This thesis concerns how attempting to measure certain fundamental properties
of a physical theory can lead us to new insights. The idea of general probabilistic
models is introduced and developed, in order to provide a way to be able to
consider and compare different theories. A measure of incompatibility of two
observables is defined, through the amount of smearing needed to make them
jointly measureable. This measure is then used to characterise the degree of
incompatibility that exists in a given theory. Quantum theory is then shown
to be as incompatible as any other theory, but only in a very course grained
sense. A related way of measuring the strength of incompatibility of a pair
of observables is then shown to put a bound on a measure of the strength of
non-local correlations. The notion of steering, or remote ensemble preparation,
is then shown to be a sufficient condition for the saturation of that bound.
Examples are considered that demonstrate that the given sufficient notion of
steering is not necessary, and it is proposed how the measures can be modified
in cases where the link does not hold. The idea of formulating measures of
error and disturbance associated with a measurement device is discussed. The
notion of a direct test of error or disturbance is used to analyse current proposed
measures, and show their shortcomings.
This thesis concerns how attempting to measure certain fundamental properties
of a physical theory can lead us to new insights. The idea of general probabilistic
models is introduced and developed, in order to provide a way to be able to
consider and compare different theories. A measure of incompatibility of two
observables is defined, through the amount of smearing needed to make them
jointly measureable. This measure is then used to characterise the degree of
incompatibility that exists in a given theory. Quantum theory is then shown
to be as incompatible as any other theory, but only in a very course grained
sense. A related way of measuring the strength of incompatibility of a pair
of observables is then shown to put a bound on a measure of the strength of
non-local correlations. The notion of steering, or remote ensemble preparation,
is then shown to be a sufficient condition for the saturation of that bound.
Examples are considered that demonstrate that the given sufficient notion of
steering is not necessary, and it is proposed how the measures can be modified
in cases where the link does not hold. The idea of formulating measures of
error and disturbance associated with a measurement device is discussed. The
notion of a direct test of error or disturbance is used to analyse current proposed
measures, and show their shortcomings.
The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.
This thesis is concerned with several topics related to concept of incompatibility of quan- tum observables. The operational description of quantum theory is given, in which incom- patibility is expressed in terms of joint measurability. A connection between symmetric informationally complete positive operator-valued measures and mutually unbiased bases is given, and examples of this connection holding based on investigations in Mathematica are presented. An extension of the Arthurs-Kelly measurement model is then given, where the measured observable is calculated, thereby generalising the results given previously in the literature. It is shown that in the case of prior correlations between measurement probes there exists the possibility that a measurement of both probes leads to marginal observables with smaller statistical spread than if measurements are performed on the individual probes. This concept is then highlighted by considering two probe states that allow for this reduction in spread, and the required conditions for success are given. Fi- nally, error-error relations for incompatible dichotomic qubit observables are considered in the case of state-dependent and independent error measures. Quantities that arise in the state-independent measures case, which were previously presented geometrically, have been given operational meaning, and optimal approximating schemes in both cases are compared. Limitations regarding the state-dependent optimal approximations, and experimental work built upon this construction are also discussed.
We introduce a new way of quantifying the degrees of incompatibility of two observables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions, some of which will be answered here. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have if arbitrary pairs of observables are considered. If one adopts a more refined measure of the degree of incompatibility, for instance, by restricting the comparison to binary observables, it turns out that there are probabilistic theories whose inherent degrees of incompatibility are greater than that of quantum mechanics.
Steering is one of the most counter intuitive non-classical features of bipartite quantum system, first noticed by Schrödinger at the early days of quantum theory. On the other hand, measurement incompatibility is another non-classical feature of quantum theory, initially pointed out by Bohr. Recently, Quintino et al. [Phys. Rev. Lett. 113, 160402 (2014)] and Uola et al. [Phys. Rev. Lett. 113, 160403 (2014)] have investigated the relation between these two distinct non-classical features. They have shown that a set of measurements is not jointly measurable (i.e., incompatible) if and only if they can be used for demonstrating Schrödinger-Einstein-Podolsky-Rosen steering. The concept of steering has been generalized for more general abstract tensor product theories rather than just Hilbert space quantum mechanics. In this article, we discuss that the notion of measurement incompatibility can be extended for general probability theories. Further, we show that the connection between steering and measurement incompatibility holds in a border class of tensor product theories rather than just quantum theory.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite-dimensional general probabilistic theories (GPTs). The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the largest incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPTs. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the superquantum nonclassical effects that were observed on these spaces.