Geometrical and Topological Aspects of Quantum Information Systems

Acknowledgements

Since the censoring red pen of my supervisor Danny does not have access to the Acknowledgments, here I am allowed to be as poetical and flowery as I like. Ergo forgive me if, more often than I should, I will fall off the cliff of scientific standards.

First of all I sincerely need to express my gratitude to all the people who told me that moving to Australia was to stupidest idea ever. I am glad you have been honest, and I am glad I did not listen to you. And thank you Eva for letting me go, after all.

The biggest thank you goes to the two people who offered (and gave) me the chance of being here, my supervisors A/Prof. Daniel Terno and A/Prof. Gavin Brennen.

Danny, you have been a great boss. Thanks from the bottom of my heart for the support you provided me during these years. Apart from being a passionate and amazing physicist, you pushed me to find my own ways, encouraged me to explore new ideas and gave me space to work on what I liked the most. I am sure that more than once you would have loved to put your hands around my neck and firmly close your fingers, but you never did it. In fact, you have always been ready to help me when in need, to kick me when lazy and to congratulate me those few times I did something good. From you I learnt to love the rigor in working and not to fear taking the wrong way (for a short while) when researching. And if I was the only freak in the office who got excited about foundations, that is definitely your fault. Thanks for everything!

Gavin, thanks for being always patient and available to explain again, and again, and again, anything I would not feel confident about. I have always been impressed by the apparently unlimited extension of your knowledge about any concept in Physics. It was easy to see how much passion you put in your work and this motivated me to improve and do my best all the time. Thanks also for sending me around the world. It has been a real privilege working and partying with you!

Along the duration of my studies I have met some great people, whom I have been fortunate enough to work with. Let me say few words about them.

To start, I would like to thank Prof. Kwek for inviting me to visit his group in Singapore and taking me to eat any sort of delicious food while discussing great physics. And I am equally grateful to Prof. Murao for inviting me to pay a visit to her group in Tokyo.

It is also my pleasure to thank Dr. Sai Vinjanampathy for the great time spent together at CQT and in the various Hawker centers of Singapore. Witnessing our discussions shifting from Star Trek to warp drive in general relativity was a clear manifestation of the beauty of being nerds. Equally kind has been Michal Hajdusek, who organized my visit to Tokyo, providing me with financial support: Dude, it was great talking physics, but it was even greater lowering the already pretty low level of the Celt one beer after the other. You guys are legends!

Nicolas ”Doc” Menicucci, you have been an example both at the whiteboard and on the dance floor. Doing calculations via Skype at 4 a.m. with you is something that, one day, I will proudly tell my kids. You taught me the importance of perfection and I will always keep in mind your lesson: It makes no sense to upload it on the arxiv if it is not going to be the best paper out there. And needless to say, thanks for sharing your best pick-up lines with me.

A warm thank you to all the admin people who constantly helped me fixing any sort of bureaucratic trouble. Especially Lisa: It is a miracle that you never threw something at me after the -th time I would give you again the wrong module. Also thanks to Michele for her constant (and patient) help with tickets and flights!

There are some special people I wish to thank: They are Aharon the Dude, Andrea and Hossein, Cristina culo gordo, Franceschina, Eleonora la Dude, Luchino and Marion, Mauro il Dude, Paolino, His darkness Saatana and Thorn. You have been like brothers and sisters to me. I always knew that, no matter what, with you guys around I would have been safe and sound. Together we shared so much foolery, laughters, sometimes tears, and I am extremely grateful to all of you guys for that.
Allow me to be particularly sentimental: Aharon, thanks for your friendship (in the shape of suggestions, bad jokes, physics, coffee, beer and food). It always soothed the mess that distance and uncertainty sometimes caused in my mind. Franceschina, I know that although you expressed disgust any time I hugged you too tight, you liked every bit of it. Thanks for offering me a roof and a cot when I needed it the most! Mauro, it has been phenomenal making this journey together. We have shared too much beer, meat and mead to tell it all here in a few lines. Good luck with the rest of the adventure, and hold me a dry spot under the bridge please.

Everybody knows about the tragedy of finding a proper house in Australia111If you do not, well, read “He died with a falafel in his hand” by J. Birmingham.. However, in these years I have been extremely lucky to share low standards of living and poor hygienic conditions with some amazing characters, who undoubtedly made the whole experience of living in Sydney more special. Therefore, let me thank Safi, Josh, Kamin, Gabriela and especially Marika, who took care of me like a younger brother. And a special tribute goes to Lili, who helped me to overcome my dislike of cats.

I should ideally thank so many more people for too many reasons. Anyone I met in the last years gave me something special and in a way made me the person I am today. However, allow me to name “few” of these beings who are particularly important to me: Alika, ChrisMcMahon, Christoph, Doozie, Enzo, Federica, Geraldine, Helena, Ivan, Jacopo, Johann and Elna, Kayla and Xavier, Michael and Louise, Nieke, Nora and Jack, Richard and Marija, Robert, Sukhi, Valentina A, Valentina B, Valentina D, Vikesh, and the rest of the quantum guys. Thanks folks for being part of this adventure.

Thanks to my friends in Italy who have never forgotten me while I have been away and have always forgiven my unjustifiable long pauses in replying to emails and phone calls: I hug you all Alberto, Castagna, Chiara, Damiano, Elena B, Elena G, Eleonora, Francesco and Lucia, Gianni, Giulia, the Goitre, Lara, and Stefania.

Every time I visit the Bacic family in Brisbane and the Aso family in Tokyo I feel welcomed like a son, and it is an amazing feeling indeed. I thank you for your generosity! Knowing that I have an extended family across the borders of this planet is like sweet popcorn for my soul.

Thank you Orly, for both the great and the disastrous time spent together. You taught me the value of honesty and integrity the hard way and I shall not forget it.

It is people like Ai Leen that make the world shivering with hope. Thanks for being so caring, idealistic and never ready to compromise. You are a wonderful small little being, please promise me you will never change, no matter what.

Thanks Sabrina for making these last months here in Sydney special. I could have easily drowned into the deep sea of despair, but you managed to hold me up with a strong grasp and an enchanting smile. Some things happen unexpectedly, and you surely are a surprising but wonderful gift.

Last, but certainly not least, I wish to thank my family, for all the support and unconditional love you gave me in these years. My mum, Federica, who won her fears and flew all the way down under to see with her own eyes the places I have been telling her about. My dad, Gianpiero, who will be ecstatic to have my burden back on his shoulders. Nonna Irma, who innocently asked me how long it takes to get to Sydney by train. Thanks also to my brother Alessandro, my aunties Renata and Chiara and my cousins. A thought always goes to my grandparents who are not here anymore: Nonno Beppe, Nonna Nella and Nonno Severino, I know how proud you are that your fool of a grandson made it to the end. I dedicate this work to you.



List of Publications

  • A. Brodutch, T.F. Demarie and D.R. Terno
    Photon polarization and geometric phase in general relativity.
    Phys. Rev. D 84, 104043 (2011).

  • D. Rideout, T. Jennewein, G. Amelino-Camelia, T.F. Demarie, B.L. Higgins, A. Kempf, A. Kent, R. Laflamme, X. Ma, R.B. Mann, E. Martin-Martinez, N.C. Menicucci, J. Moffat, C. Simon, R. Sorkin, L. Smolin, and D.R. Terno
    Fundamental quantum optics experiments conceivable with satellites – reaching relativistic distances and velocities.
    Class. Quantum Grav. 29, 224011 (2012).

  • T.F. Demarie
    Pedagogical introduction to the entropy of entanglement for Gaussian states (2012). (Pre-print arXiv:1209.2748).

