[table]capposition=top

Certification of entangled quantum states and quantum measurements in Hilbert
spaces of arbitrary dimension

Author: Shubhayan Sarkar
Supervisor: Dr. Remigiusz Augusiak

Centrum Fizyki Teoretycznej Polskiej Akademii Nauk
A thesis submitted in partial fulfillment of the requirements for the degree
of
Doctor of Philosophy in Physics

September 2022

Dedicated to my parents, my mother Swapna Sarkar and father Soumen Sarkar,
for making me who I am today.

### Abstract

The emergence of quantum theory at the beginning of 20$-th$ century has
changed our view of the microscopic world and has led to applications such as
quantum teleportation, quantum random number generation and quantum
computation to name a few, that could never have been realised using classical
systems. One such application that has attracted considerable attention lately
is device-independent (DI) certification of composite quantum systems. The
basic idea behind it is to treat a given device as a black box that given some
input generates an output, and then to verify whether it works as expected by
only studying the statistics generated by this device. The novelty of these
certification schemes lies in the fact that one can almost completely
characterise the device (up to certain equivalences) under minimal physically
well-motivated assumptions such as that the device is described using quantum
theory. The resource required in most of these certification schemes is
quantum non-locality.

A lot of work has recently been put into finding DI certification schemes for
composite quantum systems. Most of them are however restricted to lower-
dimensional systems, in particular two-qubit states. In this thesis, we
consider the problem of designing general DI schemes that apply to composite
quantum systems of arbitrary local dimensions. First, we construct a fully DI
certification scheme, also known as self-testing, that allows us to certify
generalised Greenberger-Horne-Zeilinger (GHZ) states of arbitrary local
dimension shared among any number of parties from the maximal violation of a
certain family of Bell inequalities, for two parties, the generalised GHZ
state represents the two-qudit maximally entangled state. Importantly, this is
the first instance where such states can be certified using only two
measurements per party which is in fact the minimal number of measurements
required to observe quantum non-locality.

While a substantial progress has been recently made in designing device-
independent certification schemes, most of these schemes are concerned with
entangled quantum states. At the same time the problem of certification of
quantum measurements remains largely unexplored. In particular, a general
scheme allowing one to certify any set of incompatible quantum measurements
has not been proposed so far. As designing such a scheme within the DI setting
is certainly a difficult task, here we consider a relaxation of the Bell
scenario known as the one-sided device-independent (1SDI) scenario. In this
scenario, we have an additional assumption that one of the parties is trusted
and the measurements performed by this party are known. We propose a scheme
for certification of a general class of projective measurements, termed here
“genuinely incompatible". To this end, we construct a family of steering
inequalities that are maximally violated by any set of genuinely incompatible
measurements. Interestingly, mutually unbiased bases belong to this class of
measurements. Finally, in the 1SDI scenario, we construct a family of steering
inequalities, whose maximal violation can be used to certify any pure
entangled bipartite state using the minimal number of two measurements per
observer. Building on this result, we then provide a method to certify any
rank-one extremal measurement, including non-projective measurements on the
untrusted side.

Interestingly, self-testing of entangled states and measurements can be
harnessed to propose schemes to certify that the outcomes of measurements
performed on quantum states are perfectly random in the sense that they can
not be predicted by an external party. This makes our scheme suitable for
quantum cryptographic tasks. We first show that one can generate randomness in
a fully DI way using projective measurements from composite quantum states of
arbitrary local dimension. Later in the 1SDI scenario, we construct a scheme
to certify the optimal amount of randomness that can be generated using a
quantum system of any dimension and non-projective extremal measurements,
which is twice the amount one can generate using projective measurements.

### Streszczenie

Powstanie teorii kwantów na początku XX wieku zmieniło nasz pogląd na świat
mikroskopowy i doprowadziło do powstania takich zastosowań efektów kwantowych
jak teleportacja kwantowa, kwantowe generowanie liczb losowych i obliczenia
kwantowe. Żadne z nich nie mogło by zostać zrealizowane w ramach fizyki
klasycznej. Jednym z takich zastosowań, które przyciągnęło ostatnio wiele
uwagi, jest certyfikacja złożonych układów kwantowych w wersji niezależnej od
urządzeń (ang. device-independent). Podstawową jej ideą jest traktowanie
danego urządzenia jak czarną skrzynkę, która po podaniu danych wejściowych
generuje dane wyjściowe, a następnie wykorzystane obserwowanych statystyk do
sprawdzenia, czy urządzenie to działa zgodnie z oczekiwaniami. Nowatorskość
schematów certyfikacji tego typu polega na tym, że można prawie całkowicie
scharakteryzować urządzenie (z dokładnością do pewnych równoważności) przy
minimalnych, dobrze umotywowanych fizycznie założeniach, takich jak to, że
urządzenie działa zgodnie z zasadami fizyki kwantowej. Zasobem kwantowym
wykorzystywanym przez większość tych schematów certyfikacji jest nielokalność
Bella.

Wiele pracy włożono ostatnio w stworzenie schematów certyfikacji w wersji
niezależnej od urządzeń dla złożonych układów kwantowych. Większość z nich
stosuje się jednak do układów o relatywnie niskich wymiarach lokalnych, w
szczególności do stanów dwukubitowych. W tej pracy rozważamy problem
projektowania ogólnych schematów, które mają zastosowanie do układów
kwantowych o dowolnych wymiarach lokalnych. Najpierw konstruujemy w schemat
certyfikacji, znany również jako samotestowanie, który pozwala certyfikować
uogólnione stany Greenbergera-Horne’a-Zeilingera (GHZ) o dowolnym wymiarze
lokalnym dzielony przez dowolną liczbę obserwatorów na podstawie maksymalnego
łamania pewnej rodziny nierówności Bella; w szczególnym przypadku dwóch
podukładów stan GHZ sprowadza się do stanu maksymalnie splątanego dwóch
kuditów. Co ważne, jest to pierwszy przypadek, w którym takie stany kwantowe
mogą być certyfikowane przy użyciu tylko dwóch pomiarów przez każdego z
obserwatorów, co jest w rzeczywistości minimalną liczbą pomiarów wymaganą do
zaobserwowania nielokalności Bella.

Choć w ostatnich latach dokonano znacznego postępu w projektowaniu schematów
certyfikacji w wersji niezależnej od urządzeń, większość z nich dotyczy
splątanych stanów kwantowych. Jednocześnie problem certyfikacji pomiarów
kwantowych pozostaje w dużej mierze niezbadany. Brakuje w szczególności
ogólnego schematu pozwalającego na certyfikację dowolnego zestawu
niekompatybilnych pomiarów kwantowych. Ponieważ stworzenie takiego schematu w
scenariuszu niezależnym od urządzeń jest trudnym zadaniem, w rozprawie
rozważamy pewien uproszczony scenariusz, znany jako scenariusz jednostronnie
niezależny od urządzeń (1SDI). W tym scenariuszu czynimy dodatkowe założenie,
że jedno z urządzeń pomiarowych jest w pełni scharakteryzowane i wykonuje
znane pomiary kwantowe. Proponujemy schemat certyfikacji ogólnej klasy
pomiarów rzutowych, określanych tu jako prawdziwie niekompatybilne. W tym celu
konstruujemy rodzinę nierówności sterowania (ang. steering inequalities),
których maksymalna wartość kwantowa osiągana jest przez dowolny zbiór
prawdziwie niekompatybilnych pomiarów. Co ciekawe, do tej klasy pomiarów
kwantowych zaliczają się te, które odpowiadają bazom wzajemnie
niejednoznacznym. Wreszcie, w scenariuszu 1SDI, konstruujemy rodzinę nie-
równości sterowania, których maksymalne łamanie może być wykorzystane do
certyfikacji dowolnego czystego, splątanego stanu dwucząstkowego przy użyciu
minimalnej liczby dwóch pomiarów wykonywanych przez obu obserwatorów.
Opierając się na tym wyniku, podajemy następnie metodę certyfikacji dowolnego
ekstremalnego pomiaru kwantowego rzędu jeden, włączając w to pomiary
nierzutowe.

Co ciekawe, metody samotestowania stanów oraz pomiarów kwantowych mogą być
wykorzystane do stworzenia schematów poświadczania, że wyniki pomiarów
wykonywanych na stanach kwantowych są losowe w tym sensie, że nie mogą być
przewidziane przez żadnego zewnętrznego obserwatora. To sprawia, że nasze
wyniki stają się użyteczne w zadaniach kryptograficznych. Najpierw pokazujemy,
że ze splątanych stanów kwantowych o dowolnym wymiarze lokalnym można
generować losowość w scenariuszu niezależnym od urządzeń używając pomiarów
rzutowych. Następnie, w scenariuszu 1SDI, konstruujemy schemat poświadczający
optymalną ilość losowości, która może być wygenerowana przy użyciu układu
kwantowego o dowolnym wymiarze i nierzutowych pomiarów ekstremalnych, która
równa jest dwukrotności maksymalnej ilości losowości, którą można wygenerować
przy użyciu pomiarów rzutowych.

### Declaration

The work described in this thesis was undertaken between October 2018 and
April 2022 while the author was a doctoral student under the supervision of
Prof. Remigiusz Augusiak at the Center for Theoretical Physics, Polish Academy
of Sciences. All the required coursework was completed between October 2018
and July 2020 at the Institute of Physics, Polish Academy of Sciences. No part
of this thesis has been submitted for any other degree at the Center for
Theoretical Physics, Polish Academy of Sciences or any other scientific
institution.

This thesis is based upon four different research works carried on during the
doctoral studies all of which are listed below.

  1. 1.

Shubhayan Sarkar, Remigiusz Augusiak, Self-testing of multipartite GHZ states
of arbitrary local dimension with arbitrary number of measurements per party,
Phys. Rev. A 105, 032416 (2022) [1].

  2. 2.

Shubhayan Sarkar, Jakub J. Borkała, Chellasamy Jebarathinam, Owidiusz Makuta,
Debashis Saha, Remigiusz Augusiak, Self-testing of any pure entangled state
with minimal number of measurements and optimal randomness certification in
one-sided device-independent scenario, arXiv:2110.15176 (2021) [2].

  3. 3.

Shubhayan Sarkar, Debashis Saha, Remigiusz Augusiak, Certification of
incompatible measurements using quantum steering, arXiv:2107.02937 (2021) [3].

  4. 4.

Shubhayan Sarkar, Debashis Saha, Jędrzej Kaniewski, Remigiusz Augusiak, Self-
testing quantum systems of arbitrary local dimension with minimal number of
measurements, npj Quantum Information 7, 151 (2021) [4].

In addition to the work presented in this thesis, the author has also carried
out the following research tasks all of which are listed below.

  1. 1.

Shubhayan Sarkar, Chandan Datta, Saronath Halder, Remigiusz Augusiak, Self-
testing composite measurements and bound entangled state in a unified
framework,  arXiv:2301.11409 [quant-ph] (2023) [5].

  2. 2.

Shubhayan Sarkar, Certification of the maximally entangled state using non-
projective measurements,  arXiv:2210.14099 [quant-ph] [6].

  3. 3.

Jakub Jan Borkała, Chellasamy Jebarathinam, Shubhayan Sarkar, Remigiusz
Augusiak, Device-independent certification of maximal randomness from pure
entangled two-qutrit states using non-projective measurements, Entropy 24(3),
350 (2022) [7].

  4. 4.

Shubhayan Sarkar, Debashis Saha, Probing measurement problem of quantum theory
with an operational approach, arXiv:2107.08447 (2021) [8].

  5. 5.

Shubhayan Sarkar, Universal notion of classicality based on ontological
framework, arXiv:2104.14355 (2021) [9].

  6. 6.

Colin Benjamin, Shubhayan Sarkar, Emergence of Cooperation in the
thermodynamic limit, Chaos, Solitons & Fractals 135, 109762 (2020) [10].

  7. 7.

Colin Benjamin, Shubhayan Sarkar, Triggers for cooperative behavior in the
thermodynamic limit: a case study in Public goods game, Chaos 29, 053131
(2019) [11].

  8. 8.

Shubhayan Sarkar, Colin Benjamin, Quantum Nash equilibrium in the
thermodynamic limit, Quantum Inf. Process. 18, 122 (2019) [12].

  9. 9.

Shubhayan Sarkar, Colin Benjamin, Entanglement makes free riding redundant in
the thermodynamic limit, Physica A: Statistical Mechanics and its Applications
521, 607 (2019) [13].

  10. 10.

Shubhayan Sarkar, Entropy as a bound for expectation values and variances of a
general quantum mechanical observable, International Journal of Quantum
Information, 16 1850036 (2018) [14].

  11. 11.

Shubhayan Sarkar, Chandan Datta, Can quantum correlations increase in a
quantum communication task?, Quantum Inf Process 17, 248 (2018) [15].

### Acknowledgements

First and foremost, I would like to express my deepest gratitude to my
supervisor, Remigiusz Augusiak. This thesis would not have been completed
without his support and constant encouragement. He always helped me whenever I
was stuck in academic as well as administrative problems. At all times, he was
understanding and willing to help.

Furthermore, I feel indebted to thank Center for Theoretical Physics of the
Polish Academy of Sciences for providing me with the research environment and
also to the administrative team, whom I troubled a lot during my studies. Not
to mention, they were a constant support and helped me even with my personal
difficulties.

This thesis would not have been possible without the help of my group members
Debashis Saha, Gautam Sharma, Jakub J. Borkała, Chellasamy Jebarathinam and
Owidiusz Makuta and also Jędrzej Kaniewski, who were involved in the research
tasks that finally led to this thesis. I would like to thank my friends
Chandan Datta, Rubina Ghosh, Kaustav Sengupta, Sudeep Sarkar, Saubhik Sarkar,
Suparna Saha, Ishika Palit, Suhani Gupta, Kiran Saba, Saikruba Krishnan,
Rafael Santos, Julius Serbenta, Grzegorz Rajchel-Mieldzioc and Michele Grasso.

I would like to thank my family specially my parents who believed in me and
were with me in my ups and downs. Also special thanks to Bohnishikha Ghosh for
being a constant support.

Finally, I would also like to thank the Foundation for Polish Science for
their support through the First Team project (No First TEAM/2017-4/31) co-
financed by the European Union under the European Regional Development Fund.
The grant not just financed my stay in Poland but also provided opportunities
to visit various institutes across Europe to share my research works and learn
from the leading experts in the field.

###### Contents

  1. 1 Introduction
  2. 2 Technical introduction
    1. 2.1 Basics of quantum information theory
      1. 2.1.1 States in quantum theory
      2. 2.1.2 Dynamics in quantum theory
      3. 2.1.3 Measurements in quantum theory
      4. 2.1.4 Purification of quantum states and measurements
    2. 2.2 Quantum non-locality
      1. 2.2.1 Bell non-locality
      2. 2.2.2 Quantum steering
      3. 2.2.3 Applications of quantum non-locality
    3. 2.3 Device-independent certification
      1. 2.3.1 Self-testing
      2. 2.3.2 One-sided device independent certification
      3. 2.3.3 Randomness certification
  3. 3 Certification of multipartite entangled states of arbitrary local dimension
    1. 3.1 Introduction
    2. 3.2 Family of Bell inequalities
      1. 3.2.1 SATWAP Bell inequalities
      2. 3.2.2 ASTA Bell inequalities
    3. 3.3 Self-testing
    4. 3.4 Randomness certification
    5. 3.5 Conclusions and discussions
  4. 4 Certification of incompatible measurements
    1. 4.1 Introduction
    2. 4.2 Family of steering inequalities
      1. 4.2.1 Classical bound
      2. 4.2.2 Quantum bound
    3. 4.3 Genuinely incompatible observables
    4. 4.4 1SDI certification
      1. 4.4.1 Exact certification of GI observables
      2. 4.4.2 Weaker certification
      3. 4.4.3 Robust certification
    5. 4.5 Conclusions and discussions
  5. 5 Certification of any pure bipartite entangled state and optimal randomness using quantum steering
    1. 5.1 Introduction
    2. 5.2 Family of steering inequalities
      1. 5.2.1 Classical bound
      2. 5.2.2 Quantum bound
    3. 5.3 ISDI certification of all pure bipartite entangled states
    4. 5.4 Certification of all rank-one extremal POVM
    5. 5.5 Optimal randomness certification
    6. 5.6 Conclusions and Discussions
  6. 6 Concluding remarks
    1. 6.1 Summary of the thesis
    2. 6.2 Open questions for further exploration
  7. A Some general mathematical facts
  8. B Proofs of some observations relevant to Chapter 3
  9. C Proofs of some observations relevant to Chapter 5

##  Chapter 1 Introduction

The advent of quanta by Planck in 1900 for describing black-body radiation
marked as the beginning of “quantum era of theoretical physics". With the
increasingly precise experiments on microscopic objects, it was soon realised
that the structure of atom was not describable with classical physics but
required a new non-classical description. In this pursuit,
Schr$\mathrm{\ddot{{o}}}$dinger in 1923, using the ideas of Bohr and De-
Broglie that microscopic objects might possess both wave and particle
characteristics, came up with a wave equation describing the microscopic
world, which is now known as the Schr$\mathrm{\ddot{{o}}}$dinger’s equation.
It is excellently successful in predicting the results of the experiments
performed till date. However, the Schr$\mathrm{\ddot{{o}}}$dinger equation
involves an object known as a wavefunction, whose meaning was hugely debated
and was believed to be just a mathematical object without any physical
significance. It was the work of Max Born that established that the
wavefunction contains information about the probability of the system being at
a particular position in space. This can be considered as one of the biggest
paradigm shifts in the human understanding of nature, as the microscopic world
might be inherently unpredictable, and the maximum information that can be
gained is the probability of the system being at a particular state. Many
prominent physicists did not buy this idea and even led Einstein to one of his
famous quotes, “God does not play dice". In fact, the unpredictable nature of
the microscopic world can be considered as the foundational philosophy behind
“quantum theory".

