UNRUH-DEWITT QUANTUM COMPUTING:

REALIZING QUANTUM SHANNON THEORY WITH QUANTUM FIELDS

BY

ERIC W. ASPLING

BS, Binghamton University, 2018

DISSERTATION

Submitted in partial fulfillment of the requirements for

the degree of Doctor of Philosophy in Physics

in the Graduate School of

Binghamton University

State University of New York

2024

© Copyright by Eric William Aspling 2024

All Rights Reserved

Accepted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Physics

in the Graduate School of

Binghamton University

State University of New York

2024

03/29/2024

Stephen Levy, Chair

Department of Physics and Astronomy, Binghamton University

Michael Lawler, Faculty Advisor

Department of Physics and Astronomy, Binghamton University

Masatsugu Sei Suzuki, Member

Department of Physics and Astronomy, Binghamton University

Pegor Aynajian, Member

Department of Physics and Astronomy, Binghamton University

Mengen Wang, External Examiner

Department of Electrical and Computer Engineering, Binghamton University

Thomas Hartman, External Member

Department of Physics, Cornell University

## Abstract

Qubit-field quantum transduction provides numerous advantages to quantum
computing, such as device-specific error-correcting codes, efficient
scalability, and effective entanglement generation. An all-to-all connected
bus of qubits implanted around the outside of a topological insulator, allowed
to interact with the edge state, is a promising arena for transduction with
flying fermionic qubits. Unruh–DeWitt detectors have allowed quantum
information scientists to model entanglement properties of qubit-field
interactions in many settings in a field known as Relativistic Quantum
Information (RQI). Unruh–DeWitt detectors are useful tools to realize quantum
Shannon theory, a subset of the theory of quantum communication, in condensed
matter systems, aptly named Unruh–DeWitt quantum computers. These systems will
provide quantitative measurements of communication in quantum materials that
utilize coherent states for bosonic and fermionic fields. In this thesis,
emphasis is placed on the well-studied theory of Tomonaga-Luttinger liquids,
as the bosonization of a helical Luttinger liquid provides a pedagogical arena
to construct RQI channels of fermionic systems. Multiple experimentally
realizable systems are proposed, and design constraints are constructed to
ensure maximum channel capacity. Furthermore, we elucidate the strength of
these quantum channels using measurements from quantum Shannon theory such as
coherent information, dephasing formalism, diamond distance and universality
of Unruh–DeWitt quantum logic gates.

I dedicate this work to my loving wife Amanda, who sacrificed and endured many
years to help me achieve this dream.

## Acknowledgements

I am forever grateful for the opportunities I have been given and Binghamton
University. I want to thank my mentors and advisors Michael Lawler, Bruce
White, and Charles Nelson. All of whom gave me room to explore the curiosity
that inspired me to go back to school. Without your mentorship this would not
be possible.

I would also like to thank my classmates and peers, Dennis Dempsey, Chris
Renskers, Mabrur Ahmed, Saba Kharabadze, and Justin Kulp and many more from my
communities at Binghamton and Cornell. The endless proof reading,
conversations, and encouragement was more than I could have asked for. Without
your friendship and collaboration, I couldn’t have finished this degree.

Lastly, I’d like to thank my family. My beautiful and loving wife Amanda, my
amazing children Ava, Xander, Emery, and Lillia, my supportive and encouraging
parents, Scott and Cathy, and my brother Steev, who always encouraged my
dreaming. I was lucky to have such a strong support system to help me complete
this degree. I cannot begin to name the ways I received support for they are
far to numerous. With all my heart, Thank you.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation
    2. 1.2 Quantum Troubles With Infinities
    3. 1.3 Unruh–DeWitt Detectors and Quantum Computers
    4. 1.4 Unruh–DeWitt Quantum Communication Theory
    5. 1.5 Physical Realizations
      1. 1.5.1 Case One: Luttinger Liquids and Quantum Dots
      2. 1.5.2 Case Two: Doped TMDs
    6. 1.6 Outline of Results.
  2. 2 The Unruh-DeWitt Detector: The Quantum Computing Brand
    1. 2.1 History and Definitions
    2. 2.2 Field Observables and Monopole Moments
      1. 2.2.1 The Field and Conjugate Momentum
    3. 2.3 Coherent States Developing Constraints
      1. 2.3.1 Coherent States
      2. 2.3.2 Unruh–DeWitt Quantum Gates
      3. 2.3.3 UDW State Transfer Channel
  3. 3 Abelian Bosonization of a Helical Luttinger Liquid
    1. 3.1 Motivations
      1. 3.1.1 Kondo Model Bosonization of Fermions
      2. 3.1.2 Bosonizing the Hamiltonian
      3. 3.1.3 A Zoo of interactions
    2. 3.2 Charge and Spin Bosonization
    3. 3.3 Coherent States and Their Problems.
  4. 4 Quantum Information and Quantum Computing
    1. 4.1 Introduction to Information Channels
    2. 4.2 Classical Information Channels
    3. 4.3 Quantum Information Channels
    4. 4.4 Determining if a Quantum Channel is Entanglement Breaking
      1. 4.4.1 The Quantum Channel
      2. 4.4.2 Separable States
      3. 4.4.3 Breaking Entanglement
  5. 5 Applications of Shannon Theory to UDWQCs
    1. 5.1 The Skeleton of a Quantum Computer
    2. 5.2 A Brief Review of the System
    3. 5.3 UDWQC Quantum Channels
    4. 5.4 Possible Gates For Our Quantum Computer
      1. 5.4.1 Gate Formations of Interest
      2. 5.4.2 Left- and Right-Moving Gates
      3. 5.4.3 Dirac-type Current Hamiltonians
    5. 5.5 Unruh–DeWitt Bosonic Dephasing-like Channels
      1. 5.5.1 The UDW Channel as a Bosonic Dephasing Channel
    6. 5.6 Applications of a UDW Channel From a Bosonic Dephasing perspective.
      1. 5.6.1 Canonical Bosonic Dephasing Channel
      2. 5.6.2 Cross-Talk Noise
  6. 6 Universal UDWQC with Analogous Quantum Logic Gates
    1. 6.1 Canonical Quantum Logic Gates
      1. 6.1.1 Projectors and Quantum Logic Gates
      2. 6.1.2 Quantum Computing Operations from Logic Gates
      3. 6.1.3 Intermediate Qubit
    2. 6.2 Introducing Fields into the Channels
      1. 6.2.1 Coherent Basis States of the Field
      2. 6.2.2 Encoding and Decoding and the UDW QST
      3. 6.2.3 SWAP gate with fields
    3. 6.3 Applications in Quantum Shannon Theory: Diamond Norm
    4. 6.4 Universal Quantum Computing
      1. 6.4.1 Forms of a single CNOT channel
  7. 7 Future Work
    1. 7.1 summary of results
    2. 7.2 Future Work and Open Problems
    3. 7.3 Entanglement and the DiVincenzo Criteria
    4. 7.4 A Completed Field Thoery
      1. 7.4.1 The problems with Impurity Scattering.
      2. 7.4.2 The Running Coupling
      3. 7.4.3 Nonlinear Couplings
      4. 7.4.4 Fermionic Coherent States
    5. 7.5 Connections to Cosmology
      1. 7.5.1 Path Integration and Quantum Shannon Theory
    6. 7.6 Conclusion
  8. A Classical Shadow Tomography
    1. A.1 Classical Shadows
  9. B Point-Splitting for a New Hamiltonian
    1. B.1 The $\hat{\psi}^{\dagger}\hat{\psi}$ coupling
    2. B.2 For a $\hat{\psi}^{\dagger}\partial_{z}\hat{\psi}$ interaction
  10. C Constraining the Momentum Displacement Operator