  • T.F. Demarie and D.R. Terno
    Entropy and entanglement in polymer quantization.
    Class. Quantum Grav. 30, 135006 (2013).

  • T.F. Demarie, T.Linjordet, N.C. Menicucci and G.K. Brennen
    Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators (2013). (To appear in New J. Phys., Pre-print arXiv:1305.0409).

Abstract

Geometry and topology play key roles in the encoding of quantum information in physical systems. Ability to detect and exploit geometrical and topological invariants is particularly useful when dealing with transmission, protection and measurement of the fragile quantum information. In this Thesis, we study quantum information carriers associated with discrete or continuous degrees of freedom that live on various geometries and topologies.

In the first part of this study, intrigued by the possibility of implementing quantum communication protocols in space, we analyze the effects of a gravitational field on the polarization of photons. In this case, we investigate discrete degrees of freedom moving on a continuous space-time: Specifically, we look at the geometrical description of the problem. We find that for closed trajectories, in both static and general space-times, the amount of rotation, or phase, caused by the action of gravity is independent of the reference frame chosen to define the polarization vector. We also prove that similarly to other instances of a geometric phase, its value is given by the integral of the (bundle) curvature over the surface that is encircled by the trajectory.

In the second part we study a new approach to topological quantum information by using Gaussian states to construct a system that exhibits topological order. We describe a (Gaussian) continuous-variable state analog to Kitaev surface codes prepared using quantum harmonic oscillators on a two-dimensional discrete lattice, which has the distinctive property of needing only two-body nearest-neighbor interactions for its creation. We show that although such a model is gapless, it satisfies an area law. Its ground state can be simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster-state, which does not exhibit topological order. A universal signature of topologically ordered phases is the topological entanglement entropy. Due to low signal to noise ratio it is extremely difficult to observe the topological entanglement entropy in qubit-based systems, and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to reveal the order. We prove that for our continuous-variable model the topological entanglement entropy can be observed simply via quadrature measurements, in contrast with qubit-based systems. This provides a practical path to observe topological order in bosonic systems using current technology.

In the third and last part we study the well-definiteness of the concept of entropy in a scheme alternative to Schrödinger quantization: Polymer quantization. The kinematical construction of the Hilbert space in polymer quantization is based on the discretization of the real line, which is treated as a one-dimensional graph with discrete topology. On such a setup, which resembles the more complicated construction of loop quantum gravity, we analyze whether the values of the quantum entropy computed in the Schrödinger and polymer quantization coincide or not. We study the convergence of the entropies of physically equivalent states in unitarily inequivalent representations of the Weyl-Heisenberg algebra and derive a general bound to relate the values of entropy.

Contributions

The author and his collaborators have made the respective contributions to the results presented in this work. They are listed following the structure of the Thesis as presented in the outline.

Relativistic quantum information

These results are based on the papers published in Phys. Rev. D 84, 104043 (2011) and Class Quantum Grav. 29, 224011 (2012). The idea of deriving a gauge invariant phase for closed paths came from A/Prof. Daniel Terno. I did the calculations for the explicit form of the phase. The discussions on quantum experiments in space were initially suggested by A/Prof. Thomas Jennewein and Prof. Raymond Laflamme. A/Prof. Terno and I contributed to section 3 of the review paper, which accounts for the relativistic effects in quantum information theory. We performed the analysis of relative magnitudes of the effects within the parameters of the QEYSSAT mission.

Gaussian states

I have written a pedagogical review on Gaussian states, arXiv:1209.2748, where I focus on the entanglement properties of such states, analyzing in particular the form of the covariance matrix and symplectic eigenvalues for a pair of coupled quantum harmonic oscillators.

Continuous-variable topological order

The idea of detecting topological order using continuous-variable systems was the result of fruitful discussions between A/Prof. Gavin Brennen and myself. I proposed to calculate the topological entanglement entropy for a lattice of oscillators using the Gaussian states formalism and Dr. Nicolas Menicucci introduced us to the graphical calculus for Gaussian states. I derived the form of the covariance matrix for the continuous-variable physical surface code state and did all the numerical simulations and graphs presented here. A/Prof. Brennen and I calculated the gap for the surface code state, while A/Prof. Brennen derived the bound for the decay of the correlations. I proposed to test topological order using the topological logarithmic negativity and did the related calculations. A/Prof. Brennen, Dr. Menicucci and I constructed the noise model and derived the related covariance matrix. Dr. Menicucci and I derived the analytical bound for the topological mutual information, while A/Prof. Brennen suggested how to derive the bound for the topological entanglement entropy and I did the calculations. The experimental proposals at the end of chapter 6 are due to the visionary ideas of Dr. Menicucci and his collaborators. These results are presented in arXiv:1305.0409v3, while the manuscript has been submitted to the New Journal of Physics.

Entropy in polymer quantization

The idea of studying the behavior of entropy in different quantization schemes is due to A/Prof. Terno and was the original project of my PhD. I first proposed to use Gaussian states as an example and did the calculations for the entropy of two coupled oscillators quantized in the polymer scheme, which then helped us to derive the precise entropy bonds. A/Prof. Terno introduced me to the representation theory of canonical commutation relations that were the basis for the proofs of the entropy bounds, which have been derived jointly by the two of us. These results have been published in Class. Quantum Grav. 30, 135006 (2013).

Contents

Chapter 1 Introduction

Quantum mechanics was introduced at the beginning of the last century as a radical new theory, defining the space where physics takes place and the rules that physical systems are required to obey at the very cold and small scales [11, 12, 13, 14, 15, 16, 17, 18]. This astonishing novel concept, together with the newly born theory of relativity [19, 20, 21, 22, 23], turned the rational idea of classical physics on its head, completely changing our understanding of the laws of physics. This was the first quantum revolution. Despite the innumerable experimental confirmations of quantum theory, foundational questions such as the resolution of the measurement problem (the macroscopic transition from quantum to classical) remain [24].

The “second quantum revolution” kicked off at the end of the last century when researchers started to wonder wether it was possible to obtain a better understanding of the laws of the quantum world looking at information theory through the lenses of quantum mechanics (and vice-versa). While the seminal proposals for a quantum computer are familiarly attributed to Feynman [25] and Deutsch [26], it was not until Shor’s factorization algorithm [27], which gave the shivers to every cryptographers on the planet, that these ideas led to a new view of the role played by information in physics, giving birth to the fascinating field of quantum information. As a result, the past 25 years have been a prosperous time for the physicists interested in understanding how to exploit the quantum mechanical properties of nature in order to accomplish computational and operational tasks that would be incredibly onerous in terms of classical physical resources, if not practically impossible [28, 29, 30, 31]. Today, in parallel with discoveries in quantum information theory, experimentalists have reached an extraordinary level of control over single and composite quantum systems [32], opening up the doors to the world of large scale fully quantum devices and the related applications and implications [33, 31].

In this Thesis, we pursue investigations to some of the open problems in present-day quantum information and quantum computation theory, analyzing geometrical and topological properties of both discrete and continuous quantum information systems. Roughly speaking this work is divided in three parts, preceded by a general introduction to the concepts of quantum mechanics, quantum information, relativity and loop quantum gravity.

The first part, Chapter 3, deals with photons in curved space-time [34, 35]. While photons can be used as physical realizations for qubits [31], the basic elements in quantum information, it is still not entirely understood how the presence of a gravitational field affects the information stored in the polarization of a single photon [1]. Taking into account these effects is particularly important when designing schemes of quantum communication between Earth and orbiting satellites [2]. Moreover, future tests of general relativity involving quantum systems will rely on precise estimations of the relativistic effects on the quantum degrees of freedom [36]. After a brief account of the relativistic description of the system, we will focus on polarization rotation in closed trajectories on a curved space-time, detecting interesting invariants and suggesting possible experimental applications for these effects.