With even more experiments probing the microscopic regime, it was soon
realised that quantum theory is the best description of this regime, even when
the theory leads to counter-intuitive phenomena and paradoxes. With later
works of pioneers like Heisenberg, Bohr, Von-Neumann and Dirac, to name a few,
we arrive at four postulates on which quantum theory is based on [cf. [16,
17]].

  1. 1.

Any state of the system is represented by a density matrix $\rho$ acting on
some Hilbert space $\mathcal{H}$.

  2. 2.

The time evolution of the quantum system is generated by completely positive
and trace preserving (CPTP) maps.

  3. 3.

A system composed of two subsystems $A$ and $B$ is described using a quantum
state belonging to the Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
where $\mathcal{H}_{A},\mathcal{H}_{B}$ are the Hilbert spaces associated with
the subsystems $A,B$ respectively.

  4. 4.

Any measurement to observe the state of the system $\rho$ is represented by a
set of positive operators $M=\\{E_{k}\\}$ that act on the Hilbert space
$\mathcal{H}$. Here $k$ represents the outcome of the measurement and
$\sum_{k}E_{k}=\mathbb{1}_{\mathcal{H}}$. The probability of observing the the
outcome $k$ is denoted by $p(k|M,\rho)$ and can be computed as,

$\displaystyle p(k|M,\rho)=\mathrm{Tr}(E_{k}\rho).$ (1.1)

All the notions presented here will later be revisited again in details.

An important step towards the understanding of quantum theory was put forward
by Einstein, Podolski and Rosen in 1935 in their seminal paper [18], where it
was claimed that quantum theory is incomplete. It was later speculated by
physicists like Bohm and Von-Nuemann that an underlying classical-like theory
would be able to predict all the quantum effects and would remove all the
counter-intuitiveness of the theory. This remained only a theoretical idea
until Bell’s pioneering work in 1964 [19]. He put forward a way to test
whether quantum theory is inherently different from classical physics. It was
later confirmed by experiments that any classical description is insufficient
to explain some quantum phenomena [20, 21, 22, 23].

This led to the understanding that there can exist tasks in which strategies
involving quantum states and measurements would perform much better than
classical strategies, giving birth to the field of quantum information theory.
In this thesis, we explore one of the recently contrived areas in quantum
information theory, namely device-independent quantum protocols [24] focusing
particularly on device-independent certification of quantum states,
measurements and intrinsic randomness that can be generated using quantum
systems.

The idea of device-independent certification has gained much attention lately,
mainly due to their implications in quantum cryptography as well as
foundations of quantum theory. Let us say that two spatially separated labs
are given a device whose inner mechanism is unknown and thus they can be
treated as black boxes. If an input is provided to this device, it produces an
output. Based on various inputs and outputs, one can construct the statistics.
Assuming that these devices satisfy the rules of quantum theory, any such
statistics needs to be explained via quantum measurements acting on some
quantum state. The basic idea behind device-independent certification is that
using only the statistics obtained by performing local measurements on a
composite quantum system, one can characterise this system and also the
measurements up to certain equivalences.

The strongest device-independent certification scheme is known as self-testing
where no assumptions on the devices are made apart from the fact that they are
governed by quantum theory. The idea of self-testing was first introduced by
Mayers and Yao in 1998 [25, 26] in the cryptographic context as a way of
certifying that the parties share a desired state. Since then, numerous self-
testing schemes for composite quantum systems and measurements have been
introduced. For instance Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36]
provide schemes for certification of pure bipartite entangled states that are
locally qubits. Then, Refs. [37, 38, 39, 40, 41, 42, 43, 44, 45] provide
methods to certify multipartite states of local dimension two. A few works
Refs. [46, 47, 48, 49, 50] provide certification schemes of entangled states
of local dimension higher than two. In [47] (see also [48]) a strategy for
certification of every pure bipartite entangled state was introduced.

In Chapter 3 of this thesis, we provide a general scheme that self-tests the
generalised Greenberger-Horne-Zeilenger (GHZ) state [51] of arbitrary local
dimension shared among arbitrary number of parties. Restricting to two
parties, the above state is, in fact, the maximally entangled state of
arbitrary local dimension. Unlike the previous self-testing scheme [47] that
allows for certification of arbitrary dimensional states, we use the minimum
possible number of measurements per party, that is, two. This is particularly
useful from an application point of view, as it reduces the experimental
effort necessary to implement our scheme. With the aim to self-test
measurements, we generalise this scheme to arbitrary number of measurements
per party. An important application of our self-testing scheme is towards
device-independent certification of genuine randomness. Sources generating
genuine randomness are useful in many areas such as numerical computation or
cryptography. We provide a scheme for certifying the maximum amount of
randomness that can be extracted from a quantum system of arbitrary local
dimension using projective measurements. This chapter is based on two of our
works, [4] and [1]. The first one proves self-testing statement for the
maximally entangled state of two qudits based on the maximal violation of the
SATWAP Bell inequality introduced in [52]. The second paper generalises this
result to the GHZ state shared among arbitrary number of parties based on the
maximal violation of ASTA Bell inequality introduced in [53].

Any quantum device consists mainly of two parts: a source that generates a
quantum state and a measuring device that performs a measurement on this
state. While there has been considerable progress in designing self-testing
schemes for quantum states, the avenue for certification of quantum
measurements remains largely unexplored. Recently, in [54] and [7], the
authors introduced a way of self-testing the bases of the set of two-outcome
and three-outcome observables respectively. Apart from a few results [55, 47,
56, 57, 58, 46, 59, 60, 34], there is no general method allowing to certify
any set of incompatible measurements.

To simplify the problem, depending on the physical scenario, one can make some
assumptions about the quantum devices. In this thesis, we consider a scheme in
which one of the parties is trusted, known as one-sided device-independent
(1SDI) scenario. The resource that one needs here to certify quantum states or
measurements is known as quantum steering, which is another form of quantum
non-locality [61, 62]. The quantum steering scenario is similar to the Bell
scenario, apart from the fact that one of the parties is trusted, or
equivalently, that the measurement device of the trusted party performs known
measurements. The first 1SDI certification scheme was introduced in 2016 [63]
(see also [64]). Their scheme allows for certification of the two-qudit
maximally entangled state and two binary outcome measurements.

In Chapter 4 of this thesis, we provide the first certification scheme that
applies to a general family of incompatible projective measurements which we
termed “genuinely incompatible". An important class of projective measurements
that are well-studied in quantum information theory are those corresponding to
mutually unbiased bases. For instance, they are crucial for the security of
quantum cryptographic tasks [65, 66, 67, 68, 69]. Also, there is a close link
between mutually unbiased bases and quantum cloning [70, 71, 72] (see [73],
for a review on mutually unbiased bases). As a matter of fact, we show that
mutually unbiased bases are also genuinely incompatible, and thus we provide a
certification scheme of mutually unbiased bases of arbitrary dimension. For
two observables corresponding to mutually unbiased bases, we also find the
robustness of our scheme against experimental errors, which makes it relevant
for practical applications. This chapter is based on our work [3].

Recently, quantum non-locality has been realised as an effective way to
generate genuine randomness and device-independently verify that there is no
intruder who has access to it [74, 75, 76]. The maximum amount of randomness
that one can, in principle, obtain from a quantum system of a dimension $d$ is
$2\log_{2}d$ bits. A long-standing question in quantum information theory is
whether one can find protocols that can be used to certify this maximal amount
of randomness.

In Chapter 5 of this thesis, we again consider the one-sided device-
independent scenario. We begin by devising a 1SDI scheme that can be used to
certify any pure bipartite entangled state. Importantly, our scheme utilises
only two measurements per party, which is the minimal number of measurements
necessary to observe quantum steering. Moreover, these measurements are
independent of the state to be certified. This makes our scheme much easier to
implement in experiments. Using these results, we provide a scheme for
certification of any rank-one extremal generalised measurements. This allows
us to finally certify the maximal amount of randomness that one can generate
from a quantum system of any dimension. We further show for some dimensions
that the amount of randomness is independent of the amount of entanglement in
the quantum system.This chapter is based on our work [2].

Before presenting the main results of this thesis, in Chapter 2 we introduce
the relevant technical concepts and notions which will be required throughout
this thesis. The thesis ends with concluding remarks and a list of open
problems in Chapter 6.

##  Chapter 2 Technical introduction

###  2.1 Basics of quantum information theory

Let us begin by elaborating on the four postulates of quantum theory presented
in Introduction.

####  2.1.1 States in quantum theory

######  Definition 1 (Pure quantum states).

Pure quantum states are normalised vectors belonging to a Hilbert space
$\mathcal{H}$ and are denoted by ${|{\psi}\rangle}$. The normalisation
condition ensures that

$\displaystyle\langle\psi{|{\psi}\rangle}=1.$ (2.1)

In general, the state ${|{\psi}\rangle}$ can belong to Hilbert space of
infinite dimension. However in this work, we consider only finite-dimensional
Hilbert spaces. Any pure quantum state ${|{\psi}\rangle}\in\mathcal{H}$ such
that the dimension of the Hilbert space is $d$, can be expressed using a set
of $d$ linearly independent vectors,
$\mathcal{B}=\\{{|{e_{i}}\rangle}\\}_{i=0}^{d-1}$ known as a basis. Now, any
quantum system belonging to a $d-$dimensional Hilbert space is known as qudit.
A two dimensional quantum system is known as qubit. An important basis that is
widely used in quantum information theory and will be extensively used in this
work is known as the computational or standard basis, represented by
$\mathcal{B}_{c}=\\{{|{i}\rangle}\\}_{i}$, where

$\displaystyle{|{0}\rangle}=\begin{pmatrix}1\\\ 0\\\ 0\\\ \vdots\\\
0\end{pmatrix},\quad{|{1}\rangle}=\begin{pmatrix}0\\\ 1\\\ 0\\\ \vdots\\\
0\end{pmatrix},\quad\ldots\quad{|{d-1}\rangle}=\begin{pmatrix}0\\\ 0\\\ 0\\\
\vdots\\\ 1\end{pmatrix}.$ (2.2)

Note that the elements in the basis $\mathcal{B}_{c}$ are orthogonal, that is,
$\langle i{|{j}\rangle}=\delta_{i,j}$ where

$\displaystyle\delta_{i,j}=\begin{cases}1\quad\text{if}\quad i=j\\\
0\quad\text{if}\quad i\neq j\end{cases}.$ (2.3)

In certain situations, considered also in this thesis, one needs to use
density matrices denoted by $\rho$ to describe a quantum system. These are
matrices acting on the Hilbert space $\mathcal{H}$ that satisfy the following
properties

$\displaystyle\mathrm{Tr}(\rho)=1,\quad\rho\geq
0,\quad\text{and}\quad\rho=\rho^{\dagger},$ (2.4)

that is, they are normalised and positive semi-definite. For any pure state
${|{\psi}\rangle}$, the corresponding density matrix is given by
$\rho_{p}={|{\psi}\rangle}\\!{\langle{\psi}|}$.

######  Definition 2 (Mixed states).

Quantum states that can be written as convex combination of other quantum
states, are known as mixed states, that is,

$\displaystyle\rho_{m}=\sum_{i}p_{i}{|{\psi_{i}}\rangle}\\!{\langle{\psi_{i}}|}.$
(2.5)

For a note, a particular decomposition of $\rho_{m}$ (2.5) is not unique.

Let us now consider the scenario when a system consists of two subsystems $A$
and $B$. Let $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ denote the Hilbert spaces
of $A$ and $B$ respectively. According to the postulates of quantum theory the
Hilbert space of their joint system is given by
$\mathcal{H}_{AB}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. In such composite
systems, we can have two major classes of states, separable and entangled. Let
us begin with pure separable states also known as product states.

######  Definition 3 (Product states).

Consider a pure state
${|{\psi_{AB}}\rangle}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. We call it a
product state if it can be written as
${|{\psi}\rangle}_{AB}={|{\psi}\rangle}_{A}\otimes{|{\psi}\rangle}_{B}$ where
${|{\psi}\rangle}_{A}\in\mathcal{H}_{A}$ and
${|{\psi}\rangle}_{B}\in\mathcal{H}_{B}$.

######  Definition 4 (Separable States [77]).

A mixed state $\rho_{sep}$ acting on $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
is called separable if it can be written as convex mixture of product states.

Thus, any separable state can be written as

$\displaystyle\rho_{sep}=\sum_{i}p_{i}{|{\psi_{i}}\rangle}_{A}{\langle{\psi_{i}}|}\otimes{|{\phi_{i}}\rangle}_{B}{\langle{\phi_{i}}|}$
(2.6)

where ${|{\psi_{i}}\rangle}_{A}\in\mathcal{H}_{A}$ and
${|{\phi_{i}}\rangle}_{B}\in\mathcal{H}_{B}$. Let us now look at a class of
states that do not exist in classical physics.

######  Definition 5 (Entangled States).

Quantum states that are not separable as defined in Def. 4 are classified as
entangled states.

For example, when both the Hilbert spaces of $A$ and $B$ is $\mathbbm{C}^{2}$,
then the quantum state

$\displaystyle{|{\psi_{AB}}\rangle}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}_{A}{|{0}\rangle}_{B}+{|{1}\rangle}_{A}{|{1}\rangle}_{B}\right)$
(2.7)

is entangled.

A convenient way to express any pure bipartite state, that is quantum systems
consisting of only two subsystems, ${|{\psi}\rangle}_{AB}\in\mathcal{H}_{AB}$
is by using the Schmidt decomposition [17]. Let us say Hilbert space of
subsystem $A$, denoted by $\mathcal{H}_{A}$ is of dimension $m$ and Hilbert
space of subsystem $B$, denoted by $\mathcal{H}_{B}$ is of dimension $n$, then
any state ${|{\psi}\rangle}_{AB}$ can be written as

$\displaystyle{|{\psi}\rangle}_{AB}=\sum_{i=1}^{\min\\{m,n\\}}\lambda_{i}{|{e_{i}}\rangle}\otimes{|{f_{i}}\rangle}$
(2.8)

such that $\lambda_{i}\geq 0$ with $\sum_{i}\lambda_{i}^{2}=1$ and
$\\{{|{e_{i}}\rangle}\\},\ \\{{|{f_{i}}\rangle}\\}$ are set of orthonormal
vectors defined on $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ respectively111For
ease of mathematical notation, most of the times the symbol “$\otimes$” will
be dropped..

Let us now consider quantum systems consisting of $N$ subsystems where $N$ is
any positive integer greater than two such that the local Hilbert spaces are
denoted by $\mathcal{H}_{i}$ for $i=1,2,\ldots,N$. Then such a state is
described by a density matrix $\rho_{N}$ acting on
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\ldots\otimes\mathcal{H}_{N}$. We
can straightforwardly generalise the notion of separability and entanglement
to the multipartite states as done above. This completes the classification of
quantum states. We now move on to characterising dynamics in quantum theory.

####  2.1.2 Dynamics in quantum theory

Within quantum theory, any evolution in general is represented by completely
positive and trace preserving (CPTP) maps. However, in this work we restrict
ourselves to a family of maps that are known as unitary transformations or
unitary matrices.

######  Definition 6 (Unitary transformation).

A unitary transformation $U$ is a mapping from a Hilbert space $\mathcal{H}$
to $\mathcal{H}$ that preserves the distance between any two vectors belonging
to $\mathcal{H}$.