###### List of Figures

  1. 1.1 Figures (a-d) show a simplistic view of our quantum bus. Qubits (such as qubits A and B) are placed around the edges of a Luttinger liquid. The potential wall is lowered, as in Fig. (b) and the qubits are able to interact with the topological edge states. Figures (c) and (d) indicate how scaling up this system can be done easily by adjusting where the bulk of our fields lives through raising and lowering the potential barrier.
  2. (a) Qubit A can access all qubits on the system via left- and right-movers.
  3. (b) Opening up qubits will cause interactions with the left- and right-movers.
  4. (c) Adding another block of qubits restricts of our left- and right-movers to paths (1) or (2).
  5. (d) Opening the bulk between the blocks gives Qubits B and C direct access to the entirety of the qubits.
  6. 2.1 Operators $\hat{U}_{A\varphi}$ and $\hat{U}_{\varphi B}$ are employed to encode and decode quantum information onto and off of the field $\hat{\varphi}$ thus transferring the quantum information from qubit A to qubit B.
  7. 4.1 Graphical representation of H(x) for our fair-coin flipping system. Notice the maximum entropy occurs when our probability $x=1/2$.
  8. 4.2 A classical information channel that shows a flip operation acting on a coin. The coin is initially in state of heads but the flip operation changes this to tails.
  9. 4.3 Alice’s message is encoded into the elements $x_{i}$ where it is passed through some noisy channel $\mathcal{N}$ and later deciphered by Bob.
  10. 4.4 SWAP channels exchange quantum information between qubits as demonstrated with a reference qubit (Carla).
  11. 5.1 Encoding and Decoding quantum information onto and off of the field $\hat{\varphi}$.
  12. 5.2  Introducing another UDW gate acts as additional information entering the channel and subsequently will introduce cross-talk noise to the original channel in Eq. 2.24.
  13. 5.3 Comparing the coherent information of channels with and without noise, it is clear to see that the noisy channel $\Xi_{A\rightarrow B,N}(\lambda_{\varphi,b})$ reaches a channel capacity of one at a slower rate. At $b=1$ the noisy signal is the highest, this is due to a signal boost in the unwanted states. Regardless, we can see in (d) an increase in the lower bound of coherent information for small values of $\gamma_{\varphi}$. Code for simulations can be found at Ref. [1].
  14. (a) As $b\rightarrow 0$ the coupling as well as the noise is turning off.
  15. (b) As $b\rightarrow\infty$ it takes over the dephasing process for smaller values of $\gamma_{\varphi}$
  16. (c) As the noise is turning off the quantum capacity is increasing.
  17. (d) For large $b$ and small values of $\lambda_{\varphi}$ the additional interaction can result in better quantum capacity.
  18. 6.1 Two CNOT gates form a State Transfer Gate
  19. 6.2 Two CNOT gates form a State Transfer Gate as shown with reference qubits.
  20. 6.3 Proper SWAP circuit between qubits.
  21. 6.4 Information is swapped between Alice and Bob through Frank.
  22. 6.5 Information is transferred from Alice to Bob through Frank as demonstrated with a quantum channel $\mathcal{N_{A\rightarrow B}}$.
  23. 6.3 Applications in Quantum Shannon Theory: Diamond Norm
  24. 6.8 Information is transferred from Alice to Bob through Frank as demonstrated with a quantum channel $\mathcal{N_{A\rightarrow B}}$.
  25. 6.4.1 Forms of a single CNOT channel
  26. 7.1 All-to-All connectivity means that our Qdots are fully connected. One path here is shown by red arrows. There exists a shortest path from $t_{A}$ to $t_{C}$ for information defined on a Finsler Metric for the qubits but also on a 1+1 spacetime between them.
  27. A.1 Diagrammatic description of classical shadows showing a linear relationship between $S_{N}[\rho]$ and estimator $\sigma_{N}[\rho]\approx\rho$. (a) The full-density matrix $\rho$ can be approximated by summing over reduced classical shadows with a coefficient that grows exponentially in the number of remaining qubits. (b) A classical shadow $S_{N}[\rho]$ is obtained by summing over $N$ measurement outcomes on random Pauli bases. For a given $N$, a reduced density matrix, that involves smaller coefficients in the expansion, can be approximated more accurately compared to the full density matrix.
  28. A.2 Schematic overview of our study. (a) Classical shadows(CS) of ground states and time-averaged classical shadows(TACS) from dynamics of a time-reversal invariant GHZ state are generated using quantum simulation. (b) The classical data from quantum simulation is then fed into diffusion maps, an unsupervised machine learning algorithm to learn the phases. Credit to Gaurav Gyawali
  29. A.3  A sketch of the expected 1DTFIM phase diagram at finite $T$ and finite $N$ as a function of internal energy $E$ and transverse magnetic field $h_{x}$. This diagram is a modification of the canonical ensemble representation of the phase diagram in Ref. [3], adapted to the microcanonical ensemble. The spectrum is mirror symmetric about $E=0$ due to the chiral symmetry $\mathcal{C}=ZYZYZY...$
  30. A.4 Phase identification from dynamics data. The eigenvalue spectrum (inset) obtained from diffusion maps shows the two largest eigenvalues corresponding to the two dominant diffusion coordinates dc1 and dc2 (the trivial point $k=0$, is not shown). The TACS data points largely fall on a two-stranded curve parameterized by $h_{x}$ in this 2D reduced diffusion space. The quantum critical region (in green) coincides with the inflection point neighborhood of the curve, with points on the left strand belonging mostly to the ordered phase while points on the right strand belong to the disordered phase. Credit to Mabrur Ahmed.
  31. A.5 Interpretation of dc1 and dc2 for microcanonical dynamics of 1DTFIM. The divergent behavior of dc2 qualitatively matches the xx component of the susceptibility, computed using 100k shot TACS data for a 10-site 1DTFIM, denoted by orange circles (b) dc1 matches the Bayesian inference estimate for the second Renyi entropy per site ($S_{2}/n$) in the thermodynamic limit ($n=\infty$). Bayesian inference is performed on $n$-body entropies for $n=1-5$, also computed using the 10-site dataset. Credit to Mabrur Ahmed.

##  Chapter 1 Introduction

###  1.1 Motivation

Among the resources that measure the effectiveness of a good quantum computer
is the ability to generate and propagate entanglement. Quantum coherent
information and quantum capacity have become standard methodologies for
exploring this value [4, 5, 2]. However applications of quantum communication
in quantum materials will come in many forms, some of which require different
interaction Hamiltonians. Thus, a prescription method for numerically
simulating the parameters that determine the ability of the physical system to
carry out quantum communication is necessary, if these systems are to be
realized in a laboratory setting.

The past few decades have led to fruitful collaborations in Quantum Field
Theory (QFT) and Condensed Matter (CM) systems, so naturally, one might expect
to find this overlap in quantum computing as well [6, 7, 8, 9, 10]. However,
with the difficulties of decoherence present in quantum wires [11], there has
been little to no progress in advancing technologies that would allow such a
collaboration.

The seemingly obvious application of QFT interactions for Quantum Information
Science (QIS) purposes, is in the arena of photonics, where there has been a
large effort in 1-D quantum state transport through waveguides [12, 13, 14,
15, 16, 17]. On the other hand, lesser-known fermionic application provides a
means to probe quantum communication in quantum materials [18] utilizing QFT
applications of condensed matter systems. Further extensions to these models
include QIS in Valleytronics[19], Moìre pattern materials[20], and doped
Transition Metal Dichalcogenides[21, 22, 23], all of which qualify as models
that could be used to couple spin-qubits to quantum fields for transduction
purposes, but introduce new technologies for quantum wires.

Regardless, It remains a challenge to theoretically evaluate Quantum
Information (QI) rigorously in quantum materials[24]. Topologically protected
edge states generated by the Anomalous Quantum Hall Effect (AQHE) give rise to
phenomenon such as the edge states of LLs, transport along graphene
nanoribbons, and transport on the edges of Moiré patter materials [18]. These
materials offer promising arenas for carrying out quantum computing [24, 25].
However, to benchmark these computers theoretical tests of entanglement
processing is necessary.

When information is processed through a classical computer, every step of the
way, we can measure and conclude how this information has changed. Quantum
mechanics forbids this knowledge during quantum computing processes. Once the
classical input has entered the quantum circuit and has transformed into
quantum information, any measurement of that system will cause entanglement
decoherence of the measured qubits and subsequently some, if not all, of the
quantum information will be lost[26]. The act of measuring will revert our
quantum information back into classical information and without knowledge of
the full Quantum State Tomography of the system one cannot hope to know how
the information has been changed [5].

