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.
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.
Abstract For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by N linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an N -dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle.
Abstract A general theory of preparational uncertainty relations for a quantum particle in one spatial dimension is developed. We derive conditions which determine whether a given smooth function of the particle’s variances and its covariance is bounded from below. Whenever a global minimum exists, an uncertainty relation has been obtained. The squeezed number states of a harmonic oscillator are found to be universal : no other pure or mixed states will saturate any such relation. Geometrically, we identify a convex uncertainty region in the space of second moments which is bounded by the inequality derived by Robertson and Schrödinger. Our approach provides a unified perspective on existing uncertainty relations for a single continuous variable, and it leads to new inequalities for second moments which can be checked experimentally.
We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.
This thesis addresses two forms of quantum uncertainty. In part I, we focus on preparation uncertainty, an expression of the fact that there are sets of observables for which the induced probability distributions are not simultaneously sharp in any state. We exactly characterise the preparation uncertainty regions for several finite dimensional case studies, including a new derivation of the preparation uncertainty region for the Pauli observables of qubits, and two qutrit case studies which have not previously been addressed in the literature. We also consider the variance based preparation uncertainty for position and momentum observables for the well known “particle in a box” system. We see that the appropriate momentum observable is not given by the spectral measure of a self-adjoint operator, although the position observable is. The box system lacks the phase-space symmetry used to determine the free particle and particle on a ring systems so determining the box uncertainty region is rather more difficult than in these cases. We give upper and lower bounds on the boundary of the uncertainty region, and show that our upper bound is exact in an interval.
In part II we turn our attention to measurement uncertainty, exploring the space of compatible joint approximations to incompatible target observables. We prove a general theorem, which shows that, for a broad class of figures of merit, the optimal compatible approximations to covariant targets are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for covariant observables, since the space of covariant compatible approximations is smaller than the space of all compatible approximations. We employ this theorem to derive measurement uncertainty regions for three mutually orthogonal Pauli observables, and for the quantum Fourier pair acting in any finite dimension.
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.
Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.