A unitary matrix $U$ acting on $\mathcal{H}$ is characterised by the following
properties,

$\displaystyle UU^{\dagger}=\mathbb{1}\quad\text{or}\quad U^{-1}=U^{\dagger},$
(2.9)

that is, the inverse of any unitary matrix is equal to its conjugate
transpose. Interestingly, any CPTP map can be realised as a unitary map acting
on some higher dimensional system [78]. In fact, the dynamics of closed
quantum systems are reproduced by unitary maps. Another class of CPTP maps
that is relevant for this work is known as isometry [79].

######  Definition 7 (Isometry).

An isometry is generalisation of a unitary matrix, mapping one Hilbert space
$\mathcal{H}$ of lower dimension to some other Hilbert space
$\mathcal{H^{\prime}}$ of higher dimension in a way that preserves the
distance between any two vectors belonging the Hilbert space $\mathcal{H}$.

We now move on to characterising measurements in quantum theory.

####  2.1.3 Measurements in quantum theory

Any measurement $M$ with $m$ outcomes in quantum theory is defined by the set
of positive operators $\\{E_{k}\\}$ that act on some Hilbert space
$\mathcal{H}$ for $k=0,1,2,\ldots,m-1$. These positive operators are
hermitian, that is, $E_{k}=E_{k}^{\dagger}$ and sum up to identity
$\sum_{k}E_{k}=\mathbb{1}_{\mathcal{H}}$ which is a consequence of the fact
that the probabilities of outcomes must sum up to $1$. Due to this,
measurements in quantum theory are also referred to as positive-operator
valued measures or simply POVM’s. The probability of observing the outcome $k$
if the measurement $M$ has been performed on a state $\rho\in\mathcal{H}$ is
given by,

$\displaystyle p(k|M,\rho)=\mathrm{Tr}(E_{k}\rho).$ (2.10)

It is clear to see from the above expression that $\sum_{k}p(k|M,\rho)=1$. Let
us now try to understand the idea of normalisation of the state as discussed
in the previous subsection 2.1. Suppose that state of the quantum system is
${|{\psi}\rangle}$. Now, we consider a two-outcome measurement $M_{2}$ with
measurement elements

$\displaystyle E_{0}={|{\psi}\rangle}\\!{\langle{\psi}|},\quad\text{and}\quad
E_{1}=\mathbb{1}-{|{\psi}\rangle}\\!{\langle{\psi}|}.$ (2.11)

Then the probability of observing the system in the quantum state
${|{\psi}\rangle}$ must be $1$ or equivalently, the measurement must always
output the $0^{th}$ outcome. Thus, we have that

$\displaystyle
p(0|M_{2},{|{\psi}\rangle}\\!{\langle{\psi}|})=|\langle\psi{|{\psi}\rangle}|^{2}=1.$
(2.12)

Consequently, the quantum state must be normalised so that the total
probability of finding the system in any quantum state is one.

Quantum measurements for a given Hilbert space form a convex set. Now, we look
at a special class of POVM’s that lie at the boundary of this set.

######  Definition 8 (Extremal POVM’s).

Any POVM that can not be written as a convex combination of other POVM’s is
called extremal.

As was proven in [80], the measurement operators of any rank-one extremal POVM
can be written as $E_{k}=\lambda_{k}\Pi_{k}$, where

$\displaystyle\Pi_{k}={|{\psi_{k}}\rangle}\\!{\langle{\psi_{k}}|}$ (2.13)

and $\lambda_{k}\geq 0$. Moreover, $E_{k}^{\prime}s$ are linearly independent
for all $k$. It was further proven in [80] that for POVM’s that act on a
Hilbert space $\mathcal{H}$ of dimension $d$, an extremal POVM can have atmost
$d^{2}$ outcomes. Any other POVM can be written as convex combination of
extremal POVM’s. A special class of extremal POVM’s, are known as projective
measurements.

######  Definition 9 (Projective measurements).

Projective measurements are POVM’s where the measurement elements are
represented by projectors $\Pi_{i}$ such that
$\Pi_{i}\Pi_{j}=\delta_{i,j}\Pi_{i}$.

For any rank-one projective measurement, the corresponding measurement
elements are given by
$E_{k}=\Pi_{k}={|{\psi_{k}}\rangle}\\!{\langle{\psi_{k}}|}$. Along with the
condition that $\sum_{k}\Pi_{k}=\mathbb{1}_{\mathcal{H}}$ and
$\Pi_{i}\Pi_{j}=\delta_{i,j}\Pi_{i}$, it can be concluded that
$\\{{|{\psi_{k}}\rangle}\\}$ for all $k^{\prime}s$ form a complete basis of
the Hilbert space $\mathcal{H}$ which the measurement acts on. The state of
the system after the measurement is performed on it is known as the post-
measurement state. For instance, let us consider the measurement $\\{E_{k}\\}$
and a quantum state $\rho$, then the post-measurement state $\rho_{pm}^{a}$
after the outcome $a$ is observed is given by,

$\displaystyle\rho_{pm}^{a}=\frac{\sqrt{E_{a}}\rho\sqrt{E_{a}}}{\mathrm{Tr}(E_{a}\rho)}\qquad\forall
a.$ (2.14)

An equivalent way to represent measurements in quantum theory is by using
quantum observables. Let us first consider the simplest scenario where the
measurements have only two outcomes and are projective. The quantum observable
$\mathbb{A}$ corresponding to such a measurement $M=\\{\Pi_{0},\Pi_{1}\\}$ is
represented by,

$\displaystyle\mathbb{A}=\Pi_{0}-\Pi_{1}.$ (2.15)

From the above expression, we can conclude that such quantum observables are
hermitian and unitary,

$\displaystyle\mathbb{A}=\mathbb{A}^{\dagger},\quad\text{and}\quad\mathbb{A}\mathbb{A}^{\dagger}=\mathbb{1}.$
(2.16)

One can define the expectation value of $\mathbb{A}$ in terms of the
probabilities of obtaining outcome $0,1$ as

$\displaystyle\langle\mathbb{A}\rangle_{\rho}=\mathrm{Tr}(\mathbb{A}\rho)=p(0|M,\rho)-p(1|M,\rho).$
(2.17)

The above construction of quantum observables can also be generalised to two-
outcome POVM’s where $\mathbb{A}=E_{0}-E_{1}$. However, such an observable is
not unitary. This construction of quantum observables was generalised to
arbitrary outcome POVM’s in [46]. We refer them here as generalised
observables. Consider a $d-$outcome measurement. One defines generalised
expectation values $\langle\mathbb{A}^{(l)}\rangle_{\rho}$ for
$l=0,1,\ldots,d-1$ as the Fourier transform of the probabilities $p(k|M,\rho)$
as,

$\displaystyle\langle\mathbb{A}^{(l)}\rangle_{\rho}=\sum_{k=0}^{d-1}\omega^{lk}p(k|M,\rho)\quad\text{for}\quad
l=\\{0,1,\ldots,d-1\\}$ (2.18)

where $\omega=e^{2\pi i/d}$ is the $d-th$ root of unity. Notice that using
inverse Fourier transform, we can obtain the probabilities from the
expectation values as,

$\displaystyle
p(k|M,\rho)=\frac{1}{d}\sum_{l=0}^{d-1}\omega^{-lk}\langle\mathbb{A}^{(l)}\rangle_{\rho}\quad\text{for}\quad
k=\\{0,1,\ldots,d-1\\}$ (2.19)

Using the above definition (2.18), analogous to the two-outcome case, one can
define the following expression for the corresponding observables
$\mathbb{A}^{(l)}$ in terms of the measurement elements $E_{k}$

$\displaystyle\mathbb{A}^{(l)}=\sum_{k=0}^{d-1}\omega^{lk}E_{k}\quad\text{for}\quad
l=\\{0,1,\ldots,d-1\\}.$ (2.20)

Equivalently the measurement elements $E_{k}$ can be obtained from the quantum
observables $\mathbb{A}^{(l)}$ by considering the inverse Fourier transform,

$\displaystyle
E_{k}=\frac{1}{d}\sum_{l=0}^{d-1}\omega^{-lk}\mathbb{A}^{(l)}\quad\text{for}\quad
k=\\{0,1,\ldots,d-1\\}.$ (2.21)

Some relevant properties about the observable $\mathbb{A}^{(l)}$ can be
obtained from Eq. (2.20) such as,

$\displaystyle\mathbb{A}^{(d)}=\mathbb{1},\quad\text{and}\quad\mathbb{A}^{(d-l)}=\mathbb{A}^{(-l)}=\mathbb{A}^{(l){\dagger}}$
(2.22)

along with

$\displaystyle\mathbb{A}^{(l)}\mathbb{A}^{(l){\dagger}}\leq\mathbb{1},\qquad\text{and}\qquad\mathbb{A}^{(l){\dagger}}\mathbb{A}^{(l)}\leq\mathbb{1},$
(2.23)

where to obtain the relations (2.22) we used the fact that
$\sum_{k}E_{k}=\mathbb{1}$ and $E_{k}=E_{k}^{\dagger}$. The relation (2.23)
was proven in [46]. An important result about generalised observables that are
unitary was also proven in [46].

######  Fact 1.

Consider the generalised observables $\mathbb{A}^{(l)}$ as defined in (2.20).
These observables are unitary, that is,

$\displaystyle\mathbb{A}^{(l)}\mathbb{A}^{(l)\dagger}=\mathbb{1},\quad\text{and}\quad\mathbb{A}^{(l)}=\mathbb{A}^{l}\quad\forall
l$ (2.24)

if and only if the corresponding measurement is projective, that is,
$E_{k}E_{k^{\prime}}=\delta_{k,k^{\prime}}E_{k}$ for every $k,k^{\prime}$.

This fact will be used extensively throughout this thesis. Often at times, the
generalised observables $\mathbb{A}^{(l)}$ would be referred to as
measurements as they contain all the information to reconstruct the actual
measurement. Consider now a multipartite system and on each subsystem a local
measurement $M_{i}=\\{E_{i,k}\\}$ is performed where $i=1,2,\ldots,N$ denotes
the subsystems. The probability to obtain outcomes $k_{1},k_{2},\ldots k_{N}$
when the measurements are performed on a state $\rho_{N}$ acting on
$\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\ldots\otimes\mathcal{H}_{N}$,

$\displaystyle
p(k_{1},k_{2},\ldots,k_{N}|\rho_{N})=\mathrm{Tr}(E_{1,k_{1}}\otimes
E_{2,k_{2}}\otimes\ldots E_{N,k_{N}}\rho_{N})$ (2.25)

The notion for the observables can also be generalised to the multipartite
case

$\displaystyle\mathbb{A}_{1}^{(l_{1})}\otimes\mathbb{A}_{2}^{(l_{2})}\otimes\ldots\otimes\mathbb{A}_{N}^{(l_{N})}=\sum_{k_{1},k_{2},\ldots,k_{N}=0}^{d-1}\omega^{l_{1}k_{1}+l_{2}k_{2}+\ldots
l_{N}k_{N}}E_{1,k_{1}}\otimes E_{2,k_{2}}\otimes\ldots\otimes E_{N,k_{N}}.$
(2.26)

for every $l_{1},l_{2},\ldots,l_{N}$.

An important distinction between any classical and quantum theories is the
existence of quantum measurements that are mutually incompatible. Two
projective measurements $M_{1}$ and $M_{2}$ are mutually incompatible, if
these two measurements do not commute. For example, consider the Pauli
observables $\sigma_{z}$ and $\sigma_{x}$ corresponding to projective
measurements that acts on qubits in $z$ and $x$ direction respectively

$\displaystyle\sigma_{z}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad\text{and}\quad\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ (2.27)

are incompatible. For a review of incompatibility of quantum measurements
refer to [81, 82].

####  2.1.4 Purification of quantum states and measurements

Another important concept for multipartite quantum systems is the extraction
of the local quantum state for each subsystem given some global quantum state.
For simplicity, we review it here for bipartite quantum systems but the
concept can be straightforwardly generalised to the multipartite case. Let us
suppose the global quantum state is given by $\rho_{AB}$, then the local
quantum states $\rho_{A}$ and $\rho_{B}$ are given by

$\displaystyle\rho_{A}=\mathrm{Tr}_{B}(\rho_{AB})\quad\text{and}\quad\rho_{B}=\mathrm{Tr}_{A}(\rho_{AB})$
(2.28)

where $\mathrm{Tr}_{B}$ and $\mathrm{Tr}_{A}$ denote partial trace over the
subsystem $B$ and $A$ respectively and are computed as,

$\displaystyle\mathrm{Tr}_{x}(\rho_{AB})=\sum_{j}{\langle{j_{x}}|}\rho_{AB}{|{j_{x}}\rangle}\qquad(x=A,B),$
(2.29)

where $\\{{|{j_{x}}\rangle}\\}$ is any orthonormal basis than spans the
Hilbert space $\mathcal{H}_{x}$ of the subsystem $x$ with $x=A,B$. For any
separable state $\rho_{sep}$ [cf. Def. 4] using Eq. (2.6), the local quantum
states of the subsystem $A$ can be simply computed as

$\displaystyle\rho_{A}=\mathrm{Tr}_{B}(\rho_{sep})=\sum_{i}p_{i}{|{\psi_{i}}\rangle}_{A}{\langle{\psi_{i}}|}\otimes\mathrm{Tr}\left({|{\phi_{i}}\rangle}_{B}{\langle{\phi_{i}}|}\right).$
(2.30)

Using the fact that
$\mathrm{Tr}\left({|{\phi_{i}}\rangle}_{B}{\langle{\phi_{i}}|}\right)=1$, we
arrive at

$\displaystyle\rho_{A}=\mathrm{Tr}_{B}(\rho_{sep})=\sum_{i}p_{i}{|{\psi_{i}}\rangle}_{A}{\langle{\psi_{i}}|}.$
(2.31)

Analogously, the local quantum state of the subsystem $B$ is given by,

$\rho_{B}=\mathrm{Tr}_{A}(\rho_{sep})=\sum_{i}p_{i}{|{\phi_{i}}\rangle}_{B}{\langle{\phi_{i}}|}.$
(2.32)

Computing the local quantum states of subsystem $A$ and $B$ when they are
entangled is not as straightforward as separable states. For an example, let
us consider a pure bipartite entangled state of local dimension $d$ given by,

$\displaystyle{|{\psi}\rangle}_{AB}=\sum_{i=0}^{d-1}\lambda_{i}{|{i}\rangle}_{A}\otimes{|{i}\rangle}_{B}$
(2.33)

where $\\{{|{i}\rangle}\\}$ is the computational basis (2.3) such that
$\lambda_{i}>0$ and $\sum_{i}\lambda_{i}^{2}=1$. For a note, any pure
entangled bipartite state of local dimension $d$ can be written as (2.33) up
to some local unitary transformation. The local quantum state corresponding to
the subsystem $\rho_{A}$ and $\rho_{B}$ are given by

$\displaystyle\rho_{A}=\sum_{i}\lambda_{i}^{2}{|{i}\rangle}\\!{\langle{i}|}_{A},\quad\text{and}\quad\rho_{B}=\sum_{i}\lambda_{i}^{2}{|{i}\rangle}\\!{\langle{i}|}_{B}.$
(2.34)

Any mixed quantum state [cf. Def. 2] can be realised as a pure state by adding
an additional ancillary system to this quantum state. This is known as
Stinespring’s dilation [78]. A possible purification of any mixed quantum
state $\rho$ admitting the form given in (2.5) can be written as,

$\displaystyle{|{\psi_{mA}}\rangle}=\sum_{i}\sqrt{p_{i}}{|{\psi_{i}}\rangle}_{m}\otimes{|{e_{i}}\rangle}_{A}$
(2.35)

where $\\{{|{e_{i}}\rangle}_{A}\\}$ is an orthonormal basis over the Hilbert
space of the ancillary system $A$. The mixed state $\rho_{m}$ can be extracted
from this purified state as

$\displaystyle\rho_{m}=\mathrm{Tr}_{A}\left({|{\psi_{mA}}\rangle}\\!{\langle{\psi_{mA}}|}\right).$
(2.36)

On similar lines, any POVM can be realised as a projective measurement that
acts on some higher dimensional system, known as Naimark’s dilation [83]. Any
$n$-outcome POVM $M=\\{E_{k}\\}$ that acts on the Hilbert space $\mathcal{H}$
can be realised using a projective measurement $\Pi_{k}$ and an isometry [cf.
Def. 7] $V_{iso}:\mathcal{H}\rightarrow\mathcal{H}\otimes\mathcal{H}_{A}$
where $\mathcal{H}_{A}$ is some finite dimensional Hilbert space corresponding
to some ancillary system $A$ [16, 84], as

$\displaystyle E_{k}=V_{iso}\Pi_{k}V_{iso}^{\dagger}\quad\forall k.$ (2.37)

Here, the projectors $\Pi_{k}$ are given by

$\displaystyle\Pi_{i}=\mathbb{1}_{\mathcal{H}}\otimes{|{i}\rangle}\\!{\langle{i}|}_{A}\quad\forall
i$ (2.38)

such that $\\{{|{i}\rangle}\\}$ is the computational basis and the isometry
$V_{iso}$ is given by,

$V_{iso}=\sum_{k=1}^{n}\sqrt{E_{k}}\otimes{|{i}\rangle}_{A}.$ (2.39)

This completes the introduction of basic formalism of quantum information
theory that is relevant to this work. We now move on to introducing concepts
more central to this thesis.