Chapters 4 - 6 account for the second part of the Thesis. We first introduce the notions of Gaussian states [3, 37] and topological order [38] and then combine them together into the new concept of continuous-variable topological order [5]. In general, topology refers to the ability of relating systems whose shape look different at a first glance. By analyzing common geometrical properties, one extracts quantities that are used to characterize the global properties of otherwise very different-looking systems. Topological quantum computation is the area of quantum information that studies the topological properties of a special class of two-dimensional system, also called topologically ordered [39, 40]. These systems are particularly fascinating because they can support anyons [41], unusual quasi-particles that are neither bosons nor fermions and whose very non-local properties enable the implementation of fault-tolerant quantum computation via braiding operations [42, 43]. Performing quantum computation using anyons does not require the physical implementation of quantum gates, whose action is now replaced by the spatial exchange of anyons. While very appealing from a theoretical point of view, the experimental realization of topological systems based on qubits (or qudits) is highly challenging, due to the level of control needed to build a lattice of interacting spins [5].

We present here an alternative approach that allows for a much simpler realization in the laboratory with today’s technology. Starting from the toric code state, defined as the ground state of an highly-entangled lattice of qubits, we introduce its continuous-variable surface code state analog, replacing qubits with light field modes, which are described mathematically by Gaussian states. This choice is motivated both by the easy mathematical formalism that characterizes Gaussian states, which allows for a straightforward description of the state in terms of its first and second statistical moments [44], and the availability of well-developed experimental techniques [45]. Exploiting the Gaussianity of the state, one can prove that the continuous-variable surface code exhibits topological order by detecting a non-zero topological entanglement entropy, which is the distinctive feature of topological phases [6, 7]. Remarkably, this can be achieved simply performing measurements of the quadrature operators of the light modes. Furthermore, this system requires only two-body nearest-neighbor interactions for its creation, and hence opens up to scalable implementations using, for instance, quantum optics [46, 47] or circuit-QED technology [48, 49].

Since the identification of topological order in topological quantum computation theory is strictly related to the entropic properties of the system [40], in the third and last part of this Thesis, Chapter 7, there will be space to investigate the well-definiteness of the concept of entropy in different quantization schemes. In particular, we will be dealing with polymer quantization [50], a representation of the Weyl-Heisenberg group unitarily inequivalent to the standard Schrödinger quantization [51, 52], which was introduced as a toy model to study the kinematic and the dynamic of 1-dimensional quantum systems in analogy with the more complicated setup of Loop Quantum Gravity [53, 54, 55].

While there are ways to demonstrate that physical observables in Schrödinger and polymer quantizations converge to the same value in the appropriate limit [56, 57], analysis of the quantum entropy in the two schemes was missing. Specifically, it was not clear for a quantum state whose observable predictions in the two unitarily inequivalent quantization schemes are close, whether the values of the entropy will also be close or not [4].

Here we give an answer to this question, establishing the conditions for entropy convergence. Gaussian states happen to be particularly useful also in this context. By looking at the entropy of two coupled quantum harmonic oscillators, it is relatively easy to construct an example that confirms our findings and allows us to show explicitly how the corrections introduced by the discretization of the physical space affect the value of the entropy.

1.1 Outline

Our intent is to give a uniform flow to the discussion, alternating necessary background and new results. This Thesis has three main parts, with four appendices presenting detailed calculations and additional useful theoretical background. Briefly, this is the outline of the work.

  • General introduction to the concepts of the Thesis:  
    Chapter 2 summarizes the essential facts of quantum mechanics, quantum information, relativity and quantum gravity needed for an effective comprehension of this Thesis. We first start with some basic structural ideas of quantum mechanics. Then we review the properties of entropy for classical and quantum systems, and introduce entanglement together with ways to quantify it in pure and mixed states. After introducing the stabilizer formalism, this Chapter is concluded with general concepts of relativistic quantum information and a short section about loop quantum gravity.

  • Part 1 - Geometric phase of photons:  
    The first part of the Thesis is based on the results from [1] and [2]. We discuss the gauge-invariance of the polarization phase for photons traveling along closed trajectories in closed paths.

    • Chapter 3: Here we start explaining the relativistic effects on the states of a photon and their dependence on the choice of a well-suited reference frame. Then we introduce the geometric meaning of polarization phase through the derivation of an alternative equation for the phase in a static space-time, written in terms of the bundle connection. One can then prove that in a closed trajectory the change in polarization depends only on the geometrical properties of the space-time and not on the choice of reference frame. The Chapter is concluded with some remarks about experimental applications of quantum information in space.

  • Part 2 - Topological order with Gaussian states:  
    The second part is based on [3, 5] and analyses the interplay between Gaussian states and topologically ordered phases.

    • Chapter 4: In this Chapter we rigorously introduce the mathematical description of Gaussian states, starting with the Hilbert space and phase space representations of such states, and continuing with the state evolution under symplectic transformations. Then a step-by-step derivation of the entanglement entropy formula for pure Gaussian states is presented, together with an example. To conclude, we introduce the graphical calculus for Gaussian states and give some examples of graphical representations of common experimental Gaussian procedures. This Chapter is based on the review [3].

    • Chapter 5: This Chapter serves as an introduction to the most common physical codes used in the theory of topological quantum computation. After a brief account of quantum lattice systems, we give a concise explanation of the cluster-states and surface code states, explaining how to detect topological order by means of appropriate witnesses, both in the case of pure and mixed states.

    • Chapter 6: After the two chapters of introduction and motivation, here we present the results from [5]. We start this Chapter introducing the continuous-variable states that are used in the analysis, carefully describing all the properties of the physical continuous-variable surface code state. This is the analog of the toric code state, constructed by means of interacting light modes instead of qubits and described by Gaussian states formalism. We use the graphical calculus for Gaussian states to derive an explicit form of the covariance matrix of the state and from that we calculate the value of the topological entanglement entropy, which proves that the state exists in a topological phase. We conclude with the analysis of a noise model for the surface code and describe possible experimental implementations for this model.

  • Part 3 - Entropy in polymer quantization:  
    The third and last part of this Thesis follows the results from [4], where we studied entropy in different quantization schemes.

    • Chapter 7: We start explaining the concept of algebraic quantum theory: this is done introducing -algebras and posing the conditions for the physical equivalence of two representations of the same algebra. After reviewing polymer quantization, a representation of the Weyl-Heisenberg algebra, we use the properties of entropy to analyze its convergence in two different quantization schemes, specifically polymer and Schrödinger quantization. We then derive a general bound that relates entropies of physically equivalent states in unitarily inequivalent representations and conclude with an example where we employ Gaussian states to calculate the bipartite entanglement entropy of two coupled oscillators in polymer quantization.

  • Conclusions and appendices:

    • Chapter 8 presents the results of this work in a concise manner, highlighting connections between different areas and open questions.

    • Appendix A is a short summary of differential geometry. It contains explanations of all the concepts used to derive the photon polarization equations in Chapter 3.

    • In Appendix B we first set the rules and then derive the nullifier sets that are used in Chapter 6 to describe the continuous-variable code states.

    • An upper bound for the topological entanglement entropy in the CV surface code is derived in Appendix C, together with an explicit proof of the lower bound formula for the topological mutual information. These results are based on [5].

    • Appendix D presents explicit calculations of the canonical operators expectation values in polymer quantization. These are used to calculate the value of the entropy in the coupled harmonic oscillators example illustrated in Chapter 7.

