namely, a probability function formulation of the theory. In this general framework two extra axioms are stated, reflecting the ideas of the uncertainty principle and the complementarity principle, respectively. The quantal features of these axioms are explicated. The sufficiency of the state system guarantees that the observables satisfying the uncertainty principle are unbounded and noncompatible. The complementarity principle implies a non-Boolean proposition structure for the theory. Moreover, nonconstant complementary observables are always noncompatible. The uncertainty principle and the complementarity principle, as formulated in this work, are mutually independent. Some order is thus brought into the confused discussion about the interrelations of these two important principles. A comparison of the present formulations of the uncertainty principle and the complementarity principle with the Jauch formulation of the superposition principle is also given. The mutual independence of the three fundamental principles of the quantum theory is hereby revealed.

Unitary measurements of discrete quantities in quantum mechanics

The pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given. Probabilistic aspects of measurements related to the so‐called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined. The problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus. As a by‐product a generalization of the Wigner–Araki–Yanase theorem is given.

Measurement uncertainty relations

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases, the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.

Heisenberg uncertainty for qubit measurements

Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenbergs error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a new general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of +/-1 valued observables A,B are tightly bounded from below by a quantity that measures the degree of incompatibility of A and B. The claim of a violation of Heisenberg is shown to fail as it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.

Coexistence and Joint Measurability in Quantum Mechanics

This talk is a survey of the question of joint measurability of coexistent observables and it is based on the monograph Operational Quantum Physics (Busch et al., Springer-Verlag, Berlin, 1997) and on the papers (Lahti et al., Journal of Mathematical Physics39, 6364–6371, 1998; Lahti and Pulmannova, Reports on Mathematical Physics39, 339–351, 1997; 47, 199–212, 2001).

Characterizations of the canonical phase observable

In this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables. We also show that any contractive weighted shift operator defines a unique phase observable, and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit.