Creation, Characterization, and Manipulation of Quantum Entanglement in a
Photonic System

Ashutosh Singh

RAMAN RESEARCH INSTITUTE - BANGALORE

A thesis submitted for the degree of
DOCTOR OF PHILOSOPHY
to
JAWAHARLAL NEHRU UNIVERSITY - NEW DELHI

Thesis submitted: 22 Jan, 2020
Copyright ©Ashutosh Singh (2020). All rights reserved.

Dedication
This thesis is dedicated to my parents Shri Om Prakash Singh and Smt. Lalsa
Singh for their endless love, unwavering support, encouragement, and ardent
faith in me. I can hardly describe my thanks and appreciation for you in
words. You have been my source of inspiration and motivation. Thank you so
much for everything!

## Declaration

The scientific work reported in this thesis is based on the research carried
out by me at Quantum Information and Computing lab, Light and Matter Physics
Group, Raman Research Institute-Bangalore, India, with the able support of my
supervisor and collaborators from India and abroad. No part of this thesis has
been submitted elsewhere for any other degree or qualification and it is all
my own work unless referenced to the contrary in the text. This thesis has
been run through the plagiarism checker software Turnitin.

Prof. Urbasi Sinha (Supervisor) |  |  |  Ashutosh Singh (Student)
---|---|---|---

## Certificate

This is to certify that the thesis titled “Creation, characterization and
manipulation of quantum entanglement in a photonic system" submitted by
Ashutosh Singh for the award of the degree of Doctor of Philosophy of
Jawaharlal Nehru University is his original work. This work has not been
submitted to any other University for the award of any other degree or
diploma.

Prof. Ravi Subrahmanyan Director and Center Chairperson Raman Research Institute, Bengaluru-560080, India. |  |  |  Prof. Urbasi Sinha Supervisor
---|---|---|---

## Acknowledgements

The research work presented in this thesis would not have been possible
without the help and support of many individuals, whom I would like to thank
here.

First and foremost, I would like to thank and convey my sincere gratitude to
Prof. Urbasi Sinha for being my Ph.D. supervisor. I thank her for giving me a
start in this field, for her guidance and for providing state-of-the-art lab
facilities at RRI. Our long insightful discussions in the weekly group
meetings and individual project meetings helped me shape my project and Ph.D.
Due credits to her mentorship, immense support whether financial or otherwise,
and the freedom that she gave me to learn and explore different avenues of
research in quantum information science and complete the thesis.

I would like to thank Prof. Ravi Subramanian (Director, RRI) and Prof. Reji
Philip for being part of my thesis advisory committee. Their comments and
suggestions in the annual reviews helped me keep my research on track and
complete it in a stipulated time. I would like to thank Prof. Dipankar Home
(Bose Institute, Kolkata) for being my external examiner in the Ph.D.
comprehensive exam. His cheerful disposition always made discussions very
pleasant. I have learned a lot from him through the collaborative work that we
did together and insightful discussions on the foundations of quantum
mechanics and quantum information during his visits to RRI. Thanks to all the
LAMP faculties who taught me various quantum optics and quantum information
courses in the first year of Ph.D. coursework. My heartfelt thanks to Prof. R.
Srikanth for stimulating and enlightening discussions.

My special thanks and sincere gratitude to Prof. A.R.P. Rau (Louisiana State
University, USA) for collaborating with us and having stimulating and
insightful discussions in the formulation of this project as well as on
quantum information during his annual visits at RRI. Thanks to him for being
an amazing host, arranging the lab visits, and showing me the places around in
Louisiana during my visit to LSU, USA in March 2019. My sincere thanks to
Prof. Aephraim M. Steinberg (University of Toronto, Canada) for sharing his
maximum likelihood estimation code for quantum state tomography.

I would like to thank Siva Pradyumna, visiting student fellow at RRI, for the
discussions we had during the formulation of the project and the collaborative
work we did together. Thanks to Mr. Ijaz Ahamad (IISER-Mohali) for the
collaborative work on the entanglement measures for two-qubit pure states.
Thanks to my labmates and friends - Debadrita, Animesh, Pradeep, Karthik,
Siva, Sudhi, Siman, Surya, Sourav, Kaushik, Prathwiraj, Nagalakshi, Rakshita,
Rishabh, Sanchari, Abhishek, Pradosh, and Karthik H S - who made my life at
RRI enjoyable. My special thanks to my labmate Surya N. Sahoo for his generous
help on various occasions and for stimulating discussions.

A token of thanks is due to V S Naresh - administrative officer, accounts
section, purchase section, Er. Mohamed Ibrahim and his team from mechanical
engineering workshop, Er. G. B. Suresh - in charge estate and buildings, Er.
Munees from the electrical dept, Jacob Rajan - IT dept, Savitha M. D. -
secretary to the LAMP dept, Vidya and Radha from RRI administration, library
staff, canteen staff, Shiva and Manju for keeping my office desk spick and
span, and everyone who helped me in some or the other ways during the course
of my Ph.D. journey. I would like to thank the vibrant sports community at RRI
which helped me de-stress and relax when felt exhausted after work and allowed
me to forget the problems and socialize with other students.

My special thanks to my uncle Prof. O.N. Singh (IIT-BHU) for his constant push
and support during my Ph.D. and my fiancée Dr. Monika Singh for her unwavering
support and encouragement during highs and lows of my life. My heartfelt
thanks to my parents and family members without whose encouragement and
support this would not have been possible. In the end, I would like to thank
the almighty for this wonderful life full of happiness and opportunities.

“Creation, characterization, and manipulation of quantum entanglement in a
photonic system"

Ashutosh Singh

A thesis submitted for the degree of Doctor of Philosophy
to
Jawaharlal Nehru University

Synopsis

Quantum superposition and entanglement are the quintessential characteristics
of quantum mechanics and play an invaluable role in most quantum information
processing protocols. In the existing literature, they have often been the
subject of intense debate due to their counter-intuitive characteristics.
While the former refers to the possibility of a quantum system simultaneously
being in two (or more) different states, the latter can be seen as the
superposition of the states of two (or more) different systems (or two
properties of the same system) in such a correlated manner that their joint
state cannot be factorised into the state of individual systems. Quantum
entanglement is a non-classical correlation shared among quantum systems,
which is now seen as an indispensable resource in quantum computation and
communication.

Technological advancement over the last few decades has witnessed a rapid
growth in research explorations in the field of quantum information science.
This is fuelled by its direct impact on the upcoming quantum technologies with
commercial applications. Exponential speed up of certain tasks in quantum
computation as well as enabling some other tasks in quantum information
processing such as teleportation, superdense coding, ensuring unconditional
security through quantum key distribution protocols, etc., are some of the
attractive advantages offered by quantum information science which are either
impossible or less efficient in the realm of classical information. Most of
these quantum advantages are manifestations of the superposition principle and
quantum entanglement which appear only in the framework of quantum mechanics.

The goal of quantum information processing is to prepare a set of qubits;
quantum mechanical two-level systems (or qudits, d-level systems), in a known
coherent superposition state and to perform arbitrary computation with them.
This requires the implementation of an arbitrary unitary matrix on the Hilbert
space of the qubits and finally qubit states are readout to get the result of
computation. In a physical system, we don’t get hold of the unitary matrix
directly, rather it is constructed through the Hamiltonian in the Hilbert
space of the qubits. If $H(t)$ is the Hamiltonian in the system Hilbert space
then corresponding unitary matrix is generated as

$U=\exp\left(-\frac{\iota}{\hbar}\int_{0}^{t}dt^{\prime}H(t^{\prime})\right),$
(1)

where $H\in H_{S}$; $H_{S}$ being the Hilbert space of the system.

Evolution of the system $\rho_{s}(0)$ is given by

$\rho_{s}(t)=U\rho_{s}(0)U^{\dagger}$ (2)

In a real physical system, however, the Hamiltonian that we engineer need not
be acting only on the system Hilbert space but also on the environment and in
that case, we get

$U=\exp\left(-\frac{\iota}{\hbar}\int_{0}^{t}dt^{\prime}[H(t^{\prime})+\delta
H(t^{\prime})]\right),$ (3)

where $H\in H_{S},\leavevmode\nobreak\ \delta H(t^{\prime})\in H_{S}\otimes
H_{E}$, $H_{E}$ is the Hilbert space of the environment.

When Hamiltonian acts on both, the system as well as the environment, it
generates correlation between them. If we look at the state of the system
alone by tracing out the environment degrees of freedom, the correlations are
lost and it appears as decoherence on the quantum system. Thus, quantum
decoherence is a ubiquitous and unavoidable phenomenon arising because of
entanglement between quantum systems and their environment.

Like classical correlations, entanglement also decays with time in the
presence of noise in the ambient environment. More precisely, the presence of
decoherence in the computing devices and communication channels, due to the
unavoidable and irreversible interaction between the system and environment,
causes the degradation in entanglement present in the system as the
computation evolves or particles propagate. The decay of entanglement depends
on the initial state as well as the amount and type of noise (amplitude
damping, phase damping, etc.) acting on the system. In the presence of
multiple stochastic noises, decoherence of individual qubits follows the
additive law of relaxation rates whereas the decay of entanglement does not.
In fact, entanglement may not decay asymptotically at all, and disentanglement
can happen in finite time. Entanglement Sudden Death (ESD) is the phenomenon
wherein a multipartite entangled state disentangles in finite time even when
individual qubits decohere only asymptotically in time due to noise. ESD has
been experimentally demonstrated in atomic and photonic systems.

Since quantum entanglement is an indispensable resource in quantum information
processing, manipulation that prolongs entanglement will help realize
protocols that would otherwise suffer due to short entanglement decay times.
Entanglement purification and distillation schemes could possibly recover the
initial correlation from the ensemble of noise-degraded correlated states so
long as the system has not completely disentangled. Therefore, delay or
avoidance of ESD is important. For this purpose, local NOT operation in the
computational basis on one or both qubits ($\sigma_{x}\otimes\mathbf{I}$,
$\mathbf{I}\otimes\sigma_{x}$, or $\sigma_{x}\otimes\sigma_{x}$) has been
proposed to combat the amplitude damping noise. Here, we report an all-optical
implementation of the NOT operations; a theoretical proposal as well as the
experimental demonstration, that can hasten, delay, or avert ESD, all
depending on when it is applied during the process of decoherence for the
polarization entangled photonic qubit system. We give analytical expressions
for the probabilities corresponding to the hastening, delay, and avoidance of
ESD which depend only on the parameters of the initial state density matrix of
the system. These results are then experimentally verified in a suitably
planned experiment.

Our proposal has an advantage over other known decoherence suppression schemes
such as decoherence suppression using weak measurement and quantum measurement
reversal, and delayed-choice decoherence suppression. There, as the strength
of weak interaction increases, the success probability of decoherence
suppression decreases. In our scheme, however, we can manipulate the ESD, in
principle, with unit success probability as long as we perform the NOT
operation at the appropriate time. Delay and avoidance of ESD, in particular,
will find applications in the practical realization of quantum information and
computation protocols that might otherwise suffer due to a short lifetime of
entanglement. Also, it will have implications towards such control in other
physical systems. A major advantage of the manipulation of ESD in a photonic
system is that one has complete control over the damping parameters, unlike in
most atomic systems, and hence provides a perfect test bed for such a study.
The experiment that I report in this thesis is important for practical noise
engineering in quantum information processing. Now, I will outline the
structure of my thesis.

In this thesis, I will present our work on creation and characterization of
quantum entanglement in a photonic system, and manipulation of its dynamics in
the presence of an amplitude damping channel. When a two-qubit entangled
system passes through an amplitude damping channel, some initial entangled
states follow the asymptotic decay of entanglement whereas others undergo the
sudden death of entanglement. The class of states which undergo ESD are then
manipulated by local NOT operations ($\sigma_{x}$) on one or both qubits. This
thesis is divided into seven chapters as given below.

In chapter one, I discuss the notations and basic concepts of quantum
information that shall be helpful in understanding the rest of the thesis. An
introduction to quantum entanglement along with the entanglement measures is
given. Next, evolution equation of the open quantum system is discussed using
the Kraus operator formalism. Then the basics of non-linear optics such as
birefringence, second order non-linearity, phase matching condition, and
spontaneous parametric down-conversion are presented.

In chapter two, I discuss the theoretical framework for, and experimental
implementation of a type-I polarization entangled photon source based on
spontaneous parametric down conversion. The entangled state is characterized
by Quantum State Tomography (QST) and Maximum Likelihood Estimation (MLE). The
concept of QST is discussed and it is motivated as to how a density matrix
reconstructed by linear inversion of the QST data may not satisfy the
requirements of a physical state; i.e., normalization, positivity, and
hermiticity. The process of obtaining a physical state from the above density
matrix using MLE, is discussed.

In chapter three, I address the issue of quantitative non-equivalence of
different entanglement measures for two-qubit pure states. Given a non-
maximally entangled state, an operationally significant question is to
quantitatively assess as to what extent the state is away from the maximally
entangled state, which is of importance in evaluating the efficacy of the
state for its various uses as a resource. It is this question which is
examined in this paper for two-qubit pure entangled states in terms of
different entanglement measures like Negativity (N), Logarithmic Negativity
(LN), and Entanglement of Formation (EOF). Although these entanglement
measures are defined differently, to what extent they differ in quantitatively
addressing the earlier mentioned question has remained uninvestigated.
Theoretical estimate in this paper shows that an appropriately defined
parameter characterizing the fractional deviation of any given entangled state
from the maximally entangled state in terms of N is quite different from that
computed in terms of EOF with their values differing up to $\sim 15\%$ for
states further away from the maximally entangled state. Similarly, the values
of such fractional deviation parameters estimated using the entanglement
measures LN and EOF, respectively, also strikingly differ among themselves
with the maximum value of this difference being around $23\%$. This analysis
is complemented by illustration of these differences in terms of empirical
results obtained from a suitably planned experimental study. Thus, such
appreciable amount of quantitative non-equivalence between the entanglement
measures in addressing the experimentally relevant question considered in the
present paper highlights the requirement of an appropriate quantifier for such
intent. We indicate directions of study that can be explored towards finding
such a quantifier.

In chapter four, I present the theoretical proposal for the manipulation of
entanglement sudden death in an all optical set up. Evolution of a two-qubit
entangled state is studied in the presence of an Amplitude Damping Channel
(ADC). Two levels of the qubits are realized by the H- and V-polarization
states of the photon and ADC is mimicked by the action of a half-wave plate
(HWP) acting only on the V-polarization component of the photon in a displaced
Sagnac interefrometer (DSI). As a result, polarization and momentum modes of
the photon get entangled. When momentum modes of the photon are traced out, it
acts as ADC. We see that the entangled state
$|\psi\rangle=\alpha|HH\rangle+\beta|VV\rangle$ undergoes ESD for
$|\alpha|<|\beta|$ and asymptotic decay for $|\alpha|>|\beta|$ in the presence
of an ADC. To manipulate the ESD, we again use a $\text{HWP}@45^{o}$ but
acting on both H- as well as V-polarization components and hence mimics NOT
($\sigma_{x}$) operation. The complete theoretical analysis of the experiment
is given and results of the manipulation are compared with the ESD results. We
see the phenomenon of hastening, delay, and avoidance of ESD for either cases
when NOT operation is applied on only one or both the qubits.

In chapter five, I present our experimental results on ESD followed by its
manipulation in the photonic system. This requires preparation and
characterization of a non-maximally entangled state. These entangled photons
are then passed through the DSI. ADC is artificially introduced using a HWP
acting only on the vertical polarization component of the entangled state in
the DSI. Initially, ESD experiment is performed by setting the ADC HWPs at
non-zero angles and then reconstructing the state by QST and MLE for each
setting and entanglement is computed from the density matrix. Next, ESD
manipulation experiment is set up and for each setting of the ADC HWP, NOT
operation is performed and output modes are appropriately traced out.
Resulting state is reconstructed using QST and MLE; and then entanglement is
computed from the density matrix. The results of ESD and manipulation
experiments are compared with the simulation results.

In chapter six, I extend the theoretical proposal for manipulation of ESD in a
$2\otimes 2$ system to $2\otimes 3$ and $3\otimes 3$ entangled system. We find
that these higher dimensional systems can also be manipulated similar to the
$2\otimes 2$ systems and the phenomenon of hastening, delay, and avoidance of
ESD also occur as seen in $2\otimes 2$ systems. Our study, thus, indicates the
possibility of controlled manipulation of higher dimensional entangled quantum
systems evolving in a noisy environment.

In chapter seven, I conclude with the summary of the thesis followed by future
scope of work.

Prof. Urbasi Sinha (Supervisor) |  |  |  Ashutosh Singh (Student)
---|---|---|---

## List of publications

  1. 1.

Ashutosh Singh, Siva Pradyumna, A. R. P. Rau, and Urbasi Sinha, “Manipulation
of entanglement sudden death in an all-optical setup," J. Opt. Soc. Am. B 34,
681-690 (2017).

  2. 2.

Urbasi Sinha, Surya Narayan Sahoo, Ashutosh Singh, Kaushik Joarder, Rishab
Chatterjee, Sanchari Chakraborti, Invited review article: “Single-Photon
Sources", Opt. Photon. News 30 (9), 32-39 (2019).
ArXiv version: “Single photon sources: ubiquitous tools in quantum information
processing," https://arxiv.org/abs/1906.09565 (2019).

  3. 3.

Ashutosh Singh, Ijaz Ahamed, Dipankar Home, and Urbasi Sinha, “Revisiting
comparison between entanglement measures for two-qubit pure states," J. Opt.
Soc. Am. B 37, 157-166 (2020).

  4. 4.

Ashutosh Singh and Urbasi Sinha, “Entanglement protection in higher-
dimensional systems", Phys. Scr. 97, 085104 (2022).

Prof. Urbasi Sinha (Supervisor) |  |  |  Ashutosh Singh (Student)
---|---|---|---

###### Contents

  1. Declaration
  2. Certificate
  3. Acknowledgements
  4. 1 Introduction and basics
    1. 1.1 Motivation
    2. 1.2 Outline of the thesis
    3. 1.3 Basics of quantum mechanics and quantum information
      1. 1.3.1 Concept of state vector and density matrix
      2. 1.3.2 Pure and mixed states
      3. 1.3.3 Bit and Qubit
      4. 1.3.4 Partial trace and reduced density matrix
      5. 1.3.5 Partial transpose
      6. 1.3.6 Fidelity
    4. 1.4 Quantum entanglement
      1. 1.4.1 Separable and entangled states
      2. 1.4.2 Basic requirements for an entanglement measure
      3. 1.4.3 Measures of entanglement
    5. 1.5 Time evolution of the quantum systems
      1. 1.5.1 Evolution of the closed quantum systems
      2. 1.5.2 Evolution of the open quantum systems
    6. 1.6 Basics of birefringence and nonlinear optics
      1. 1.6.1 Anisotropy and birefringence
      2. 1.6.2 Non-linearity
    7. 1.7 Spontaneous Parametric Down-Conversion
      1. 1.7.1 Phase matching technique
    8. 1.8 References
  5. 2 Preparation and characterization of polarization entangled photon source
    1. 2.1 Quantum state tomography (QST)
      1. 2.1.1 Stokes parameter representation of polarized light
      2. 2.1.2 Single-qubit tomography
      3. 2.1.3 Maximum likelihood estimation
      4. 2.1.4 Two-qubit tomography
    2. 2.2 Preparation of the entangled photon source
    3. 2.3 Decoherence due to walk-off
      1. 2.3.1 Temporal walk-off
      2. 2.3.2 Spatial walk-off
    4. 2.4 Temporal walk-off pre-compensation
      1. 2.4.1 Partial temporal compensation
      2. 2.4.2 Complete temporal compensation
    5. 2.5 Effect of spectral filtering on entanglement
    6. 2.6 Visibility of entanglement
    7. 2.7 References
  6. 3 Revisiting comparison between entanglement measures
    1. 3.1 Introduction: Background and Motivation
    2. 3.2 Theoretical study of the deviation of any given state from the MES using different EMs
    3. 3.3 Deviation of the experimental states from MES using different EMs
    4. 3.4 Extending the analysis to higher dimensional systems
      1. 3.4.1 Bipartite pure qutrit states
      2. 3.4.2 Tripartite pure qubit states
    5. 3.5 Outlook
    6. 3.6 Concluding Remarks
    7. 3.7 References
  7. 4 Manipulation of entanglement sudden death in an all-optical setup
    1. 4.1 Introduction
    2. 4.2 Experimental set up for ESD and its analysis
    3. 4.3 Manipulation of ESD using the NOT operation on both qubits
    4. 4.4 Effect of the NOT operation applied on only one of the qubits
    5. 4.5 Some analytical expressions
    6. 4.6 Results and Discussion
    7. 4.7 Conclusions and future work
    8. 4.8 An alternative approach to analyze the ESD and manipulation experiments
      1. 4.8.1 Analytical expressions for X-state with non-zero entries in the other diagonal terms
      2. 4.8.2 Analytical expressions for the X-state in Reference [35]
    9. 4.9 References
  8. 5 Experimental demonstration of ESD and its manipulation
    1. 5.1 Introduction
    2. 5.2 Setting up the ESD experiment
      1. 5.2.1 ESD with one HWP vs. two HWPs
      2. 5.2.2 Alignment of DSIs using He-Ne Laser and coincidence optimization
      3. 5.2.3 Coincidence optimization and QST
      4. 5.2.4 Instability and drift in the DSI
    3. 5.3 ESD experiment and results
      1. 5.3.1 Results
    4. 5.4 ADE experiment and results
      1. 5.4.1 Results
    5. 5.5 Attempts towards manipulation of ESD using NOT operation on both the qubits
      1. 5.5.1 Setting up the ESD manipulation experiment
      2. 5.5.2 Aligning ESD manipulation setup using He-Ne laser
      3. 5.5.3 Spatial mode tracing and constraints
      4. 5.5.4 Coincidence optimization and QST
      5. 5.5.5 Spectral filtering consideration
      6. 5.5.6 Current status and way forward
    6. 5.6 References
  9. 6 Entanglement protection in higher-dimensional systems
    1. 6.1 Introduction
    2. 6.2 Physical Model
    3. 6.3 X-type qubit-qutrit entangled system: State-I
      1. 6.3.1 ESD in the presence of ADC
      2. 6.3.2 $\sigma_{x}$ applied to qubit and $F_{01}$ applied to qutrit
      3. 6.3.3 $F_{01}$ applied to qutrit only
    4. 6.4 X-type qubit-qutrit entangled system: State-II
      1. 6.4.1 $\sigma_{x}$ applied to qubit and $F_{01}$ applied to qutrit
      2. 6.4.2 $\sigma_{x}$ operation applied to qubit only
      3. 6.4.3 $F_{01}$ applied to qutrit only
      4. 6.4.4 $F_{02}$ applied to qutrit only
      5. 6.4.5 $F_{201}$ applied to qutrit only
    5. 6.5 Generalization to qutrit-qutrit system
    6. 6.6 Summary and discussion
    7. 6.7 References
  10. 7 Summary and future scope
  11. A Availability of analysis code files and experimental data
  12. B List of Abbreviations