###  2.2 Quantum non-locality

In the seminal work [18] published in 1935, Einstein, Podolsky and Rosen (EPR)
pointed out to one of the key features of quantum theory which makes it
inherently different from classical physics. In this work, two spatially
separated quantum systems were considered on which two parties named Alice,
and Bob can perform local measurements. The joint quantum state of both the
systems were considered to be entangled. It was concluded in this work that
quantum theory is incomplete. It was later hypothesised that there could exist
some hidden variables that can not be directly observed but the knowledge of
which would remove the paradoxes within quantum theory. In other words, these
hidden variables would provide a classical explanation of the observed quantum
phenomenon. In the subsequent years it was further noted by
Schr$\mathrm{\ddot{o}}$dinger [61] from the EPR work, that Alice can in fact
affect the quantum state of the system which is spatially separated from her
by just performing measurements on her part of the system. The problem was
debated upon for decades but mostly from a philosophical perspective without
much consensus. We would elaborate on these ideas with details in the
subsequent subsections.

The problem was revisited again by Bell in 1964 [19, 85]. With the improved
tools of quantum theory, Bell was able to devise a way that can put an end to
the debate whether there exist any hidden variables that can provide a
classical description of quantum theory. It turned out that there exist some
set of joint probabilities that Alice and Bob can observe that can not be
reproduced by local hidden variables [see below]. It was later confirmed in
numerous experiments such as [20, 21, 22, 23], that it is indeed the case that
quantum theory is intrinsically different from classical physics. This not
just led to paradigm shift in foundation of physics but also led to enormous
development in the newly born field of quantum information theory. We now
proceed towards understanding Bell’s solution to the EPR problem and the
discovery of quantum non-locality also known as “Bell non-locality".

####  2.2.1 Bell non-locality

Figure 2.1: Simplest Bell scenario: Two parties Alice and Bob are located at
separated labs. A source sends one subsystem to Alice and the other to Bob.
Inside their labs, Alice and Bob perform two two-outcome measurements each on
the received subsystem. After the experiment is complete, they compare their
results to construct the joint probability distribution $p(a,b|x,y)$.

We begin by considering the arguments by Einstein, Podolsky and Rosen in
details. Consider two parties, namely Alice and Bob in two different spatially
separated labs. Each of them receive a subsystem corresponding to an entangled
quantum state such that knowing the position or momentum of either of the
subsystem gives the information about the position and momentum of the
counterpart. Now, Alice measures the position of her subsystem and Bob
measures the momentum of his subsystem. EPR argued that for each of the
subsystem one has information about its position as well as momentum which was
forbidden in quantum theory as position and momentum are non-commuting
measurements. This led them to conclude that there is an inherent
inconsistency in quantum theory and it is incomplete. They argued that such a
phenomenon should not exist which seems “non-local" in the sense that one can
know the state of a far away system based on the experiment done locally on
its counterpart even when there is no interaction between them. Due to this,
in the subsequent years it was hypothesised by physicists like Bohm and Von-
Neumann that there might exist some hidden variables that would remove this
inconsistency and provide a local description of quantum theory.

Building on this idea, Bell in his original work [19] described a similar
scenario. There are two parties Alice and Bob in two different labs that are
spatially separated from each other. Both of them receive a quantum system
from the preparation device. In their respective labs, Alice and Bob can
perform two local measurements on their part of the system. During this
operation they are not allowed to communicate among each other or
equivalently, they are not supposed to know each others measurement choice or
the outcomes of the measurements. The scenario is schematically depicted in
Fig. 2.1. The experiment is repeated enough number of times to gather
statistics corresponding to the joint probability distributions
$\vec{p}=\\{p(a,b|x,y)\\}$ where $p(a,b|x,y)$ denotes the probability of
obtaining outcome $a,b$ when Alice and Bob perform $x,y$ measurement
respectively. One usually refers $\vec{p}$ as “correlations".

Let us now again consider the hypothesis that there might exist some hidden
variables that would provide a consistent local description of quantum theory.
Let us denote those hidden variables by $\lambda$. In presence of such hidden
variables the joint probability distribution can be expressed as

$\displaystyle
p(a,b|x,y)=\sum_{\lambda\in\Lambda}p(a,b|x,y,\lambda)p(\lambda)$ (2.40)

where $\Lambda$ denotes the set of the hidden variables $\lambda$, and
$p(\lambda)$ denotes the probability with which a particular $\lambda$ occurs.
Here, $p(a,b|x,y,\lambda)$ denotes the probability of occurrence of outcome
$a,b$ given the input $x,y$ and the hidden variable $\lambda$. The above
statement points to the fact that we do not have access to $\lambda$ and can
only observe the probabilities averaged over it.

Let us now concentrate on $p(a,b|x,y,\lambda)$. From rule of conditional
probabilities $p(a,b)=p(b|a)p(a)$ where $p(b|a)$ denotes the the conditional
probability of occurrence of $b$ given $a$, we have

$\displaystyle p(a,b|x,y,\lambda)=p(a|x,\lambda)p(b|a,x,y,\lambda).$ (2.41)

Now, the assumption of “locality" or “local realism" states that outcome of
Bob does not depend on the outcome of Alice or her measurement choice

$\displaystyle p(b|a,x,y,\lambda)=p(b|y,\lambda).$ (2.42)

Thus, for any local hidden variable, we have that

$\displaystyle
p(a,b|x,y,\lambda)=p(a|x,\lambda)p(b|y,\lambda)\qquad\forall\lambda.$ (2.43)

Consequently, any joint probability distribution admitting a local hidden
variable model must be of the form

$\displaystyle
p(a,b|x,y)=\sum_{\lambda\in\Lambda}p(a|x,\lambda)p(b|y,\lambda)p(\lambda).$
(2.44)

Now, using these joint probabilities, one can construct a functional of the
form

$\displaystyle\mathcal{B}=\sum_{a,b,x,y}c_{a,b,x,y}p(a,b|x,y)$ (2.45)

where $c_{a,b,x,y}$ are some real numbers. For ease of understanding we refer
here to a functional presented by Clauser, Horne, Shimony and Holt (CHSH) [86]
in which the cofficients in (2.45) are chosen as

$\displaystyle c(a,b,x,y)=\begin{cases}1\quad\text{if}\quad
a\oplus_{2}b=x.y\\\ -1\quad\text{otherwise}\end{cases}$ (2.46)

where $a\oplus_{2}b$ represents $a+b$ modulo $2$. Usually the CHSH Bell
functional is represented in the expectation value picture as

$\displaystyle\mathcal{B}_{CHSH}=\langle A_{0}\otimes B_{0}\rangle+\langle
A_{0}\otimes B_{1}\rangle+\langle A_{1}\otimes B_{0}\rangle-\langle
A_{1}\otimes B_{1}\rangle.$ (2.47)

Assuming that the joint probabilities can be described by a local hidden
variable model (2.44), one arrives at an upper bound of the CHSH expression
(2.47) given by

$\displaystyle\mathcal{B}_{CHSH}\leq 2.$ (2.48)

The maximum value that one can achieve for a Bell expression using local
hidden variable models is usually referred to as the “classical bound" or the
“local bound" and is denoted by $\beta_{L}$. It turns out that there exist
some states and measurements in quantum theory that violate this bound. To
give an example let us consider that the preparation device in Fig. 2.1
prepares the two-qubit maximally entangled state

$\displaystyle{|{\psi}\rangle}_{AB}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}_{A}{|{0}\rangle}_{B}+{|{1}\rangle}_{A}{|{1}\rangle}_{B}\right).$
(2.49)

Assume then that on her share of their state Alice and Bob perform the
following measurements

$\displaystyle A_{0}=\sigma_{z}\qquad\text{and}\qquad A_{1}=\sigma_{x},$
(2.50)

and,

$\displaystyle
B_{0}=\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}}\qquad\text{and}\qquad
B_{1}=\frac{\sigma_{z}-\sigma_{x}}{\sqrt{2}}.$ (2.51)

respectively. It follows that for this choice of the state and measurements,
the value of the CHSH expression (2.47) is

$\displaystyle\mathcal{B}_{CHSH}=2\sqrt{2}.$ (2.52)

One thus concludes that there exist joint probability distributions in quantum
theory that violate the notion of local realism. Contrary to EPR, Bell showed
that even if there exists some hidden variables that can not be observed
directly, they are insufficient to give a local explanation of some
predictions of quantum theory. This is known as “Bell non-locality".

As a matter of fact, the value $2\sqrt{2}$ in Eq. (2.52) is the maximum value
attainable using quantum state and measurements. This was proven by Tsirelson
in [87], and is referred as “Tsirelson bound" or simply “quantum bound" and is
denoted by $\beta_{Q}$.

An additional constraint in the above experiment as mentioned before is that
there is no communication between Alice and Bob during the experiment. This
restricts the joint probability distributions to be “no-signalling" [88], that
is,

$\displaystyle
p(a|x)=\sum_{b}p(a,b|x,y)=\sum_{b}p(a,b|x,y^{\prime})\qquad\forall
y,y^{\prime}$ (2.53)

and

$\displaystyle
p(b|y)=\sum_{a}p(a,b|x,y)=\sum_{a}p(a,b|x^{\prime},y)\qquad\forall
x,x^{\prime}.$ (2.54)

Assuming that the joint probability distributions are no-signalling, one can
even outperform quantum theory, that is,

$\displaystyle\mathcal{B}_{CHSH}=4$ (2.55)

which is also the algebraic bound of the Bell functional $\mathcal{B}_{CHSH}$.
This is also known as the “no-signalling bound" and is denoted by
$\beta_{NS}$.

Figure 2.2: Generalised Bell scenario: $N$ parties denoted by $A_{i}$ for
$i=1,2,\ldots,N$ are located at separated labs. A source sends one subsystem
to each of the labs. Inside their labs, each party performs $m$ number of
$d-$outcome measurements on the received subsystem. After the experiment is
complete, they compare their results to construct the joint probability
distribution $p(a_{1},a_{2},\ldots,a_{N}|x_{1},x_{2},\ldots,x_{N})$.

The above scenario can be straightforwadly generalised to the multipartite
scenario where there is a preparation device, that sends subsystems to $N$
spatially separated parties denoted by $A_{i}$ for $i=1,2,,\ldots,N$. Each of
the parties can now perform $m-$measurements each of which are $d-$outcome
where $m,d$ are arbitrary positive integers strictly greater than $1$. The
scenario is depicted in Fig. 2.2.

The joint probability distributions in such a scenario is denoted by

$\vec{p}=\\{p(a_{1},a_{2},\ldots,a_{N}|x_{1},x_{2},\ldots,x_{N})\\},$ (2.56)

where $a_{k}=\\{0,1,\ldots,d-1\\}$ denotes the outcome when $A_{k}$ performs
the measurement $x_{k}=\\{1,2,\ldots,m\\}$. It is important to note here that

$\displaystyle\vec{p}\in\mathbb{R}^{(md)^{N}},$ (2.57)

where $\mathbb{R}^{(md)^{N}}$ denotes the real vector space of dimension
$(md)^{N}$. The Bell functional in the multipartite case can be written as

$\displaystyle\mathcal{B}_{m,d,N}=\vec{c}\cdot\vec{p}$ (2.58)

where $\vec{c}=\\{c_{a_{1},a_{2}\ldots a_{N},x_{1},x_{2},\ldots,x_{N}}\\}$ and
$\vec{c}\in\mathbb{R}^{(md)^{N}}$.

Figure 2.3: A two-dimensional representation of the set of correlations. The
local set is a subset of the quantum set which is a subset of the no-
signalling set. The no-signalling and local set are polytopes, that is, every
point inside these sets can be written as a convex combination of the vertices
of the polytope. On the other hand, the quantum set is not a polytope but
convex.

The collection of joint probability distributions that admit a local hidden
variable are known as the “set of local correlations" or simply “local set"
denoted by $\mathcal{L}_{m,d,N}$. Similarly, one can define “set of quantum
correlations" or simply “quantum set" denoted by $\mathcal{Q}_{m,d,N}$ and
“set of no-signalling correlations" or “no-signalling set" denoted by
$\mathcal{N}_{m,d,N}$ referring to joint probability distributions that admit
a quantum and no-signalling models respectively. One can understand Bell’s
non-locality arising from the fact that the local set lies strictly inside the
quantum set. All these sets are convex in nature, that is, convex combination
of different elements of the set is also an element belonging to this set.
Elements of the set that can not be written as a convex combination of other
elements are known as "extremal points" of the set and any other element in
the set can be written as a convex combination of these extremal points. By
definition, the local set lies inside the quantum set that lies inside the no-
signalling set [89],

$\displaystyle\mathcal{L}_{m,d,N}\subseteq\mathcal{Q}_{m,d,N}\subseteq\mathcal{N}_{m,d,N}.$
(2.59)

It follows from the work of Bell [19] and then Popescu and Rohrlich [88] that
these inclusions are strict in the minimal scenario [m=d=N=2]. This is
schematically represented in Fig. 2.3. Now, Bell inequalities can be
understood as a hyper plane that cuts the quantum set into two different
parts. The first part consists of correlations that admit a local model and
the second part consists the rest. We now move on to a different notion of
non-locality known as quantum steering.

####  2.2.2 Quantum steering

The idea of quantum steering was conceived by Schr$\mathrm{\ddot{o}}$dinger in
1935 but was formalised after almost $70$ years in 2007 by Wisemen et. al.
[62]. We begin by describing the simplest quantum steering scenario described
in [90] that consits of two spatially separated parties Alice and Bob. A
preparation device sends one subsystem to Alice and another subsystem to Bob.
Alice sends input $y=0,1$ to Bob based on which Bob performs a measurement on
his subsystem and gets an outcome $b=0,1$ which is sent back to Alice.
Contrary to Bell scenario, Alice is trusted which means that Alice has full
control of her lab and can perform a tomography on her subsystem. The
experiment is repeated enough number of times such that Alice can reconstruct
the state of the subsystem $\sigma_{b|y}$ that acts on Alice’s Hilbert space
$\mathcal{H}_{A}$ for every $y,b$. As a matter of fact these states are un-
normalised and they form a set known as an assemblage, denoted by
$\\{\sigma_{b|y}\\}$. The scenario is schematically represented in Fig. 2.4.

Similar to the idea of local hidden variables in Bell non-locality, one can
define a notion which every classical description of the experiment must
abide. In the quantum steering scenario presented above, the classical notion
corresponds to the assemblage $\\{\sigma_{b|y}\\}$ admitting a local hidden
state model. One can understand this as, the assemblage observed by Alice only
depends on some local state $\rho_{\lambda}$ that might be classically
correlated with Bob,

$\displaystyle\sigma_{b|y}=\sum_{\lambda\in\Lambda}p(\lambda)p(b|y,\lambda)\rho_{\lambda}.$
(2.60)

Figure 2.4: Simplest quantum steering scenario: Two parties Alice and Bob are
located at separated labs. A source sends one subsystem to Alice and the other
to Bob. Bob performs two two-outcome measurements on the received subsystem
which might change the state of the subsystem of Alice. Alice is trusted and
can perform a tomography on her subsystem and reconstruct the assemblage (the
set of unnormalised states) and can figure out whether the assemblage is
steerable or not.

Here $p(b|y,\lambda)$ represents the probability of obtaining outcome $b$ by
Bob given input $y$ and hidden variable $\lambda$ and $p(\lambda)$ denotes the
probability distribution of the hidden variables. If the assemblage admits a
local hidden state model then it is non-steerable from Bob to Alice. On the
other hand, the assemblage is called steerable from Bob to Alice if no hidden
state model can be constructed to express $\\{\sigma_{b|y}\\}$ as in (2.60).
The way to detect whether the assemblage is steerable or not, is to use a
steering functional of the form

$\displaystyle W=\sum_{b}\sum_{y}\mathrm{Tr}(F_{b|y}\sigma_{b|y}).$ (2.61)

where the coefficients $F_{b|y}$ are some positive semi-definite operators
acting on $\mathcal{H}_{A}$.