It is well understood, in quantum computing, that encoding information (such
as spin) via entanglement of particles, and measurements of the entanglement
as it propagates through the quantum circuit happen stochastically through
interactions with the environment[4]. Quantum error correcting codes have
shown that highly entangled states provide better information propagation[27,
6, 28, 29]. Therefore, a key aspect to building quantum computing systems
resides in the ability to generate and maintain entanglement between qubits.

Furthermore, transduction of this information via quantum fields can be
utilized similarly to classical information, for long range communication,
with the restriction of 1-D motion. Field-mediated quantum gates play a
natural role in the theory of quantum computing and quantum networking,
specifically in the regime of ground-to-satellite communication [30, 31, 32,
33, 34, 35, 36]. Long-range communication of photons in non-inertial frames of
reference, such as the gravitational well of the earth, indicate the need for
relativistic theories to explore relativistic effects [37, 38]. It is easy to
bog oneself down in the details of these relativistic theories. Formulating
and presenting comparisons of channels utilized in RQI and canonical channels
allows practitioners to forego the details of these theories.

###  1.2 Quantum Troubles With Infinities

Quantum Mechanics has the unusual way of working out beautifully in some
infinite limits, and completely breaking down in others. For example, to
achieve unitarity in measurement theory, one must sample infinite
measurements, a complication that does not exist in the Everettian
interpretation of quantum mechanics[39]. However, in a laboratory setting,
infinite measurements do not exist, yet generalizations of quantum state
tomography, such as the theory of classical shadows, are able to approximate
unknown states with calculable errors[40, 41, 42]. So long as limits of
infinity are not able to be evaluated, the true quantum state tomography of a
system can only ever be approximated with these errors. A brief overview with
applications that show the strength of classical shadows are presented in
Appx. A.

Another source of complications with quantum mechanics and infinity, is in
regards to a relativistic setting. QFT introduced a perturbative methodology,
that recovered (special) relativistic invariance, where accurate dynamics were
modeled through summing over infinite interactions between continuous fields
on Lorentz invariant backgrounds. These infinite interactions yielded infinite
energies that require renormalization to match experimental observations[43,
44, 45, 46]. Despite these complications, QFT proves to be one of the most
accurate descriptions of reality and connecting the formalism of QFT to
quantum computing has been a high-priority task for the last few decades
through simulations [7] or transduction.

Throughout this project we identify multiple ways that infinite limits work
well to realize the theory, but are not realistic in a laboratory setting. We
describe the ways that our systems have side-stepped these problems through
effective field theory limitations and non-perturbative solutions. However, a
theory of renormalization, complete with the structure of RG flow in the Kondo
model, is necessary to overcome the historical issues presented within the
following models. As this thesis is an introduction to QIS properties of a
novel system we save this investigation for future work.

###  1.3 Unruh–DeWitt Detectors and Quantum Computers

Over the past several decades one approach to unite the fields of QI and QFT
has been accomplished utilizing tools from cosmology, namely Unruh–DeWitt
detectors. As a subset of the field Relativistic Quantum Information (RQI),
Unruh-DeWitt (UDW) detectors model interactions between spin structures and
quantum fields [2, 47, 48, 49, 50, 51, 52, 53, 54].

Originally proposed as two-state systems coupled to quantum fields, UDW
detectors are used to model the detection of radiation in accelerating/non-
inertial frames of reference[55, 56, 57]. Extensions of this model have shown
that spin-qubits coupled to scalar fields for encoding and decoding quantum
information, can be used to formalize quantum information channels and measure
quantum Shannon properties such as quantum capacity[2, 48, 18, 58].

As mentioned above the two necessary components of quantum computing are
entanglement generation and propagation. UDW detector models have realized
both. Firstly, the generation of entanglement has been shown through extensive
work in a process called entanglement harvesting[48, 52, 47, 59, 60, 61, 62,
63]. More simplistically, this can be demonstrated through singular unitary
logic gate operations that will be formalized in Chapter 2.

Secondly, entanglement propagation via. state transfer[58] has been
demonstrated through aforementioned quantum Shannon theory measurements such
as von Neumann entropy, quantum coherent information and quantum capacity[2].
This process is understood to happen through applications of multiple
controlled unitaries [47] and therefore can draw some analogy to the canonical
state transfer channel which is constructed using two CNOT gates[5, 64].
Moreoever, we demonstrate other quantum logic gates that can be constructed
using UDW unitary gate operations such as the SWAP gate. Extensions of this
kind can lead to other insights in regards to quantum communication, such as
the diamond distance norm (also known as the diamond norm)[5, 4, 65].

###  1.4 Unruh–DeWitt Quantum Communication Theory

The diamond norm is a comparison measure of distinguishability between two
quantum channels and becomes especially useful when constructing novel quantum
channels that may have unique benefits, and relating them to well known
quantum channels that may prove to be cumbersome [66, 67, 4]. A diamond norm
of zero indicates that the channels in question result in the same output
density matrix with zero distinguishability. A product of having UDW quantum
channels written in a language analogous to canonical quantum computing, is
that the diamond norm can measure the “idealness” of a UDW quantum channel.
The UDW channels are set up to specifically model qubit-field interactions,
but as we will show, in limits of strong coupling this diamond distance proves
to demonstrate that the quantum state transfer channel, associated with UDW
detectors is identical to that of the canonical state transfer.

Formalizing UDW quantum gates as analogous quantum computing gates has other
benefits in quantum Shannon theory. One of the accomplishments of this thesis,
is the recognition that the UDW state transfer channel can be constructed in a
bosonic dephasing channel perspective. Sociologically, dephasing channels are
reserved to describe the decoherence in multipartite systems through the
systematic elimination of non-zero off-diagonals in the density matrix.
However, we will show that the dephasing of the off-diagonals in the coherent
state denstiy matrix, the computational basis of a quantum field, are exactly
the intention of a functional UDWQC. For this reason we use terms such as
“dephasing like” to avoid any sociological confusing.

RQI channels have a reputation for being tedious and lengthy in calculation.
With correlation functions (vacuum expectation values) of eight or more vertex
operators in some of the most simplistic situations. To make matters worse, if
one were to consider noisy interactions in RQI quantum channels this number of
vertex operators grows in size. These lengthy correlators can be not only
operationally complicated, there are nontrivial computational considerations.
However, as we will show in Chapter 5, in bosonic dephasing formalism, noisy
calculations can be simplified by adjusting the dephasing parameter, a
functional parameter in terms of three parameters; smearing, switching, and
coupling between field and qubits. This simplification avoids all extensions
to the complicated correlation functions, but the noise-free correlators will
still need to be calculated regardless.

###  1.5 Physical Realizations

There are extensions to the model proposed in this thesis via two potential
systems for realizing the study of quantum information flow in quantum
materials: CdTe-HgTe-CdTe heterostructures in the quantum spin Hall phase [68,
69] and doped Transition Metal Dichalcogenides (TMDs) and their applications
in either the anomalous or spin quantum-Hall regime [70, 71, 22, 23]. These
systems contain spin qubits coupled to topologically protected edge states and
allow a theoretical analysis combining the formalism of UDW detectors with the
bosonization of fermionic systems. This theory is restricted by the ability to
engineer these systems for exploring quantum information flow through quantum
fields. Building on the previous work of Simidzija et al[2, 47, 48], we show
that a non-zero channel capacities are possible in our model. Furthermore, we
provide pedagogical approach to evaluate quantum Shannon theoretic values with
numerical simulation. Through this pedagogy, one can extend our model to
evaluate these properties of the above experimental realizations.

####  1.5.1 Case One: Luttinger Liquids and Quantum Dots

The first extension of the UDWQC is through coupling quantum dots (Qdots) of
CdTe to a helical LL of HgTe [72]. A cartoon representation of this, given by
Fig. 1.1 depicts how the Qdots, isolated by a potential well, can be forced to
interact with the topologically protected edge states of the LL by lowering
the potential barrier that separates the two. Modelling the Qdots as Kondo
impurities, we recreate the two-state system common in the UDW detector
formalism.

(a) Qubit A can access all qubits on the system via left- and right-movers.

(b) Opening up qubits will cause interactions with the left- and right-movers.

(c) Adding another block of qubits restricts of our left- and right-movers to
paths (1) or (2).