1.1.1 Original results

This is a brief list of the original results presented in this Thesis. They have been divided according to the relevant sections as explained in the outline.

Relativistic quantum information

The calculations and results about the gauge invariance of the geometric phase presented in Section 3.2.1 are original and were first published in [1]. The ideas for experimental applications of relativistic quantum information are reviewed in [2].

Continuous-variable topological order and Gaussian states

While Chapter 4 consists of a general review of known results on Gaussian states, the example in 4.2.2 is original and was first presented in [3].  
The descriptions of the physical CV codes in Section 6.3 are corrected versions of other descriptions appeared in the literature before. All the simulations, calculations and results in Section 6.3 and 6.4 are novel, together with the entropy bounds presented in Appendix C and they were first derived in [5].

Entropy in polymer quantization

The calculations and results in Chapter 7 are mostly original, apart from the polymer quantization description in Section 7.3. They were first derived in [4].

Chapter 2 Thesis’ Prolegomena

This Thesis starts with a minimal review of all the relevant mathematical definitions and physical features that will be used in the following Chapters111We acknowledge [58] for inspiring the title of this Chapter.. We begin with a short introduction to quantum mechanics, defining the basic concepts of Hilbert space, quantum states, physical observables and measurements [31, 59]. Then we move to information theory, focusing in particular on the concept of entropy and information [60]. First we offer a parallel view of different information measures both in classical and quantum physics, secondly we specifically list the required properties of the quantum entropy. We then introduce the paradigmatic notion of entanglement [61], clarifying the meaning of bipartite entanglement and presenting a number of quantities that are normally used to quantify the quantum correlations shared between two systems [62]. The next notion to be described is the stabilizer formalism [63], a method that can be used to simplify the description of quantum many-body systems on a lattice. This method is important for us because these are the systems that will be employed in the discussion of topological order for continuous-variables [5] in Chapter 6.

After that, we take our quantum systems up to space, where in addition to the quantum theory we also need the laws of relativistic quantum information [34, 2]. There, we investigate the behavior of quantum information enclosed into the degrees of freedom of photons traveling a curved space-time [1]. Therefore, in this Introduction we succinctly describe the physics of massless point-like particles in special relativity, in order to simplify the subsequent extension to general relativity in Chapter 3.

Eventually, we conclude this introductory Chapter spending few words about quantum gravity and specifically loop quantum gravity [54, 64, 65], one of the most promising attempts to derive a theory that encloses both quantum mechanics and general relativity. Loop quantum gravity is one of the main motivations to examine entropy in different quantization schemes [50, 4], as we do in Chapter 7.

2.1 Basic elements of quantum mechanics for quantum information

The protagonist of quantum information is the quantum bit, or qubit [31]. As the name suggests, the qubit is the conceptual extension of a classical bit to the quantum world, where physical states can exist in a unique condition known as superposition of states. Classically, a bit is the basic unit of information, which physically corresponds to a system that can exist in two different states, traditionally called zero and one. Start with a (fair) coin, and flip it in the air. It will fall showing you either the head () or the tail (). This is the street version of a bit. The concept of qubit is built upon similar considerations, but it is now implemented by a 2-level quantum state and exhibits some extra features that cannot be found in classical physics. Before clarifying what the last sentence means, let me explain briefly part of the terminology we will adopt.

Traditionally (any quantum mechanics book is a good source for the details presented here, see for example [31, 59, 66]), quantum mechanics tells us that a (quantum) system is described by a unit vector, or state, that belongs to an Hilbert space. Hilbert spaces are a class of -dimensional complex vector spaces with certain additional properties. First of all they are equipped with a positive definite scalar product,

(2.1)

which is a continuous map conventionally anti-linear in the first argument and linear in the second argument. Using the inner product one can introduce the concept of norm of an element as

(2.2)

Furthermore, a Hilbert space is a complete inner product space, meaning that all the Cauchy sequences in converge to a value in .

A qubit state lives in a 2-dimensional Hilbert space spanned by some (vector) basis that we choose to label

(2.3)

in analogy with the classical description of the bit states , where the symbol is called a ket and is the standard depiction for a (pure) state of a quantum system. As mentioned before, quantum mechanics actually tells us more: A qubit can exist in states that are linear superpositions of the basis states, such as

(2.4)

where the absolute value of the coefficients, namely amplitudes, ( and ) correspond to the normalized probabilities of finding the state in one of the two possible outcomes , after an appropriate measurement. For future references, we call the basis , with the established name of computational basis. In Eq.(2.4) lies (part of) the real power of quantum computation. Quantum states can exist as linear combinations of other quantum states, weighted with different amplitudes. Notice also that, although the amount of information that can be extracted from a qubit is exactly the same of a classical bit, one bit, the quantum superposition principle or more generally the linearity of quantum mechanics, allows us to perform pre-measurement operations on qubits that are in no way possible for classical systems.

In the ket formalism, the inner product is rewritten as

(2.5)

where the element is called bra and is the vector dual to the ket . Duality is a basic concept in algebra and in differential geometry. It will be discussed with more rigor later in the introduction and in Appendix A. To generalize the inner product, introduce an orthonormal basis defined on some Hilbert space and consider any two vectors , , then we have that

(2.6)

The following concept we require to introduce is the representation of physical observables in quantum mechanics. In the quantum regime, physical (classical) observables, as position, momentum and energy, are associated to (quantum) operators that act on quantum states. When one represents quantum states by kets, any observable operator is given by a hermitian matrix , where and is the Hermitian adjoint of . Every Hermitian matrix has real eigenvalues, and for an operator they correspond to the values that the associated physical observable can acquire after a measurement. The expectation value of some observable in the state is given by

(2.7)

and if , the quantities

(2.8)

are the matrix elements of the observable .

The temporal evolution of a closed quantum system from a time to a time is given by some unitary operator , such that

(2.9)

and , where is the unit operator. The unitarity condition immediately implies that . Unitary operators can be seen as gates acting on one or many-qubit state, evolving it accordingly to their properties. Among the most important 1-qubit gates are the Pauli operators:

(2.10)

For instance, it is trivial to see that the operator swaps the computational basis vectors

(2.11)

The Pauli matrices, together with the unit operator , taken with multiplicative factors form a group closed under matrix multiplication [67], called the qubit Pauli group. Its extension to -qubits is at the basis of the stabilizer formalism [31] and the generalization [68] to quantum-modes [45], to be specified later, plays a fundamental part in our future treatment of continuous-variable topological phases in Chapter 6.

It is worth pointing out that an alternative basis to the computational basis is the conjugate basis

(2.12)

and the two basis are related by the Hadamard operation;

(2.13)

Measurements in quantum mechanics are a tricky business. If we write the desired observable on a orthogonal basis

(2.14)

then the outcome of a measurement can solely be one of the eigenvalues of the linear decomposition. Specifically, the probability to obtain for a state is

(2.15)

The measurement operators are also known as projectors, and they satisfy the completeness equation and . After a measurement with outcome , the state of the system, which initially was collapses into the state

(2.16)

with if .

States that can be labelled by a ket, as the ones treated so far, are called pure states. All the required information for their description is encoded in . Whenever we are ignorant about the description of a state, we are only allowed to treat it statistically as an ensemble of pure states with different probability amplitudes. This is the density matrix of a system, defined by

(2.17)

with probability of the system to be in . If there exists a base where , then and the state is pure. For pure states, the density matrix language is analogous to the ket language. If this condition is not satisfied, then the state is mixed and cannot be represented by a ket.