Let us now consider the first steering inequality proposed in [90]. The
scenario considered there is minimal in the sense that there are only two
parties and Bob performs only two measurements. The matrices $F_{b|y}$ of the
steering functional were chosen as,

$\displaystyle F_{0|0}={|{0}\rangle}\\!{\langle{0}|},\qquad
F_{1|0}={|{1}\rangle}\\!{\langle{1}|}$ (2.62)

and

$\displaystyle F_{0|1}={|{+}\rangle}\\!{\langle{+}|},\qquad
F_{1|0}={|{-}\rangle}\\!{\langle{-}|},$ (2.63)

where ${|{+}\rangle}=\frac{{|{0}\rangle}+{|{1}\rangle}}{\sqrt{2}}$ and
${|{-}\rangle}=\frac{{|{0}\rangle}-{|{1}\rangle}}{\sqrt{2}}$. Consequently,
the simplest steering functional can be represented as

$\displaystyle
W_{2,2,2}=\mathrm{Tr}\left({|{0}\rangle}\\!{\langle{0}|}\sigma_{0|0}\right)+\mathrm{Tr}\left({|{1}\rangle}\\!{\langle{1}|}\sigma_{1|0}\right)+\mathrm{Tr}\left({|{+}\rangle}\\!{\langle{+}|}\sigma_{0|1}\right)+\mathrm{Tr}\left({|{-}\rangle}\\!{\langle{-}|}\sigma_{1|1}\right).$
(2.64)

For assemblages that admit a local hidden state model, one can find an upper
bound to the steering functional (2.61) as

$\displaystyle W_{2,2,2}\leq 1+\frac{1}{\sqrt{2}}.$ (2.65)

This is known as “local hidden state bound" or simply “local bound" and is
again denoted by $\beta_{L}$. The quantum bound $\beta_{Q}$ of the steering
inequality $W_{2,2,2}$ is $2$ and for instance can be achieved when the
preparation device prepares the maximally entangled state (2.49) and Bob
performs the Pauli z and Pauli x measurements

$\displaystyle B_{0}=\sigma_{z},\qquad B_{1}=\sigma_{x}.$ (2.66)

The scenario can be straightforwardly generalised to the case when Bob
performs arbitrary number of measurements $m$ with arbitrary number of
outcomes $d$. The steering functional in this case admits a form

$\displaystyle
W_{m,d,2}=\sum_{y=1}^{m}\sum_{b=0}^{d-1}\mathrm{Tr}(F_{b|y}\sigma_{b|y}).$
(2.67)

Comparing Bell scenario to quantum steering scenario, in Bell scenario one
obtains joint probability distributions from the experiment and constructs the
Bell functional by suitably choosing the real coefficients $c_{a,b,x,y}$. In
quantum steering scenario the experiment generates the assemblage and the
steering functional is constructed by suitably choosing the positive semi-
definite matrices $F_{b|y}$. However, it is always possible to map quantum
steering to the Bell scenario and express the steering functional in terms of
expectation values of joint observables or equivalently joint probability
distributions $\vec{p}$ similar to Bell functional. This idea will be
particularly useful for this thesis.

Let us consider the steering functional $W_{m,d,2}$ (2.67) with the
coefficients $F_{b|y}$ being positive, Hermitian and summing up to the
identity for all $y$, that is,

$F_{b|y}\geq 0,\qquad
F_{b|y}=F_{b|y}^{\dagger},\qquad\sum_{b}F_{b|y}=\mathbb{1}.$ (2.68)

Consequently, $\\{F_{b|y}\\}$ is a valid quantum measurement. As discussed in
subsection 2.1.3, one can equivalently represent a quantum measurement by
using $d$ generalised observables $A_{k|y}$ (2.21), that is,

$F_{b|y}=\frac{1}{d}\sum_{b=0}^{d-1}\omega^{-bk}A_{k|y}$ (2.69)

with $k=0,\ldots,d-1$ and $\omega=\mathrm{exp}(2\pi i/d)$. Let us now consider
that Bob performs the measurements $\\{N_{b|y}\\}$ on some state $\rho_{AB}$.
Then, the assemblage $\sigma_{b|y}$ can be expressed as

$\sigma_{b|y}=\mathrm{Tr}_{B}[(\mathbb{1}_{A}\otimes N_{b|y})\rho_{AB}].$
(2.70)

Again, we represent the measurements $\\{N_{b|y}\\}$ using $d$ generalised
observables $B_{l|y}$ as

$N_{b|y}=\frac{1}{d}\sum_{b=0}^{d-1}\omega^{-bl}B_{l|y}.$ (2.71)

Now, we are ready to express the steering functional (2.67) in terms of
expectation values. First using the observation $\eqref{Akx1}$, the steering
functional (2.67) can be written as

$\displaystyle W_{m,d,2}$ $\displaystyle=$
$\displaystyle\sum_{b}\sum_{y}\mathrm{Tr}(F_{b|y}\sigma_{b|y})$ (2.72)
$\displaystyle=$ $\displaystyle\sum_{b}\sum_{y}\mathrm{Tr}(F_{b|y}\otimes
N_{b|y}\rho_{AB}).$

Now, using observations (2.69) and (2.71), we arrive at

$\displaystyle\sum_{b=0}^{d-1}\sum_{y=1}^{N}\mathrm{Tr}(F_{b|y}\sigma_{b|y})=\frac{1}{d}\sum_{k=0}^{d-1}\sum_{y=1}^{N}\left\langle
A_{k|y}\otimes B_{-k|y}\right\rangle.$ (2.73)

For a detailed review on quantum steering, refer to [91]. In the next part, we
briefly discuss the applications of quantum non-locality.

####  2.2.3 Applications of quantum non-locality

Apart from the foundational aspects, quantum non-locality has has given rise
to enormous number of applications in computation, communication and
information theory. Here we first give a brief account of various applications
of Bell non-locality and then quantum steering.

Suppose, Alice and Bob are spatially separated and both of them receive
$n-$bit strings denoted by $x,y$ and Bob wants to compute a function $f(x,y)$.
The minimum amount of information needed by Bob from Alice to compute $f(x,y)$
is known as communication complexity of $f(x,y)$. It has been shown that for
certain functions $f(x,y)$, Bell non-locality reduces the communication
complexity when compared to classical communication between Alice and Bob
[92]. Another important application of Bell-nonlocality is in the field of
quantum cryptography. The earliest connection between quantum non-locality and
cryptography was realised by Herbert in 1975 [93]. However, the breakthrough
was achieved by Ekert in his seminal paper in 1991 [94] that showed that Bell-
nonlocality serves as a way to generate secure key among two spatially
separated parties. The protocol was based on the maximal violation of CHSH
inequality which makes it physically impossible for some external attacker to
know the key shared between Alice and Bob. As a consequence this protocol can
be realised in a device-independent way, that is, without assuming any details
about the inner working of the device apart from the fact that it is governed
by quantum theory. In fact device-independent quantum key distribution (DIQKD)
has been one of the key applications of quantum non-locality [95, 24, 96, 97,
98, 99, 100, 101, 74, 102, 103]. Using Bell inequalities one can even find the
minimum dimension of a quantum system required for obtaining certain
correlations [104]. Since Bell inequalities can be violated using only
entangled states, they serve as an important tool to detect entanglement [105,
106, 107, 108]. Bell non-locality has been identified as a way to generate
genuine randomness [109, 76, 110, 75, 111, 112, 113, 114, 115, 116, 117, 118,
119, 120, 54].

Mayers and Yao in [25, 26] realised that correlations obtained by performing
local measurements on spatially separated systems can used for device-
independent certification of quantum states and measurements. This was termed
as self-testing. It was later realised that in fact Bell-nonlocality allows
one to self-test quantum states and measurements [27, 28, 31, 37, 38, 55, 57,
47]. In the subsequent sections, we discuss device-independent certification
and randomness generation in details.

Quantum steering is also useful in quantum cryptography as was realised in
[121, 122]. In a practical scenario where one of the parties is trusted, for
instance, in a bank-client relationship where the bank can be considered to be
trusted, one can construct one-sided device-independent protocols. Since, then
there have been numerous protocols for quantum key distribution using quantum
steering [123, 124, 125, 126, 127]. Quantum steering is also useful in tasks
like secret sharing, where a referee sends an encrypted message to multiple
players which can be decoded only if the players work together [128, 129, 130,
131]. Also, randomness can be certified using quantum steering to be secure
and genuine [132, 133, 134, 135]. Quantum steering has also shown advantage in
tasks like subchannel discrimination, that is, how well one can distinguish
different branches of a quantum channel [136, 137]. Further, quantum steering
can be used for certification of quantum states and measurements [63, 64, 138,
139].

This completes the analysis of quantum non-locality relevant to this thesis.
In the next section we introduce the idea of device-independent certification
and in particular self-testing and one-sided device certification.

###  2.3 Device-independent certification

In recent years, the field of device-independent schemes has gained a lot of
interest. The novelty of such schemes lies in the fact that one can predict
some properties about the system by looking only at the statistics generated
by this system. For instance, consider that we are given a device that is
promised to produce entangled photons. The natural question is how we verify
that the device works as promised and whether we should trust the
manufacturer. One way is to break the device and check the source. Another way
is not to break the device but perform a quantum tomography on the state
generated by the device. But one needs the knowledge of quantum optics and
also would have to trust his measurement device to infer the state. However,
using Bell inequalities, we can verify the device without breaking it or
without trusting any other device. For this, as depicted in Fig. 2.5, the
entangled subsystems are allowed to be far enough such that they are spatially
separated. Now, two local measurements are performed on each of these arms.
Here the verifiers do not have any knowledge about the measurement device but
choose two inputs freely corresponding to the two measurements and record
their outputs. The experiment is repeated enough number of times to collect
joint probability distribution for different inputs and outputs. If this
distribution violates some Bell inequality, it can be concluded that the
device generates entangled photons.

(a)

(b)

Figure 2.5: (a) Device-dependent scenario: To verify the device, one has to
assume the inner workings of the device. Here, if it is known that the device
consists a photon source that passes through a $\chi^{2}$ crystal, then the
device can produce entangled photons. (b) Device-independent scenario: The
inner workings of the device are unknown. To verify whether the device
generates entangled photons, one needs to perform a Bell test by measuring the
joint correlation statistics $p(a,b|x,y)$.

Thus, quantum nonlocality serves as a way to infer properties about a device
without knowing the inner workings of it apart from the fact that it is
governed by quantum theory. Such device-independent schemes provide maximum
security in cryptographic scenarios where there might be some hidden
mechanisms that are uncontrollable or the device leaks out some information
that can be used by some external attacker. However, in practical scenarios
one can make some well justified assumptions on the device. In this thesis, we
focus on two different types of device-independent certification, first, fully
device-independent certification or self-testing and second, one-sided device
independent certification.

####  2.3.1 Self-testing

Let us now consider the following task: Instead of detecting some property
about the state such as entanglement, we want to characterise the state and
measurement which generates the observed statistics with the device being
treated as a black box. This is the strongest form of device-independent
certification and is known as self-testing. The idea of fully device-
independent certification or self-testing was first introduced by Mayers and
Yao in 1998 [25, 26]. The scenario for self-testing is same as Bell scenario
Fig. 2.1. There are two independent observers Alice and Bob who are spatially
separated from each other and can freely choose their inputs. Also, they are
not allowed to communicate among each other during the experiment. However,
the natural assumption here is that the device behaves according to quantum
theory, that is, every statistics is generated by some quantum measurement
acting on some quantum state. Another important assumption here is that the
preparation device always prepares the same state and there is no correlation
between the measurement device and the preparation device. From the experiment
Alice and Bob obtain the joint probability distribution $\\{p(a,b|x,y)\\}$
where $a,b$ denote the outcomes when they choose the measurements labelled by
$x,y$, respectively.

Now, consider that the preparation device generates the state $\rho$ and the
measurements performed by Alice and Bob in the observable picture as $A_{a|x}$
and $B_{b|y}$ [see Sec. 2.1.3] respectively for all $a,b,x,y$. For simplicity,
from here on we refer the observables $A_{a|x}$ and $B_{b|y}$ as measurements.
Using the joint probability distribution $\\{p(a,b|x,y)\\}$, Alice and Bob
want to characterise the state $\rho$ sent by the preparation device and also
their measurements represented in the observable picture as $A_{a|x}$ and
$B_{b|y}$ for all $a,b,x,y$. It is important here to note that one can
identify such quantum realisations only up to the equivalences under which the
probability distribution remains invariant. Also, notice that in the scenario
presented above, one can not certify the source to be producing a unique mixed
state. The reason being that any mixed state can be decomposed in terms of
pure states. Let us say that the target or the ideal state that is expected to
be produced by the source is denoted by ${|{\tilde{\psi}}\rangle}$ and the
target measurements that are expected to be performed are denoted in the
observable form as $\tilde{A}_{a|x}$ for Alice and $\tilde{B}_{b|y}$ for Bob.
Then, there are two major equivalences under which the probability remains
invariant,

  1. 1.

An additional state might be attached to the target state on which the
measurements act trivially, that is, the actual state and measurements in the
device can be

$\displaystyle\rho={|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}\otimes\sigma$
(2.74)

such that $\sigma$ acts on some unknown but finite dimensional Hilbert space
$\mathcal{H}^{\prime\prime}$ and,

$\displaystyle A_{a|x}=\tilde{A}_{a|x}\otimes\mathbb{1}^{\prime\prime}\quad
B_{b|y}=\tilde{B}_{b|y}\otimes\mathbb{1}^{\prime\prime}.$ (2.75)

  2. 2.

The actual state and measurements might be unitarily rotated with respect to
the target state, that is,

$\displaystyle\rho=U_{A}\otimes U_{B}\
{|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}\ U_{A}^{\dagger}\otimes
U_{B}^{\dagger},$ (2.76)

and

$\displaystyle A_{a|x}=U_{A}\tilde{A}_{a|x}U_{A}^{\dagger},\quad
B_{b|y}=U_{B}\tilde{B}_{b|y}U_{B}^{\dagger}$ (2.77)

where $U_{A}:\mathcal{H}_{A}\rightarrow\mathcal{H}_{A},\
U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ are local unitary
transformations on the subsystem of Alice and Bob respectively.

Given these two equivalences, we now present the self-testing definition that
is relevant for this thesis.

######  Definition 10 (Self-testing).

Consider the above Bell experiment with Alice and Bob performing measurements
$A_{a|x}$ and $B_{b|y}$ on a state $\rho_{AB}$ and observing correlations
$\\{p(a,b|x,y)\\}$. The state $\rho_{AB}$ and measurements $A_{a|x}$ and
$B_{b|y}$ are certified to be the target state ${|{\tilde{\psi}}\rangle}$ and
target measurements $\tilde{A}_{a|x}$ and $\tilde{B}_{b|y}$ from
$\\{p(a,b|x,y)\\}$ if:

  1. 1.

The Hilbert space of Alice and Bob decompose as

$\displaystyle\mathcal{H}_{A}=\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{A^{\prime\prime}},\qquad\mathcal{H}_{B}=\mathcal{H}_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$
(2.78)

where the target state ${|{\tilde{\psi}}\rangle}$ belongs to
$\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}}$ and
$\mathcal{H}_{A^{\prime\prime}}$ and $\mathcal{H}_{B^{\prime\prime}}$
represents the auxiliary Hilbert space of Alice and Bob respectively.

  2. 2.

There exists a unitary $U_{B}:\mathcal{H}_{B}\to\mathcal{H}_{B}$ and
$U_{A}:\mathcal{H}_{A}\to\mathcal{H}_{A}$ such that the state is

$U_{A}\otimes U_{B}\ \rho_{AB}\ U_{A}^{\dagger}\otimes
U_{B}^{\dagger}={|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}_{A^{\prime}B^{\prime}}\otimes\sigma_{A^{\prime\prime}B^{\prime\prime}},$
(2.79)

where $\sigma_{A^{\prime\prime}B^{\prime\prime}}$ is some state acting on the
Hilbert space
$\mathcal{H}_{A^{\prime\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$.

  3. 3.

The measurements are certified as

$U_{A}\,A_{a|x}\,U_{A}^{\dagger}=\tilde{A}_{a|x}\otimes\mathbb{1}_{A^{\prime\prime}},\qquad
U_{B}\,B_{b|y}\,U_{B}^{\dagger}=\tilde{B}_{b|y}\otimes\mathbb{1}_{B^{\prime\prime}}.$
(2.80)

for all $a,b,x,y$. Here,
$\mathbb{1}_{A^{\prime\prime}},\mathbb{1}_{B^{\prime\prime}}$ are identities
acting on the Hilbert space $\mathcal{H}_{A^{\prime\prime}}$ and
$\mathcal{H}_{B^{\prime\prime}}$ respectively.

