Fundamental study of weak values and post-selected measurements

: from geometry and quantum foundations to quantum information and open systems

Abstract

The second quantum revolution has arrived, and quantum physics is revealing an increasing number of practical applications every day. At the heart of quantum physics lies a foundational question: the measurement problem. What exactly happens when we measure quantum systems? Despite the fact that quantum theory provides a framework for measurements, many scientists and philosophers continue to debate the consequences of measuring to this day.
The focus of this thesis is on weak measurements—a type of quantum measurement where the interaction strength between the measuring device and the system being studied is very small. After the weak interaction, post-selection takes place, requiring a projective measurement and filtering on the desired final state. Ultimately, the shift in the ancilla’s wavefunction is proportional to the weak value—a complex and unbounded number. In recent years, weak measurements have garnered significant attention due to their amplification power and fundamental properties. In this study, we specifically focus on the polar description of weak values. Initially, we delve into the geometrical properties of weak values. The argument of the weak value corresponds to a geometric phase associated with the symplectic area of a triangle formed by the geodesics between the pre-selected state, a state involving the observable, and the post-selected state. To enhance visual comprehension, we also apply the Majorana description to the three states involved in the weak value. Our analysis shows then that the argument of the weak value is related to the sum of two solid angles on the Bloch sphere.
Next, we examine the modulus of weak values. We demonstrate that weak values can be expressed as the expectation value of a non-normal operator. We prove that weak values can differ from the observable’s eigenvalues only if the operator is non-normal. Moreover, we establish strong correlations between the Henrici departure from normality—a parameter that indicates how far a matrix deviates from normality—and the modulus of the weak value in the strong amplification regime. These findings shed further light on the nature of weak values and their relationship with non-normal operators.
Looking at the weak value from a different angle, we can express it as the expectation value of the observable using a pseudo-Hermitian projector. We further demonstrate that, by modifying the Hilbert space standard metric, weak values can be described as expectation values in an indefinite metric space, where the metric has signature (−1,1, . . . ,1). We establish a link between this space and non-classical logics and find that the emerging logic is paraconsistent and paracomplete. These new insights provide a fresh perspective that may inspire further research in the field of quantum foundations.
In quantum mechanics, a state can never be truly isolated and constantly interacts with its environment. In this study, we explore the effects of weak measurements with dissipation between the weak interaction and post-selection. Our investigation reveals that as the dissipation duration increases towards infinity, weak values tend to the expectation value of the observable in the preselected state. However, we observe anomalous weak values, even at infinite dissipation time, in cases where the ground state is degenerate. By examining the system at short dissipation times with weak measurements in the amplification regime, we can extract valuable information about the dissipative dynamics, including the dissipation rate and whether the system is Markovian or
non-Markovian.
Quantum computing is among the most sought-after applications of quantum physics. In this study, we explore a protocol for implementing quantum algorithms using modular values. Specifically, we apply this protocol to the Deutsh-Jozsa algorithm, the Grover algorithm, and the phase estimation protocol. To assess the feasibility of our approach, we report experimental results obtained from the IBM quantum computer.
In summary, weak measurements hold great promise for both their amplification properties and their potential to deepen our understanding of fundamental quantum properties. We believe that our work will serve as a foundation for future studies exploring the polar description of weak values and their application to quantum foundations, as well as investigations into the effects of dissipation on weak measurements. Additionally, our exploration of modular values in quantum computing holds significant potential for advancing the field and driving progress towards practical quantum technologies.

Date of Award	13 Sept 2023
Original language	English
Awarding Institution	University of Namur
Sponsors	ARC (Actions de recherche concentrées)
Supervisor	Yves Caudano (Supervisor), Timoteo Carletti (Co-Supervisor), Alexandre Mayer (President), Alexandre Matzkin (Jury), John Martin (Jury), Dominique Lambert (Jury) & Pérola Milman (Jury)