In the density matrix representation the conservation of probability is given by the obvious relation

(2.18)

and each density matrix is a positive operator such that, given a ket , the following inequality always holds

(2.19)

The expectation value of an observable with respect to is expressed by

(2.20)

while the action of an unitary operator on the density matrix is

(2.21)

and the time evolution of obeys the following equation

(2.22)

with time-dependent Hamiltonian of the system.

Let me recall that this description of quantum mechanics, named after Heisenberg [12, 13], is completely equivalent to the formulation introduced by Schrödinger [15], where kets are replaced by wave functions and physical observables are equivalent to differential operators.

2.2 Entropy and its properties

Statistical thermodynamics, quantum mechanics and information theory promoted entropy from an auxiliary variable of the mechanical theory of heat [69, 70] to one of the most important quantities in science [71]. In the field of information theory, entropy plays a most fundamental role: It quantifies the amount of uncertainty, or lack of information, the observer has over a system [31, 72]. Classically, entropy is a concept associated to the probability distribution of a classical variable, while in the quantum regime entropy is a function of the density matrix of the state [60]. In the following, we will first introduce different classical statistical functions and then present their quantum counterparts.

2.2.1 Classical world

In classical information theory, the basic measure of uncertainty is the Shannon entropy [73]. Given a random variable , the Shannon entropy quantifies how much we do not know about the variable before we learn its value. Less prosaically, if the variable can take different values, each of them with a certain probability , such that is the probability distribution of the variable , then the Shannon entropy is defined as

(2.23)

and it is maximized by a uniform distribution with all equal .

The Shannon entropy is the lower parametrical limit of a one parameter family of entropies known as Rényi- entropies [73, 60], defined by

(2.24)

In the limit , one recovers the Shannon entropy for . Fruitfully, for each value of the Rényi entropy vanishes for and it acquires its maximum value for the uniform distribution.

Given two variables and with probability distributions and , there exists a measure that quantifies how different these distributions are. This is called the relative entropy [72]

(2.25)

and it is related to the rate at which, in the limit of very large sampling, one can safely identify if we are sampling from the or distribution.

The last concept we wish to introduce at this point is a quantity that tells us something about how much information two distributions have in common. This is the mutual information [31]. Classically, the mutual information is a quantity associated to the probability distributions of two classical variables and :

(2.26)

Intuitively, the mutual information describes how much information about we learn by measuring the value of . If the two quantities are not correlated, then it simply reduces to the Shannon entropy of .

2.2.2 Quantum world

The quantum analog of the Shannon entropy is the von Neumann entropy, or quantum entropy [74]. It is constructed replacing the classical probability distribution with the closest quantum concept, i.e. the density matrix (2.17) of the quantum state, generically . Thus, given , its von Neumann entropy is defined as

(2.27)

where is the set of the eigenspectrum of . From this last definition, it is trivial to realize that for a pure state since the von Neumann entropy is always identically zero. Therefore, only non-pure states have non-vanishing von Neumann entropy, whose greatest value is given by the maximally mixed state ().

The quantum version of the relative entropy for two states is defined as [73]

(2.28)

In analogy with the classical case, it says something about the statistical distinguishability between the two states and . Operationally, larger the value of the relative entropy, greater is the amount of information we can extract when performing a measurement aimed at distinguish the two states.

In complete analogy with the classical case, one can construct a family of quantum Rényi entropies for with [73],

(2.29)

where the von Neumann entropy is given by the limit. Rényi entropies are particularly appealing because the full family contains information about the (entanglement) spectrum of the state , which is more than the information given by the only entanglement entropy. Furthermore, Rényi entropies showed to be useful in the treatment of topologically ordered systems, offering an alternative way to identify topological phases [75].

The quantum mutual information for two states and is defined substituting the Shannon entropy in the classical definition with the von Neumann entropy [31],

(2.30)

The quantum mutual information is the most used measure to quantify correlations in mixed states, since it captures the total amount of information between the two subsystems, both in the classical and quantum case. It will be useful when dealing with mixed states of topologically ordered quantum systems.

Before proceeding, it is necessary to say something about the spaces where a state is defined. So far we have been dealing with density matrices implicitly associated to an Hilbert space . However, we will mostly consider situations where states are defined on the tensor product of two or more Hilbert space. In this sense, systems made of subsystems each associated to an individual Hilbert spaces are described by states that live in the composite product space

(2.31)

For instance, if lives in , then the subsystem defined on is described by the reduced density matrix . This is derived tracing out all the degrees of freedom associated to :

(2.32)

and, conversely, .

2.2.3 Everything you always wanted to know about the von Neumann entropy but were afraid to ask

At this point you should be convinced that quantum entropy is a rather significant and interesting quantity. It is not by any means an observable of the system, i.e. there is no quantum mechanical operator whose expectation value is the entropy. Quantum entropy is instead a functional of the state, just like in classical theory. From Eq.(2.27) it is evident that for a generic quantum state one has

(2.33)

Moreover, only if the rank of is finite and vice-versa.

We will now list some of the basic properties of the von Neumann entropy [60] that will be useful hereinafter. A good starting point is to notice that the entropy of a quantum state is invariant under unitary evolution of the state. For a unitary transformation , this property translates into

(2.34)

If one considers a direct product state given by

(2.35)

then the total entropy is equivalent to the sum of the entropies of the single constituents:

(2.36)

This property is called additivity of the entropy. If we now limit ourselves to two Hilbert spaces such that (but the following is easily generalizable to more dimensions) then we can introduce another property of the entropy called subadditivity.

To define subadditivity, consider the joint state state of two lower dimensional density matrices as in Eq.(2.32), with

(2.37)

Then the following inequalities hold

(2.38)
(2.39)

where the latter is known as the triangle inequality.

An interesting immediate consequence of this is that for a global pure state , the quantum entropy of a subsystem of can still be non-zero and quantifies the degree of entanglement, a special kind of correlations unique to quantum systems. From subadditivity, another property of the entropy follows. This is the concavity [76], which implies that the total entropy of a mixed state is always bigger or equal to the weighted sum of the single entropies of the composing elements, i.e. for , then

(2.40)

Physically, this last inequality is telling us that by sampling from a mixture of state we also lose information by not knowing from which state we sample. Using concavity, it is possible to derive an upper-bound for [77]. This will be used later when discussing the convergence of entropy in different quantization schemes, namely

(2.41)

where is the Shannon entropy of the probability distribution . In the special case where the states are all one-dimensional projections, a fancy way to say that they can be written as , then Eq.(2.41) reduces to

(2.42)

The equality in (2.41) is satisfied when the have support on orthogonal subspaces.

For infinitely dimensional Hilbert spaces, entropy as a function of is generally a discontinuous quantity and small variations of the density matrix can induce a discontinuous jump in the value of the entropy. For finite-dimensional systems this change is bounded by the Fannes’ inequality [78]: Taken two density matrices

(2.43)

where is the trace distance between the two density matrices. A continuity property for the entropy is the lower semicontinuity. Given a sequence of density matrices that weakly converge to the density matrix

(2.44)

i.e. all matrix elements satisfy , then the entropy is upper bounded by

(2.45)

Furthermore, the weakly convergence for density matrices also implies that

(2.46)

When the trace distance relation above (2.46) holds, together with the convergence of the energy expectation values,

(2.47)

then the following continuity property is satisfied

(2.48)

The continuity conditions can be relaxed when we consider bounded Hamiltonians,

(2.49)

for all such that , in which case the entropy is automatically continuous in the sense of Eq.(2.48) for all density matrices associated to finite energy [60], i.e.