One natural assumption that we make in the above definition is that the local
states of Alice and Bob are full-rank. This comes from the fact that Alice and
Bob can only characterise the part of the quantum measurements that act on the
quantum state. Notice, that in general the unitaries appearing in the second
equivalence can be replaced by isometries [see Def. 7]. Another important
equivalence pointed out in [140], that is not stated above is that the actual
quantum state and measurements can not be distinguished from the conjugate of
the target state and measurements, that is,

$\displaystyle\rho={|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}^{*},\quad
A_{a|x}=\tilde{A}_{a|x}^{*},\quad B_{b|y}=\tilde{B}_{b|y}^{*}.$ (2.81)

However, this equivalence is not relevant for this thesis (it will be
clarified when analysing the results of this thesis). For other definitions of
self-testing refer to [141].

As discussed before, the set of joint probability distributions is convex. It
is important to note here that one can only certify quantum states and
measurements from probability distribution that lie at the boundary of this
set, that is, extremal probability distributions [142]. The reason is that any
point inside this set can be represented as convex combination of the extremal
points which does not correspond to a unique probability distribution and thus
can not be achieved by only a particular state and measurements.

Now, It turns out that if one utilises the maximal violation of the Bell
inequalities, then the desired self-testing of quantum state and measurements
can be achieved by collecting less statistics compared to the case when one
uses tomography. For instance, both Alice and Bob are required to choose at
least four inputs corresponding to a tomographically complete set of
measurements to certify any two qubit state. However, employing Bell
inequalities one can certify two qubit states using only two measurements per
party. The maximal violation of a Bell inequality is achieved by the joint
probability distributions lying at the boundary of the quantum set as shown in
Fig. 2.6. However, it might happen that the Bell functional touches the
boundary of the quantum set at more than one point and one can have some
weaker self-testing statement where one certifies a family of states or
measurements [143, 144, 145]. To certify some particular quantum state and
measurements, the Bell violation should point to a unique probability
distribution. As a consequence, to self-test any quantum realisation, we need
to ensure the maximum violation of the Bell inequality and then we need to
show that the probability distribution that gives the maximum violation is
generated by unique quantum state and measurements up to the equivalences as
suggested before. Thus, in the definition of self-testing [cf. Def. 10], we
can replace the statement “observing correlations $\\{p(a,b|x,y)\\}$" with
“observing the maximal violation of a Bell inequality $\mathcal{B}$".

In a physical experiment, we can never achieve the maximal violation of a Bell
inequality but some value which is close to this violation. From an
experimental perspective, it is thus necessary to understand self-testing in
the presence of noise and whether the proposed certification schemes are
robust against experimental imperfections.

######  Definition 11 (Robust self-testing).

Consider the above Bell experiment with Alice and Bob performing the
measurements $A_{a|x}$ and $B_{b|y}$ respectively on a quantum state
$\rho_{AB}$. Assume that the value of a given Bell expression $\mathcal{B}$
for the observed correlations satisfy

$\displaystyle\mathcal{B}\geq\beta_{Q}-\epsilon.$ (2.82)

Alice and Bob can robustly certify a target state ${|{\tilde{\psi}}\rangle}$
and target measurements $\tilde{A}_{a|x}$ and $\tilde{B}_{b|y}$ from the
observed Bell value, if there exists a unitary
$U_{B}:\mathcal{H}_{B}\to\mathcal{H}_{B}$ and
$U_{A}:\mathcal{H}_{A}\to\mathcal{H}_{A}$ such that the state is

$\left|\left|U_{A}\otimes U_{B}\rho_{AB}U_{A}^{\dagger}\otimes
U_{B}^{\dagger}-{|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}_{A^{\prime}B^{\prime}}\otimes\sigma_{A^{\prime\prime}B^{\prime\prime}}\right|\right|_{2}\leq
f_{1}(\epsilon),$ (2.83)

where $\sigma_{A^{\prime\prime}B^{\prime\prime}}$ is some state acting on the
Hilbert space
$\mathcal{H}_{A^{\prime\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$ and the
measurements are

$\displaystyle||U_{A}\,A_{a|x}U_{A}^{\dagger}-\tilde{A}_{a|x}\otimes\mathbb{1}_{A^{\prime\prime}}||_{2}$
$\displaystyle\leq$ $\displaystyle f_{2}(\epsilon),\quad\text{and}$
$\displaystyle||U_{B}\,B_{b|y}\,U_{B}^{\dagger}-\tilde{B}_{b|y}\otimes\mathbb{1}_{B^{\prime\prime}}||_{2}$
$\displaystyle\leq$ $\displaystyle f_{3}(\epsilon).$ (2.84)

for all $a,b,x,y$ and
$\mathbb{1}_{A^{\prime\prime}},\mathbb{1}_{B^{\prime\prime}}$ are identities
acting on to the support of Alice’s and Bob’s subsystem $\rho_{A},\rho_{B}$
respectively. Further, when $\epsilon$ goes to $0$, the functions
$f_{1}(\epsilon),f_{2}(\epsilon)$ and $f_{3}(\epsilon)$ must vanish.

For a note, the $||X||_{2}$ denotes the Hilbert-Schmidt norm of an operator
$X$, and is defined as $||X||_{2}=\sqrt{\mathrm{Tr}(X^{\dagger}X)}$. For two
unitary matrices $A$ and $\tilde{A}$, we have that

$\displaystyle||A-\tilde{A}||_{2}=2\left[\mathrm{Tr}(\mathbb{1})-\mathrm{Re}\mathrm{Tr}(A\tilde{A}^{\dagger})\right].$
(2.85)

The above statement can be understood as, if one observes a value $\epsilon$
lower than the maximal violation in a Bell experiment, then overlap between
the state shared between the parties and the target state up to some
equivalences is bounded from below by a function of $\epsilon$. One can arrive
at a similar conclusion for the measurements.

##### Ways to do self-testing

There are several works that provide self-testing statements using numerical
approaches based on semi-definite programs [42, 43, 49, 50]. However, these
methods can not be used to provide self-testing statements for higher
dimensional states due to computational requirements. In other words, there
does not exist any numerical scheme that can certify entangled states of
arbitrary local dimension in an efficient way. Due to this, in this work we
focus on analytical methods of self-testing.

There are a few analytical techniques that have been explored for the task of
self-testing but most of these techniques work when the state to be certified
is a two-qubit state. For instance, the initial self-testing schemes [36, 38,
45] were based on Jordan’s lemma [74, 146] that says that if two Hermitian
matrices with eigenvalues $\pm 1$ act on a Hilbert space $\mathcal{H}$, then
these matrices decompose as a direct sum of matrices acting on Hilbert space
of dimension less than or equal to two. Another such method is using the swap
gate that serves as an isometry that maps the non-ideal state to the target
state [27, 28, 30, 31, 47, 40].

Figure 2.6: The CHSH functional $\mathcal{B}_{CHSH}=2$ represents a facet of
the local set of correlations $\mathcal{L}_{2,2,2}$. In other words, CHSH
functional is a hyperplane that represents one of the face of the local
polytope. Interestingly, the CHSH functional $\mathcal{B}_{CHSH}=2\sqrt{2}$
touches the quantum set at a single extremal point making it particularly
useful for self-testing.

In this thesis, we follow another approach to derive self-testing statements
that is based on “sum of squares (SOS) decomposition" of the Bell operator.
This involves decomposing the Bell operator in terms of some positive
operators. This method was first explored for self-testing of any pure
entangled two-qubit state in [29]. Based on this method, some self-testing
schemes were provided in [39, 143, 54] that can be used to certify multi-qubit
states and subspaces. A scheme to self-test two-qutrit maximally entangled
state was proposed in [46] that employed the method of SOS decomposition of
the Bell operator. In [46, 52] family of Bell inequalities were constructed
using this technique, the maximal violation of which was achieved by maximally
entangled state of arbitrary local dimension.

Let us now elaborate on the SOS decomposition of a Bell operator. Any Bell
operator (2.45) can be represented as

$\displaystyle\hat{\mathcal{B}}=\sum_{a,b,x,y}c_{a,b,x,y}M_{a|x}\otimes
N_{b|y}$ (2.86)

where $M_{a|x},N_{b|y}$ are measurement operators corresponding to the input
$x$ and output $a$ of Alice and input $y$ and output $b$ of Bob respectively.
If $\beta_{Q}$ is the quantum bound of the Bell expression
$\langle\hat{\mathcal{B}}\rangle$ where $\hat{\mathcal{B}}$ is given in
(2.86), then let us assume that the positive semi-definite operator
$\beta_{Q}\mathbb{1}-\hat{\mathcal{B}}$ can be decomposed in the following way

$\displaystyle\beta_{Q}\mathbb{1}-\hat{\mathcal{B}}=\sum_{i}P_{i}^{\dagger}P_{i}.$
(2.87)

Note that the right hand side of the above equation consists of positive
operators $P_{i}^{\dagger}P_{i}$ composed of the operators $M_{a|x},N_{b|y}$.
Given a generic Bell operator, it is not that straightforward to find such
decompositions. One can numerically find it using the Navascués-Pironio-Acín
(NPA) hierarchy [147, 148, 149] which is based on semi-definite programming.
Let us now consider that the state ${|{\psi}\rangle}$ achieves the maximal
violation of a Bell inequality, then the left hand side of (2.87) vanishes and
thus

$\displaystyle\sum_{i}||P_{i}{|{\psi}\rangle}||^{2}=0.$ (2.88)

All the terms are positive in the above sum which allows to conclude that
$||P_{i}{|{\psi}\rangle}||=0$ for all $i$. Thus,

$\displaystyle\forall i\qquad P_{i}{|{\psi}\rangle}=0.$ (2.89)

This simple expression contains all the information about the state and
measurements that give rise to the maximal violation of the corresponding Bell
inequality. Often, these relations can be used to derive self-testing results.

Since, this technique is central to the results presented in this thesis we
would provide a simple example of self-testing based on SOS decomposition.
Before proceeding let us state an important fact that was proven in Ref. [46]
which is central to some of the results presented in this thesis.

######  Fact 2.

Consider two unitary matrices $R_{0},R_{1}$ that act on a finite-dimensional
Hilbert space $\mathcal{H}$ satisfying $R_{0}^{d}=R_{1}^{d}=\mathbb{1}$. If
$R_{0}$ and $R_{1}$ satisfy the relation $R_{0}R_{1}=\omega R_{1}R_{0}$, then
$dim(\mathcal{H})=d.t$ for some positive integer $t$ and there exists a
unitary $U:\mathcal{H}\rightarrow\mathcal{H}$ such that

$\displaystyle UR_{0}U^{\dagger}=Z_{d}\otimes\mathbb{1},\quad\text{and}\quad
UR_{1}U^{\dagger}=X_{d}\otimes\mathbb{1},$ (2.90)

where $Z_{d},X_{d}$ are the $d-$dimensional generalisation of the Pauli
matrices $\sigma_{z},\sigma_{x}$ (2.27) given by,

$\displaystyle
Z_{d}=\sum_{i=0}^{d-1}\omega^{i}{|{i}\rangle}\\!{\langle{i}|},\qquad
X_{d}=\sum_{i=0}^{d-1}{|{i+1}\rangle}\\!{\langle{i}|}.$ (2.91)

Let us now consider the CHSH inequality (2.47), which for our convenience is
scaled down by the factor $\frac{1}{\sqrt{2}}$, and consider its operator form

$\displaystyle\hat{\mathcal{B}}_{CHSH}=\frac{1}{\sqrt{2}}\left(A_{0}\otimes
B_{0}+A_{0}\otimes B_{1}+A_{1}\otimes B_{0}-A_{1}\otimes B_{1}\right).$ (2.92)

We assume here that the measurements of Alice and Bob are projective and thus,
$A_{x}^{2}=B_{y}^{2}=\mathbb{1}$ for all $x,y$. The local and quantum bound of
this Bell inequality is $\sqrt{2}$ and $2$ respectively (2.52) and thus the
SOS decomposition of the corresponding shifted Bell operator is given by

$\displaystyle
2\mathbb{1}-\hat{\mathcal{B}}_{CHSH}=\frac{1}{2}\left(P_{0}^{\dagger}P_{0}+P_{1}^{\dagger}P_{1}\right)$
(2.93)

where,

$\displaystyle
P_{0}=\mathbb{1}-A_{0}\otimes\frac{B_{0}+B_{1}}{\sqrt{2}}\quad\text{and}\quad
P_{1}=\mathbb{1}-A_{1}\otimes\frac{(B_{0}-B_{1})}{\sqrt{2}}.$ (2.94)

Let us now suppose that a state $\rho_{AB}$ attains the quantum bound of the
above Bell operator. Then, we can always add an ancillary system $E$ to purify
this state. Let us denote this purified state by ${|{\psi_{ABE}}\rangle}$ such
that $\rho_{AB}=\mathrm{Tr}_{E}({|{\psi}\rangle}\\!{\langle{\psi}|}_{ABE})$.
Then, from Eq. (2.94) the following relations must hold,

$\displaystyle\left[A_{0}\otimes\frac{B_{0}+B_{1}}{\sqrt{2}}\otimes\mathbb{1}_{E}\right]{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}$
(2.95)

and,

$\displaystyle\left[A_{1}\otimes\frac{B_{0}-B_{1}}{\sqrt{2}}\otimes\mathbb{1}_{E}\right]{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}$
(2.96)

Using the fact that $A_{0}$ are unitary and Hermitian, we have from Eq. (2.95)
that

$\displaystyle\frac{B_{0}+B_{1}}{\sqrt{2}}\otimes\mathbb{1}_{AE}{|{\psi_{ABE}}\rangle}=A_{0}\otimes\mathbb{1}_{BE}{|{\psi_{ABE}}\rangle}$
(2.97)

Let us recall that the measurements can be characterised only on the support
of the local reduced states $\rho_{A}$ and $\rho_{B}$ and consequently, in the
proof we assume that the local states of Alice and Bob are full-rank. Also,
From here on, for convenience we drop the term $\mathbb{1}_{E}$. Now,
multiplying by $A_{0}\otimes\mathbb{1}_{B}$ on both the sides of Eq. (2.97) ,
we have that

$\displaystyle\left[\mathbb{1}_{A}\otimes\frac{B_{0}+B_{1}}{\sqrt{2}}\right]A_{0}\otimes\mathbb{1}_{B}{|{\psi_{ABE}}\rangle}=A_{0}^{2}\otimes\mathbb{1}_{B}{|{\psi_{ABE}}\rangle}$
(2.98)

where we used the fact that measurements acting on subsystem $A$ and $B$
commute. Using (2.97) and the fact that $A_{0}^{2}=\mathbb{1}$, we have that

$\displaystyle\mathbb{1}\otimes\left[\frac{B_{0}+B_{1}}{\sqrt{2}}\right]^{2}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$
(2.99)

Taking a partial trace over the subsystems $A,E$, we have that

$\displaystyle\left[\frac{B_{0}+B_{1}}{\sqrt{2}}\right]^{2}\rho_{B}=\rho_{B}.$
(2.100)

The fact that $\rho_{B}$ is full-rank and thus invertible allows us to
conclude from the above equation that

$\displaystyle\left[\frac{B_{0}+B_{1}}{\sqrt{2}}\right]^{2}=\mathbb{1}_{B}.$
(2.101)

Similarly, one obtains from (2.96) that

$\left[\frac{B_{0}-B_{1}}{\sqrt{2}}\right]^{2}=\mathbb{1}_{B}.$ (2.102)

Expanding the the above two equations (2.101) and (2.102), we have that

$B_{0}^{2}+B_{1}^{2}+\big{\\{}B_{0},B_{1}\big{\\}}=2\mathbb{1}_{B},\quad\text{and}\quad
B_{0}^{2}+B_{1}^{2}-\big{\\{}B_{0},B_{1}\big{\\}}=2\mathbb{1}_{B}$ (2.103)

where $\big{\\{}B_{0},B_{1}\big{\\}}=B_{0}B_{1}+B_{1}B_{0}$ denotes the anti-
commutator of $B_{0},B_{1}$. Now, using the fact that
$B_{0}^{2}=B_{1}^{2}=\mathbb{1}_{B}$, we obtain that

$\displaystyle\big{\\{}B_{0},B_{1}\big{\\}}=0.$ (2.104)