(d) Opening the bulk between the blocks gives Qubits B and C direct access to
the entirety of the qubits.

Figure 1.1: Figures (a-d) show a simplistic view of our quantum bus. Qubits
(such as qubits A and B) are placed around the edges of a Luttinger liquid.
The potential wall is lowered, as in Fig. (b) and the qubits are able to
interact with the topological edge states. Figures (c) and (d) indicate how
scaling up this system can be done easily by adjusting where the bulk of our
fields lives through raising and lowering the potential barrier.

Developing the technology necessary to achieve this a formidable problem.
Luttinger liquids have only been realized experimentally in recent years in
HgTe and developing a system that maintains stable edge states that reliably
interact with impurities will be very challenging. Furthermore, in this thesis
we focus on impurity scattering of the fermionic free theory. This is not a
practical interaction block for realizing this theory physically, though
extensions to other interaction blocks of the Tomonaga-Luttinger liquid are
straightforward as demonstrated in Chapter 3. Instead, we aim to show the
effectiveness of quantum Shannon theory on our model, which is system
independent.

####  1.5.2 Case Two: Doped TMDs

Doped Monolayer TMDs are a far more stable and allow for first principle
investigations into the aforementioned interests of the CdTe-HgTe-CdTe model.
Dopants, such as vanadium atoms, display signs of mid-gapped, optically
excitable [23] states, which makes them good candidates for qubits.
Furthermore, doped monolayer TMDs such as Vanadium Tungsten Disulfide
($\text{V-WS}_{2}$), display room temperature ferromagnetic moments at
$~{}4\%$ doping[73]. We can infer the qubit nature of the dopants by examining
the ferromagnetic properties using experimental techniques such as cantilever
magnetometry.

The stability of doped atoms as qubits would allow for bilayer stacking and
interlayer communication between dopants as a first approach to examine the
spin-qubit communication in a UDWQC setting. In the future, replacing one of
the layers with a material such as MoTe, a material that experiences a
topologically protected edge state, will allow for investigations of qubit-
field transduction for quantum communication over distances in these fermionic
systems.

With these two experimental setups, near-term investigations into UDWQCs
provide insights into fermionic flying qubits, thermalization of quantum
materials, and general transduction properties of qubit-field systems.

###  1.6 Outline of Results.

In Chapter 2, we outline the idealized model we aim to study throughout this
program: Kondo-like impurities coupled to the topological edge states of the
Luttinger liquid model. While these models are well-understood, our approach
introduces important subtleties such as rapid switching functions, as seen in
Eq. 2.9, and Gaussian-like smearing. These terms are necessary to utilize the
current UDW program for RQI purposes and may potentially alter the physics of
the standard system.

In Sec. 2.3.2, we provide an overview of the RQI program outlined in Ref. [2].
This involves systematically developing UDW quantum gates designed to achieve
the Quantum State Transfer (QST) operation, as discussed in Sec. 2.3.3. The
QST channel facilitates the unidirectional transfer of quantum information
from one state to another. Chapter 4 offers detailed insights into the QI
aspects of this process, including scenarios involving entanglement-breaking
channels with unitaries similar to those described in Eq. 2.19.

In Chapter 3, we explore the process of bosonization as it applies to
Luttinger liquids and introduce the entire Tomonaga-Luttinger liquid model. We
highlight how different interaction blocks apply to various physical systems
of interest beyond our idealized model. Furthermore, we illustrate the
bosonization procedure using point-splitting methods in Sec. B and extend this
procedure to other blocks of the Hamiltonian, as outlined in Table 3.1. If one
intends to apply the results of this thesis to physical systems, the
utilization of this bosonized block-Hamiltonian will likely be necessary.

In Chapter 5, we present our first major result of the thesis: Our idealized
model of a Kondo-like impurity coupled to a helical Luttinger liquid can be
bosonized to generate the same unitaries as the QST channel. With this
formalism in mind we begin to explore our UDW quantum gates and the
applications to quantum Shannon theory. We revisit the pedagogy explained in
Chapter 2, but this time with the intent to show the similarities it possesses
to that of bosonic dephasing channels. In fact, one can demonstrate, as we do
in Eqs. 5.18 and 5.19, that the parameters of the UDW model can be wrapped up
into a dephasing parameter which accomplishes the usual goal of minimizing
off-diagonals in the coherent state density matrix.

Three key findings emerge through the lens of the UDW gate model resembling
dephasing processes. Firstly, in Sec. 5.6.1 we show that interacting with the
environment following the usual dephasing prescription introduces no
additional noise to the system so long as we remain working with field-
mediated channels. Secondly, in this dephasing-like perspective, introducing
additional detectors in a form of cross-talk interactions do not necessarily
contribute field-observables to the already cumbersome correlator in Eq. 2.25.
Instead one can utilize the dephasing parameter to account for the effects of
additional detectors, significantly reducing computational resources. Lastly,
evaluating these cross-talk channels have provided insights into unwanted
additional noisy interactions. One surprising result is that it is possible to
craft such a cross-talk interaction that increases the lower bound of the
coherent information as demonstrated in Fig. 5.3. These results can be found
in detail in Ref. [58].

In Chapter 6, we elucidate the projector form of UDW quantum gate formalism
and compare it to that of canonical logic gates. Under this prescription, we
are able to generate UDW quantum logic gates such as CNOT, Hadamard $H$, $S$,
and $T$-gates (See Sec. 6). This specific set of gates has the well-known
property of being universal. In other words, one can utilize these quantum
gates to generate any operation on a quantum computer. To prove the
effectiveness of our new UDW logic gates, we turn once again to the tools of
quantum Shannon theory, namely the diamond distance.

The diamond distance is a measure of distinguishability between quantum
information channels. A diamond distance of one indicates that the channels
are completely distinguishable, whereas a value of zero indicates
indistinguishability[4]. Our first step to show effectiveness is to compare
this value directly to the coherent information of our QST channel as these
results were previously shown. As one can see in Figs 6.7 and 6.7, the diamond
norm captures this indistinguishability quality of our UDW field-mediated QST
channel when compared to the qubit-qubit QST channel. We further utilize the
diamond distance metric to verify our field-mediated CNOT gates in several
arrangements in Figs. 6.10 and 6.10. The primary results of this chapter can
be found in Ref. [1].

A surprising consequence of the culmination of work thus far is that with the
introduction of universal quantum computing gates, we have theoretically
achieved all seven of the DiVincenzo criteria of quantum computing [74]. These
criteria, laid out in Chapter 7, are well-known requirements of a system to
carry out quantum computations. With these criteria in place, we describe
future work to realize options for first-principle investigations.

We close this thesis by describing in detail current and future projects that
aim to tackle several of the main open problems introduced throughout this
work. Furthermore, we demonstrate the wide range of impact this work seemingly
influences, including topics like faster-than-light signaling and black hole
evaporation.

##  Chapter 2 The Unruh-DeWitt Detector: The Quantum Computing Brand

###  2.1 History and Definitions

Simulations in quantum field theory (QFT) have long been expected to be a
practice of quantum computers (QC)[6, 7, 8, 9, 10]. Sociologically, this means
carrying out lattice QFT calculations using standard qubits. We hope to
address a different style of QC+QFT. Namely, field-mediated QC with UDW
detectors as two level systems for readouts. However, a physically realizable
experiment has many difficulties to get through. Including faster-than-light
signaling and information loss via entanglement decoherence [75, 2, 50, 76].
Massless Dirac fermions evaluated in UDW detector models have been a promising
endeavor for relativistic quantum information processes [77, 78, 79]. Right-
and left-movers of Helical Luttinger Liquids (HLL) (which have been
experimentally realized on 1+1 dimensional cylindrical space-times[68, 71,
69]), provide the right flavor of flying qubit to carry out these
computations, but offer a difficult arena for calculations as we’ll show in
Chapter 3.