(2.50)

2.3 My name is Verschränkung, but you can call me Entanglement

In 1935, the EPR trio Einstein, Podolsky and Rosen, realized that quantum description allows for local operations performed on part of a global system to affect the state of another part of the system indefinitely separated in space [79]. Whereas the ghost of the verschränkung, the term coined by Schrödinger to describe this wonder of Nature, haunted Einstein till the end of his life, today entanglement is widely recognized as the distinctive trademark of quantum mechanics and arguably the main ingredient in quantum information processing [61].

Beyond its elusive nature, entanglement reveals itself as a non-local more-than-classical kind of correlations among two (or more) quantum states. Practically, we say that a state is entangled if it is not separable [31]. A pure state is separable if it can be written as the direct product of states of the subsystems,

(2.51)

where and . Physically, separability has a direct connection to locality, meaning that different parts of a pure separable state can be prepared locally and the outcomes of local measurements on the subsystems are independent. The easiest example, for the two-qubit Hilbert space , consists of the Bell basis,

(2.52)

These states are entangled, because they cannot be written as the product of two states

(2.53)

Similarly, a mixed state is separable if its global density matrix can be written as a convex sum of product states,

(2.54)

as shown in [80]. If this is not possible, then the state is entangled.

Entanglement might well be considered the quintessence of quantum mechanics, or its quantessence222I apologize for the terrible joke.. Apart from being rather interesting as a phenomenon per se, entanglement proved to be an additional physical resource exclusive of the quantum regime. One of the first and most fascinating examples where entanglement was used to achieve something otherwise impossible in classical physics was quantum teleportation [29]. Following that successful application, it was understood that entanglement could be used to perform tasks within quantum information, such as quantum computing, super-dense coding and quantum cryptography [28, 61].

Interestingly enough, it is extremely challenging to quantify and identify correlations, quantum or classical, between subsystems of a larger system and a unique characterization of the degree of entanglement of a system (assuming this last sentence makes physical sense) still does not exist [62]. The classification of correlations and the appropriate choice of measures goes beyond the scopes of this Thesis, so they are not treated here. However, we still need to define a few measures of entanglement that will be used in the following Chapters. A criterion to determine whether a quantity is a meaningful entanglement witness or not was derived in [81]. It states that any quantity that does not increase on average under local operations and classical communication (LOCC) is an entanglement monotone, and can be meaningfully used to quantify entanglement. The physical reason behind this idea is that entanglement cannot increase when we limit ourselves to operations performed locally (a mathematical motivation follows from Nielsen’s majorization theorem, see [82]).

In the following section we start with the quantification of bipartite entanglement in the case of pure states and then extend the same concept to mixed states.

2.3.1 Bipartite entanglement of pure states

For pure states, entanglement manifests itself as disorder in the subsystems of the entangled system. Consider the pure state , divide it into two complementary subsystems and and hand them to the most revered quantum couple of all times, Alice and Bob (a strictly not necessary but always enjoyable step to do). The preferred measure to quantify the amount of entanglement between any bipartition held by the two lovers333I believe that every quantum physicist likes to think that between Alice and Bob there is something more than just entanglement (although things get kinky when Eve gets involved). is the entanglement entropy, or degree of entanglement, which is equivalent to the von Neumann entropy of any of the two partitions (see [31, 61, 3])

(2.55)

If the subsystems and are in a product state, then no entanglement is present and hence . Otherwise, the quantum correlations along the cut induce a positive value of the von Neumann entropy. Although the entanglement entropy is an appropriate measure of entanglement for pure states, it does not have any particular physical interpretation for mixed states.

2.3.2 Negativity and logarithmic negativity

While quantum mutual information takes in account all kind of correlations, a measure that manages to capture solely the quantum correlations for mixed states is the negativity [83], which is based on Peres’ criterion to determine if a mixed state is entangled between and . The criterion says that for any (bipartite) separable state, taken the general form of the density matrix

(2.56)

then the partial transposition of with respect to ,

(2.57)

is always positive definite [84]. Consequently, the condition is sufficient to assert that the state is entangled. Mathematically, this is equivalent to saying that the state is entangled if the partial transpose of the density matrix has at least one negative eigenvalue. This property can be used to construct a quantitative measure of the degree of entanglement for mixed states: For a generic , the negativity of a subsystem is defined as

(2.58)

where the term is the trace norm of the partial transpose of . The negativity is therefore equal to the absolute value of the sum of all the negative eigenvalues of the partial transpose (whose sum can be different to 1) and it is zero if and its complement are not entangled.

A second entanglement monotone is the logarithmic negativity [83], simply given by

(2.59)

This quantity proved to be very useful since it is an upper bound to the entanglement entropy for all pure states

(2.60)

and it can be computed in a simpler way, since it requires the full spectrum of the density matrix instead of the reduced one that is usually more difficult to extract.

2.3.3 Beyond bipartite entanglement

Entanglement is not only a property of bipartite systems, but it reveals itself also in the more complicated setup of multi-partite systems. Nowadays, despite a very large literature on this topic [62], we still do not have a meaningful quantification and characterization of multi-partite entanglement [85].

In between classical correlations and entanglement, lies another measure of correlations, the quantum discord. For the bipartite state , discord is defined as the difference between quantum mutual information and classical correlations

(2.61)

where is the maximization of the classical correlations measure over all possible POVM measurements , explicitly

(2.62)

and is the difference between von Neumann entropy and conditional entropy. Originally, discord was introduced in [86, 87] as an attempt to discriminate between the purely classical correlations and the quantum ones. Since its introduction, the nature of discord has been widely investigated and many physical interpretations have been given, especially in terms of computational advantages for states with positive discord but zero entaglement [88]444Just before posting this Thesis on the arXiv, a new work by Gheorghiu and Sanders [89] suggests that non-zero discord is a quantifier for noisy measurements rather than the flagpole of the quantum-classical border..

Although we will not use these concepts in this Thesis, it is important to be aware that quantifying correlations in quantum information theory is still a controversial and open topic. A nice introduction to the argument of multi-partite entanglement and additional references can be found in [90], while we suggest to look at [91] for a complete review about quantum discord and related measures.

2.4 Stabilizer formalism

As we have seen, operations in quantum mechanics can be rather difficult to understand, especially from a physical point of view. For this reason, when dealing with graph states [85] and lattice quantum many-body systems, particular classes of systems that can be described in terms of the geometrical properties of the underlying pattern of the interactions, we will make use of a method known as stabilizer formalism [63]. This allows to describe quantum states in terms of the action of certain special operators, called stabilizers. For a generic state , we call stabilizer any operator such that [31]

(2.63)

Then the stabilizer set of a quantum state composed of subsystems, is defined as the set of stabilizer operators that have as eigenvector with eigenvalue . The stabilized state is uniquely determined if the stabilizer set is generated by exactly independent stabilizer generators. If is generated by elements, then it does not stabilize a single state but rather a dimensional subspace of the global Hilbert space of the system for qubits and for qudits of dimension . When we discuss the toric code [38] in 5.2.1, it will be shown that the Hamiltonian ground state is not unique but four-degenerate because the toroidal structure of the system lattice causes two stabilizers to be linearly dependent by the others, see 5.2.1.

The stabilizer formalism is a powerful tool to describe topologically ordered systems such as quantum double models [39]. The ground state of each of these models is in a topological phase and the defining Hamiltonian is constructed as a linear combination of the elements composing the ground state subspace stabilizer set, as explained in Chapter 5. Most importantly, the stabilizer formalism allows for a simpler description of the evolution of the state. Given the stabilizer condition from Eq.(2.63), under a unitary transformation of the state, , transforms as in order to preserve its stabilizer status . Note that the transformation under the action of is opposite from the Heisenberg evolution of the observables under the same unitary . In fact, when we evolve stabilizers we are not modeling the evolution of observables, but rather evolving the old stabilizers into new stabilizers for the new state. Hence, the unitary evolution applied to the stabilizer must counteract that applied to the state in order to maintain the stabilizer’s role as such [5].