###### List of Figures

  1. 1.1 Difference between a classical bit and qubit illustrated using a switch [Image credit: Ref. 12]. A classical bit always remains in one of the two possible orthogonal states (‘0’, or ‘1’) whereas a qubit can exist in a coherent superposition of the two orthogonal states as well.
  2. 1.2 Poincare sphere representation of a qubit. All points at the suraface of Poincare sphere represent pure states whereas those lying inside the spehere represent mixed state. Point at the center of the sphere is maximally mixed.
  3. 1.3 Schematic diagram of an open quantum system.
  4. 1.4 Non-collinear phase matching condition for the SPDC process where a pump photon having energy $\hbar\omega_{p}$ gets absorbed and two lower energy photons $\hbar\omega_{s}$ (signal) and $\hbar\omega_{i}$ (idler) get emitted. The $\mathbf{k_{p},\leavevmode\nobreak\ k_{s}}$ and $\mathbf{k_{i}}$ are the momentum vectors of the pump, signal, and idler, respectively.
  5. 1.5 Schemtaic of a type-I non-collinear SPDC process resulting in concentric cones. A pair of photons collected from diamtrically opposite sides of the ring will be polarizatiion entangled. (Image Credit: S. N. Sahoo)
  6. 1.6 Type-I SPDC ring in collinear (left) and non-collinear (right) configurations imaged by Andor CMOS camera.
  7. 1.7 Schematic of type-II non-collinear SPDC process resulting in intersecting cones [27].
  8. 1.8 Type-II SPDC ring in collinear (left) and non-collinear (right) configurations imaged by Andor CMOS camera. In the overlapping region of the two rings, one cannot comment whether which ring the photon belongs to, hence a pair of photons collected from these regions will be polarization entangled.
  9. 1.9 Wave surfaces for a negative uniaxial crystal. The blue curves show the wave surfaces at fundamental frequency and red curves show the wave surfaces for down-converted frequency. The unit vector $\hat{i}$ points in the direction of index matched wave normals, and $\theta_{pm}$ is the phase matching angle.
  10. 2.1 Poincare sphere representation of polarization state of light depicting the stokes parameter [Image credit: Ref. 2].
  11. 2.2 Schematic of the experimental set up for quantum state tomography of a single qubit.
  12. 2.3 Three digrams depict measurement along three linearly independent bases ($|R\rangle,\leavevmode\nobreak\ |D\rangle,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ |H\rangle$ from left to right) which locate the state of single qubit (represented by as an open circle in the Poincare sphere) in the Hilbert space [Image credit: Ref. 5].
  13. 2.4 Schematic of the experimental set up for quantum state tomography of two-qubit state.
  14. 2.5 Schematic of the experimental apparatus (not to scale) for preparation of SPDC based type-I polarization entangled photon source using two crystal geometry and characterization using quantum state tomography. Different symbols have the following meaning: P, polarizing beam splitter; Q, quarter wave plate; H, half wave plate; NLC, non-linear crystal; TC, temporal compensator; L, plano-convex lens; M, mirror; BPF, bandpass filter; AL, aspheric lens, SMF, single mod fiber; SPAD, single-photon avalanche diode; and TT, time tagger unit or coincidence module.
  15. 2.6 SPDC ring of type-I polarization entangled photon source imaged by Andor CMOS camera. Photon at the diametrically opposite sides of the cone are entangled.
  16. 2.7 The two-qubit entangled state reconstructed through QST. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state had $81.4\%$ fidelity with the ideal state, concurrence = 0.669, and purity = 0.709.
  17. 2.8 Entangled photon generation and propagation inside the crystals. For clarity, only signal photon trajectories are drawn. The “o" and “e" indicate ordinary and extraordinary rays, respectively; $L_{c}$ is the length of each crystal [Image credit: Ref. 14].
  18. 2.9 The two-qubit entangled state reconstructed through QST with partial temporal compensation. The left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $86.97\%$ with the ideal state, concurrence = 0.764, and purity = 0.776.
  19. 2.10 The two-qubit entangled state reconstructed through QST with partial temporal compensation. The left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $93.46\%$ with the ideal state, concurrence = 0.909 and purity = 0.891.
  20. 2.11 The two-qubit entangled state reconstructed through QST with partial temporal compensation. The left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $95.03\%$ with the ideal state, concurrence = 0.717 and purity = 0.929.
  21. 2.12 The two-qubit entangled state reconstructed through QST with perfect temporal compensation and narrow spectral filtering using 810-10 nm band pass filters for pump HWP at $23.1^{\circ}$. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $97.86\%$ with the ideal state.
  22. 2.13 The two-qubit entangled state reconstructed through QST with nearly perfect temporal precompensation and narrow spectral filtering using 810-10 nm band pass filters for pump HWP at $13.1^{\circ}$. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $97.88\%$ with the ideal state.
  23. 2.14 Result of the entanglement visibility measurement in $|HH\rangle,\leavevmode\nobreak\ |VV\rangle,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ |LL\rangle$ bases. Note that when this data was taken, soure fidelity was about $\sim 88\%$.
  24. 3.1 A comparison of different entanglement measures with respect to the state parameter $c_{0}$ for two-qubit pure states. Here, C, N, LN, EOF, and EE denote Concurrence, Negativity, Logarithmic Negativity, Entanglement of Formation, and Entanglement Entropy, respectively.
  25. 3.2 This figure illustrates the variation of different entanglement measures with respect to the state parameter $c_{0}$.
  26. 3.3 This figure shows how the fractional deviations of a given state from the maximally entangled state calculated using different entanglement measures N, LN and EOF vary with respect to the state parameter $c_{0}$.
  27. 3.4  This figure illustrates the differences in the fractional deviations of the respective states from the maximally entangled state calculated using different entanglement measures N, LN and EOF by varying the state parameter $c_{0}$.
  28. 3.5 This figure shows how the absolute values of the derivatives of different entanglement measures; 2N, LN and EOF with respect to state parameter $c_{0}$, vary with $c_{0}$.
  29. 3.6 Schematic of the experimental apparatus (not to scale) for preparation of SPDC based type-I polarization entangled photon source using two-crystal geometry and characterization using quantum state tomography. Different symbols have the following meaning: P, polarizing beam splitter; Q, quarter wave plate; H, half wave plate; NLC, non-linear crystal; TC, temporal compensator; L, plano-convex lens; M, mirror; BPF, bandpass filter; AL, aspheric lens; SMF, single mode fiber; SPAD, single-photon avalanche diode; and TT, time tagger unit or coincidence module.
  30. 3.7 Representative 3D plot of the experimentally reconstructed density matrix for state-I with P=0.966 and 2N=0.964.
  31. 3.8 Representative 3D plot of the experimentally reconstructed density matrix for state-II with P=0.962 and 2N=0.759.
  32. 3.9 Representative 3D plot of the experimentally reconstructed density matrix for state-III with P=0.960 and 2N=0.544.
  33. 3.10  This figure illustrates that the curve for the distance measure $D$ provides an upper bound to the fractional deviation curves for N and LN, but not for EOF, since there are certain states for which the fractional deviation curve for EOF is not upper bounded by the curve related to $D$.
  34. 3.11 This figure shows that the fractional deviation curve for the quantity $B_{V}$ provides an upper bound to the other fractional deviation curves corresponding to the quantities N, LN and EOF, signifying that in terms of the amount of Bell-CHSH violation, any given state is further away from the maximally entangled state than that estimated using different entanglement measures.
  35. 4.1 The proposed experimental set up for ESD in the presence of ADC in a photonic system.
  36. 4.2 The proposed experimental set up for manipulation of ESD involving NOT operation on both the qubits of a bipartite entangled state in the presence of ADC in a photonic system.
  37. 4.3 Plot of Negativity N vs. ADC probability ($p,p^{\prime}$) for ESD using two HWPs for a bipartite entangled state. The plot implies ESD for $p=0,\leavevmode\nobreak\ p^{\prime}=0.5$ and $p^{\prime}=0,\leavevmode\nobreak\ p=0.5$. For intermediate values of $p\leavevmode\nobreak\ (p^{\prime})$, the curvature reflects the non-linear relation between them as per (17), the action of two HWPs oriented at $\theta$ and $\theta^{\prime}$ applied one after another is not equivalent to that of one HWP oriented at $\theta+\theta^{\prime}$.
  38. 4.4 Plot of Purity P vs. ADC probability ($p,p^{\prime}$) for ESD using two HWPs for a bipartite entangled state. The state (4.7) is initially pure, gets mixed at intermediate ADC probabilities and becomes pure again at $p=1$ or $p^{\prime}=1$.
  39. 4.5 Plot of Negativity N vs. ADC probability ($p_{n},p^{\prime}$) such that NOT operation is applied on both the qubits at $p=p_{n}$ for manipulation of ESD of a bipartite entangled state. The NOT operation leads to hastening for $0.375<p_{n}<0.5$, delay for $0.1667<p_{n}<0.375$, and avoidance of ESD for $0\leq p_{n}\leq 0.1667$.
  40. 4.6 Plot of Negativity N vs. ADC probability ($p_{n},p^{\prime}$) such that NOT operation is applied on only one of the qubits at $p=p_{n}$ for manipulation of ESD of a bipartite entangled state. The NOT operation leads to hastening for $0.4<p_{n}<0.5$, delay for $0.1667<p_{n}<0.4$, and avoidance of ESD for $0\leq p_{n}\leq 0.1667$.
  41. 4.7  Plot shows the effect of NOT operation on manipulation of ESD at various $p_{n}$ values for density matrix parameters $u=0.2,|v|=0.4$. The plot for ESD, Eq. (4.17) and a rendering of (4.18), for every $p$, giving the value of $p^{\prime}_{0}$, the compounding of them giving the flat line $p_{end}=0.5$ are shown by dotted red curve and dot-dashed red line. The plot for manipulation of ESD using NOT on both or only one qubit are shown by dashed and solid blue curves, respectively. For $0\leq p_{n}\leq 0.1667$, $p_{end}=1$ means avoidance of ESD, the $p_{B}$ value for single and double NOT coinciding for these particular parameters but $p_{B}$ is in general different. For $0.1667<p_{n}<0.4$ ( $0.1667<p_{n}<0.375$) the dashed (solid) blue curve lies above 0.5, implies delays of ESD for double (single) NOT manipulation of ESD. The dashed (solid) blue curve dipping below 0.5 for $0.375<p_{n}<0.5$ ($0.4<p_{n}<0.5$) for manipulation using double (single) NOT operation implies hastening of ESD in this range.
  42. 4.8 Plot of $p_{end}$ vs. $p_{n}$ for manipulation of ESD using NOT operation on one (solid blue curve) or both (dashed blue curve) the qubits of a bipartite entangled state in the presence of ADC for density matrix parameters $u=0.14,|v|=0.347$. The NOT operation leads to avoidance for $0\leq p_{n}\leq 0.1228$ ($0\leq p_{n}\leq 0.1715$) with disentanglement happening at $p_{end}=1$ in this range, and delay of ESD for $0.1228<p_{n}<0.4035$ ($0.1715<p_{n}<0.4035$) as $p_{end}$ curve lies above the $p_{end}=0.4035$ dotted red ESD line for single (double) NOT operation. There is no hastening of ESD for this particular choice of parameters for single or double NOT operation.
  43. 4.9 Plot of $p_{end}$ vs. $p_{n}$ for manipulation of ESD using NOT operation on one (solid blue curve) or both (dashed blue curve) the qubits of a bipartite entangled state in the presence of ADC for density matrix parameters $u=0.2,|v|=0.15$. The NOT operation when applied on only one of the qubits leads to avoidance of ESD for $0<p_{n}<0.0274$ with disentanglement happening at $p_{end}=1$ in this range, and delay for $0.0274<p_{n}<0.1875$ as the solid blue curve lies above the $p_{end}=0.1875$ dotted red ESD line. The NOT operation when applied on both the qubits leads to delay for $0<p_{n}<0.1875$ as the dashed blue curve lies above the $p_{end}=0.1875$ dotted red ESD line. There is no hastening of ESD for single or double NOT operation and no avoidance of ESD for double NOT operation for this particular choice of parameters.
  44. 5.1 Experimental setup for demonstrating ESD in the photonic system [1]. Polarization entangled photons are sent to two DSI where a half-waveplate ($HWP1\leavevmode\nobreak\ \text{at}\leavevmode\nobreak\ \theta$) acts on $|V\rangle$ polarization only. Upon tracing out the spatial modes; mode $|a\rangle$ (transmitted, top) and $|b\rangle$(reflected, bottom), of the interferometer at PBS, the HWP action mimics ADC.
  45. 5.2 A piture of the experimental setup for demonstrating ESD in the photonic system captured from our laboratory.
  46. 5.3 Overlap of the clockwise and counter-clockwise propagating Gaussian beams at the output port (spatial mode $|a\rangle$) of the DSI in collinear configuration captured by DataRay beam profiler.
  47. 5.4 Interference due to clockwise and counter-clockwise propagating Gaussian beams on a polarizer at $45^{\circ}$ at the output port $|a\rangle$ of the DSI in non-collinear configuration captured by DataRay beam profiler.
  48. 5.5 The two-qubit entangled state reconstructed through QST with perfect temporal compensation and narrow spectral filtering using 810-10 nm band pass filters for pump HWP at $22.5^{\circ}$. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $95.50\%$ with the ideal state.
  49. 5.6 The two-qubit product state $|HH\rangle$ reconstructed through QST with narrow spectral filtering using 810-10 nm band pass filters for pump HWP at $45^{\circ}$. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state has a fidelity of $99.67\%$ with the ideal state.
  50. 5.7 Coincidence measured in $|VV\rangle\langle VV|$ bases vs. time (100s iterations) for a maximally entangled state input to DSIs. It clearly shows short time scale oscillation and long term drift.
  51. 5.8 Coincidence measured in $|LL\rangle\langle LL|$ bases vs. time (100s iteration) for maximally entangled state input to the DSIs. It shows both both short time scale as well as long time scale oscillations.
  52. 5.9 The horizontal axis denote the number of 100s iterations for which data was recorded and vertical axis denotes corresponding changes in singles counts on the left side, right side, and the coincidence counts per iteration.
  53. 5.10 The plot of PCC vs. time delay. Since each coincidence count measurement as well as temperature data was recorded in the steps of 100 s , hence PCC is calculated at the interval of 100 s.
  54. 5.11 The two-qubit entangled state reconstructed through QST with nearly perfect temporal compensation and spectral filtering using 810-10 nm band pass filters for pump HWP at $13.1^{\circ}$. The left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state had a fidelity of $97.88\%$ with the ideal state.
  55. 5.12 Plot of Concurrence vs. ADC probability $p$ shows the pheonomenon of ESD. The initial state $|\psi_{i}\rangle=\alpha|HH\rangle+\beta|VV\rangle$ with $|\beta|=4|\alpha|$ undergoes ESD (solid blue shaded line) as concurrence drops to zero at $p=0.48$. The red dots indicate experimental data points and vertical bars represent error in state reconstruction due to statistical fluctuation in the counts.
  56. 5.13 Plot of purity vs. ADC probability p for the ESD experiment. The average purity of the initial state $|\psi_{i}\rangle=\alpha|HH\rangle+\beta|VV\rangle$ with $|\beta|=4.12|\alpha|$ drops from 0.959(3) to a minimum value of 0.353(4) at the time of ESD and then starts increasing again after this point. The red dots indicate experimental data points and vertical bars represent error in state reconstruction due to statistical fluctuation in the counts.
  57. 5.14 The two-qubit entangled state reconstructed through QST with perfect temporal compensation and narrow spectral filtering using 810-10 nm band pass filters for pump HWP at $13.1^{\circ}$. Left (right) figure shows the real (imaginary) part of the density matrix. The reconstructed state had a fidelity of $98.71\%$ with the ideal state.
  58. 5.15 Plot of concurrence vs. ADC probability p for the ADE experiment. The state $|\psi_{i}\rangle=\alpha|HH\rangle+\beta|VV\rangle$ with $|\alpha|=4.12|\beta|$ undergoes ADE (solid blue shaded curve) as concurrence drops to zero only at $p=1$. The red dots indicate experimental data points and vertical bars represent error in state reconstruction due to statistical fluctuation in the counts.
  59. 5.16 Plot of purity vs. ADC probability p for ADE experiment. The average purity of the initial state: $|\psi_{i}\rangle=\alpha|HH\rangle+\beta|VV\rangle$ with $|\alpha|=4.12|\beta|$, drops from 0.97(1) to a minimum value of 0.80(1) and then starts increasing again and finally end up as a pure state when the joint state of the two-qubits becomes $|HH\rangle$. The red dots indicate experimental data points and vertical bars represent error in state reconstruction due to statistical fluctuation in the counts.
  60. 5.17 Experimental setup for demonstrating manipulation of ESD using NOT operation on both the qubits of a bipartite entangled state in the presence of an ADC in a photonic system [2]. Entangled photons are prepared on the left side which are then fed into two DSIs (top and bottom) where ADC is mimicked using HWPs acting on V-polarization component and NOT operations are applied by a HWP at $45^{\circ}$ on both H- as well as V-polarization components for the manipulation of ESD and system is evolved in the ADC afterwards. In the end, output spatial modes are traced out and QST is performed (right side).
  61. 5.18 QST result with ESD manipulation setup for ADC settings (0,0,0).
  62. 5.19 QST results for pump HWP at $22.5^{\circ}$ with 810-03nm BPF in the $L_{1}R_{1}$ couplers for ADC HWPs at (0,0,0) in the DSIs.
  63. 6.1 Two-level atom (qubit), and a three-level atom (qutrit) in V-configuration. The parameters $p$, $p_{1}$, and $p_{2}$ denote the decay probabilities between the levels $|1\rangle_{S}\rightarrow|0\rangle_{S}$ of the qubit, and $|1\rangle_{S}\rightarrow|0\rangle_{S}$, and $|2\rangle_{S}\rightarrow|0\rangle_{S}$ levels of the qutrit, respectively. The qutrit is taken to be in V-configuration such that the dipole transitions are allowed only between the levels $|1\rangle_{S}\leftrightarrow|0\rangle_{S}$ and $|2\rangle_{S}\leftrightarrow|0\rangle_{S}$.
  64. 6.2 Scheme for prolonging the entanglement in the presence of ADC. Initial state $\rho_{i}$ undergoes ADC decoherence then NOT operations are performed at $t=t_{n}$ and system is left to evolve in the ADC. The clock measuring the time is reset at $t=t_{n}$ and time after the application of NOT operation is measured by $t^{\prime}=t-t_{n}$. Such a splitting of time is essential for later discussion.
  65. 6.3 Plot of Negativity vs. ADC probability for $0\leq x<1/3$ for the entangled state (6.11). The system undergoes ADE for $0\leq x\leq 0.20$, and ESD for $0.20<x<1/3$.
  66. 6.4 Plot of Negativity vs. ADC probability ($p,p^{\prime}$) for $x=0.25$ for the state (6.11). It undergoes ESD at $p=0.6168$ for $p^{\prime}=0$ and vice-versa. For a non-zero $p$, ESD occurs along the curved path in the ($p,p^{\prime}$) plane.
  67. 6.5 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ for ESD (red curve) and its manipulation (green curve) such that the NOT operation ($\sigma_{x}$) is applied on the qubit and trit-flip operation $\mathbb{F}_{01}$ applied on the qutrit at $p=p_{n}$ for $x=0.25$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.0615$, delay for $0.0615<p_{n}<0.1641$, and hastening of ESD for $0.1641<p_{n}<0.6168$ as the green curve lies below red curve in this range.
  68. 6.6 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in ESD (red curve) and its manipulation (green curve) such that trit-flip operation $\mathbb{F}_{01}$ is applied on the qutrit at $p=p_{n}$ for $x=0.25$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.2941$, delay for $0.2941<p_{n}<0.6168$, and hastening of ESD does not occur is this case.
  69. 6.7 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ for ESD (red curve) and its manipulation (green curve) such that the NOT operation ($\sigma_{x}$) is applied on the qubit and trit-flip $\mathbb{F}_{01}$ applied on the qutrit at $p=p_{n}$ for $x=0.5$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.3586$, delay for $0.3586<p_{n}<0.4177$, and hastening of ESD for $0.4177<p_{n}<0.8452$ as green curve lies below the red curve in this range.
  70. 6.8 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in ESD (red curve) and its manipulation (green curve) such that NOT operation is applied only on the qubit at $p=p_{n}$ for $x=0.5$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.2309$, delay for $0.2309<p_{n}<0.2964$, and hastening of ESD for $0.2964<p_{n}<0.8452$ as green curve lies below the red curve in this range.
  71. 6.9 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation $\mathbb{F}_{01}$ is applied on the qutrit at $p=p_{n}$ for $x=0.5$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.7143$, and delay of ESD for $0.7143<p_{n}<0.8452$ as green curve lies above the red curve but less than one in this range. The hastening of ESD does not occur in this case.
  72. 6.10 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation $\mathbb{F}_{02}$ is applied on the qutrit at $p=p_{n}$ for $x=0.5$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.2032$, delay for $0.2032<p_{n}<0.2693$, and hastening of ESD for $0.2693<p_{n}<0.8452$ as green curve lies below red curve in this range.
  73. 6.11 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in ESD (red curve) and its manipulation (green curve) such that only trit-flip operation ($\mathbb{F}_{201}$) is applied on the qutrit at $p=p_{n}$ for $x=0.5$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.2059$, delay for $0.2059<p_{n}<0.2676$, and hastening of ESD for $0.2676<p_{n}<0.8452$ as green curve lies below the red curve in this range.
  74. 6.12 Plot shows the non-linear curvature in $p^{\prime}$ vs. $p\leavevmode\nobreak\ or\leavevmode\nobreak\ p_{n}$ in NSD (red curve) and its manipulation (green curve) such that the trit-flip operation ($\mathbb{F}_{01}$) is applied on only one of the qutrits at $p=p_{n}$ for $x=0.25$. The action of LUOs give rise to avoidance for $0\leq p_{n}\leq 0.0238$, delay for $0.0238<p_{n}<0.3636$, and hastening of NSD does not occur in this case.