Unruh–DeWitt detector models were originally designed to detect radiation,
apparent only in an accelerating frame of reference [55, 56]. However, the
same framework accurately represents stationary qubits coupled to quantum
fields. Through the smearing and switching functions of the UDW detector
model, quantum information can be propagated and generated through the fields
and collected by another detector in a process known as entanglement
harvesting [80]. The implementation of UDW models into quantum computing has
led to insights in information transformation in quantum computing. To
understand the relevance of detectors in quantum computing we start from the
ground up. We’ll need to establish a Hamiltonian in a form that is friendly to
quantum computing. To do this we will spend a fair deal of this chapter
dealing with definitions, complications, and resolutions that make this system
possible.

###  2.2 Field Observables and Monopole Moments

The process of encoding and decoding information onto fields in a quantum
circuit, like that of figure 2.1, requires understanding the system and its
constraints. In this thesis, we are restricted to 1+1 dimensional studies to
avoid complications such as the effects of Huygens’ principle which is
reserved for higher dimensional spacetimes and results from the radial motion
of quantum information through processes of thermalisation [2]. Furthermore,
considering topological edge states of helical luttinger liquids behave
similar to photonic wave guides for the right- and left-movers, our treatment
can be extended, without loss of generality to photonics. This fact further
exemplified in Chapter. 3 where we demonstrate that at the level of field-
mediated quantum channels, we can treat fermions and bosons as equivalent
through the process of bosonization[81].

Figure 2.1: Operators $\hat{U}_{A\varphi}$ and $\hat{U}_{\varphi B}$ are
employed to encode and decode quantum information onto and off of the field
$\hat{\varphi}$ thus transferring the quantum information from qubit A to
qubit B.

####  2.2.1 The Field and Conjugate Momentum

Let’s begin by establishing some definitions and procedures. Starting with the
scalar field and the conjugate momentum in the interaction picture,

$\displaystyle\hat{\varphi}(x,t)$ $\displaystyle=\int\frac{dk}{\sqrt{4\pi
k}}[\hat{a}(k)e^{i(\omega t-kx)}+\hat{a}^{\dagger}(k)e^{-i(\omega t-kx)}]$
(2.1) $\displaystyle\hat{\pi}(x,t)$ $\displaystyle=\int\frac{dk}{\sqrt{4\pi
k}}[-ik\hat{a}(k)e^{i(\omega t-kx)}+ik\hat{a}^{\dagger}(k)e^{-i(\omega
t-kx)}]$ (2.2)

where $\hat{a}^{\dagger}(\hat{a})$ are the usual creation (annihilation)
operators that follow the standard bosonic commutation relations
$[\hat{a}_{i},\hat{a}_{j}]=[\hat{a}^{\dagger}_{i},\hat{a}^{\dagger}_{j}]=0$
and $[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=\delta_{i,j}$. When working with
qubit-field interactions, it is common procedure to work with smeared out
observables $\hat{\mathcal{O}}$ with form

$\hat{\mathcal{O}}=\int dx(F_{1}(x\hat{\varphi}(x,t)+F_{2}(x)\hat{\Pi}(x,t))$
(2.3)

with smearing functions $F_{1}$ and $F_{2}$. Moving forward we will pick up
the common notation that

$\hat{\mathcal{G}}(F)=\int dxF(x)\hat{\mathcal{G}}(x,t).$ (2.4)

For the 1+1 HLL, the smearing fuctions $F_{1}$ and $F_{2}$ are assumed to be
equal. This not the case generally.

Sticking with the system of HLL coupled to Kondo impurities we can write down
a naive detector interaction Hamiltonian defined by

$\hat{H}_{\text{int}}(t)=\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)J_{\alpha,z}\hat{\mu}_{\alpha}(t)\otimes(\hat{\psi}^{\dagger}_{+}(y)\hat{\psi}_{+}(y)-\hat{\psi}^{\dagger}_{-}(y)\hat{\psi}_{-}(y)).$
(2.5)

Here $\hat{\psi}_{-}$ ($\hat{\psi}_{+}$) denote our spin-down, left-moving
(spin-up, right-moving) fermions (We discuss the structure of these fields in
Chapter 3). This interaction term assumes the Qdot limit where factors of
$e^{\pm 2ik_{F}y}$ have been suppressed as we restrict
$\sigma>>\frac{1}{k_{F}}$ [82]. Notice that Eq. 2.5 has many familiar
properties, but the introduction of the smearing and switching functions are
common in UDW formalism[83]. Since this model includes a Kondo impurity, some
may recognize our two-level system, which we treat as the qubit as,

$\displaystyle J_{\alpha,z}\hat{\mu}_{\alpha}(t)$
$\displaystyle=J_{xz}\hat{S}_{x}(t)+J_{yz}\hat{S}_{y}(t)+J_{zz}\hat{S}_{z}(t)$
(2.6) $\displaystyle
J_{xz}e^{i\hat{H}^{\prime}_{0}t}\hat{S}_{x}e^{-iH^{\prime}_{0}t}$
$\displaystyle=\frac{J}{2}(\hat{S}_{+}e^{-i\Omega t}+\hat{S}_{-}e^{+i\Omega
t})\equiv J\hat{\mu}(t)$ (2.7)

where $\hat{S}_{i}$ is the projector onto some spin state and $\Omega$ is some
real valued number that indicates a non-trivial energy difference between spin
states. The second equality follows by choosing $J_{\alpha,z}=JX_{\alpha}$ to
point along a new $\hat{X}$ direction and time dependence generated by
Hamiltonian
$\hat{H^{\prime}}_{0}=-g\hat{\mu}_{B}\vec{B}\cdot\vec{S}\equiv\Omega\hat{S}_{Z}$
with $\hat{Z}$ perpendicular to $\hat{X}$. We’ve added the magnetic field to
show that this system is a two-level system given by,

$\hat{\mu}(t)={\ket{1}\\!\\!\bra{0}}e^{-i\Omega
t}+{\ket{0}\\!\\!\bra{1}}e^{i\Omega t}$ (2.8)

It should be noted here that coupling to both the field and the conjugate
momentum separately is necessary to ensure that our channel is not
entanglement breaking. We will demonstrate that the resulting channel has a
zero channel capacity which is consistent with the literature. Furthermore,
the energy gap $\Omega$ cannot be vanishing, or the interaction will not
happen.

Our formulation takes the usual convention of a HLL and implants it into the
UDW model to describe a quantum channel between distant spin-qubits
communicating via Dirac fermions. It allows for an elegant promotion to a
quantum circuit model by constructing the Hamiltonian in the following way

$\hat{H}^{(\text{r},\nu)}_{\text{int}}(t)=J\chi(t)\hat{\mu}(t)\otimes\hat{\mathcal{G}}(F)_{\nu
r}^{\dagger}\hat{\mathcal{G}}(F)_{\nu r}.$ (2.9)

where $\nu$ specifies the interacting spin-qubit ($\nu\in\\{A,B\\}$ in Fig.
2.1), $r\in\\{\pm\\}$. The delta-like switching function $\chi(t)$, can be
seen as the instance of time $t_{\nu}$ of interaction for qubit $\nu$ such
that we can express the coupling as $J_{\nu}=J\chi(t_{\nu})$ . We can then
construct a set of unitary operators $\hat{U}_{A\mathcal{G}}$ and
$\hat{U}_{\mathcal{G}B}$ that take the form

$\hat{U}_{\nu
O}=\exp{(-iJ_{\nu}\hat{\mu}_{\nu}(t)\otimes\hat{\mathcal{G}}(F)_{\nu
r}^{\dagger}\hat{\mathcal{G}}(F)_{\nu r}}.$ (2.10)

Since we wish to work in a strong coupling regime, our choice of field
interactions that exist in the literature is far less. One that is commonly
used is a scalar boson field that couples linearly to the two state moment
$\mu(t_{\nu})$ [48]. In chapter 3 we discuss other possibilities for exploring
gates including strong coupling of quadratic interactions which is not well
understood in the literature. For now we establish the common formalism of the
usual scalar fields of the strongly coupled UDW detector system.

