Universita` degli Studi di Padova Dipartimento di Fisica e Astronomia \G. Galilei" Dipartimento di Ingegneria dell'Informazione Corso di Laurea Magistrale in Fisica Direct Reconstruction of the Quantum Density Operator via Measurements of Arbitrary Strength Author: Supervisor: Giulio Foletto Prof. Giuseppe Vallone September 2017 UNIVERSITA` DEGLI STUDI DI PADOVA Dipartimento di Fisica e Astronomia \G. Galilei" Dipartimento di Ingegneria dell'Informazione Corso di Laurea Magistrale in Fisica Direct Reconstruction of the Quantum Density Operator via Measurements of Arbitrary Strength by Giulio Foletto Abstract One of the fundamental problems of experimental quantum physics is the determination of the state of a system, which provides its accurate description and contains all the available information about it. The standard solution is quantum state tomography, a technique that employs a large number of measurements of complementary observables to reconstruct a general density operator. Recently, new protocols have been proposed which allow to obtain the matrix elements one by one with fewer and less complicated observations by exploiting so-called weak measurements, that perturb the system less than ideal projections at the cost of being less precise. This thesis presents an evolution of these schemes that eschews the weakness of the measurement, thus gaining in the accuracy of the results and in practical feasibility. We realized an experiment that uses these methods to reconstruct the density operator of the polarization state of light in single photons regime. Acknowledgements I would like to thank first my supervisor Prof. Giuseppe \Pino" Vallone for having given me the opportunity to work on this project and for having provided direction and much needed help. Not only he made this work possible, that would be true of any thesis supervisor, but he made it enjoyable with his passion and energy. He and Prof. Paolo Villoresi are at the helm of a top notch research group in one of the most promising fields of physics and technology, but their merit is also the creation of a great team of people with whom my collaboration has been easy and fun. On this note I would also like to thank everyone working at the LUXOR laboratory for the help and for having accepted me as part of the family. In particular, Luca Calderaro has been fundamental for the experimental part of this thesis and I thank him for having introduced me to the lab life and for having taught me most of what I now know about the technical aspects of quantum optics. Finally I thank my family, friends and girlfriend for having put up with me in the last few months, during which most of my replies were \Sorry, I'm at the lab right now" or \Back to writing my thesis, bye!". Despite my love for physics, without you this work and the five years before it would have been much more difficult. ii Contents Abstract ii Acknowledgements ii Table of Contents iii 1 Introduction 1 1.1 Mixed States and Density Operators . .2 1.2 Purification of a Mixed State . .4 1.3 The Bidimensional Case: The Qubit . .5 1.4 Quantum State Tomography . .7 2 Weak Measurements 9 2.1 Weak Values and the Ancilla Measurement Scheme . .9 2.2 The Qubit Pointer . 11 2.3 Applications of the Weak Value . 13 2.3.1 Amplification through Postselection . 13 2.3.2 The Meaning of the Weak Value and the Quantum Three-Box Paradox . 14 2.3.3 Average Trajectories of Photons in a Double Slit Experiment . 15 2.3.4 Monogamy of Bell's Inequalities Violations . 16 2.4 Direct Weak Measurement of a Pure Quantum State . 16 2.5 Direct Weak Measurement of a General Quantum State . 18 2.5.1 Direct Weak Reconstruction of the Density Operator . 19 2.5.2 Direct Weak Reconstruction of the Dirac Distribution . 20 2.6 Normalization of the Results . 20 3 Direct State Reconstruction via Strong Measurements 23 3.1 Evolution of the Basic Protocol to Couplings of Arbitrary Strength . 23 3.2 Evolution of the Advanced Protocols . 24 3.2.1 The Density Operator Protocol . 24 3.2.2 The Dirac Distribution Protocol . 25 3.2.3 A Summary of the two Protocols . 26 3.3 Comparison with Weak Reconstruction Protocols . 27 3.3.1 The Inherent Bias of Weak Measurements . 27 3.3.2 The Role of Experimental Errors . 28 3.4 Comparison with QST . 30 3.5 Measuring Products of Non-Commuting Observables with Strong Couplings 32 iii iv 4 The Experiment 35 4.1 The Source of Entangled Photons . 36 4.1.1 Correlation: The SPDC process . 36 4.1.2 Superposition: The Sagnac Interferometer . 38 4.1.3 Performance and Polarization Stability . 40 4.1.4 Temporal Coherence of the Photons . 41 4.2 The Measurement System . 43 4.2.1 Design . 43 4.2.2 Practical Realization . 45 4.2.3 Phase Calibration and Stability . 