###### List of Tables

  1. 2.1 QST settings for projection onto different bases for single qubit.
  2. 2.2 QST settings for projection onto different bases for two-qubit system.
  3. 3.1 Differences between the respective fractional deviation parameters for different EMs for two-qubit pure states given as percentage values.
  4. 3.2 Comparison tables for the properties of experimentally prepared two-qubit entangled states and its deviation from the intended maximally entangled state. Different Q-values are reported as percentage quantity.
  5. 3.3 Differences in the percentage deviations of entanglement measures of a given state from the value corresponding to the maximally entangled two-qutrit pure state.
  6. 5.1 This table summarizes the results of different ADC settings on the output mode population in the ESD-manipulation experiment when NOT operation is applied on both the qubits.
  7. 5.2 This table summarizes the results of different ADC settings on the input polarization in ESD-manipulation experiment when NOT operation is applied on both the qubits. Out of these only five settings (S.N. 1, 2/3/5, 4, 7, and 8/6 ) give different results.
  8. 6.1 Different combination of local unitary operations applied on the initial state-I (Eq. 6.11) and state-II (Eq. 6.16) resulting in avoidance (A), delay (D), and hastening (H) of ESD. Due to symmetry in the population of initial entangled state, LUOs $\sigma_{x}\otimes\mathbb{F}_{102}$ and $\mathbb{I}_{2}\otimes\mathbb{F}_{102}$ applied on the either states lead to same effect as $\sigma_{x}\otimes\mathbb{F}_{01}$ and $\mathbb{I}_{2}\otimes\mathbb{F}_{01}$, respectively

##  Chapter 1 Introduction and basics

###  1.1 Motivation

Quantum Information Processing (QIP) exploits the fundamentals laws of quantum
mechanics for information processing, communication, computation, metrology,
and sensing [1, 3]. The fundamental difference which makes QIP more efficient,
attractive, and fascinating is its ability to use entangled states and the
information processing of the more general input states in the form of
superposition of different possible eigenstates of a quantum system as opposed
to the information processing via classical systems which can handle only
bits. Moreover, two distant parties which may be space-like separated can
share arbitrary and unknown quantum information without physically sending the
quantum system in which the information is stored using a prior shared
entanglement between them [1, 3]. Further, no-cloning theorem and uncertainty
principle ensures that arbitrary and unknown quantum information can neither
be copied nor eavesdropped without leaving behind a trace for the same.

The possibility of preparation and controlled manipulation of multi-partite
superposition states with exquisite precision has opened up the doors for
quantum computation which is fundamentally equivalent to doing massive
parallel computation with classical systems. All the aforementioned quantum
advantages require maintenance of quantum coherence (well defined phase
relationship) [4] among different parts of the multi-partite systems for large
time scales as well as over long distances. It means that the interaction of
the quantum systems with their environment need to be minimized. Ideally,
system should be isolated from the environment to avoid loss of quantum
coherence due to interaction of these systems with the environment.

However, practical realization of an ideal isolated quantum system may not be
possible and systems will inevitably interact with its environment leading to
decoherence [5, 6] in the system. The term decoherence refers to a dynamical
process through which a quantum mechanical system looses the phase
relationship between its different parts. Decoherence is an unavoidable and
irreversible phenomenon arising because of entanglement between system and
environment due to interaction. It is the biggest obstacle towards practical
realization of exponential speedup promised for certain tasks is quantum
computation. In the presence of local stochastic noises in the environment,
system doesn’t only loose the single particle coherence, even the non-local
coherence (entanglement) degrades in the multi-particle system [6, 7]. Since
entanglement is the quintessential resource for QIP protocols, protecting
entanglement against the detrimental effects of the noise is important. There
are several schemes to protect entanglement from decoeherence such as weak
measurement and reversal, dynamical decoupling, quantum bang-bang protocol,
delayed choice decoherence suppression [6], etc.

In this thesis, we propose another scheme [8] to protect quantum entanglement
in the presence of an amplitude damping channel (ADC). We know that under the
action of an ADC, excited state of a two-level atomic system decays down to
ground state emitting the energy to the environment. Thus, as time passes,
excited state population decreases and ground state population increases.
Therefore, any process which flips the population between ground and excited
states of the quantum system is expected to give some interesting results on
the system dynamics. Motivated by this idea, we propose a local unitary
operation (NOT operation, $\sigma_{x}$) on the individual qubit of a two-qubit
entangled state during the process of decoherence and study its impact on
entanglement preservation in a noisy environment. It turns out that the action
of NOT operation on one or both qubits preserves entanglement for longer
duration [8] for states which would have otherwise become separable in finite
time due to the action of ADC. Here, we present the results based on
simulation as well as taken some steps to verify them experimentally in a two-
qubit entangled state evolving in the presence of an ADC in a photonic system.

Consider a practical scenario where Charlie prepares a bipartite entangled
state for some QIP task and he has to send the entangled particles to Alice
and Bob through a quantum channel which is noisy and can potentially cause
disentanglement before the particles reach the two parties. In this scenario,
we ask the following question: given a bipartite entangled state which would
undergo finite time disentanglement (also known as Entanglement Sudden Death
or ESD) in the presence of an ADC, can we alter the time of disentanglement by
some suitable local unitary operations during the process of decoherence? To
answer this question, we explore not only qubit-qubit system but higher-
dimensional systems such as qubit-qutrit and qutrit-qutrit system as well. If
such systems undergo ESD, we propose a set of local unitary operations such
that when applied on the subsystems during the process of decoherence, these
operations can manipulate the disentanglement dynamics, in particular, delay
the time at which ESD occurs. Depending on the combination of local unitary
operations, and the time of their application, this method is shown to be able
to hasten, delay or completely avoid the ESD even in higher-dimensional
systems. Such an entanglement protection scheme will not only facilitate the
aforementioned task but will also find application where two parties, say
Alice and Bob, share an entangled pair for some quantum information processing
task and they know a-priori that ADC is present in the environment, and
therefore they can decide whether they are faced with the prospect of ESD.
Then, they can locally apply suitable local unitary operations at appropriate
time to delay or avoid the ESD. Our proposed scheme for preserving
entanglement longer will also find application in entanglement distillation
protocols.

###  1.2 Outline of the thesis

This thesis deals with creation, characterization, and manipulation of quantum
entanglement in a photonic system. It has two main parts which deal with
static and dynamic properties of entanglement, respectively, in bipartite
systems. It is organized in a way that the current introductory chapter deals
with several basic and technical concepts which are required for understanding
the research work reported in this thesis.

In our exploration of the static properties of entanglement, we experimentally
prepare a polarization entangled photon source using Spontaneous Parametric
Down-Conversion (SPDC) and reconstruct the state (density matrix) via Quantum
State Tomography (QST) and Maximum Likelihood Estimation (MLE) [9]. For
characterization of the experimentally prepared states and quantification of
entanglement, we use different computable measures of entanglement. In this
process, we make some non-trivial observations on the comparison of
entanglement measures for two-qubit pure states and indicate directions for
further study in higher-dimensional systems [10]. In order to comprehend this
aspect of the thesis, a basic understanding of the concepts such as SPDC based
single and entangled photon sources, quantification of entanglement using
different entanglement measures, as well as QST and MLE are essential.

In our investigation of entanglement dynamics, we study the evolution of
entanglement in a noisy environment. The noise under consideration is a local,
identical but independent Markovian noise acting on each qubit. It is found
that under the action of an Amplitude Damping Channel (ADC), an initially
entangled state can either undergo Asymptotic Decay of Entanglement (ADE) or
finite time disappearance of entanglement, also known as Entanglement Sudden
Death (ESD) [7]. For the state undergoing ESD, we propose a local unitary
operation (more specifically NOT operation, also known as $\sigma_{x}$) on one
or both qubits such that depending on the time of application of NOT
operation, ESD can be hastened, delayed or avoided [8]. The phenomenon of ESD
and ADE are then experimentally demonstrated for polarization entangled
photonic qubit system in a Displaced Sagnac Interferometer (DSI). This is
followed by a discussion on our attempts towards experimental demonstration of
manipulation of ESD in such a system. In order to appreciate this portion of
the thesis, we require to understand the evolution of open quantum systems, in
general, and entangled states, in particular. The overarching theme of this
thesis being entanglement, it is important to have a basic understanding of
the same; both theoretically as well as experimentally and some basic concepts
of quantum information.

In the end, we theoretically extend the idea of manipulation of ESD to higher-
dimensional systems such as $2\otimes 3$ and $3\otimes 3$ systems. By choosing
suitable measure(s) of entanglement for this purpose, we study the evolution
of entanglement in the presence of ADC and phenomenon of ESD is observed. Like
$2\otimes 2$ systems, we propose a set of local unitary operations for qubit
as well as qutrit subsystems wherein we indicate more than one possible
choices of local unitaries for qutrit. We then demonstrate the phenomenon of
hastening, delay, and avoidance of ESD in these systems as well.

To start with, we will discuss the basic terminology, notation, and some basic
concepts of quantum information to set the stage for the rest of the thesis.
This chapter begins by motivating the subject of this thesis which is followed
by a primer on the concept of state vector and density matrices, concept of
pure and mixed states, followed by a detailed introduction to classical bit
and quantum-bit (qubit). Then, we introduce the concept of quantum
entanglement and its measures for pure and mixed states. Next, evolution of
closed quantum systems is introduced which is followed by open quantum system
dynamics using the Kraus operator formalism. The chapter ends with the
discussion of the basics of birefringence, nonlinear optics, and spontaneous
parametric down conversion (SPDC).

  * •

Let us begin the introduction with some basic concepts of quantum mechanics
and quantum information.

###  1.3 Basics of quantum mechanics and quantum information

####  1.3.1 Concept of state vector and density matrix

In quantum mechanics, the state of an isolated physical system is described by
a state vector, a unit vector in the Hilbert space, which completely
characterizes the physical properties of the system. The simplest quantum
mechanical system is a qubit, which resides in a two-dimensional Hilbert space
$\mathbb{C}^{2}$. If $|0\rangle$ and $|1\rangle$ are taken as the basis
vectors for the Hilbert space, then the general state vector can be written
[1] as

$|\Psi\rangle=\alpha|0\rangle+\beta|1\rangle,$ (1.1)

where $\alpha$ and $\beta$ are, in general, complex numbers satisfying the
normalization condition for the state vectors: $|\alpha|^{2}+|\beta|^{2}=1$.

The state vectors for quantum systems are idealized descriptions which
represent only pure states and not the statistical (incoherent) mixtures -
ensemble comprising of many pure states with weighted probabilities which are
frequently encountered in experiments. The density matrix or density operator
is an alternate and more general description of quantum systems for both pure
as well as mixed states [1, 3]. All the accessible information of a quantum
system is encoded in the density matrix.

Consider an ensemble of systems all in the same state $|\Psi\rangle$, also
known as pure state. The density matrix $\hat{\rho}$ for such a pure state
$|\Psi\rangle$ is defined as the outer product of the wave function and its
conjugate as given below [1, 3].

$\hat{\rho}:=|\Psi\rangle\langle\Psi|\leavevmode\nobreak\ .$ (1.2)

Let us consider an ensemble of n-systems where not all of them are in the same
state but $n_{i}$ no. of systems are in the state $|\Psi_{i}\rangle$, etc.,
where $\sum_{i}n_{i}=n$. The probability of finding an individual system in
the state $|\Psi_{i}\rangle$ of such a ensemble is given by $p_{i}=n_{i}/n$.
In such a situation, when all the systems of the ensemble
$\\{|\Psi_{i}\rangle\\}$ are not in the same state, but a statistical mixture
of the states $\\{|\Psi_{i}\rangle\\}$ with respective probabilities $p_{i}$
then it is said to be in mixed state [1, 3]. The density matrix of such a
ensemble is given by

$\hat{\rho}:=\sum_{i}p_{i}|\Psi_{i}\rangle\langle\Psi_{i}|.$ (1.3)

Any physical density matrix satisfies the following three cardinal properties:

$\displaystyle\begin{split}\text{Tr}[\hat{\rho}]&=1\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{(Normalization)}\\\
\hat{\rho}^{{\dagger}}&=\hat{\rho}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(Hermiticity)}\\\
\text{det}[\hat{\rho}]&\geq 0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(Positive
semidefiniteness)}\\\ \end{split}$ (1.4)

A pure state density matrix, in addition, satisfies the condition:
$\hat{\rho}^{2}=\hat{\rho}$, i.e., it is also a projector.

####  1.3.2 Pure and mixed states

Consider an ensemble of particles in the states $\\{|\Psi_{i}\rangle\\}$. If
all the particles are in the same state then the ensemble is represented by a
pure state. For a pure state, $\hat{\rho}^{2}=\hat{\rho}$, and thus

$\text{Tr}[\hat{\rho}^{2}]=1\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(For pure state)}.$ (1.5)

When all the particles of the ensemble $\\{|\Psi_{i}\rangle\\}$ are not in the
same state, but a statistical mixture of the states $\\{|\Psi_{i}\rangle\\}$
with respective probabilities $p_{i}$ then it is said to be in mixed state and
its density matrix is given by

$\hat{\rho}=\sum_{i}p_{i}|\Psi_{i}\rangle\langle\Psi_{i}|.$ (1.6)

For a mixed state $\hat{\rho}^{2}\neq\hat{\rho}$ and thus

$\text{Tr}[\hat{\rho}^{2}]<1\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(For mixed state)}.$ (1.7)

The term $\text{Tr}[\hat{\rho}^{2}]$ is defined as the  purity of a state. It
can take a minimum value of $1/d$ for maximally mixed state, where $d$ is the
dimensionality of the system, and $1$ for pure state.

####  1.3.3 Bit and Qubit

Classical bit, also known as bit, is the basic unit of digital information in
classical information theory, which can take logical values ‘0’ or ‘1’. These
two possibilities are mutually exclusive. Because of its binary nature, it can
be encoded into the property of any physical system which has two stable
states, e.g., ON or OFF state of a transistor, direction of magnetic domain in
a magnetic media, etc. On the other hand, quantum bit or qubit is the quantum
mechanical analogue of the classical bit which can exist in an arbitrary
superposition of the two possible states, say $|0\rangle$ and $|1\rangle$. The
term ‘qubit’ was first introduced by Schumacher in 1995 [11]. The general
state of a qubit is given by

$|\Psi\rangle=\alpha|0\rangle+\beta|1\rangle,$ (1.8)

where $\alpha$ and $\beta$ are in general complex, and
$|\alpha|^{2}+|\beta|^{2}=1$.

An example of a classical and quantum bit is illustrated using a switch [12]
is shown in Fig. 1.1 below. Quantum superposition is the key difference
between the two. A classical bit always remains in one of the two possible
orthogonal states (‘0’, or ‘1’) whereas a qubit can exist in a coherent
superposition of the two orthogonal states as well.

Figure 1.1: Difference between a classical bit and qubit illustrated using a
switch [Image credit: Ref. 12]. A classical bit always remains in one of the
two possible orthogonal states (‘0’, or ‘1’) whereas a qubit can exist in a
coherent superposition of the two orthogonal states as well.

The state of a qubit is an element of two-dimensional Hilbert space. The
states $|0\rangle$ and $|1\rangle$ are the eigenstates of the Pauli matrix
$\sigma_{z}$ which form the basis vectors of this space, and known as
computational basis states.

$|0\rangle=\left(\begin{array}[]{cc}1\\\
0\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ |1\rangle=\left(\begin{array}[]{cc}0\\\
1\end{array}\right).$ (1.9)

If the normalization is written implicitly into the coefficients, and global
phases are ignored, then Eq. (1.8) can be rewritten as

$|\Psi\rangle=\cos\left(\frac{\theta}{2}\right)|0\rangle+\exp(i\phi)\sin\left(\frac{\theta}{2}\right)|1\rangle,$
(1.10)

where $0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$.