Now, as stated in Fact 2, for two unitary matrices $B_{0},B_{1}$ with the
additional property that $B_{i}^{2}=\mathbb{1}_{B}$ for $i=0,1$ and satisfy
the condition (2.104), there exist local unitary transformation
$V_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ such that

$V_{B}\ B_{0}\
V_{B}^{\dagger}=\sigma_{z}\otimes\mathbb{1}_{B^{\prime\prime}},\quad\text{and}\quad
V_{B}\ B_{1}\ V_{B}^{\dagger}=\sigma_{x}\otimes\mathbb{1}_{B^{\prime\prime}}$
(2.105)

where $\mathbb{1}_{B^{\prime\prime}}$ acts on the auxiliary Hilbert space
$\mathcal{H}_{B^{\prime\prime}}$ of some finite dimension and
$\sigma_{z},\sigma_{x}$ are Pauli matrices (2.27). This also shows that the
Hilbert space of Bob decomposes as
$\mathcal{H}_{B}=\mathbb{C}^{2}\otimes\mathcal{H}_{B^{\prime\prime}}$. Now,
consider a unitary matrix $W=W^{\prime}\otimes\mathbb{1}_{B^{\prime\prime}}$
such that $W^{\prime}$ is also unitary and can be expressed as a matrix
written in the $2-$dimensional computational basis
$\\{{|{0}\rangle},{|{1}\rangle}\\}$ as

$\displaystyle
W^{\prime}=\begin{pmatrix}\cos\left(\frac{\pi}{8}\right)&\sin\left(\frac{\pi}{8}\right)\\\
\sin\left(\frac{\pi}{8}\right)&-\cos\left(\frac{\pi}{8}\right)\end{pmatrix}.$
(2.106)

The unitary matrix $W^{\prime}$ transforms the Pauli observables
$\sigma_{z},\sigma_{x}$ to the ideal ones (2.51) and thus from Eq. (2.105) we
arrive at

$U_{B}B_{0}U_{B}^{\dagger}=\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}}\otimes\mathbb{1}_{B^{\prime\prime}}\qquad\text{and}\qquad
U_{B}B_{1}U_{B}^{\dagger}=\frac{\sigma_{z}-\sigma_{x}}{\sqrt{2}}\otimes\mathbb{1}_{B^{\prime\prime}},$
(2.107)

where $U_{B}=WV_{B}$. Now, to find Alice’s observables $A_{i}$ we consider an
analogous SOS decomposition of the CHSH operator given by

$\displaystyle
2\mathbb{1}-\hat{\mathcal{B}}_{CHSH}=\frac{1}{2}\left(Q_{0}^{\dagger}Q_{0}+Q_{1}^{\dagger}Q_{1}\right)$
(2.108)

where,

$\displaystyle Q_{0}=\mathbb{1}-\frac{A_{0}+A_{1}}{\sqrt{2}}\otimes
B_{0}\quad\text{and}\quad Q_{1}=\mathbb{1}-\frac{A_{0}-A_{1}}{\sqrt{2}}\otimes
B_{1}.$ (2.109)

Again, for any state ${|{\psi_{ABE}}\rangle}$ that maximally violates the CHSH
inequality, from Eq. (2.109) the following relations must hold,

$\displaystyle\frac{A_{0}+A_{1}}{\sqrt{2}}\otimes
B_{0}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}$
(2.110)

and

$\displaystyle\frac{A_{0}-A_{1}}{\sqrt{2}}\otimes
B_{1}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$
(2.111)

As concluded for Bob’s observables, the Hilbert space of Alice decomposes as
$\mathcal{H}_{A}=\mathbb{C}^{2}\otimes\mathcal{H}_{A^{\prime\prime}}$ and
there exist local unitary transformation
$U_{A}:\mathcal{H}_{A}\rightarrow\mathcal{H}_{A}$ such that

$U_{A}\ A_{0}\
U_{A}^{\dagger}=\sigma_{z}\otimes\mathbb{1}_{A^{\prime\prime}},\quad\text{and}\quad
U_{A}\ A_{1}\ U_{A}^{\dagger}=\sigma_{x}\otimes\mathbb{1}_{A^{\prime\prime}}$
(2.112)

where $\mathbb{1}_{A^{\prime\prime}}$ acts on the auxiliary Hilbert space
$\mathcal{H}_{A^{\prime\prime}}$.

After determining the form of the measurements, we can finally certify the
quantum state ${|{\psi_{ABE}}\rangle}$ that achieves the maximum violation of
the CHSH Bell inequality. Since,
$\mathcal{H}_{A}=\mathbb{C}^{2}\otimes\mathcal{H}_{A^{\prime\prime}}$ and
$\mathcal{H}_{B}=\mathbb{C}^{2}\otimes\mathcal{H}_{B^{\prime\prime}}$, we can
decompose the state ${|{\psi_{ABE}}\rangle}$ as

$U_{A}\otimes
U_{B}{|{\psi_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}=\sum_{i,j=0,1}{|{i}\rangle}_{A^{\prime}}{|{j}\rangle}_{B^{\prime}}{|{\psi_{ij}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}$
(2.113)

where
${|{\psi_{ij}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}\in\mathcal{H}_{A^{\prime\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}\otimes\mathcal{H}_{E}$.
For convenience, we drop the subscripts from the state. Now, plugging in the
certified measurements (2.107) and (2.112) in the SOS relation (2.95)

$\sigma_{z}\otimes\sigma_{z}\otimes\mathbb{1}_{A^{\prime\prime}B^{\prime\prime}E}{|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}.$
(2.114)

Substituting the general form of the state (2.113) and simplifying, we obtain
that

$\displaystyle\sum_{i,j=0,1}(-1)^{ij}{|{i}\rangle}_{A^{\prime}}{|{j}\rangle}_{B^{\prime}}{|{\psi_{ij}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}=\sum_{i,j=0,1}{|{i}\rangle}_{A^{\prime}}{|{j}\rangle}_{B^{\prime}}{|{\psi_{ij}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}.$
(2.115)

By projecting the above equation on to
${\langle{i}|}_{A^{\prime}}\\!{\langle{j}|}_{B^{\prime}}$ for all $i,j$ such
that $i\neq j$, we arrive at the following condition

$\displaystyle{|{\psi_{ij}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}=0\quad
s.t.\quad i\neq j$ (2.116)

and thus the state that achieves the maximal violation of the CHSH inequality
simplifies to

${|{\tilde{\psi}_{ABE}}\rangle}={|{0}\rangle}_{A^{\prime}}{|{0}\rangle}_{B^{\prime}}{|{\psi_{00}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}+{|{1}\rangle}_{A^{\prime}}{|{1}\rangle}_{B^{\prime}}{|{\psi_{11}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}.$
(2.117)

Now, we consider the other SOS relation (2.96) and plug into it the certified
measurements (2.107) and (2.112). This gives us

$\displaystyle\sigma_{x}\otimes\sigma_{x}\otimes\mathbb{1}_{A^{\prime\prime}B^{\prime\prime}E}{|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}.$
(2.118)

Now, substituting the simplified state (2.117), we obtain that

${|{1}\rangle}_{A^{\prime}}{|{1}\rangle}_{B^{\prime}}{|{\psi_{00}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}+{|{0}\rangle}_{A^{\prime}}{|{0}\rangle}_{B^{\prime}}{|{\psi_{11}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}={|{0}\rangle}_{A^{\prime}}{|{0}\rangle}_{B^{\prime}}{|{\psi_{00}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}+{|{1}\rangle}_{A^{\prime}}{|{1}\rangle}_{B^{\prime}}{|{\psi_{11}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}.$
(2.119)

Again projecting the above equation on to
${\langle{0}|}_{A}^{\prime}{\langle{0}|}_{B}^{\prime}$, we arrive at the
condition that

$\displaystyle{|{\psi_{00}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}={|{\psi_{11}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}.$
(2.120)

Thus, the state which achieves the maximal violation of the CHSH inequality is
given by

$\displaystyle U_{A}\otimes
U_{B}{|{\psi_{ABE}}\rangle}=\frac{1}{\sqrt{2}}({|{0}\rangle}_{A^{\prime}}{|{0}\rangle}_{B^{\prime}}+{|{1}\rangle}_{A^{\prime}}{|{1}\rangle}_{B^{\prime}})\otimes{|{aux}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}$
(2.121)

where
${|{aux}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}=\sqrt{2}{|{\psi_{00}}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}$.
This completes the proof that the maximal violation of CHSH Bell inequality
can be used to certify the two-qubit maxmally entangled state. In Chapter 3,
we generalise this proof to self-test maximally entangled state of arbitrary
local dimension. We now move on to presenting the idea of one-sided device
independent certification.

####  2.3.2 One-sided device independent certification

Let us now consider the following task inspired from a real-world scenario
consisting of a bank that wants to securely send information to its clients.
The security of the Bank can be considered to be strong and chances of attack
directly on the bank is much lower. However, individual clients are prone to
intruders who want to steal their information. In such a situation, bank can
be considered to be trusted and the clients are untrusted.

Based on the above example, let us consider a scenario where there are two
parties Alice and Bob and a preparation device which sends one system to Alice
and other to Bob. Here Alice is trusted, that is, she can perform a full
tomography on her subsystem. Bob now performs measurements on his subsystem
which might steer or affect Alice’s subsystem. This scenario is in fact the
quantum steering scenario. Observing some specific joint probability
distribution $\\{p(a,b|x,y)\\}$ allows one to certify the preparation device
as well the measurements performed by the untrusted Bob up to the equivalences
as discussed in the previous subsection.

######  Definition 12 (One-sided device independent certification (1-SDI
certification)).

Consider the above experiment with Alice and Bob performing measurements
$A_{a|x}$ and $B_{b|y}$ on a state $\rho_{AB}$ and observing correlations
$\\{p(a,b|x,y)\\}$ along with the fact that Alice’s measurements $A_{a|x}$ are
known and act on Hilbert space $\mathcal{H}_{A}$. The state $\rho_{AB}$ and
measurements $B_{b|y}$ are certified to be the target state
${|{\tilde{\psi}}\rangle}$ and target measurements $\tilde{B}_{b|y}$ from
$\\{p(a,b|x,y)\\}$ if:

  1. 1.

The Hilbert space of Bob decomposes as

$\displaystyle\mathcal{H}_{B}=\mathcal{H}_{B^{\prime}}\otimes\mathcal{H}_{B^{\prime\prime}}$
(2.122)

where the target state ${|{\tilde{\psi}}\rangle}$ belongs to
$\mathcal{H}_{A}\otimes\mathcal{H}_{B^{\prime}}$ and
$\mathcal{H}_{B^{\prime\prime}}$ represents the auxiliary Hilbert space of
Bob.

  2. 2.

There exists a unitary $U_{B}:\mathcal{H}_{B}\to\mathcal{H}_{B}$ such that the
state is

$\Big{(}\mathbb{1}_{A}\otimes U_{B}\Big{)}\rho_{AB}\left(\mathbb{1}\otimes
U_{B}^{\dagger}\right)={|{\tilde{\psi}}\rangle}\\!{\langle{\tilde{\psi}}|}_{AB^{\prime}}\otimes\sigma_{B^{\prime\prime}},$
(2.123)

where $\sigma_{B^{\prime\prime}}$ is some state acting on the Hilbert space
$\mathcal{H}_{B^{\prime\prime}}$.

  3. 3.

The measurements are certified as

$U_{B}\,B_{b|y}\,U_{B}^{\dagger}=\tilde{B}_{b|y}\otimes\mathbb{1}_{B^{\prime\prime}}.$
(2.124)

where $\mathbb{1}_{B^{\prime\prime}}$ is identity acting on the Hilbert space
$\mathcal{H}_{B^{\prime\prime}}$.

Analogous to self-testing, any correlation that can be used for 1SDI
certification must belong to the boundary of the quantum set. As discussed
before, the quantum correlations that achieve the maximal violation of a
steering inequality are extremal or lie at the boundary of the quantum set. If
one employs the maximum violation of a steering inequality to certify the
quantum realisations, one needs to measure much lesser number of correlations
than the standard procedure of observing at least the full tomographically
complete set of correlations. One can thus provide an analogous definition of
1SDI certification by replacing the line “observing correlations
$\\{p(a,b|x,y)\\}$" by “observing the maximal violation of a steering
inequality" in Def. 12. Again, we can define robust 1SDI analogously to Def.
11 imposing that Alice is trusted.

##### Ways to do 1SDI certification

The first results on 1SDI certification were derived in [63, 64] where the
idea of swap isometry was employed to certify the maximally entangled state
and Pauli observables. In [139] any pure bipartite entangled state was
certified by using the subspace method as was done in [47]. There is also a
numerical method [150] that aims to characterise the steering assemblages but
again the limitation being that higher dimensional assemblages can not be
certified in an efficient way using this method.

In chapter 4 in this thesis, we develop an analytical method for 1SDI
certification. We explore the fact that the algebraic bound of the steering
inequality gives some relations which can be solved to find the desired
certification. Here we give a short proof to highlight this idea using the
simplest steering inequality (2.61).

Let us consider the simplest steering functional (2.61) and express it in the
correlation picture (2.73) as discussed above

$\displaystyle W_{2,2,2}=\langle\sigma_{z}\otimes
B_{0}\rangle+\langle\sigma_{x}\otimes B_{1}\rangle.$ (2.125)

Note that for convenience, we scaled the inequality (2.61) by a factor of $2$
and also removed the term corresponding to
$\mathbb{1}_{A}\otimes\mathbb{1}_{B}$. Achieving the maximal quantum value
implies that

$\displaystyle\langle\sigma_{z}\otimes B_{0}\rangle+\langle\sigma_{x}\otimes
B_{1}\rangle=2.$ (2.126)

Since, $B_{0}$ and $B_{1}$ are valid observables, both the terms in the above
expression are upper bounded by $1$. Consequently, one can achieve the quantum
bound iff each of the expectation value in the above expression (2.126) is
$1$,

$\displaystyle\sigma_{z}\otimes
B_{0}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}$
(2.127)

and,

$\displaystyle\sigma_{x}\otimes
B_{1}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$
(2.128)

Here, as done for self-testing, we introduce an external system $E$ to purify
the state shared between Alice and Bob. For convenience, we will drop
$\mathbb{1}_{E}$ from further calculations. Let us recall that the
measurements can be characterised only on the support of the local reduced
states $\rho_{A}$ and $\rho_{B}$. Thus, without loss of generality we assume
that these local states are full-rank. Now multiplying (2.127) with
$\sigma_{x}\otimes B_{1}$ on both the sides and then using (2.128), we arrive
at

$\displaystyle\sigma_{x}\sigma_{z}\otimes
B_{1}B_{0}{|{\psi_{ABE}}\rangle}=\sigma_{x}\otimes
B_{1}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$ (2.129)

Similarly, multiplying (2.128) with $\sigma_{z}\otimes B_{0}$ on both the
sides and then using (2.127), we arrive at

$\displaystyle\sigma_{z}\sigma_{x}\otimes
B_{0}B_{1}{|{\psi_{ABE}}\rangle}=\sigma_{z}\otimes
B_{0}{|{\psi_{ABE}}\rangle}={|{\psi_{ABE}}\rangle}.$ (2.130)

Now, using the fact that $\sigma_{z}\sigma_{x}+\sigma_{x}\sigma_{z}=0$ in the
above equation, we have that

$\displaystyle\sigma_{x}\sigma_{z}\otimes
B_{0}B_{1}{|{\psi_{ABE}}\rangle}=-{|{\psi_{ABE}}\rangle}.$ (2.131)

Now, adding (2.129) and (2.131), we have that

$\displaystyle\sigma_{x}\sigma_{z}\otimes\left(B_{0}B_{1}+B_{1}B_{0}\right){|{\psi_{ABE}}\rangle}=0.$
(2.132)

Since, $\sigma_{x},\sigma_{z}$ are invertible we get

$\displaystyle\mathbb{1}_{A}\otimes\left(B_{0}B_{1}+B_{1}B_{0}\right){|{\psi_{ABE}}\rangle}=0.$
(2.133)

Now, tracing over the subsystems $A,E$ we have

$\displaystyle\left(B_{0}B_{1}+B_{1}B_{0}\right)\rho_{B}=0.$ (2.134)

Again, $\rho_{B}$ is full-rank and thus invertible which finally gives us the
desired relation

$\displaystyle B_{0}B_{1}+B_{1}B_{0}=0.$ (2.135)

Now, using Fact 2 we can conclude that the Hilbert space of Bob decomposes as
$\mathcal{H}_{B}=\mathbb{C}^{2}\otimes\mathcal{H}_{B^{\prime\prime}}$ and
there exist a local unitary transformation
$U_{B}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{B}$ such that

$\displaystyle
U_{B}B_{0}U_{B}^{\dagger}=\sigma_{z}\otimes\mathbb{1}_{B^{\prime\prime}},\quad\text{and}\quad
U_{B}B_{1}U_{B}^{\dagger}=\sigma_{x}\otimes\mathbb{1}_{B^{\prime\prime}}.$
(2.136)

Now, we find the state ${|{\psi_{ABE}}\rangle}$ that gives the maximum
violation of the steering inequality (2.125). As was done for self-testing, we
consider a general state of the form remembering that Alice is trusted

$\mathbb{1}_{A}\otimes
U_{B}{|{\psi_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}=\sum_{i,j=0,1}{|{i}\rangle}_{A^{\prime}}{|{j}\rangle}_{B^{\prime}}{|{\psi_{ij}}\rangle}_{B^{\prime\prime}E}.$
(2.137)

where
${|{\psi_{ij}}\rangle}_{B^{\prime\prime}E}\in\mathcal{H}_{B^{\prime\prime}}\otimes\mathcal{H}_{E}$.
Now, using this state and the certified measurements (2.136), we solve the
relations (2.127) and (2.128), that is,

$\displaystyle\sigma_{z}\otimes\sigma_{z}\otimes\mathbb{1}_{B^{\prime\prime}}\otimes\mathbb{1}_{E}{|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}$
(2.138)

and,

$\displaystyle\sigma_{x}\otimes\sigma_{x}\otimes\mathbb{1}_{B^{\prime\prime}}\otimes\mathbb{1}_{E}{|{\tilde{\psi}_{ABE}}\rangle}={|{\tilde{\psi}_{ABE}}\rangle}.$
(2.139)

Following exactly the same approach, as the one used for self-testing from Eq.
(2.115) to Eq. (2.121), we find that up to a local unitary transformation
$U_{B}$ we have that

$\displaystyle\mathbb{1}_{A}\otimes
U_{B}{|{\psi_{ABE}}\rangle}=\frac{1}{\sqrt{2}}({|{00}\rangle}+{|{11}\rangle})\otimes{|{aux}\rangle}_{B^{\prime\prime}E}$
(2.140)

This completes our analysis of one-sided device independent certification.
Now, we proceed towards another important avenue for device-independent
protocols namely device-independent certification of genuine randomness.

