Preparation and Measurement Uncertainty in Quantum Mechanics

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 simulability and incompatibility in quantum theory and other operational theories

Quantum theory is a particularly important instance of an operational theory.By looking at quantum theory from the perspective of more abstract operational framework one is able to study its properties in a wider context. This allows us to identify some of the physical features characteristic of quantum theory and it helps us to understand what makes quantum theory special among other theories. From the information-theoretic point of view this might give us insight into the foundations behind the advantages of quantum information processing over its classical counterpart. In this thesis, based on Publications I – VI, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories and how operational theories can be compared to each other, in doing so establishing quantum theory intuitively as an operational theory among other theories. The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We look at the components of simulability, analyzing and demonstrating them in quantum theory as well as various toy theories. This gives us structural information that differentiates quantum theory from other theories. We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead toother physically feasible devices. We demonstrate different types of restrictions by classifying them and analysing their structure. As a second application we see how the simulation of measurements relates to joint measurability, i.e. compatibility of measurements, and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-freeinformation principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories

On the Measurement of Quantum Work: Operational Aspects

Work is one of the cornerstones of classical thermodynamics. However, a direct transfer of this concept to quantum systems has proved problematic, especially for non-equilibrium processes. Unlike in the classical case, quantum work cannot be defined unambiguously. Depending on the specific setting and the imposed assumptions, different definitions are well motivated. In particular, in quantum thermodynamics, a clear distinction must be made between the measurement, storage, and use of work, since these three facets of the concept are not necessarily compatible with each other. The present thesis is mainly concerned with the measurement aspect. With the help of illustrative scenarios several approaches to quantum work measurements, their advantages and drawbacks are discussed. The focus will be on the question to what extent quantumness plays a decisive role in such scenarios, both in a qualitative and quantitative sense. Based on the gedankenexperiment of a Szilárd machine a criterion is proposed which can be used to verify genuine quantum correlations between the work medium in a heat engine and its thermal environment. In a Szilárd scenario a Maxwell's demon determines the state of the work medium and uses this information to extract work. We split this model into a bipartite setting. The demon only has access to the environment and, thus, can only indirectly measure the state of the work medium. By sharing the acquired information with another agent, the latter can extract work. The question of the quantumness of the experiment can then be reduced to the question of the maximum attainable work in the context of a suitable quantum steering scenario. For the constructed setting a bound for the work output achievable for classical correlations between the engine and the environment is derived. Work extraction beyond this classical limit thus proves the quantum nature of the machine. The verification of non-classical correlations by means of quantum steering is motivated by the fact that such a scenario reflects the typical asymmetry of a thermodynamic setup. While the machine itself is considered to be controllable and characterized in detail, no requirements are imposed on the correlated environment and the measurements performed on it. Consequently, this verification of a truly quantum heat engine is semi-device-independent. In a second scenario, the compatibility of average work and work fluctuations in a driven system is discussed. Fluctuation theorems play an important role in classical non-equilibrium thermodynamics. The best-known example is the Jarzynski equality. This equation establishes a connection between the free energy difference of two equilibrium states and the fluctuating work measured in a non-equilibrium process. A transfer of the Jarzynski equality to quantum systems succeeds most simply if the work definition is based on a so-called two-point measurement scheme. This approach determines the work as the difference of two projective energy measurements. The disadvantage of this definition is the unavoidable disturbance of the quantum state by the measurement, which makes a determination of the correct average work impossible. By means of a generalized two-point measurement scheme, it is shown how this contradiction between fluctuating and average work can be overcome. The approach is based on the concept of joint measurability. Unsharp measurements with a smaller disturbance of the quantum state can be measured jointly and allow for the determination of the correct average work. Nevertheless, the connection between measured fluctuations and the change of free energy can be preserved by means of a modified Jarzynski equality, as elucidated in this thesis. Even though the two-point measurement scheme - both in its projective form and in the generalized variant presented in this thesis - satisfies a Jarzynski equality, the operationality and the associated experimental significance are to be assessed differently than in the classical case. In classical thermodynamics, the Jarzynski relation can be used practically to determine, for example, the change of free energy in RNA molecules. However, it is crucial for such an experiment that the non-equilibrium work can actually be measured without requiring detailed knowledge of the system under consideration. In contrast, the two-point measurement scheme defines work as the energy difference of the system between the beginning and the end of the process. Crucially, for the measurement of these energies the Hamiltonians have to be known and the free energy difference could therefore be calculated directly from this knowledge without reference to the Jarzynski equality. Thus, the operationality of the quantum Jarzynski relation differs fundamentally from its classical counterpart. In this thesis we develop a measurement scheme which, in principle, allows us to employ a quantum version of the Jarzynski equation without knowledge of the Hamiltonians. The crucial point is to include the apparatus that drives the system out of equilibrium in the quantum picture and to define the work measurement on that very apparatus. Such a work measurement can only be meaningfully defined as a quantum expectation value and work fluctuations cannot directly be measured, in contrast to the classical case. The work along a classical microstate trajectory can be determined in a single run. The trajectory itself does not need to be known for this purpose; its existence is sufficient. Quantum trajectories do not exist unless they are objectified by a measurement. It is shown how measurements on the environment of the system can provide information about the trajectories. A conditioning of the measured work on these trajectories then allows for the determination of work fluctuations in the quantum system. For these fluctuations an inequality is conjectured whose limit is given by the classical Jarzynski equation. Numerical results support the conjecture. A proof is still missing. By means of the presented framework, the free energy difference of a quantum system can, in principle, be determined without knowledge of the underlying Hamiltonian. However, as is shown, this requires an optimization over several external parameters, since the inequality in general provides only an upper bound. Thus, the operationality of the model enforces a quantum disadvantage. The methods presented in this thesis can be applied to various scenarios in quantum thermodynamics. Especially the framework for work measurements on an external apparatus offers an alternative to common approaches when the system under investigation and especially its Hamiltonian is not known in advance. The focus on operationality will help to better understand to what extend the work quantities defined and measured in quantum thermodynamic systems differ from the classical concept of work

The Poincaré Half-Plane for Informationally-Complete POVMs

It has been shown in previous papers that classes of (minimal asymmetric) informationally-complete positive operator valued measures (IC-POVMs) in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states. The latter states may also be derived starting from the Poincaré upper half-plane model H . To do this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates, some of the eigenstates of which are the sought fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen–Specker theorem

The Poincaré Half-Plane for Informationally-Complete POVMs

International audienceIt has been shown in previous papers that classes of (minimal asymmetric) informationally-complete positive operator valued measures (IC-POVMs) in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states. The latter states may also be derived starting from the Poincaré upper half-plane model H . To do this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates, some of the eigenstates of which are the sought fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen–Specker theorem