If we consider a point with ($\theta,\phi$) as defined above and $r=1$ as the
spherical polar coordinates ($r,\theta,\phi$) on a sphere then corresponding
point on the Poincare sphere represents a state of the quantum system whose
density matrix is given by

$\hat{\rho}=\frac{1}{2}(\mathbb{I}_{2}+\vec{r}.\vec{\sigma}),$ (1.11)

where $\vec{r}$ is the Poincare vector given by

$\vec{r}=\left(\begin{array}[]{c}r_{D}\\\ r_{R}\\\
r_{H}\end{array}\right)=\left(\begin{array}[]{c}\sin(\theta)\cos(\phi)\\\
\sin(\theta)\sin(\phi)\\\ \cos(\theta)\end{array}\right),\leavevmode\nobreak\
\vec{\sigma}=\left(\begin{array}[]{c}\sigma_{x}\\\ \sigma_{y}\\\
\sigma_{z}\end{array}\right),$ (1.12)

and, $\mathbb{I}_{2}$ is the identity matrix, and $\sigma_{i}$s are the Pauli
matrices as given below.

$\displaystyle\leavevmode\nobreak\
\mathbb{I}_{2}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\leavevmode\nobreak\
\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\leavevmode\nobreak\
\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right).$ (1.13)

Three components of $\vec{r}$ ($r_{D}$, $r_{R}$, $r_{H}$) can be viewed as the
coordinates in the three dimensional polarization space and give the
polarization component of the photon in diagonal/anti-diagonal (D/A),
right-/left-circular (R/L), and horizontal/vertical (H/V), bases,
respectively. A spin-1/2 particle such as electron, polarization state of a
photon, etc., are good examples of a qubit. If we represent the basis states
$|H\rangle\equiv|0\rangle$ and $|V\rangle\equiv|1\rangle$, then the general
polarization state of a photon can be written as Eq. (1.10) or (1.11) and it
can be represented on the Poincare sphere as shown in the Fig. 1.2.

$\theta$$\phi$$|H\rangle$$|V\rangle$$|D\rangle$$|A\rangle$$|R\rangle$$|L\rangle$$|\Psi\rangle$$x$$y$$z$
Figure 1.2: Poincare sphere representation of a qubit. All points at the
suraface of Poincare sphere represent pure states whereas those lying inside
the spehere represent mixed state. Point at the center of the sphere is
maximally mixed.

Note that Eq. (1.11) with $|\vec{r}|=1$ captures all the pure states which lie
on the surface of the Poincare sphere. All the states in Eq. (1.11) with
$|\vec{r}|<1$ are mixed states and they lie inside the sphere, and maximally
mixed state ($\mathbb{I}/2$) lies at the center of the sphere. It is important
to bear in mind that the orthogonal states lie $180^{\circ}$ apart, i.e., on
the diametrically opposite sides of the sphere. The $|H\rangle$ and
$|V\rangle$ polarization states lie on the opposite sides
($\theta=180^{\circ}$ apart) of the pole, and $|D\rangle$, $|R\rangle$,
$|A\rangle$, and $|L\rangle$ lie on the equator at $\phi=90^{\circ}$ apart
from the adjacent ones. These polarization states can be expressed in terms of
the bases states $|H\rangle$ and $|V\rangle$ as follows:

$\displaystyle|D\rangle=\frac{|H\rangle+|V\rangle}{\sqrt{2}},\leavevmode\nobreak\
\leavevmode\nobreak\
|A\rangle=\frac{|H\rangle-|V\rangle}{\sqrt{2}},\leavevmode\nobreak\
\leavevmode\nobreak$ (1.14)
$\displaystyle|R\rangle=\frac{|H\rangle+i|V\rangle}{\sqrt{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ |L\rangle=\frac{|H\rangle-i|V\rangle}{\sqrt{2}}.$

Such a visualization of a polarization qubit on the Poincare sphere will be
very convenient and intuitive to understand the unitary transformations on the
Hilbert space of a qubit and projective measurements on different bases set
for quantum state tomography in next chapter.

####  1.3.4 Partial trace and reduced density matrix

Consider a composite quantum system characterized by the density matrix
$\hat{\rho}^{(AB)}$ comprising of two subsystems A and B. Suppose we are
interested in only one of the subsystems; say A, then this situation is
captured by the reduced density matrix. For subsystem A, it is obtained by
tracing out [3] the subsystem B as follows:

$\hat{\rho}^{(A)}=\text{Tr}_{B}[\hat{\rho}^{(AB)}].$ (1.15)
$\text{Tr}_{B}(|a_{i}\rangle\langle a_{j}|\otimes|b_{k}\rangle\langle
b_{l}|)\equiv|a_{i}\rangle\langle a_{j}\text{Tr}(|b_{k}\rangle\langle
b_{l}|)=|a_{i}\rangle\langle a_{j}|\langle b_{k}|b_{l}\rangle.$ (1.16)

where $\\{|a_{i}\rangle,|a_{j}\rangle\\}$, and
$\\{|b_{k}\rangle,|b_{l}\rangle\\}$ are the basis vectors of the state space A
and B, respectively. Physically, partial trace is the unique operation for
obtaining the correct measurement statistics of a subsystem of the composite
quantum system.

####  1.3.5 Partial transpose

Let us consider a bipartite joint system density matrix $\hat{\rho}^{(AB)}$
with the density matrices of its individual subsystems given by
$\hat{\rho}^{(A)}=\text{Tr}_{B}[\hat{\rho}^{(AB)}]$ and
$\hat{\rho}^{(B)}=\text{Tr}_{A}[\hat{\rho}^{(AB)}]$. The elements of the
density matrix $\hat{\rho}^{(AB)}$ can be written as

$\rho_{m\mu,n\nu}=\langle
a_{m},b_{\mu}|\hat{\rho}^{(AB)}|a_{n},b_{\nu}\rangle$ (1.17)

where {$a_{m}$} and {$b_{\mu}$} are the orthonormal bases sets in the Hilbert
space of subsystems A and B, respectively.

As the name suggests, partial transpose [13] is the transpose taken with
respect to one of the subsystems of a bipartite quantum system. Using the
above representation, partial transposition with respect to subsystem A can be
written as follows:

$\text{PT}^{A}(\rho_{m\mu,n\nu})=\rho_{n\mu,m\nu}$ (1.18)

Likewise, partial transposition with respect to subsystem B can be written as
follows:

$\text{PT}^{B}(\rho_{m\mu,n\nu})=\rho_{m\nu,n\mu}$ (1.19)

This can be further illustrated by taking the explicit example of the density
matrix of a $2\otimes 2$ dimensional system and dividing it into $2\times 2$
blocks and doing transpose in each block to get the partially transposed
density matrix with respect to subsystem B.

$\begin{split}\hat{\rho}^{(AB)}&=\left(\begin{array}[]{cc|cc}\rho_{11,11}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{11,12}}&\rho_{11,21}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{11,22}}\\\
\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{12,11}}&\rho_{12,12}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{12,21}}&\rho_{12,22}\\\
\hline\cr\rho_{21,11}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{21,12}}&\rho_{21,21}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{21,22}}\\\
\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{22,11}}&\rho_{22,12}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{22,21}}&\rho_{22,22}\\\
\end{array}\right)\\\
\Rightarrow\text{PT}^{B}(\hat{\rho}^{(AB)})&=\left(\begin{array}[]{cc|cc}\rho_{11,11}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{12,11}}&\rho_{11,21}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{12,21}}\\\
\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{11,12}}&\rho_{12,12}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{11,22}}&\rho_{12,22}\\\
\hline\cr\rho_{21,11}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{22,11}}&\rho_{21,21}&\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rho_{22,21}}\\\
\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{21,12}}&\rho_{22,12}&\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}{\rho_{21,22}}&\rho_{22,22}\\\
\end{array}\right)\end{split}$ (1.20)

It is evident that the partial transpose operation interchanges the entries in
blue and magenta in each block. Partial transpose operation finds application
in deciding the (non)separability of a bipartite state using PPT criterion
(discussed in Sec. 1.4.3).

####  1.3.6 Fidelity

Fidelity is a distance measure of two quantum states. Fidelity
$F(\hat{\rho},\hat{\sigma})$ of the states $\hat{\rho}$ and $\hat{\sigma}$
measures the “closeness" of the states and it is defined as [14]

$F(\hat{\rho},\hat{\sigma}):=\text{Tr}\left[\sqrt{\sqrt{\hat{\rho}}\leavevmode\nobreak\
\hat{\sigma}\leavevmode\nobreak\ \sqrt{\hat{\rho}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \right].$ (1.21)

Fidelity of a pure state $|\psi\rangle$ and an arbitrary state $\hat{\rho}$ is
defined as

$F(|\Psi\rangle,\hat{\rho})=\text{Tr}\left[\sqrt{\langle\Psi|\hat{\rho}|\Psi\rangle|\Psi\rangle\langle\Psi|}\right]=\sqrt{\langle\Psi|\hat{\rho}|\Psi\rangle},$
(1.22)

i.e., fidelity equals the square root of the overlap between $\Psi$ and
$\hat{\rho}$.
Fidelity between two pure states $|\phi\rangle$ and $|\psi\rangle$ is defined
as

$F(|\phi\rangle,|\psi\rangle)=|\langle\phi|\psi\rangle|.$ (1.23)

It is symmetric in its inputs and a good distance measure between the quantum
states. Some useful properties of the fidelity are:

  1. (i)

$0\leq F(\rho,\sigma)\leq 1$

  2. (ii)

$F(\rho,\sigma)=F(\sigma,\rho)$

  3. (iii)

$F(\rho,\rho)=1$

  4. (iv)

$F(\rho,\sigma)=1\leavevmode\nobreak\ \Rightarrow\rho=\sigma$

  5. (v)

$F(\rho,\sigma)=0\leavevmode\nobreak\ \Rightarrow\rho\perp\sigma$

In literature, some physicists choose to define Fidelity as the overlap
squared. So, one should be careful and consistent with the fidelity
definition.

  * •

Given a bipartite entangled state $\hat{\rho}^{(AB)}$, how do we know whether
it is entangled or not (separable)? How to quantify entanglement? What are the
basic requirements for the entanglement measure? We will learn all these
things in the next section.

###  1.4 Quantum entanglement

To quote E. Schrödinger who coined the term entanglement to describe the
strange connection between the quantum systems.

  * •

“I would not call that one but rather the characteristic trait of quantum
mechanics, the one that enforces its entire departure from classical lines of
thought."

  * •

“The best possible knowledge of the whole does not include the best possible
knowledge of its parts. By the interaction the two representatives [the
quantum states] have become entangled."

####  1.4.1 Separable and entangled states

A bipartite state $\hat{\rho}^{(AB)}$ (whether pure or mixed) is said to be
separable, classically correlated or unentangled, if it can be prepared
“classically" by two parties; say Alice and Bob, by local operation and
classical communication (LOCC) alone. The state created in this manner can
contain only classical correlations. States which cannot be prepared by LOCC
alone are said to be entangled.

Mathematically speaking, a bipartite pure state $|\Psi\rangle_{AB}$ is said to
be separable if it can be written as the direct product of the states from the
subsystem Hilbert spaces, i.e.,

$|\Psi\rangle_{AB}=|\Psi\rangle_{A}\otimes|\Psi\rangle_{B}\leavevmode\nobreak\
,$ (1.24)

where
$|\psi\rangle_{AB}\in\mathcal{H}^{AB}=\mathcal{H}^{A}\otimes\mathcal{H}^{B}$,
$|\psi\rangle_{A}\in\mathcal{H}^{A}$, and
$|\psi\rangle_{B}\in\mathcal{H}^{B}$. The joint state $|\Psi\rangle_{AB}$ is
said to be entangled [15] if it is not separable.

On the other hand, a bipartite mixed state $\hat{\rho}^{(AB)}$ is said to be
separable if it can be written as the convex combination of the pure product
states from the subsystem Hilbert spaces, i.e.,

$\hat{\rho}^{(AB)}=\sum_{i}p_{i}\leavevmode\nobreak\
\hat{\rho}_{i}^{(A)}\otimes\hat{\rho}_{i}^{(B)}.$ (1.25)

where
$\hat{\rho}^{(AB)}\in\mathcal{H}^{AB}=\mathcal{H}^{A}\otimes\mathcal{H}^{B}$,
$\hat{\rho}^{(A)}_{i}=|\psi_{i}\rangle_{A}\langle\psi_{i}|\in\mathcal{H}^{A}$,
$\hat{\rho}^{(B)}_{i}=|\phi_{i}\rangle_{B}\langle\phi_{i}|\in\mathcal{H}^{B}$,
$\sum_{i}p_{i}=1$ ($p_{i}$ are non-negative coefficients), and
$|\psi_{i}\rangle_{A}$ and $|\phi_{i}\rangle_{B}$ denote the normalized states
of subsystems A and B, respectively.

The above separability criterion is not operational, as it is very difficult
to find a composition of the form (1.25) or to prove that it doesn’t exist.
Therefore, we need operational criterion and measures of entanglement,
discussed below.

####  1.4.2 Basic requirements for an entanglement measure

A measure of entanglement $E(\hat{\rho})$ must assign a non-negative value for
each state $\hat{\rho}$. A good measure of entanglement $E(\hat{\rho}_{AB})$
is required to satisfy the following criterion [15-17]:

  1. 1.

Separability: For separable states, the measure of entanglement must vanish.

$E(\hat{\rho}_{AB})=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\text{separable}\leavevmode\nobreak\ \hat{\rho}_{AB}\leavevmode\nobreak\ .$
(1.26)

  2. 2.

Normalization: The entanglement of a maximally entangled qudit-qudit system is
given by

$E(\hat{\rho}_{AB})=\log_{2}d.$ (1.27)

  3. 3.

Invariant under unitary transformation: $E(\hat{\rho}_{AB})$ is invariant
under all local unitary transformations $U_{A}\otimes U_{B}$. i.e.,

$E\left(\hat{\rho}_{AB}\right)=E\left((U_{A}\otimes
U_{B})\hat{\rho}_{AB}(U_{A}\otimes U_{B})^{\dagger}\right).$ (1.28)

It should not depend on the choice of basis.

  4. 4.

Non increasing under LOCC: On an average, $E(\hat{\rho}_{AB})$ should not
increase under local operation and classical communication (LOCC).

$E(\hat{\rho}_{AB})\geq E(\Theta(\hat{\rho}_{AB})),$ (1.29)

where $\Theta(\hat{\rho}_{AB})$ is completely positive trace preserving (CPTP)
map.

  5. 5.

Continuity: If the distance between two density matrices is vanishingly small,
the difference between their entanglement should vanish, i.e.,

$E(\hat{\rho})-E(\hat{\sigma})\rightarrow 0\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
||\hat{\rho}-\hat{\sigma}||\rightarrow 0.$ (1.30)

  6. 6.

Additivity: The entanglement contained in $n$ identical copies of a state
$\rho$ equals $n$ times the entanglement of one copy.

$E(\hat{\rho}^{\otimes n})=nE(\hat{\rho}).$ (1.31)

  7. 7.

Sub-additivity: The entanglement of the tensor product of the two states
$\hat{\rho}$ and $\hat{\sigma}$ should not exceed the sum of entanglement of
each of the states.

$E(\hat{\rho}\otimes\hat{\sigma})\leq E(\hat{\rho})+E(\hat{\sigma}).$ (1.32)

  8. 8.

Convexity: The entanglement measure should satisfy the convexity property,
i.e.