In the context of continuous-variable systems, there exists an equivalent way to express the stabilizer relations by using nullifiers [68, 92]. In analogy with Eq.(2.63), an operator is called a nullifier for a state when the relation

(2.64)

holds. When the generators of the stabilizer set are elements of a Lie group [67], then the elements of the Lie algebra that generates the Lie group compose the nullifier set of the state. Note that nullifiers transform under the same transformation rule of the stabilizers.

2.5 Relativistic quantum information

Quantum mechanics deals with physics at small length scales. A fascinating new area in physics is the study of relativistic and gravitational effects on quantum information at scales usually associated to the relativistic regime and, to a lesser degree, the use of quantum information theory in relativistic physics. This is called relativistic quantum information [34, 35] and deals with the interaction between gravity and the quantum phenomena, analyzing the evolution of quantum systems over very large distances.

Previously we introduced the concept of qubit. But so far a qubit has only been a mathematical concept, without any physical realization. In practice, a qubit can be encoded by well-chosen degrees of freedom of some physical system. For example photons, massless particles of light, are a popular implementation of qubits [93]. In fact, while it is possible to encode a 2-level quantum state into their spin degrees of freedom, they also serve well as information carriers at relativistic lengths, enabling for gedanken (and in the near future real) experiments in open space [2].

However, these degrees of freedom are not isolated, and they are generally transformed under the effect of gravity during the evolution of the information carriers. Understanding the gravitational effects on photons [1] is therefore essential. From a theoretical point of view, it helps to increase our comprehension of the physics of quantum phenomena at relativistic scales, which is particularly important with a theory of quantum gravity in mind, and practically it is a key step toward the implementation of quantum communication protocols between the Earth and satellites [2, 94, 95].

2.5.1 Physics of photons in special relativity

In this subsection we aim to give an intuition of the meaning of polarization rotation for a photon, looking at the problem in the setting of special relativity. In the following Chapter we will analyze the effects induced by general relativity, motivating how one can construct paths along which the phase introduced is independent of the choice of reference frame.

In a Minkowski space-time, the generic state of a spin-particle is given by some irreducible representation of the Poincaré group [96, 67] and can be represented by

(2.65)

where is the momentum four-vector, is the Lorentz invariant measure and symbolizes the total spin degrees of freedom. The basis states are complete and labeled by the four-momentum and the spin along a particular direction. Experimentally, a generic single photon state would correspond to a wave packet of the form in Eq.(2.65). In the following, the single photon state of interest is described by a well-defined three-momentum , since , and the helicity eigenvalues . Hence, a sharp-momentum state can be written as [34]

(2.66)

One can therefore think of the states as the computational basis of a qubit. Alternatively, it is possible to use a pair of three-vectors to label the same state, the momentum vector and the polarization vector , where and

(2.67)

The descriptions of a quantum state looked by observers associated to two different reference frames connected by a Lorentz transformation , are related by a quantum Lorentz transformation. This is a unitary representation of the Poincaré group [96, 67] that describes the transformation

(2.68)

States of definite helicity are invariant under Lorentz transformations, while momentum states are generally affected. This motivates the choice of working with sharp-momentum states: Since the basis states in Eq.(2.67) are direct products of momentum and polarization, the spin states do not entangle with momentum states, although they still acquire a phase. Note that wave packets introduce an additional hurdle. The entangling of the momentum and spin degrees of freedom forbids complete distinguishability of states with different polarizations. As explained in [34], a Lorentz transformation acts on the generic spin single-particle state with sharp momentum as

(2.69)

where is the matrix element of the representation of the Wigner little group element , related to the spin representation of the Lorentz group. In this sense a Lorentz transformation acts as a quantum gate on the particle state. The classical information stored in controls how transforms, and both control how the spin state changes.

In order to keep things simple, we are not interested in presenting all the mathematical details of the transformation rules for a massless particle and thus, with much hand-waiving, we outline the idea and show the final result straight-away. For a photon with well-defined momentum , the effects of any Lorentz transformation are equivalent to the effects of a single rotation around the -axis by an angle [97],

(2.70)

where the reference frame is defined by the standard vector, which is a unit light-like vector pointing in the -direction, mathematically 555 The standard vector is transformed into the particle momentum by a Lorentz transformation, , where , rotations and boosts [98].. For any rotation, the matrix elements are given by

(2.71)

and consequently, the photon state transforms as

(2.72)

The physical significance of this last equation is that a Lorentz transformation introduces a relative phase for a photon state written in the helicity basis,

(2.73)

and the angle is exactly that phase. If one chooses a reference frame to measure the polarization of the photon at the source, then the phase introduced after the Lorentz transformation can be considered as a consequence of the rotation of the polarization frame along the trajectory.

In a flat space-time, if the sender knows the direction of propagation of the photon it is always possible to setup a standard reference frame as the triad of 3-vectors

(2.74)

where and the two polarization vectors are respectively and . Then, in principle, one could invert the gate described by Eq.(2.72) to align the receiving detector in accordance with the detector of the sender and measure the correct polarization. We will see in the following Chapter that in general relativity there is no similar procedure to follow and it is therefore necessary to find other ways to extract the information encoded within [1].

2.6 Loop quantum gravity

The two theories we have been discussing in this introduction are the pillars of modern physics: The quantum theory, ranging from quantum mechanics [59] to quantum field theories [99], and Einstein’s theory of relativity [100]. In general, the gauge theories that describe the fundamental interactions of Nature can be quantized in a canonical way [96], i.e. starting from a classical theory and somehow promoting the classical variables to quantum field operators. More deeply, gauge theories are based on the concept of symmetry. All forces among particles in Nature are identified by a structural group, which determines the gauge invariance of the theory, in the sense that a solution of the theory equations is still a solution under the action of the gauge group [101].

Although this approach works intimately well for particle interactions, something more devious happens when attempting to canonically quantize general relativity [54, 101], mainly for two reasons [65]: First, a theory of quantum gravity seems to require a non-perturbative, or background independent, quantization, since the metric of the manifold becomes itself a dynamical variable that interacts with the presence of mass. Second, it is not a trivial task to identify the gauge group of general relativity, which is usually believed to be the group of space-time diffeomorphisms. Under these assumptions, it becomes rather hard to choose the proper physical observables of the theory [64, 55].

Loop quantum gravity (LQG) is a non-perturbative canonical quantization of gravity, where Einstein’s equations are described in terms of a SU(2) Yang-Mills gauge theory [54]. The most remarkable (and maybe not completely unexpected) result from LQG is that the geometry at the quantum level is discrete [102]. In the theory, one defines area and volume operators that have discrete spectra and minimal values. The orthonormal basis states related to the diagonalization of these observables are graphs called spin networks. The vertices of a spin network are labelled by representations of the SU(2) group, and the graph is set on a metric-independent manifold. The dynamics of the theory is specified by a sequence of (allowed) moves that transform the graph from an initial to a final quantum geometrical state. The superposition of all the possible histories determine the quantum state of a space-time [55].

With these ideas in mind, the authors in [50] introduced polymer quantization. Polymer quantization is a toy model proposed to study how semiclassical states can arise from the full theory of quantum gravity, and relies on assumptions similar to the construction of LQG. In practice, it is a representation of the Weyl algebra unitarily inequivalent to the Schrödinger’s one, which succeeded in describing the kinematics and dynamics of a one-dimensional quantum system on a discretized version of the real line. In Chapter 7 we will study quantum entropy in the context of polymer quantization, investigating whether the entropic predictions of different quantizations are in accordance.