###  2.3 Coherent States Developing Constraints

####  2.3.1 Coherent States

A quick reminder of the 1-D time-independent scalar field $\hat{\varphi}(x)$
and it’s associated conjugate momentum $\hat{\Pi}(x)$ are given as

$\displaystyle\hat{\varphi}(x)$
$\displaystyle=\int\frac{dk}{2\pi}\sqrt{\frac{v}{2\omega(k)}}[\hat{a}(k)e^{ikx}+\hat{a}^{\dagger}(k)e^{-ikx}]$
(2.11) $\displaystyle\hat{\Pi}(x)$
$\displaystyle=\int\frac{dk}{2\pi}\sqrt{\frac{\omega(k)}{2v}}[-i\hat{a}(k)e^{ikx}+i\hat{a}^{\dagger}(k)e^{-ikx}].$
(2.12)

In this continuous space, these field observables can be expressed as
displacement operators $D[\alpha(x)]$ with a point-wise coherent amplitude

$\alpha_{\varphi}(x)=\sqrt{\frac{v}{2\omega(k)}}e^{-ikx}.$ (2.13)

For convenience we have introduced subscripts to differentiate between
coherent amplitudes of the field and conjugate momentum observables. A Fourier
transform following the usual prescription

$f(k)\coloneqq\frac{1}{\sqrt{(2\pi)}}\int dxf(x)e^{ikx}$ (2.14)

allows us to express a continuous displacement operator of a scalar bosonic
field[48] as

$\exp{(\pm i\hat{\varphi})}\ket{0}=\hat{D}[\alpha(k)]\ket{0}=\exp{\left(\int
dk\left[\alpha(k)\hat{a}_{k}^{\dagger}-\alpha(k)^{*}\hat{a}_{k}\right]\right)}\ket{0}\equiv\ket{\pm\alpha}.$
(2.15)

The final equivalence is a notational convenience we will adopt for the
majority of this thesis when carrying out simulations. This notational
convenience can be understood as single mode excitation of a continuous
spectrum in a very idealized sense. For now, as we continue to introduce
structure, we proceed with the more general form.

These field observables follow the equal-time canonical commutation relations
$[\hat{\varphi}(x),\hat{\varphi}(x^{\prime})]=0,\,[\hat{\Pi}(x),\hat{\Pi}(x^{\prime})]=0,$
and $[\hat{\varphi}(x),\hat{\Pi}(x^{\prime})]=i\delta(x-x^{\prime})$ and
consequently produce displacement operators that obey commutation relations

$[\hat{a}_{\varphi}(k^{\prime}),\hat{D}(\alpha_{\varphi}(k))]=\alpha_{\varphi}(k^{\prime})\hat{D}_{\varphi}(\alpha(k)).$
(2.16)

####  2.3.2 Unruh–DeWitt Quantum Gates

A unitary operator in the form of a quantum gate can generate and/or propagate
quantum information through a quantum circuit [5]. We expect this same
consequence to be achievable with UDW detector models [2]. Incorporating our
scalar field from Eq. 2.11 we present the well known UDW detector model as

$\hat{U}_{UDW}(t_{1},t_{2})=Te^{-i\int_{t_{1}}^{t_{2}}dt\hat{\mathrm{H}}_{int}(t)}$
(2.17)

with

$\hat{\mathrm{H}}_{int}(t)=J\chi(t)\int_{\mathbb{R}}dx\
F(k)\hat{\mu}(t)\otimes\hat{\varphi}(k,t).$ (2.18)

where $\chi(t)$ and $F(k)$ are the switching and smearing functions
respectively and $J$ is an effective coupling constant that indicates what
gate is interacting at what time and $\hat{\mu}(t)$ is a two state (qubit)
detector that has the form Eq. 2.7. By using a delta-like switching function
we can indicate the time ordering of our unitary and find that it simplifies
to

$\hat{U}_{UDW}=\exp{(-iJ\hat{\mu}\otimes\hat{\varphi})}.$ (2.19)

The structure of $\hat{\mu}$ as discussed above is that of a qubit observable
and for this we can rewrite the gate by introducing projectors $\hat{P}_{s}$
onto the eigenstates of $\hat{\mu}$ as

$\hat{U}_{UDW}=\sum_{s\in\pm}\hat{P}_{s}\otimes e^{is\hat{\varphi}(F)}.$
(2.20)

In this form we see that $\hat{U}_{UDW}$ is like a controlled unitary gate, it
acts with one unitary on the field in the Hilbert of the qubit and another
unitary in the Hilbert space of the field. This is part of the “encoding”
process that imparts the quantum information from the qubit to the field. Here
we have redefined the field observables to include the smearing and coupling
of the UDW model

$\displaystyle\hat{\varphi}(F)$ $\displaystyle\coloneqq J_{\varphi}\int
dxF(k)\hat{\varphi}(k,t)$ (2.21) $\displaystyle\hat{\Pi}(F)$
$\displaystyle\coloneqq J_{\Pi}\int dxF(k)\hat{\Pi}(k,t).$ (2.22)

One of the concerns of this theory is the way in which we treat the Hilbert
space of the field. For instance, we are allowing coupling between a Hilbert
space of dimension two $\mathcal{H}_{A}$ , with an infinite dimensional
Hilbert space $\mathcal{H}_{\phi}$ in the full space of
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{\phi}\otimes\mathcal{H}_{B}$
where $\phi$ is a scalar field. In the literature, it is common to treat the
Hilbert space of the field as two-dimensional coherent state Hilbert space, a
computational basis of the quantum field. This is a result of infinitely large
coupling strength and Gaussian-like spatial smearing. In reality, these things
are non-physical. However, for significantly large coupling the extra
dimensions of the Hilbert space minimally contribute in some calculable way,
and can be accounted for as noise.

Equation 2.20 captures one of the two criteria discussed in the beginning of
this section, namely entanglement generation. To accomplish the propagation of
QI and establish metric of effectiveness of that communication we’ll need to
evaluate a Quantum State Transfer (QST)

####  2.3.3 UDW State Transfer Channel

##### Setting the Channel Up

For a comprehensive review of quantum information channels and the history and
pedagogy involved, see chapter 4. It was shown in Ref. [2], that to write down
a channel that preserves entanglement and performs a state transfer, we would
need two of the controlled unitaries presented in Eq. 2.20. This will lead to
the unitary

$\hat{U}_{\nu\varphi}=\sum_{z,x\in\pm}\hat{P}_{x}\hat{P}_{z}\otimes
e^{ix\hat{\Pi}(F)_{\nu}}e^{iz\hat{\varphi}(F)_{\nu}}$ (2.23)

where we introduce the notation that $\hat{P}_{x}$, with an $x$-index, and
$\hat{P}_{z}$, with a $z$-index, are the projection operators onto the
eigenstates of Pauli matrices $\hat{\sigma}_{x}$ and $\hat{\sigma}_{z}$
respectively and $\nu$ indicates the qubit interacting with the field at a
given time $t_{\nu}$. Figure 2.1 presents an idealized circuit diagram for
this channel which has the mathematical structure

$\Xi_{A\rightarrow
B}=\operatorname{Tr}_{A\varphi}[U_{A\hat{\varphi}}U_{\varphi
B}(\hat{\rho}_{A,0}\otimes\hat{\rho}_{\varphi}\otimes\hat{\rho}_{B,0})U^{\dagger}_{\varphi
B}U^{\dagger}_{A{\varphi}}]$ (2.24)

and can be expanded out as

$\Xi_{A\rightarrow B}=\\\ \sum_{l,l^{\prime},x_{i},z_{i}}\langle
0|e^{iz_{1}\hat{\varphi}_{A}}e^{ix_{1}\hat{\Pi}_{A}}e^{ix_{2}\hat{\Pi}_{A}}e^{iz_{2}\hat{\varphi}_{A}}\\\
\times
e^{iz_{1}\hat{\varphi}_{A}}e^{ix_{1}\hat{\Pi}_{A}}e^{ix_{2}\hat{\Pi}_{A}}e^{iz_{2}\hat{\varphi}_{A}}|0\rangle\\\
\times\braket{l^{\prime}_{z}}{\hat{P}_{-z_{1}}\hat{P}_{-x_{1}}\hat{P}_{x_{4}}\hat{P}_{z_{4}}}{l_{z}}_{C}\bra{l^{\prime}_{z}}\\\
\otimes\hat{P}_{-z_{3}}\hat{P}_{-x_{3}}\ket{+_{y}}_{B}\bra{+_{y}}\hat{P}_{x_{2}}\hat{P}_{z_{2}}.$
(2.25)

The correlator in Eq. 2.25, which is a result of the trace over the coherent
states of the field, is a tricky value to calculate. One may recognize the
correlator as a vacuum expectation value of eight vertex operators, often
utilized in conformal field theory and string theory. If utilizing coherent
states, we recognize that they will lead to non-orthogonal inner products
$|\braket{+\alpha(k)}{-\alpha(k)}|$ that contain unwanted states, that if we
could dephase away, would improve the coherent information of the channel.
Furthermore, any additional interactions on the field will only yield a larger
correlator. In chapter 5 we also explore the possibility that a dephasing
perspective may eliminate the necessity to increase the size of the correlator
for additional UDW interactions.