$E(\lambda\hat{\rho}+(1-\lambda)\hat{\sigma})\leq\lambda
E(\hat{\rho})+(1-\lambda)E(\hat{\sigma}).$ (1.33)

From the definition of entanglement, first condition is obvious: separable
states are not entangled. Second and third conditions are essential for
entanglement to be considered global property of the joint system and they
render it impossible to create and distribute the entanglement via LOCC alone.
It should be noted that whether all these requirements are necessary or not,
is still an open question.

####  1.4.3 Measures of entanglement

Some measures that fulfil most of the above requirements are discussed below.

##### (i) Schmidt decomposition

Consider a $n\otimes m$ dimensional Hilbert space $\mathcal{H}_{AB}$ defined
by Kronecker product of the two Hilbert spaces $\mathcal{H}_{A}$ and
$\mathcal{H}_{B}$. For a pure state $|\Psi\rangle_{AB}$ from
$\mathcal{H}_{AB}$, there exists an orthonormal bases
$\\{|a_{i}\rangle,\leavevmode\nobreak\ i=1,2...n\\}$ and
$\\{|b_{i}\rangle,\leavevmode\nobreak\ i=1,2...m\\}$ in the spaces
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively, such that

$|\Psi\rangle_{AB}=\sum_{i=1}^{d}\sqrt{\lambda_{i}}|a_{i}\rangle\otimes|b_{i}\rangle\leavevmode\nobreak\
;\leavevmode\nobreak\ d\leq\text{min}(n,m).$ (1.34)

where $\lambda_{i}$’s are non-negative coefficients known as Schmidt
coefficients satisfying $\sum_{i=1}^{d}\lambda_{i}=1$, and above decomposition
is known as Schmidt decomposition [1]. The bases $|a_{i}\rangle$ and
$|b_{i}\rangle$ are called Schmidt bases for A and B, respectively, and the
number of non-zero values of $\lambda_{i}$ is called Schmidt number for the
state $|\psi\rangle_{AB}$. The values $\lambda_{i}$’s are precisely the
features of the state $|\Psi\rangle_{AB}$ that remains intact when the
subsystems are subjected to local unitary transformation. For a pure state
$|\Psi\rangle_{AB}$ the $\lambda_{i}$’s are equal to the eigenvalues of the
density matrix of subsystem A, given by
$\hat{\rho}_{A}=\text{Tr}_{B}[|\Psi_{AB}\rangle\langle\Psi_{AB}|]$. The state
$|\Psi\rangle_{AB}$ is entangled [1] if it has more than one Schmidt
coefficients and separable iff there is only one Schmidt coefficient, i.e.,
$\lambda_{i}=1$.

##### (ii) Entropy of entanglement

Consider a bipartite pure state $|\Psi_{AB}\rangle$ with density matrix
$\hat{\rho}_{AB}$ ($=|\Psi_{AB}\rangle\langle\Psi_{AB}|$) in the Hilbert space
$H_{AB}$ of dimension $d_{A}\otimes d_{B}$. The quantum state of the
individual subsystem is obtained by tracing out the other subsystems such that
$\hat{\rho}_{i}=\text{Tr}_{j}[\hat{\rho}_{ij}]\leavevmode\nobreak\
;\leavevmode\nobreak\ i,j=A,B$. If the joint state $|\Psi_{AB}\rangle$ is
entangled, then the marginal density operators will be mixed, implying the
presence of quantum entanglement. Entropy of entanglement [18] uses the Von
Neumann entropy of density matrix, defined by

$S(\hat{\rho})=-\text{Tr}[\hat{\rho}\log_{2}(\hat{\rho})]\leavevmode\nobreak\
=-\sum_{i}\lambda_{i}\log\lambda_{i}.$ (1.35)

where $\lambda_{i}$’s are the eigenvalues of the density matrix $\hat{\rho}$.
For pure bipartite states, the entropy of entanglement is defined as the Von
Neumann entropy of the reduced density matrices.

$E(\hat{\rho}_{AB})=S(\hat{\rho}_{A})=S(\hat{\rho}_{B})\leavevmode\nobreak\ ,$
(1.36)

where $\hat{\rho}_{A}=\text{Tr}_{B}[\hat{\rho}_{AB}]$, and
$\hat{\rho}_{B}=\text{Tr}_{A}[\hat{\rho}_{AB}]$.

Note that von Neumann entropy is zero for a pure entangled state and non-zero
for its reduced states which makes it a measure of entanglement.

##### (iii) Entanglement of formation

Entanglement of formation [19] is an information theoretic measure of
entanglement for bipartite mixed states which quantifies the resource required
to create the given state. Consider a bipartite mixed state density matrix
$\hat{\rho}$ as given below.

$\hat{\rho}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$ (1.37)

Note that it can have infinitely many pure state decompositions.

Entanglement of formation ($\mathcal{E}$) of the state $\hat{\rho}$ is defined
as the minimum average entanglement over all possible pure-state
decompositions of $\hat{\rho}$.

$\mathcal{E}(\hat{\rho})=\text{inf}\sum_{i}p_{i}\leavevmode\nobreak\
\mathcal{E}(|\psi_{i}\rangle)\leavevmode\nobreak\ .$ (1.38)

For a maximally mixed state, entanglement of formation is trivially zero.

##### (iv) Concurrence and Tangle

Concurrence ($C$) [20] is a measure of entanglement defined for $2\otimes 2$
and $2\otimes 3$ dimensional systems. For a pure state $|\Psi\rangle_{AB}$,
concurrence is defined as

$C(|\Psi\rangle_{AB})=|\langle\Psi_{AB}|\tilde{\Psi}_{AB}\rangle|,$ (1.39)

where $\tilde{\Psi}_{AB}=\sigma_{y}\otimes\sigma_{y}|\Psi\rangle_{AB}$ is
referred as spin flipped state vector and $\sigma_{y}$ is the second Pauli
matrix.

The concurrence of a mixed entangled state $\hat{\rho}_{AB}$ is defined as

$C(\hat{\rho}_{AB})=\text{Max}\\{0,\Lambda\\},$ (1.40)

where
$\Lambda=\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}$,
and $\lambda_{i}^{\prime}s$ are the positive eigenvalues of the auxiliary
matrix $\tilde{\rho}_{AB}$ defined by

$\tilde{\rho}_{AB}=\hat{\rho}_{AB}(\sigma_{y}\otimes\sigma_{y})\hat{\rho}^{*}_{AB}(\sigma_{y}\otimes\sigma_{y}).$
(1.41)

The complex conjugation is done in the computational basis:
$|00\rangle,|01\rangle,|10\rangle,|11\rangle$. Tangle ($\tau$) is defined as
the square of concurrence, i.e.,

$\tau=[C(\hat{\rho}_{AB})]^{2}$ (1.42)

Initially, interest in Concurrence started due to its connection to the
entanglement of formation ($\mathcal{E}$) as given below

$\mathcal{E(C(\rho))}:=h\left(\frac{1+\sqrt{1+C^{2}}}{2}\right),$ (1.43)

where $h(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$.

##### (v) Peres-Horodecki criterion and Negativity

The Peres-Horodecki criterion for separability is also known as positive
partial transpose (PPT) criterion [21]. It is necessary condition for
separability of pure as well as mixed states. It turns out that for $2\otimes
2$ and $2\otimes 3$ dimensional systems, the PPT criterion is also sufficient
condition for separability.

When a positive map is applied to one of the subsystems of a composite quantum
system in a separable state, it always maps to a valid quantum state. However,
when the composite state is entangled, the same positive map, in general, does
not results in a valid density operator. This happens because a positive
operator does not behave like a completely positive operator in the presence
of quantum entanglement in the system. The partial transposition operator is
such an example of a positive but not completely positive operator.

Let the composite system density matrix be written as

$\rho_{AB}=\sum_{ijkl}p_{ij}^{kl}|i_{A}k_{B}\rangle\langle j_{A}l_{B}|,$
(1.44)

where $\\{|i_{A}\rangle\\}$, and $\\{|k_{B}\rangle\\}$ form the orthonormal
basis set for the subsystem ‘A’ and ‘B’, respectively. Then the partial
transposed density matrix, with respect to the subsystem $`A^{\prime}$, is
given by

$\rho^{T_{A}}_{AB}=\sum_{ijkl}p_{ij}^{kl}|j_{A}k_{B}\rangle\langle
i_{A}l_{B}|.$ (1.45)

If the state $\hat{\rho}_{AB}$ is separable then above Eq. (1.44) can be
rewritten as

$\hat{\rho}_{AB}^{T_{A}}=\sum_{ijkl}p_{ij}^{kl}|j_{A}\rangle\langle
i_{A}|\otimes|k_{B}\rangle\langle
l_{B}|=\sum_{ijkl}p_{ij}^{kl}(|i_{A}\rangle\langle
j_{A}|)^{\text{T}}\otimes|k_{B}\rangle\langle l_{B}|.$ (1.46)

Since $(|i_{A}\rangle\langle j_{A}|)^{\text{T}}$ are again valid density
matrices for Alice, it implies that $\hat{\rho}_{AB}^{T_{A}}\geq 0$. Same
argument can be given for partial transposition with respect to Bob’s
subsystem. In short, the partial transpose of a bipartite separable state
$\hat{\rho}_{AB}$ with respect to any subsystem is positive.

Negativity is a measure of entanglement based on the PPT criteria of
separability and it is defined as the sum of absolute values of all the
negative eigenvalues of the partially transposed density matrix with respect
to one of its subsystems.

Note that PPT criterion is necessary but not sufficient condition for
separability in higher-dimensional ($d\otimes D$, $d,D\geq 3$) systems. If a
higher-dimensional system is found to be PPT, we can’t say whether state is
entangled or separable, and in that case we can use matrix realignment method
(discussed below) for detecting some of the bound entangled states which may
not be detected by PPT criterion.

##### (vi) Matrix realignment method for detecting entanglement

This criterion for inseparability of bipartite systems of arbitrary dimensions
is motivated by Kronecker product approximation technique for density
matrices. It is based on a realigned matrix [22] obtained from the density
matrix via realignment method as discussed below.

Consider a matrix $A=[a_{ij}]$ having dimension $m\otimes n$, where $a_{ij}$
is the $i$th row and $j$th column entry of this matrix. Let us define $vec(A)$
as

$vec(A)=[a_{11},.....a_{m1},.....a_{m2},......a_{1n},......a_{mn}]^{T}\leavevmode\nobreak\
,$ (1.47)

where symbol $T$ denotes transpose.

Now, consider a block matrix B of dimension $m\times m$ with each block of
size $n\times n$. Let us define the realigned matrix $\tilde{B}$ of dimension
$m^{2}\times n^{2}$ that contains the same elements as in matrix as $B$ but in
a rearranged position as given below.

$\tilde{B}=\left[\begin{array}[]{c}vec(B_{1,1})^{T}\\\ \vdots\\\
vec(B_{m,1})^{T}\\\ \vdots\\\ vec(B_{1,n})^{T}\\\ \vdots\\\
vec(B_{m,n})^{T}\\\ \end{array}\right]\leavevmode\nobreak\ ,$ (1.48)

where $B_{i,j}$ denotes the block in the $i$th row and $j$th column.

The Singular value decomposition of B is given by

$\tilde{B}=UDV^{\dagger}=\sum_{i=1}^{q}d_{i}u_{i}v_{i}^{\dagger}$ (1.49)

where D is a diagonal matrix with elements $d_{1}\geq d_{2}\geq
d_{2}\geq.....d_{q}\geq 0$ with $q=\text{min}(m^{2},\leavevmode\nobreak\
n^{2})$, and $U={u_{1},\leavevmode\nobreak\
u_{2}.......u_{m^{2}}}\in\mathcal{C}^{m^{2}\times m^{2}}$ &
$V={v_{1},\leavevmode\nobreak\
v_{2}.......v_{m^{2}}}\in\mathcal{C}^{n^{2}\times n^{2}}$ are unitary
matrices.

where $d_{i}s$ are given by non-negative square roots of the eigenvalues of
the matrix $\tilde{B}\tilde{B}^{\dagger}$ or $\tilde{B}^{\dagger}\tilde{B}$.
In fact, rank of the matrix $\tilde{B}$ is equal to number of non-zero
singular values $d_{i}$. Inspired by this construction, Loan and Pitsianis
gave the following decomposition for $B$.

$B=\sum_{i=1}^{r}X_{i}\otimes Y_{i}\leavevmode\nobreak\ ,$ (1.50)

where $vec(X_{i})=\sqrt{d_{i}}u_{i}$ and $vec(Y_{i})=\sqrt{d_{i}}v_{i}^{*}$.

For a given density matrix $\rho_{AB}$, we can write the realigned matrix
$\tilde{\rho}_{AB}$ according to the transformation given in Eq. (1.48). As an
example, a bipartite $2\otimes 2$ density matrix $\rho_{AB}$ can be
transformed as follows:

$\rho_{AB}=\left(\begin{array}[]{cc|cc}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}\\\
\rho_{21}&\rho_{22}&\rho_{23}&\rho_{24}\\\
\hline\cr\rho_{31}&\rho_{32}&\rho_{33}&\rho_{34}\\\
\rho_{41}&\rho_{42}&\rho_{43}&\rho_{44}\\\
\end{array}\right)\Rightarrow\tilde{\rho}_{AB}=\left(\begin{array}[]{cccc}\rho_{11}&\rho_{21}&\rho_{12}&\rho_{22}\\\
\hline\cr\rho_{31}&\rho_{41}&\rho_{32}&\rho_{42}\\\
\hline\cr\rho_{13}&\rho_{23}&\rho_{14}&\rho_{24}\\\
\hline\cr\rho_{33}&\rho_{43}&\rho_{34}&\rho_{44}\\\ \end{array}\right)$ (1.51)

Realigned Negativity is then defined as

$R(\rho_{AB})=\text{max}(0,\leavevmode\nobreak\ ||\tilde{\rho}_{AB}||-1)$
(1.52)

where symbol $||.||$ denotes trace norm.

This criteria is motivated by the fact that for any separable state, trace
norm of the realigned matrix will not be greater than one. Hence, If $R(\rho)$
turns out to be greater than zero, state must be entangled. This criterion can
detect some of the bound entangled states in higher-dimensional system which
may not be detected by PPT criterion.

  * •

Given a quantum system $\rho(0)$ at time $t=0$. How does it evolve with time
in the presence of different environments? We will learn this in the next
section.

###  1.5 Time evolution of the quantum systems

In this section, we introduce the minimum basic formalism required to
understand and mathematically describe decoherence in the framework of quantum
theory. A closed quantum system is one which neither interchanges energy nor
matter with its environment. On the other hand, an open quantum system
interchanges energy or matter or both with its environment.

####  1.5.1 Evolution of the closed quantum systems

Evolution of a closed quantum system [1] in the pure state $|\psi(t)\rangle$
is given by Schr$\ddot{o}$dinger equation:

$\frac{d}{dt}|\psi(t)\rangle=-\frac{i}{\hbar}H(t)|\psi(t)\rangle,$ (1.53)

where $H(t)$ is the Hamiltonian of the system.

Evolution of the states under Schr$\ddot{o}$dinger equation is unitary and
governed by a unitary time-evolution operator $U(t,t^{\prime})$ such that it
takes the state $|\psi(t^{\prime})\rangle$ to $|\psi(t)\rangle$ as

$|\psi(t)\rangle=U(t,t^{\prime})|\psi(t^{\prime})\rangle,$ (1.54)

where $U(t^{\prime},t^{\prime})=\mathbb{I}$.