####  2.3.3 Randomness certification

One of the central requirements for any cryptographic task is access to
devices that can generate outputs not predictable by any attacker. Within
classical framework, the security of such cryptographic tasks relies majorly
on the idea that some functions can not be efficiently computed by a classical
computer. For instance, the security of the RSA protocol [151] relies on the
fact that large numbers can not be factored efficiently using classical
computers. It was shown by Shor in 1994 [152] that one could efficiently
factorise large numbers using quantum resources. Consequently, with the advent
of quantum technologies, many such classical protocols have been shown to be
prone to attackers possessing quantum computers. Thus, it is a matter of time
when the classical cryptosystems would fail and we need better protocols that
are secure against quantum attacks. This is the basis for the origin of the
field of quantum cryptography.

One of the inferences that one can make about quantum theory is that it is
inherently unpredictable. For instance, consider a source that generates the
quantum state

$\displaystyle{|{\psi_{A}}\rangle}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}+{|{1}\rangle}\right).$
(2.141)

Now, if a measurement is performed by Alice on this state in the $z-$basis,
that is,
$M=\\{{|{0}\rangle}\\!{\langle{0}|},{|{1}\rangle}\\!{\langle{1}|}\\}$, then
the outcomes occur with equal probability and are unpredictable. However, here
we require that the state and measurements are exactly same as the ideal ones.
This is device-dependent generation of randomness, that is, one has to
completely trust his devices to generate this randomness. However, let us say
that the source generates the maximally entangled state

$\displaystyle{|{\psi_{AE}}\rangle}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}_{A}{|{0}\rangle}_{E}+{|{1}\rangle}_{A}{|{1}\rangle}_{E}\right)$
(2.142)

instead of the state (2.141) such that some attacker Eve has access to
subsystem $E$. Again, the measurement is performed on the subsystem $A$ in the
$z-$basis. Even if the outcomes seems unpredictable to Alice, if Eve also
performs a measurement in the $z-$basis then she is able to guess the outcome
of Alice with certainty as depicted in Fig. 2.7. Further, Eve might also have
access to Alice’s measurement device. Thus, it is necessary to find protocols
for randomness generation in which one does not has to trust the device used
to produce randomness. This is known as device-independent certification of
randomness.

The idea of DI randomness certification schemes relies on the premise that
there is some attacker who has access to the devices that are used to generate
randomness. However, the attacker can only guess the outcome of the devices
randomly. For instance, if the device generates two outputs, then we say that
these outputs are perfectly random if chances that the attacker can guess them
is never more than fifty percent. On the other hand, if the attacker can guess
the outcome with certainty then the outputs are not random.

Figure 2.7: Alice locally in her lab gets perfectly random outcomes. However,
her subsystem can be correlated with another subsystem possessed by an
intruder Eve. Thus, locally to Alice her outcomes seem perfectly random but
Eve can perfectly guess her outcome based on her results.

Let us now elaborate on this problem in a more formal way. Let us say that
Alice performs a measurement $M$ on a system $S$ and observes an outcome $a$.
Now, the attacker Eve can perform some measurement $\\{Z_{e}\\}$ on her
subsystem and gets an outcome $e$. The outcome $e$ is the guess that Eve makes
about Alice’s outcome. Then the best average probability of guessing by Eve
$p_{guess}(E|S)$ has to be maximised over all possible measurements $z$, and
thus we arrive at the following expression as suggested in Ref. [89],

$\displaystyle p_{guess}(E|S)=\max_{z}\sum_{e}p(e|z)p(e=a|a,z)$ (2.143)

where the above quantity $p(e=a|a,z)$ denotes the probability of Eve’s outcome
$e$ to be the same as Alice’s outcome $a$ and $S$ denotes the system of Alice.
When Alice and Eve posses quantum resources, that is, let us say that the
state shared between Alice and Eve is given by $\rho_{AE}$ and Alice performs
a measurement $\\{M_{a}\\}$ and Eve performs a measurement $\\{Z_{e}\\}$ then
the above guessing probability can be written as

$\displaystyle
p_{guess}(E|\rho_{A})=\max_{Z}\sum_{a}\mathrm{Tr}(\rho_{AE}M_{a}\otimes
Z_{a})$ (2.144)

where $Z=\\{Z_{a}\\}$ and $\rho_{A}=\mathrm{Tr}_{E}(\rho_{AE})$. The amount of
randomness that can be generated by Alice is quantified by the min-entropy of
the of the guessing probability [153], that is,

$\displaystyle H_{min}(E|S)=-\log_{2}p_{guess}(E|S).$ (2.145)

Note that the above quantity is $0$ if Eve can perfectly guess Alice’s
outcome. Interestingly, the maximum amount of randomness that in principle can
be generated from a $d-$dimensional quantum system is of amount $2\log_{2}d$
bits.

Let us now discuss how quantum non-locality, in particular, self-testing
serves as a tool in designing methods for randomness certification.

Figure 2.8: Bell scenario with an intruder Eve: In the simplest Bell scenario,
we consider an additional party Eve who represents an attacker who wants to
guess Alice’s and Bob’s outputs. She can receive some subsystem from the
source correlated with the subsystem of Alice and Bob. She also might know the
inputs of Alice and Bob. She uses all these information to guess their outputs
by performing measurements on her received subsystem.

Consider again the Bell scenario but now with an additional party Eve (the
intruder) who has access to the preparation device as well as the measurement
devices of Alice and Bob. The scenario is depicted in Fig. 2.8. Now, Eve wants
to guess the measurements outcome of one of the parties, let us say, Alice. In
this case, Eve has the following information or strategies to guess the
outcomes:

  1. 1.

Eve can posses some subsystem $E$ that is correlated with the state sent by
the preparation device. As a consequence, the state shared among the parties
is defined by $\rho_{AB}=\mathrm{Tr}_{E}(\rho_{ABE})$, where
$\rho_{ABE}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{E}$
denotes the state shared between Alice, Bob and Eve. There can not be any
restriction on the dimension of the local Hilbert spaces of any of the
parties. This also allows one to consider that the state shared among the
parties to be pure as we can always add additional ancillary systems with Eve
and purify the state.

  2. 2.

Eve might have control over Alice’s and Bob’s measurement devices in the sense
that the measurements might be performed on some auxiliary system of Alice and
Bob which can interfere with their observations.

  3. 3.

Eve might perform a measurement on her subsystem given by POVM $Z=\\{Z_{a}\\}$
that acts on $\mathcal{H}_{E}$. As discussed above, the probability of
obtaining outcome $a$ from measurement performed by Eve on her share of the
joint state $\rho_{ABE}$ is the best guess of Alice’s outcome $a$.

It is also necessary for Eve to remain undetected during the attack otherwise
Alice and Bob can discard those runs and start over again. Thus, the joint as
well as local statistics of Alice and Bob should remain equivalent to the
ideal statistics as expected by them, that is,

$\displaystyle{\langle{\psi_{ABE}}|}A_{a|x}\otimes
B_{b|y}\otimes\mathbb{1}_{E}{|{\psi_{ABE}}\rangle}={\langle{\tilde{\psi}_{AB}}|}\tilde{A}_{a|x}\otimes\tilde{B}_{b|y}{|{\tilde{\psi}_{AB}}\rangle}$
(2.146)

where ${|{\tilde{\psi}_{AB}}\rangle}$ is the ideal state that is expected to
be be sent by the preparation device and $\tilde{A}_{a|x}$ and
$\tilde{B}_{b|y}$ represent the ideal measurement effects that are expected to
be performed by the respective parties. Note that there is no constraint on
Eve’s measurement as Alice and Bob do not know about Eve and the statistics
are averaged over Eve’s outcomes. Let us denote the set of strategies as
$S_{p}$ that are employed by Eve such the constraint (2.146) is satisfied.
Thus, the probability of Eve to guess Alice’s outcome for some input $x$ is
given by,

$\displaystyle
G(x,P)=\sup_{S_{p}}\sum_{a}{\langle{\psi_{ABE}}|}A_{a|x}\otimes\mathbb{1}_{B}\otimes
Z_{a}{|{\psi_{ABE}}\rangle}.$ (2.147)

This is also known as local guessing probability [154, 155, 156, 120]. Here
$P$ denotes the set of observed correlations.

Now, suppose that Alice wants to generate randomness in her lab by performing
the measurements $A_{x}$ on the state
$\rho_{A}=\mathrm{Tr}_{BE}({|{\psi_{ABE}}\rangle}\\!{\langle{\psi_{ABE}}|})$.
Now, Alice and Bob perform a Bell state using the state
${|{\psi_{ABE}}\rangle}$ and measurements $A_{x}$ and $B_{y}$ respectively,
and observe the maximal violation of a Bell inequality. For simplicity of
argument, we assume here that the measurements are projective. Let us say that
the state and measurements can be self-tested from this violation to be the
ideal ones, that is,

$\displaystyle U_{A}\otimes
U_{B}{|{\psi_{ABE}}\rangle}={|{\tilde{\psi}_{A^{\prime}B^{\prime}}}\rangle}\otimes{|{aux}\rangle}_{A^{\prime\prime}B^{\prime\prime}E}$
(2.148)

and the measurements on the local support of the state are certified to be the
ideal measurements as

$\displaystyle
U_{A}\,A_{x}\,U_{A}^{\dagger}=\tilde{A}_{x}\otimes\mathbb{1}_{A^{\prime\prime}},\qquad
U_{B}\,B_{y}\,U_{B}^{\dagger}=\tilde{B}_{y}\otimes\mathbb{1}_{B^{\prime\prime}}.$
(2.149)

Then the guessing probability of Eve (2.147) is given by

$\displaystyle
G(0,E)=\sup_{\rho_{E},Z_{a}}\sum_{a}\mathrm{Tr}(\tilde{A}_{a|0}\rho_{A^{\prime}})\mathrm{Tr}(Z_{a}\rho_{E}).$
(2.150)

where
$\rho_{E}=\mathrm{Tr}_{AB}({|{\psi_{ABE}}\rangle}\\!{\langle{\psi_{ABE}}|})$.
Using the fact $\sum_{a}Z_{a}=\mathbb{1}$, we conclude that

$\displaystyle
G(0,E)\leq\max_{a}\mathrm{Tr}(\tilde{A}_{a|0}\rho_{A^{\prime}}).$ (2.151)

Thus, Alice can securely generate at least

$\displaystyle
H_{min}(E)\geq-\log_{2}(\max_{a}\mathrm{Tr}(\tilde{A}_{a|0}\rho_{A^{\prime}}))$
(2.152)

amount of randomness. Notice that this bound is independent of any strategy of
Eve. Thus, self-testing allows one to locally generate randomness in a highly
secure way even in the presence of an intruder. This completes the subsection
on randomness and the section on technical introduction. We now move onto the
main results of this thesis.

##  Chapter 3 Certification of multipartite entangled states of arbitrary
local dimension

###  3.1 Introduction

The strongest form of device-independent certification or self-testing has
attracted considerable attention lately as it allows to obtain maximal
information up to certain equivalences about a concerned quantum system with
minimal assumptions. For instance, in Refs. [27, 28, 29, 30, 31, 32, 33, 34,
35, 36], self-testing schemes have been proposed for certification of pure
bipartite entangled states that are locally qubits. There are a few results
that allow for certification pure bipartite entangled states of local
dimension three [49, 50, 46]. A scheme for self-testing of pure bipartite
states of arbitrary local dimension was proposed in [47]. However this scheme
relies on self-testing of two-qubit states [27, 28, 29] by considering
different two-dimensional subspaces of the local state of both the parties. In
their scheme, one party has to perform three and the other has to perform four
measurements on their respective subsystems. As far as the experimental
implementation of self-testing schemes is concerned, it is thus a question of
utmost importance as to whether one can design schemes exploiting the minimal
number of measurements necessary to observe non-locality which is two per
observer.

Self-testing schemes were also designed for multipartite states, in particular
the tripartite ones, which are states shared among three parties and are
locally qubits [41, 42, 43, 44, 45]. The problem to certify states shared
among arbitrary number of parties has not been much explored with a few
exceptions [37, 38, 39] that provide a scheme to self-test $N-$qubit graph
states and the Dicke states. A scheme that applies to states of arbitrary
local dimension and shared among arbitrary number of parties was proposed in
[40]. This scheme again adapts the procedure of Ref. [47] to the multiparty
scenario that relies on the self-testing technique for the two-qubit states.
Here again, one party has to perform three and the other parties have to
perform four measurements. In Ref. [157], the authors show a proof-of-
principle demonstration that every pure entangled state can be certified.
However, in this scheme all the parties have to perform at least $d^{2}$
measurements where $d$ is the local dimension of the state. Each of the
measurements have $2^{s}$ outcomes where $s$ is chosen such that $2^{s}$ is
the smallest integer larger than $d$. All these limitations make this scheme
practically difficult to implement in experiments.

It is thus a problem of key interest to find device-independent certification
schemes that are experimentally friendly in the sense that they require
minimal resources and effort for their practical implementation. We consider
the general problem to design a scheme for self-testing of pure entangled
states of arbitrary local dimension shared among arbitrary number of parties
using Bell inequalities composed of truly $d-$outcome measurements where $d$
is any positive integer. Further, each party must perform the minimal number
of measurements possible to observe a Bell violation, which is, two.

In this Chapter, we provide a method to self-test one of most studied
multipartite states, namely the $N$-partite Greenberger–Horne–Zeilinger (GHZ)
states of local dimension $d$, or simply, generalised GHZ state

$\displaystyle{|{\text{GHZ}_{N,d}}\rangle}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}{|{i}\rangle}^{\otimes
N}$ (3.1)

with $N$ and $d$ being arbitrary integers such that $N,d\geq 2$. Notice that
the generalised GHZ state (3.1) reduces to the two-qudit maximally entangled
state (3.5) when $N=2$. For this, we utilise the Bell inequalities proposed
[53] which is a generalisation of the Bell inequalities proposed in [52]. In
this scheme, every party needs to perform only two measurements. With the view
to self-test measurements we generalise this result when each party performs
arbitrary number of measurements. The results presented below are based on our
works [4, 1].

###  3.2 Family of Bell inequalities

We consider the most general Bell scenario as depicted in Fig. 2.2. It
involves $N$ parties in spatially separated labs. Each of them receives a
subsystem from a source and performs one of $m$ $d-$outcome measurements on
it. After the experiment is complete, they combine their results to