2.7 Discussion

The short introduction to Loop Quantum Gravity concludes this introductory Chapter. While no new physics has been introduced in these pages, we covered most of the basic definitions that will be employed in the following Chapters and give a taste of the various flavors of the Thesis’ topics.

Chapter 3 Photons – Phases and Experiments

In 2.5.1 we have analyzed the meaning of polarization rotation in Minkowski space-time. In this Chapter we look at photons traveling in a general gravitational field [100, 103]. In particular, gravity causes the polarization of photons to rotate and a meaningful evaluation of the rotation can be achieved only by an appropriate definition of reference frames [36]. The problem of comparison of reference frames is not new, and it has been widely analyzed in the context of quantum information [104]. The encoding of quantum information into the degrees of freedom of a physical system always requires a choice of frame where the encoding acquires informational meaning [105]. Then, exchange of quantum information runs parallel to exchange of information about reference frames between sender and receiver. Partial knowledge of reference frames can lead to loss of communication capacity, and to mistakes in identifying the information content of a physical system [1].

Precise understanding and estimation of the change in photon polarization can be used for tests of relativity [106], by sending signals between earth and satellites in orbit. Furthermore, these effects must be accurately evaluated when dealing with the implementation of quantum protocols in space [107, 108, 109], such as quantum key distribution [30]. In principle, in a curved space-time the sender needs to fix a reference frame at each point of the trajectory, which is practically impossible, and exchange this information with the receiver in order to measure the signal appropriately. Therefore, all these implementations hide an expensive price to pay: Finding a realistic way to share a reference frame between the source of the signal and the receiver.

An answer to this problem was given in [36], where the authors introduced a natural gauge convention that fixes a set of rules to define reference frames without the need of communication between the parties. Here we present an alternative solution to the problem by determining gauge-invariant trajectories of the photons along which the polarization rotation can be precisely calculated irrespective of the choice of reference frame [1].

We begin this Chapter explaining concisely the mathematical treatment of photons in general relativity, extending the meaning of polarization rotation to curved backgrounds. We then show how one can look at this problem from a geometrical point of view and demonstrate gauge-invariant aspects of gravity-induced polarization rotation along closed trajectories both in three-dimensional static projections of the four-dimensional space-time and in the entire four-dimensional manifold. To conclude the analysis we review a number of experimental proposals aimed at testing general relativity at new scales, which are prominently based on some of the concepts derived in this Chapter.

In the discussion we will make use of abundant terminology and elements taken from differential geometry. While we use [101] and [110] as primary sources, Appendix A provides a short introduction to the relevant concepts. In the following we adopt the signature convention for the metric (A.3).

3.1 General relativitistic effects on photon states

Our kinematic description of photons in a gravitational field relies on the short wave, or geometric optics limit, assumption [111]. We assume that the wavelength of the particle is much smaller than the minimum value between the typical curvature radius and some distance taken large enough to ensure that the values of the amplitude, polarization and wavelength vary significantly in its range111Analogously, the wave period must be much shorter than the time scales involved in the process.. This allows to adopt the first post-eikonal approximation [100]: Such approximation is an expansion of the source-free Maxwell equations in empty space, based on the geometric optics limit and on the implicit assumption that the electromagnetic field is weak enough not to experience self gravitational interaction [112]. Then, the more complicated laws of wave propagation in the space-time reduce to the amplitude of the wave being transported along the photons world lines [113]. At the first-order expansion, photons are approximated to point particles that follow null trajectories with a tangent 4-momentum,

(3.1)

and carry a transversal, , space-like polarization vector,

(3.2)

The momentum and polarization vectors are parallel transported along the trajectory,

(3.3)

where a generic corresponds to the covariant derivative of in the direction of as explained in A.33. An interesting future line of research consists of analyzing the gravitational effects on the light polarization if one where to proceed using the full classical Maxwell equations instead (for an example of such a description in a uniform gravitational field, see [114]).

A space-time manifold can always be covered by patches that locally look like a Minkowski space-time [101]. Hence, at any point of the space-time, one can define an orthonormal tetrad

(3.4)

where the observer is at rest. Then the local components of the momentum and polarization vectors are defined in this tetrad.

In the following discussion we consider stationary gravitational fields. A gravitational field is stationary, or constant, if it is possible to choose a reference frame where all the components of the metric tensor are independent of the time coordinate [115]. In other words: On a stationary space-time, after a choice of reference frame (or, more poetically, after a choice of time) one can select three-dimensional space-like surfaces all equipped with the same metric. Practically, it is possible from a tetrad, using the Landau-Lifshitz formalism, to construct a triad foliating the space-time into space-like surfaces. This three-dimensional projection of the space-time is defined by the map

(3.5)

where is the four-dimensional space-time manifold and is a three-dimensional spatial space. Consequently, any four-vector is transformed into the three-vector corresponding to our choice of foliation by a push-forward map (see [101, 116] for details). This is equivalent to dropping the time-like coordinate of the four-vector. In the following we refer to this as the projection of a vector on the static surface .

Hence, the local (static) description of a photon state can be provided by the standard reference frame

(3.6)

as explained in 2.5.1. The polarization vectors specify a linear polarization basis and are dependent upon the momentum vector . One can rewrite the parallel transport equations in this local representation of the space-time mapping the four-dimensional covariant derivative to the correspondent three-dimensional expression. Then

(3.7)

where the derivation of these formulas is provided in [112, 116, 117]. The parameter is called affine parameter and determines the trajectory while the symbol D identifies the three-dimensional covariant derivative.

The polarization and momentum vector from Eq.(3.7) undergo an evolution known as gravitational Faraday rotation [118, 119, 120], which causes the momentum-polarization triad to rotate with angular velocity given by

(3.8)

This is analog to the electromagnetic Faraday effect, which explains how a polarized electromagnetic wave that travels through plasma rotates under the action of a magnetic field [121]. In contrast with the classical Faraday effect, in this case the rotation is purely a geometric effect where is the gravitoelectric field term, and plays the role of the gravitomagnetic field term. The names follow from the quasi-Maxwell form of the Einstein equation for a stationary space-time, derived from the Landau-Lifshitz formalism [116]. Both terms are related to the elements of the projected three-dimensional metric shown in A.3 as

(3.9)

where g is a three-dimensional vector with components .

3.2 Geometric phase

In this section we look at the problem of polarization rotation from a geometric perspective. Starting from the parallel transport equations, we initially derive an alternative equation for the polarization rotation for an arbitrary choice of the polarization basis. We then project it to a static space-time and show that the phase accrued by the photon state in the sense of Eq.(2.73) depends on a Machian term and on a reference frame term. The main result of this section is that the phase, along a closed trajectory on , is invariant under the choice of a different reference frame. Moreover, we extend the discussion to general four-dimensional space-times and demonstrate that for closed paths constructed ad hoc the same argument still holds.

An equation for the polarization rotation

We begin by defining at each point of the trajectory a local orthonormal tetrad (or vierbein) such that the momentum vector is locally given by

(3.10)

The temporal gauge simplifies things quite a lot, and allows us to set

(3.11)

where the local polarization basis is chosen according to some procedure. Hence, the general form of the real linear polarization four-vector at some point of the trajectory is

(3.12)

and the phase explictly appears in the formulation. We set the initial phase and thus

(3.13)

at the starting point of a trajectory. Using the parallel transport equations from Eq.(3.3), we can derive a differential equation for the polarization rotation. Since along the trajectory we generically have that