##### Choosing the Correct Parameters

Within Eq. 2.18 and Eqs. 2.21 and 2.22, there are still a few free parameters
remaining. Namely, the coupling constants $J_{\varphi}$ and $J_{\Pi}$ as well
as the smearing function $F(k)$. We therefore utilize these parameters to
remove unwanted states that contribute to the noise in the qubit-field
interactions.

These unwanted states show up in two ways, firstly during the application of
$e^{ix\hat{\Pi}}$, which when applied to a coherent state will generally
change the value of the state, and secondly with non-zero values of the inner
product $|\braket{+\alpha(k)}{-\alpha(k)}|$. In RQI literature, it is common
to redefine the coherent amplitude, given in Eq. 2.13, to include the coupling
constants and the smearing function. By doing this we can utilize equation
2.16 and the following identity

$e^{\pm i\Pi}\hat{a}_{k}e^{\mp i\Pi}=\hat{a}_{k}\mp\alpha_{\Pi}(k)$ (2.26)

to prove the relation

$\hat{\Pi}\ket{\pm\alpha(k)}=\pm\gamma\ket{\pm\alpha(k)}+e^{i\Pi}\Pi\ket{0}$
(2.27)

which in the limit of $\gamma^{2}\gg\braket{0}{\Pi^{2}}{0}$ gives us the
following constraint in (1+1) dimensions

$\left(J_{\varphi}\int
dk\,|\tilde{F}_{\nu}(k)|^{2}\right)^{2}\gg\frac{1}{2}\int
dk\,\omega(k)\,|\tilde{F}_{\nu}(k)|^{2}.$ (2.28)

This constraint specifically allows for the application of $e^{ix\Pi}$ to a
coherent state and results in a phase constant (see Appx. C for a
demonstration of this constraint.). Furthermore, constraining this phase
constant by

$\Gamma\coloneqq J_{\Pi}J_{\varphi}\int
dk\,|\tilde{F}_{\nu}(k)|^{2}=\frac{\pi}{4}\,\mathrm{mod}\,2\pi$ (2.29)

results in a Bloch rotation of the output qubit A’s states that eliminates the
first batch of unwanted states.

The second group of unwanted states, and the states directly targeted for
dephasing, follow from the non-orthogonal inner product discussed at the end
of the previous section. The inner product has the form,

$|\braket{+\alpha(k)}{-\alpha(k)}|=\exp{\left[-(J_{\varphi})^{2}\,\int\frac{dk}{2\omega(k)}\,|\tilde{F}_{\nu}(k)|^{2}\right]}.$
(2.30)

Choosing a Gaussian smearing function with Gaussian width $\sigma$, and
setting $J\gg\sigma$ renders these terms approximately zero. With these
constraints, the coherent information of the field mediated communication
channel from qubit A to qubit B in Fig. 2.1 grows toward one as we increase
the strength of the coupling. Which is demonstrated in Fig. [4] of Ref. [2],
as well as Figs. 5.3.

These recent results are well understood in the RQI community, but follow from
the linear nature of the field observables in Eqs. 2.11 and 2.12. For certain
nonlinear cases, it becomes a challenge to redefine the coherent amplitudes to
include the coupling and smearing constants.

With the introduction of the QST channel, from the RQI literature, we claim
that the UDWQC system in some idealized scenario, exhibits effective
generation and propagation of entanglement, a minimal requirement for quantum
communication. We will revisit and derive this result in detail in Chapter 5.
In the remainder of the thesis we aim to show that the this idealized scenario
has some realistic attributes that may prove to be replicable in a laboratory
setting. Furthermore, we show several theoretical advances achieved through
the UDWQC program, that may prove to further our knowledge of quantum
transduction between qubits and fields. Before we demonstrate these
theoretical breakthroughs, we first outline the procedure of bosonization that
allows us to evaluate our fermionic fields as scalar bosons, making
correlators in Eq. 2.25 far more palatable.

##  Chapter 3 Abelian Bosonization of a Helical Luttinger Liquid

###  3.1 Motivations

The UDWQC program connects spin-qubits to condensed matter systems that are
understood through the guise of QFT. The current literature on UDW detector
model gives a perturbative prescription to some fermionic systems [79], but a
non-perturbative method is still lacking. A common prescriptive method in
condensed matter that connects fermions and bosons is through the process of
bosonization [84, 85, 82, 71]. With Bosonization we need only examine
literature of bosonic interaction. Furthermore, for maximal quantum
information generation and propagation, we are required in the realm of
quantum computing, to consider only strong couplings[2, 51].

A key fact of this thesis is that we are working with systems of right- and
left-moving fermions. However, the ability to bosonize our fermions in the HLL
allows our usage of the UDW model as there is a fair bit of literature on the
UDW model specifically with scalar bosons, and we intend to continue to build
on that literature. In this chapter, we will lay down the foundations and
formalisms of bosonization. Along the way we will recover Eqs. 2.20 and 2.23
and show the fermion to boson equivalence at the level of the correlator [81].

When working in the interaction picture we follow the standard prescription of
Wick rotating our variables as

$\displaystyle z$ $\displaystyle=-i(x-vt)$ $\displaystyle\bar{z}$
$\displaystyle=i(x+vt)$ (3.1) $\displaystyle\partial_{z}$
$\displaystyle=-\frac{i}{2}\left(\frac{1}{v}\partial_{t}-\partial_{x}\right)$
$\displaystyle\partial_{\bar{z}}$
$\displaystyle=-\frac{i}{2}\left(\frac{1}{v}\partial_{t}+\partial_{x}\right)$

and therefore are working in Euclidean space. There are several key
interactions left out of this thesis that we will describe in Chapter 7 such
as the time-evolution of the field after the information is encoded. For this
section we treat the system as a free fermion field interacting with a Kondo
impurity.

####  3.1.1 Kondo Model Bosonization of Fermions

Our “naive” Hamiltonian of Eq. 2.5 that was introduced in the previous
chapter, may not have been as niave as we stated. With topological edge states
of a HLL, our right and left movers stem from the Hamiltonian

$\hat{H}_{\mathrm{int}}(t)=\sum_{k,k^{\prime},\sigma,\sigma^{\prime}}J_{\alpha,\beta}\hat{\mu}_{\alpha}(t)\cdot(\hat{c}^{\dagger}_{k,\sigma}(k)\hat{\tau}^{\sigma,\sigma^{\prime}}_{\beta}\hat{c}_{k^{\prime},\sigma^{\prime}}(k)+h.c.).$
(3.2)

Where $J_{\alpha,\beta}$ is the coupling constant, $\hat{\mu}_{\alpha}(t)$ is
some “spin-state” which will be defined in more detail later, $\hat{\tau}$ a
the Pauli matrix and $\hat{c}^{\dagger}_{\sigma}(\hat{c}_{\sigma})$ are
fermionic creations (annihilation) operators. We can split this operator into
left- and right-moving parts such that
$\hat{c}_{\sigma}=\hat{c}_{\uparrow}+\hat{c}_{\downarrow}$ and likewise for
$\hat{c}^{\dagger}_{\sigma}$. Furthermore,

$\displaystyle\hat{c}_{\uparrow}(k)$
$\displaystyle=\hat{\psi}_{\uparrow,+}e^{ik_{F}x}$ (3.3)
$\displaystyle\hat{c}_{\downarrow}(k)$
$\displaystyle=\hat{\psi}_{\downarrow,-}e^{-ik_{F}x}$

and plugging the left- and right-moving fields into Eq. 3.2 we carry out the
dot product and recover a Hamiltonian,

$\hat{H}_{\mathrm{int}}(t)=J\hat{\mu}(t)\left(\hat{\psi}_{+}^{\dagger}\hat{\psi}_{+}-\hat{\psi}_{-}^{\dagger}\hat{\psi}_{-}+\hat{\psi}_{+}^{\dagger}\hat{\psi}_{-}(e^{2ik_{F}x})-\hat{\psi}_{-}^{\dagger}\hat{\psi}_{+}(e^{-2ik_{F}x})\right)$
(3.4)

where $\hat{\psi}_{+}(\hat{\psi}_{-})$ is shorthand for
$\hat{\psi}_{+}(\bar{z})(\hat{\psi}_{-}(z))$ which are the right- and left-
moving fermion fields and we have dropped the indices $\uparrow$ and
$\downarrow$ as they have become redundant.