Substituting $|\psi(t)\rangle$ into Eq. (1.53), we get

$\frac{\partial}{\partial
t}U(t,t^{\prime})=-\frac{i}{\hbar}H(t)U(t,t^{\prime}).$ (1.55)

For an isolated system (which is not driven by any external agency)
Hamiltonian becomes time independent and Unitary operator can be written as

$U(t,t^{\prime})=exp[-\frac{i}{\hbar}H(t-t^{\prime})].$ (1.56)

If the initial state of system is a statistical mixture then it is
characterized by a density matrix $\rho$ instead of a state vector. Let the
initial state be given by

$\rho(t^{\prime})=\sum_{i}p_{i}|\psi(t^{\prime})\rangle\langle\psi(t^{\prime})|.$
(1.57)

Each state vector of the ensemble evolves as per Eq. (1.56) and thus the state
of the system at a later time ’t’ is given by

$\begin{split}\rho(t)&=\sum_{i}p_{i}U(t,t^{\prime})|\psi(t^{\prime})\rangle\langle\psi(t^{\prime})|U^{\dagger}(t,t^{\prime}),\\\
&=U(t,t^{\prime})\rho(t^{\prime})U^{\dagger}(t,t^{\prime}).\end{split}$ (1.58)

Upon differentiating the above eqn. wrt. time, we get the eqn. of motion as

$\frac{d}{dt}\rho(t)=-\frac{i}{\hbar}[H(t),\rho(t)].$ (1.59)

This is known as Liouville - von Neumann equation. It gives the evolution of a
closed quantum system $\rho$ with time.

####  1.5.2 Evolution of the open quantum systems

Let us consider a quantum system ‘S’ denoted by density matrix $\rho_{S}$ in
the Hilbert space $\mathcal{H}_{S}$ interacting with another quantum system
‘E’, called environment, having density matrix $\rho_{E}$ in the Hilbert space
$\mathcal{H}_{E}$. The state of the joint system; System + Environment (S+E),
is given by the tensor product $\rho_{S}\otimes\rho_{E}$ which resides in the
joint Hilbert space $\mathcal{H}_{S}\otimes\mathcal{H}_{E}$. The schematic of
a quantum system interacting with its environment is shown in the Fig. 1.3.

($S,\mathcal{H}_{S},\rho_{S}$)System($E,\mathcal{H}_{E},\rho_{E}$)Environment($S+E,\mathcal{H}_{S}\otimes\mathcal{H}_{E},\rho$)
Figure 1.3: Schematic diagram of an open quantum system.

The joint system (S+E) behaves as a closed quantum system and follows the
Hamiltonian dynamics, as discussed above. During the evolution, system and
environment develop some kind of correlation and consequently, if we consider
the evolution of the system alone, it becomes non-unitary and behaves as an
open quantum system [1, 2]. The Hamiltonian of the joint quantum system can be
written as

$H(t)=H_{S}\otimes\mathbb{I}_{E}+\mathbb{I}_{S}\otimes H_{E}+H_{int}(t),$
(1.60)

where $H_{S}$ and $H_{E}$ are the system and environment Hamiltonian,
respectively, and $H_{int}(t)$ is the interaction Hamiltonian acting on the
Hilbert space $\mathcal{H}_{S}\otimes\mathcal{H}_{E}$. The $\mathbb{I}_{S}$
and $\mathbb{I}_{E}$ are the identity matrices in $\mathcal{H}_{S}$ and
$\mathcal{H}_{E}$, respectively.

The reduced density matrix of the system ($\rho_{s}$) can be found by tracing
over the degrees of the freedom of the environment from the joint system
density matrix [1, 2] as follows:

$\rho_{s}=\text{Tr}_{E}(\rho).$ (1.61)

Substituting $\rho(t)$ from Eq. (1.58), we get

$\rho_{S}(t)=\text{Tr}_{E}\left(U(t,t^{\prime})\rho(t^{\prime})U^{\dagger}(t,t^{\prime})\right).$
(1.62)

Above equation gives the evolution of an open quantum system. To illustrate
this further, we consider a specific noise model of Amplitude Damping Channel
(ADC) and discuss the time evolution of a two-level system using Kraus
operator formalism.

##### Amplitude damping channel

Amplitude damping channel [1, 2] is a noise model that captures the
dissipative interaction of a quantum system with its environment. For example,
spontaneous decay of the excited state of a two-level atomic system resulting
in emission of a photon. Consider a two-level systems ‘S’ in a cavity
interacting with the vacuum mode of the electromagnetic field at absolute zero
temperature. Initially, electromagnetic field (which is treated as environment
‘E’) is assumed to be in the vacuum state. Let us label the ground and excited
states of the system by $|g\rangle_{S}$ and $|e\rangle_{S}$, respectively, and
label $|0\rangle_{E}$ and $|1\rangle_{E}$ as the vacuum state and the one
photon state of the environment, respectively.

Evolution of the system and environment is unitary and represented by
following quantum map:

$\displaystyle\begin{split}|g\rangle_{S}|0\rangle_{E}&\rightarrow|g\rangle_{S}|0\rangle_{E}\\\
|e\rangle_{S}|0\rangle_{E}&\rightarrow\sqrt{1-p}|e\rangle_{S}|0\rangle_{E}+\sqrt{p}|g\rangle_{S}|1\rangle_{E}\end{split}$
(1.63)

For an atom initially in excited state, there is a probability $(1-p)$ that
the system remains in the excited state environment in vacuum state and there
is probability $p$ that system de-excites to ground state with an excitation
in the environment. Under the Born-Markov approximation; $p=1-\exp(-\Gamma
t)$, where $t$ denotes time and $\Gamma$ is the decay rate of the two-level
atomic system.

Thus, a superposition of ground and excited states of the two-level atomic
system evolves with the environment as follows:

$(\alpha|g\rangle_{S}+\beta|e\rangle_{S})|0\rangle_{E}\xrightarrow{U_{SE}}\alpha|g\rangle_{S}|0\rangle_{E}+\beta\left[\sqrt{1-p}|e\rangle_{S}|0\rangle_{E}+\sqrt{p}|g\rangle_{S}|1\rangle_{E}\right]$
(1.64)

By tracing out the environment in the $\\{|0\rangle_{E},\leavevmode\nobreak\
|1\rangle_{E}\\}$ bases, we get the Kraus operators as follows:

$K_{\mu}=\leavevmode\nobreak\
_{E}\langle\mu|U_{SE}|0\rangle_{E},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mu=0,\leavevmode\nobreak\ 1.$ (1.65)
$\Rightarrow\leavevmode\nobreak\ K_{0}=\left(\begin{array}[]{cc}1&0\\\
0&\sqrt{1-p}\end{array}\right),\leavevmode\nobreak\
K_{1}=\left(\begin{array}[]{cc}0&\sqrt{p}\\\ 0&0\end{array}\right).$ (1.66)

The Kraus operator $K_{1}$ governs the decay of the system from
$|e\rangle_{S}$ to $|g\rangle_{S}$ and $K_{0}$ governs the evolution when
there is no decay. These operators are non-unitary operators and satisfy the
completeness condition

$\sum_{i}K^{\dagger}_{i}K_{i}=\mathbb{I}$ (1.67)

Evolution of the system $\rho_{S}(0)$ is given by

$\rho_{S}(p)=\sum_{i=0}^{1}K_{i}\rho_{S}(0)K^{\dagger}_{i},$ (1.68)

The Eq. (1.68) governs the evolution of a system $\rho_{S}(0)$ with time in
the presence of an ADC.

  * •

Let us learn some basics of birefringence and non-linearity. Birefringence
plays a very important role in most of the polarization dependent optical
components such as half- and quarter-wave plates, polarizer, polarizing beam
splitters, and in the phase matching of non-linear crystals used for
spontaneous parametric down-conversion, etc.

###  1.6 Basics of birefringence and nonlinear optics

####  1.6.1 Anisotropy and birefringence

When an electric field E is applied to a dielectric media, it causes the
separation between positive and negative charge centres creating electric
dipoles. The net dipole moment P, electric field E, and electric displacement
vector D are related as

$\textbf{D}=\epsilon_{0}\textbf{E}+\textbf{P}.$ (1.69)

When dipoles are always oriented along the direction of applied electric
field, displacement vector D is parallel to the applied electric field E,
medium is called isotropic [23]. This is the case with the media which has
same properties in all the directions.

In an isotropic media, we have

$\textbf{P}=\epsilon_{0}\chi\textbf{E}\Rightarrow\textbf{P}\parallel\textbf{E}.$
(1.70)

Thus,

$\textbf{D}=\epsilon\textbf{E}\Rightarrow\textbf{D}\parallel\textbf{E}.$
(1.71)

where $\epsilon=\epsilon_{0}(1+\chi)$ is a scalar quantity, called dielectric
permittivity, and $\chi$ is electric susceptibility.

But, if the structure is not symmetric then dipoles may orient themselves in a
direction other than field E due to restoring forces from all the directions.
As a result $\textbf{P}\nparallel\textbf{E}$. Such a medium is called
anisotropic. In such a medium, we have

$\begin{split}D_{x}=\epsilon_{xx}E_{x}+\epsilon_{xy}E_{y}+\epsilon_{xz}E_{z},\\\
D_{y}=\epsilon_{yx}E_{x}+\epsilon_{yy}E_{y}+\epsilon_{yz}E_{z},\\\
D_{z}=\epsilon_{zx}E_{x}+\epsilon_{zy}E_{y}+\epsilon_{zz}E_{z}.\end{split}$
(1.72)

Or,

$\left(\begin{array}[]{ccc}D_{x}\\\ D_{y}\\\
D_{z}\end{array}\right)=\left(\begin{array}[]{ccc}\epsilon_{xx}&\epsilon_{xy}&\epsilon_{xz}\\\
\epsilon_{yx}&\epsilon_{yy}&\epsilon_{yz}\\\
\epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}\end{array}\right)\left(\begin{array}[]{ccc}E_{x}\\\
E_{y}\\\ E_{z}\end{array}\right).$ (1.73)

Or,

$\textbf{D}=\bar{\bar{\epsilon}}\textbf{ E}.$ (1.74)

In component form,

$D_{i}=\sum_{j}\epsilon_{ij}E_{j}\leavevmode\nobreak\ ;\leavevmode\nobreak\
i,j=1,2,3.$ (1.75)

It can be shown that the $\bar{\bar{\epsilon}}$ tensor is symmetric; i.e,

$\epsilon_{xy}=\epsilon_{yx},\leavevmode\nobreak\
\epsilon_{yz}=\epsilon_{zy},\leavevmode\nobreak\ \epsilon_{xz}=\epsilon_{zx}.$
(1.76)

One can rotate the coordinate system such that the $\epsilon$ matrix becomes
diagonal, this coordinate system is known as principal axis coordinate system
[23] and $\bar{\bar{\epsilon}}$ takes the following form:

$\bar{\bar{\epsilon}}=\left(\begin{array}[]{ccc}\epsilon_{xx}&0&0\\\
0&\epsilon_{yy}&0\\\ 0&0&\epsilon_{zz}\end{array}\right).$ (1.77)

In the principal axis system, if we rewrite the diagonal elements of
$\bar{\bar{\epsilon}}$ tensor as

$\displaystyle\begin{split}&\epsilon_{x},\leavevmode\nobreak\
&&\epsilon_{y},\leavevmode\nobreak\ &&\epsilon_{z}\leavevmode\nobreak\
\leavevmode\nobreak\ &&\text{: Principal permittivity}\\\ \textrm{then}\\\
&k_{x}=\frac{\epsilon_{x}}{\epsilon_{0}},\leavevmode\nobreak\
&&k_{y}=\frac{\epsilon_{y}}{\epsilon_{0}},\leavevmode\nobreak\
&&k_{z}=\frac{\epsilon_{z}}{\epsilon_{0}}\leavevmode\nobreak\
\leavevmode\nobreak\ &&\text{: Dielectric constant}\\\
&n_{x}=\sqrt{k_{x}},\leavevmode\nobreak\
&&n_{y}=\sqrt{k_{y}},\leavevmode\nobreak\
&&n_{z}=\sqrt{k_{z}}\leavevmode\nobreak\ \leavevmode\nobreak\ &&\text{:
Refractive indices}.\end{split}$ (1.78)

Case(I): If $\epsilon_{x}=\epsilon_{y}=\epsilon_{z}$, then medium is said to
be isotropic, for example: glass.

Case(II): If $\epsilon_{x}=\epsilon_{y}\neq\epsilon_{z}$, then medium is said
to be uniaxial anisotropic medium, for example: quartz.

Case(III): If $\epsilon_{x}\neq\epsilon_{y}\neq\epsilon_{z}$, then medium is
said to be biaxial anisotropic medium, for example: mica.

####  1.6.2 Non-linearity

In the linear optics, the response of a medium to an applied electric field E
is given by

$\textbf{P}=\epsilon_{0}\bar{\bar{\chi}}\textbf{E}\leavevmode\nobreak\ .$
(1.79)

This is known as linear approximation [23]. When the applied E field becomes
very large compared to the inter-atomic fields in the medium then linear
approximation fails. The oscillators are no more harmonic, and induced
polarization is given by

$P_{i}=\epsilon_{0}\chi^{(1)}_{ij}E_{j}+\epsilon_{0}\chi^{(2)}_{ijk}E_{j}E_{k}+\epsilon_{0}\chi^{(3)}_{ijkl}E_{j}E_{k}E_{l}+.....$
(1.80)

where $\chi^{(1)}_{ij}$, $\chi^{(2)}_{ijk}$, and $\chi^{(3)}_{ijkl}$ are
linear, second-, and third-order nonlinear optical susceptibilities,
respectively. The $\chi^{(2)}_{ijk}$ plays a very important role in many
nonlinear optical phenomena, e.g. second harmonic generation, parametric
generation, etc. It has 27 elements and if a medium possesses center of
inversion symmetry then all the 27 elements vanish and such a crystal doesn’t
exhibit nonlinear effects.

  * •

We will now discuss some basic concepts related to Spontaneous Parametric
Down-Conversion and how phase matching is achieved in a birefringent medium.
This process is an important tool for the generation of single and entangled
photons.

###  1.7 Spontaneous Parametric Down-Conversion

Generally speaking, Spontaneous Parametric Down-Conversion (SPDC) [24, 25] is
the process by which a parent photon decays into arbitrary number of lower
energy daughter photons. For our purposes, we identify SPDC as a second order
non-linear optical process wherein a high energy pump photon ($\omega_{p}$),
in the presence of a non-linear medium, spontaneously splits into two lower
energy photons; historically known as signal (photon with higher frequency,
$\omega_{s}$), and idler (photon with lower frequency, $\omega_{i}$), due to
scattering by the zero point fluctuation of vacuum. This process is known as
SPDC because it is generated by quantum vacuum fields (spontaneous), initial
and final quantum mechanical states of the medium are identical and photon
energy is always conserved (parametric), and signal and idler frequencies are
lower than the pump frequency (down-conversion). It is also known as
spontaneous parametric scattering (SPS), and optical parametric generation
(OPG). An example of type-II SPDC process (discussed below) resulting in
intersecting cones is shown below.