48 4.2.4 Analysis of the Statistical Error . 50 4.3 Weakening the Coupling . 51 5 Results 55 5.1 Results at Maximum Strength . 57 5.2 Results for Variable Strength . 61 6 Conclusions 65 A Proof of the Theoretical Results 67 A.1 Measuring the Weak Value . 67 A.2 Weak Measurements of Products . 70 A.3 Derivation of the Measurement Protocols . 74 A.4 Strong Measurements of Products . 81 A.5 Details about the Simulations . 83 A.5.1 Generation of Random Density Matrices . 83 A.5.2 Simulation of the Experiment . 84 Bibliography 85 Chapter 1 Introduction At the very foundation of every physical theory stands the concept of state. A state is often a mathematical entity that, together with a physical interpretation, can describe a system and allows the determination of (at least some of) its properties. It does so by restricting such properties \by as many conditions or data as theoretically possible" [1, p. 11]. For instance, thermodynamics can define the state of an ideal gas with the variables P (pressure), V (volume) and T (absolute temperature), whereas in classical mechanics the phase space of positions and momenta is the domain of the possible states of a particle. Given perfect knowledge of a state and of the laws governing its evolution (e.g. a sequence of transformations of the gas, or the Hamilton's equations of motion), many theories can find the entire future and past of a system. Quantum physics is by nature not so deterministic. On account of the superposition prin- ciple, its states reside in a vector structure, precisely a Hilbert space [2, Ch. 2, Sec. 1], but are described by rays so that, intuitively, the sum of a state with itself does not change it. Unit vectors are conventionally chosen as representatives of such rays and contain all the information that can ideally be extracted from the particular configuration the system is in. Even with this knowledge, only the outcomes of some measurements can be predicted for sure, while others remain aleatory because of the uncertainty principle. Moreover, a distinction has to be made between these cases (pure states) and the ones in which even less information is available (mixed ones). For the latter situations, more general descriptions such as that of density operators, must be introduced. Just as with rays in a Hilbert space, with this formalism it is possible to calculate the mean values and all other statistical moments of any measurement from a single mathematical object. But then how does one find it? Quantum state tomography (QST) comes to the aid. It is a set of techniques that aim to reconstruct the state of a system by means of performing specific measurements on many identical copies of it. It has been used extensively for the purpose of characterizing devices that produce quantum systems in a predetermined state, but its importance has recently been growing in the promising fields of quantum information and communication. Here, states are used to encode and transmit a message, so that their precise knowledge can become relevant for the success of an exchange or the certification of the involved parties. Moreover, the quantum computer of the future will need verification of the states it uses as inputs for its calculations and validation for the reading of its results. In more fundamental research, QST can be used as the final step of quantum simulations to find and characterize the ground state of complicated Hamiltonians. 1 2 Introduction Up until recently, QST of d-dimensional states has involved O(d2) measurements and a linear reconstruction of the entire density operator. Yet, in 2011 Lundeen et al. proposed a scheme that can directly obtain each single element of a quantum wavefunction [3] or density operator [4] in a fixed number of measurements and a few simple calculations that in general do not depend on the nature of the system to be studied or the dimension d of its associated Hilbert space. However, this protocol is based on the concepts of weak values and weak couplings between the system and an ancilla, the measurement device that actually provides the needed information, but new theoretical work suggests that this weakness is not necessary and even counter-productive for tomography [5]. The main goal of this thesis is to expand on these ideas and show that the density operator can be directly measured within the framework of ancillae but without the use of weak couplings between systems and with better results in terms of accuracy and precision. After a brief review of the necessary formalism and of the procedures of standard QST, I will explain the fundamentals of weak values and measurements and their use in state reconstruction schemes.