The SPDC is a three-wave mixing process which utilizes the second-order non-
linear susceptibility, $\chi^{(2)}$. It satisfies energy and momentum
conservation, collectively known as phase matching condition (discussed
below). If the energies of the signal and idler photons are equal (different),
the process is known as degenerate (non-degenerate) SPDC. The daughter photons
may be correlated in energy, momentum, time and spatial degrees of freedom.
Generally, these photons posses only classical correlations and special
techniques need to be employed to make them entangled.

####  1.7.1 Phase matching technique

Consider a non-centro-symmetric uniaxial birefringent crystal. For efficient
SPDC process, it is required that the phase velocities of all the interacting
waves (pump, signal, and idler) are equal, so that the intensity of down-
converted signal can coherently build up throughout the crystal at the expense
of pump beam. If the phase velocities of the pump, signal, and idler were
different then the energies would be exchanged back and forth between them,
resulting in inefficient SPDC.

The frequencies and wave vectors of the three-wave interaction is governed by
the following energy and momentum conservation laws, known as phase matching
condition [24-26]:

$\hbar\omega_{p}=\hbar\omega_{s}+\hbar\omega_{i}\leavevmode\nobreak\ ,$
(1.81a) $\textbf{k}_{p}=\textbf{k}_{s}+\textbf{k}_{i}+\Delta\textbf{k}.$
(1.81b)

where $\textbf{k}_{p}$, $\textbf{k}_{s}$, and $\textbf{k}_{i}$ are the k
vectors corresponding the frequencies $\omega_{p}$, $\omega_{s}$, and
$\omega_{i}$ of the pump, signal, and idler photons respectively, and
$\Delta\textbf{k}$ is the phase mismatch term.

$\gamma_{s}(\omega_{s})$$\gamma_{i}(\omega_{i})$Time$\gamma_{p}(\omega_{p})$
(a) Feynman diagram

$\omega_{p}$$\omega_{s}$$\omega_{i}$ (b) Energy conservation

$\textbf{k}_{p}$$\textbf{k}_{s}$$\textbf{k}_{i}$$\Delta\textbf{k}$ (c)
Momentum conservation

Figure 1.4: Non-collinear phase matching condition for the SPDC process where
a pump photon having energy $\hbar\omega_{p}$ gets absorbed and two lower
energy photons $\hbar\omega_{s}$ (signal) and $\hbar\omega_{i}$ (idler) get
emitted. The $\mathbf{k_{p},\leavevmode\nobreak\ k_{s}}$ and $\mathbf{k_{i}}$
are the momentum vectors of the pump, signal, and idler, respectively.

Phase matching is achieved whenever $\Delta\textbf{k}=0$. When all k vectors
are parallel, it is called collinear phase matching and when the k vectors are
non-parallel, it is called non-collinear phase matching. An example of non-
collinear phase matching is shown in Fig. 1.4.

Now,

$\textbf{k}=\frac{n\omega}{c}\leavevmode\nobreak\ \hat{i}\leavevmode\nobreak\
.$ (1.82)

where, n is the refractive index at frequency $\omega$, $c$ is the velocity of
light in vacuum, and $\leavevmode\nobreak\ \hat{i}$ is the unit vector
parallel to k.

Using Eq. (1.82), under phase matching condition, Eq. (1.81b) can be rewritten
as

$n_{p}\omega_{p}\hat{i}_{p}=n_{s}\omega_{s}\hat{i}_{s}+n_{i}\omega_{i}\hat{i}_{i}.$
(1.83)

If all the three waves propagate with parallel wave normals in the crystal,
i.e. $\hat{i}_{p}=\hat{i}_{s}=\hat{i}_{i}$, then Eq. (1.83) reduces to

$n_{p}=\frac{\omega_{s}}{\omega_{p}}n_{s}+\frac{\omega_{i}}{\omega_{p}}n_{i}\leavevmode\nobreak\
.$ (1.84)

There are two different ways in which phase matching condition can be
achieved: type-I and type-II phase matching.

1) Type-I phase matching: The signal and idler, both have same polarization,
i.e. either both are $o$-rays or both $e$-rays, but orthogonal to pump
polarization. In the case of a negative uniaxial crystal ($n_{o}>n_{e}$), an
extra-ordinary polarized high energy pump photon down-converts into two
ordinary polarized low energy photons.

$\textbf{k}^{e}_{p}(\theta)\rightarrow\textbf{k}^{o}_{s}+\textbf{k}^{o}_{i}.$
(1.85)

where $\theta$ is the angle between the pump propagation direction and the
optic axis of the crystal, inside it. Whereas, in the case of a positive
uniaxial crystal ($n_{o}<n_{e}$), an ordinary polarized high energy pump
photon down-converts into two extra-ordinary polarized low energy photons.

$\textbf{k}^{o}_{p}\rightarrow\textbf{k}^{e}_{s}(\theta)+\textbf{k}^{e}_{i}(\theta).$
(1.86)

As an example, schematic of a type-I SPDC source in paired-crystal geometry
(more details can be found in the next chapter [2]) resulting in a pair of
concentric overlapping cones is shown in Fig. 1.5. The cross section of such a
SPDC cone is a ring in the non-collinear configuration and a Gaussian spot in
the collinear configuration as shown in Fig. 1.6. These images were captured
from SPDC sources in our lab by Andor CMOS camera.

Figure 1.5: Schemtaic of a type-I non-collinear SPDC process resulting in
concentric cones. A pair of photons collected from diamtrically opposite sides
of the ring will be polarizatiion entangled. (Image Credit: S. N. Sahoo)
Figure 1.6: Type-I SPDC ring in collinear (left) and non-collinear (right)
configurations imaged by Andor CMOS camera.

2) Type-II phase matching: The signal and idler have orthogonal polarizations,
i.e., if one is $e$-ray, other is $o$-ray. In the case of a negative uniaxial
crystal, an extra ordinary polarized high energy pump photon down-converts
into one ordinary and one extra-ordinary polarized low energy photons.

$\textbf{k}^{e}_{p}(\theta)\rightarrow\textbf{k}^{e}_{s}(\theta)+\textbf{k}^{o}_{i}.$
(1.87)

Whereas, in the case of a positive uniaxial crystal, an ordinary polarized
high energy pump photon down-converts into one ordinary and one extra-ordinary
polarized low energy photons.

$\textbf{k}^{e}_{p}(\theta)\rightarrow\textbf{k}^{o}_{s}+\textbf{k}^{e}_{i}(\theta).$
(1.88)

As an example, schematic of a type-II SPDC source resulting in intersecting
cones is shown in Fig. 1.7. The cross section of such a SPDC cone is a pair of
intersecting rings in the non-collinear configuration and rings just touch
each other in the collinear configuration as shown in Fig. 8. These images
were captured from SPDC sources in our lab by Andor CMOS camera.

Figure 1.7: Schematic of type-II non-collinear SPDC process resulting in
intersecting cones [27]. Figure 1.8: Type-II SPDC ring in collinear (left) and
non-collinear (right) configurations imaged by Andor CMOS camera. In the
overlapping region of the two rings, one cannot comment whether which ring the
photon belongs to, hence a pair of photons collected from these regions will
be polarization entangled.

It should be noted that phase matching condition may not be satisfied in all
uniaxial crystals. The desirable optical properties of a crystal are high non-
linearity, large birefringence, and small dispersion. Depending on the
relative orientation of the k vectors of pump, signal, and idler, phase
matching is called collinear when all k vectors are parallel to each other,
and non-collinear when the k vectors are non-parallel.

The refractive indices of the $e$-ray ($n_{e}$) and $o$-ray ($n_{o}$)
propagating in a uniaxial with a common wave normal at an angle $\theta$ to
the optic axis are given by,

$n^{(o)}=n_{o}\leavevmode\nobreak\ ,$ (1.89a)
$n^{(e)}(\theta)=\frac{n_{o}n_{e}}{[n_{e}^{2}\sin^{2}(\theta)+n_{o}^{2}\cos^{2}(\theta)]^{1/2}}\leavevmode\nobreak\
.$ (1.89b)

where $n_{0}$ and $n_{e}$ are ordinary and extraordinary refractive indices of
the crystal.

As an example, we choose BBO which is a negative uniaxial crystal with point
symmetry 3. The effective non-linear optical constants [23] of BBO for type-I
phase matching are given by

$d_{\text{eff}}=[d_{11}\cos(3\phi)-d_{22}\sin(3\phi)]\cos(\theta)+d_{31}\sin(\theta),$
(1.90)

where $\theta$ and $\phi$ are usual polar and azimuthal angles.

The Sellmier’s equation for BBO are given by

$\begin{split}n^{2}_{o}=2.7359+\frac{0.01878}{\lambda^{2}-0.01822}-0.01354\lambda^{2},\\\
n^{2}_{e}=2.3753+\frac{0.01224}{\lambda^{2}-0.01667}-0.01516\lambda^{2}.\end{split}$
(1.91)

where $\lambda$ is in microns. For $\lambda=0.405\leavevmode\nobreak\ \mu m$,
we get $n_{o}=1.611$ and $n_{e}=1.544$.

If $n_{o}$ and $n_{e}$ are known at each of the frequencies $\omega_{p}$,
$\omega_{s}$, and $\omega_{i}$ then the phase matching angle $\theta_{pm}$ can
be calculated for type-I or type-II phase matching using Eq. (1.84) and
appropriate condition as given in Eq. (1.89b). An example of phase matching
for negative uniaxial crystal is shown in Fig. 1.9.

ZY$\hat{i}$$\theta_{pm}$OOptics axis Figure 1.9: Wave surfaces for a negative
uniaxial crystal. The blue curves show the wave surfaces at fundamental
frequency and red curves show the wave surfaces for down-converted frequency.
The unit vector $\hat{i}$ points in the direction of index matched wave
normals, and $\theta_{pm}$ is the phase matching angle.

Although SPDC process produces the brightest photon sources, the efficiency of
SPDC is about 1 in $10^{9}$. Hence, we use “Non-depleted pump approximation"
for SPDC process. In the semi-classical analysis of SPDC, the pump beam is
treated classically, and signal and idler fields are quantized [25]. The
interaction Hamiltonian is then written as

$\begin{split}\hat{H}&=\int_{v}d^{3}r\leavevmode\nobreak\
\chi^{(2)}\mathbf{E}_{p}(\mathbf{r},t)\hat{\mathbf{E}}^{(-)}_{s}(\mathbf{r},t)\hat{\mathbf{E}}^{(-)}_{i}(\mathbf{r},t)+h.c.\\\
&=\frac{1}{v}\int_{v}d^{3}r\leavevmode\nobreak\
\chi^{(2)}\mathbf{E}_{p}e^{i(\mathbf{k}_{p}.\mathbf{r}-\omega
t)}\sum_{\mathbf{k}_{s},m}\hat{\mathbf{P}}_{\mathbf{k}_{s},m}\sqrt{\frac{\hbar\omega\mathbf{k}_{s}}{2\epsilon_{0}}}\hat{a}^{\dagger}_{\mathbf{k}_{s},m}e^{i(\mathbf{k}_{s}.\mathbf{r})}\sum_{\mathbf{k}_{i},n}\hat{\mathbf{P}}_{\mathbf{k}_{i},n}\sqrt{\frac{\hbar\omega\mathbf{k}_{i}}{2\epsilon_{0}}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\hat{a}^{\dagger}_{\mathbf{k}_{i},n}e^{i(\mathbf{k}_{i}.\mathbf{r})}+h.c.\end{split}$
(1.92)

where $\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i}$ are the pump, signal and
idler wave vectors, and
$\hat{\mathbf{P}}_{\mathbf{k}_{p}},\hat{\mathbf{P}}_{\mathbf{k}_{s}},\hat{\mathbf{P}}_{\mathbf{k}_{i}}$
are corresponding polarization vectors.

As soon as pump beam enters the non-linear crystal, SPDC process begins. The
phase matching condition (discussed above) ensures that the down-conversion
probability amplitudes add constructively throughout the crystal.

###  1.8 References

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##  Chapter 2 Preparation and characterization of polarization entangled
photon source

In this chapter, we discuss the experimental techniques for preparation and
characterization of a polarization Entangled Photon Source (EPS). We give a
theoretical framework for and experimental realization of a Spontaneous
Parametric Down-Conversion (SPDC) based type-I polarization-entangled photon
source. The two-crystal geometry, where optic axes of the two crystals are
orthogonal to each other, is used in type-I phase matching to produce the
polarization entangled photons by pumping it with a diagonally polarized laser
beam. This scheme causes the spatial and temporal walk-offs leading to
decoherence in such sources when Continuous Wave (CW) diode lasers with low
coherence time are used as pump. We discuss the origin of decoherence
mechanism and its compensation using specially designed crystals. The source
is characterized by complete two-qubit Quantum State Tomography (QST) - a
technique to reconstruct the state (density matrix) of an unknown ensemble of
quantum systems through a series of measurements.

To begin with, we introduce the concept of tomography in classical and quantum
systems and then give a brief introduction to Stokes parameter and how
polarization state of light can be expressed in terms of Stokes parameters.
This is followed by experimental setup for the measurement of Stokes
parameters and its relation to polarization density-matrix of qubits. Then we
give a detailed discussion on single- and two-qubit QST. The tomographic
measurements are inevitably affected by statistical errors, imperfect
measurement settings, and device imperfections and if we directly reconstruct
the density matrix from such measurements, the state may not be physical. To
avoid this issue, we use Maximum Likelihood Estimation (MLE) to ensure that
the reconstructed density matrices be necessarily physical - i.e, Hermitian,
unit trace, and positive, even with imperfect measurements.

Degree of entanglement is quantified by computing Concurrence of the
reconstructed density matrix and closeness of the reconstructed state with the
ideal/fiducial state is quantified by fidelity which is a measure of overlap
between the two states. Effect of spectral filtering on the purity and
Concurrence of two-qubit entanglement is discussed. In the end, we briefly
comment on the two-qubit entanglement characterization by measuring visibility
of entanglement and make a remark on how Bell-state fidelity can be inferred
by a smaller set of measurements in Mutually Unbiased Bases (MUBs).

###  2.1 Quantum state tomography (QST)

The word tomography has originated from greek words ‘tomos’ meaning ‘slice or
section’ and ‘graphein’ meaning ‘to write’. Thus, the word ‘tomography’ can be
loosely translated to ‘section imaging’. A very popular injury diagnosing tool
in medical science is what is known as computerized tomography scan (or CT
scan) which uses a programmed computer and rotating X-ray machine to generate
different cross-sectional images of the human body. From a set of images that
contain reduced information about the object, the complete object is
reconstructed. For example, the 3D content of a picture can be extracted from
several 2D pictures taken from different directions. This process of
generating images is known as tomography, the device used is called tomograph,
and the image generated is called tomogram. This is also the basic idea of
quantum state tomography (QST) wherein a series of measurements (known as
tomographic set) are performed in different bases onto the quantum system for
complete characterization of the quantum state.

While characterizing the classical systems, it may be possible to perform a
series of measurements on the same system, hence a single copy of the
classical system would suffice for its complete characterization. But, in the
case of quantum systems, from the measurement postulate of quantum mechanics
we know that the very act of measurement disturbs the state of the quantum
system being measured. Hence, further measurements on the same system wouldn’t
give any meaningful information. Due to this limitation, QST must be performed
on an ensemble of identically prepared systems in several stages. Therefore,