####  3.1.2 Bosonizing the Hamiltonian

We aim to express our fermionic fields in left- and right-moving bosonic
fields $\hat{\phi}$ and $\hat{\bar{\phi}}$.

$\displaystyle\hat{\phi(z)}$
$\displaystyle=\int_{k>0}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}[\hat{b}(k)e^{-kz}+\hat{b}^{\dagger}(k)e^{kz}]$
(3.5) $\displaystyle\hat{\bar{\phi}}(\bar{z})$
$\displaystyle=\int_{k>0}\frac{dk}{2\pi}\frac{1}{\sqrt{2k}}[\hat{\bar{b}}(k)e^{-k\bar{z}}+\hat{\bar{b}}^{\dagger}(k)e^{k\bar{z}}]$
(3.6)

Following the prescription in [85], we define bosonic fields $\hat{\varphi}$
and conjugate momentum $\hat{\Pi}$ defined by
$\hat{\Pi}=\frac{1}{v}\partial_{t}\hat{\varphi}$,

$\displaystyle\hat{\varphi}(x)$
$\displaystyle=\int\frac{dk}{2\pi}\sqrt{\frac{v}{2\omega(k)}}[\hat{b}(k)e^{ikx}+\hat{b}^{\dagger}(k)e^{-ikx}]$
(3.7) $\displaystyle\hat{\Pi}(x)$
$\displaystyle=\int\frac{dk}{2\pi}\sqrt{\frac{\omega(k)}{2v}}[-i\hat{b}(k)e^{ikx}+i\hat{b}^{\dagger}(k)e^{-ikx}].$
(3.8)

as well as the dual boson $\hat{\vartheta}$ such that the following relations
hold.

$\displaystyle\hat{\phi}$
$\displaystyle=\frac{1}{2}(\hat{\varphi}+\hat{\vartheta}),$
$\displaystyle\hat{\bar{\phi}}$
$\displaystyle=\frac{1}{2}(\hat{\varphi}-\hat{\vartheta})$ (3.9)

from here we evaluate the left and right mode expansion of our scalar field
and following commutation relations we develop the following expressions for
our fermion fields.

$\displaystyle\hat{\psi}_{+}(z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}e^{-i\sqrt{4\pi}\hat{\phi}(z)},$
$\displaystyle\hat{\psi}_{+}^{\dagger}(z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}e^{i\sqrt{4\pi}\hat{\phi}(z)},$ (3.10)
$\displaystyle\hat{\psi}_{-}(\bar{z})$
$\displaystyle=\frac{1}{\sqrt{2\pi}}e^{i\sqrt{4\pi}\hat{\bar{\phi}}(\bar{z})},$
$\displaystyle\hat{\psi}_{-}^{\dagger}(\bar{z})$
$\displaystyle=\frac{1}{\sqrt{2\pi}}e^{-i\sqrt{4\pi}\hat{\bar{\phi}}(\bar{z})}.$

These equations are an essential part of bosonization and will allow us to
transform the fermions of Eq. 3.2 into bosons.

When evaluating the left- and right-mover terms of the interaction Hamiltonian
we employ the method of point-splitting to find,

$\displaystyle\hat{\psi}^{\dagger}_{+}(z)\hat{\psi}_{+}(z)$
$\displaystyle=\frac{i}{\sqrt{\pi}}\partial_{z}\hat{\phi}(z)$ (3.11)
$\displaystyle\hat{\psi}^{\dagger}_{-}(\bar{z})\hat{\psi}_{-}(\bar{z})$
$\displaystyle=-\frac{i}{\sqrt{\pi}}\partial_{\bar{z}}\hat{\bar{\phi}}(\bar{z})$
(3.12)

these derivations and the derivation of the resulting Hamiltonian can be found
in Appendix B. Here we list the result

$\displaystyle\hat{H}_{\mathrm{int}}(t)$
$\displaystyle=J\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(\partial_{z}\hat{\phi}+\partial_{\bar{z}}\hat{\bar{\phi}}))$
(3.13) $\displaystyle=J\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(2\hat{\Pi}))$ (3.14)

This is a significant consequence as we are able to take our quadratic
fermionic interaction and turn it into a linear coupling with a scalar boson.
This gives us a rank-one unitary like that of Eq. 2.20. As we’ll find out in
Chapter 4 a channel made with solely this unitary is inherently entanglement
breaking. However, our bosonization process will give us some alternatives to
consider.

####  3.1.3 A Zoo of interactions

The entirety of the Tomonaga–Luttinger liquid can be expressed as

$\hat{H}=\hat{\psi}^{\dagger}(x)\left(J_{1}i\nu_{F}\partial_{x}+J_{2}\Delta\right)\hat{\psi}(x)+J_{3}(\hat{\rho}+\hat{j})+J_{4}(\hat{\psi}_{+}^{\dagger}\hat{\psi}_{-}(e^{2ik_{F}x})-\hat{\psi}_{-}^{\dagger}\hat{\psi}_{+}(e^{-2ik_{F}x}))\\\
+2J_{5}\hat{\rho}_{+}(x)\hat{\rho}_{-}(x)+J_{6}\left(\hat{\rho}_{+}(x)^{2}+\hat{\rho}_{-}(x)^{2}\right).$
(3.15)

For simplicity we introduce the shorthand
$\hat{\rho}_{+}=\hat{\psi}_{+}(x)^{\dagger}\hat{\psi}_{+}(x)$ and
$\hat{\rho}_{-}=\hat{\psi}_{-}(x)^{\dagger}\hat{\psi}_{-}(x)$ so that we can
express the electron and current densities as

$\displaystyle\hat{\rho}$
$\displaystyle=\hat{\rho}_{+}+\hat{\rho}_{-}=\frac{1}{\sqrt{\pi}}\partial_{x}\hat{\varphi}$
(3.16) $\displaystyle\hat{j}$
$\displaystyle=\hat{\rho}_{+}-\hat{\rho}_{-}=\frac{1}{\sqrt{\pi}}\hat{\Pi}.$
(3.17)

The electron density Eq. 3.16 is not time-reversal symmetric when combined
with our Kondo impurity spin structure. Thus, this interaction is not
physical. Nonetheless, it is clear to see that a Hamiltonian that includes
both densities can be constructed as a toy model to demonstrate a LL
Hamiltonian that realizes nonzero channel capacity.

We have in the Tomonaga–Luttinger liquid plus Kondo impurity model, block
interactions that are relevant to particular condensed matter systems. For an
example we consider the term $J_{3}\rho$ we maybe be describing a Helical
Luttinger Liquid such as HgTe. Therefore, evaluating the interaction blocks of
the Hamiltonian will provide insights to individual systems.

Bosonization of Eq. 3.15 is presented in Table 3.1. We propose that some of
these interaction blocks provide nonzero channel capacities that could be
experimentally realized. Through the prescription method provided in this
thesis, one can pick and choose components to realize these experiments, but
not all of these blocks are made equally.

Interaction Type |  Fermion Form |  Boson Form |  Coupling type
---|---|---|---
Electron Density + Fermion Density |  $\hat{\psi}_{+}^{\dagger}\hat{\psi}_{+}-\hat{\psi}_{-}^{\dagger}\hat{\psi}_{-}+\hat{\psi}_{+}^{\dagger}\hat{\psi}_{+}+\hat{\psi}_{-}^{\dagger}\hat{\psi}_{-}$ |  $\frac{1}{\sqrt{\pi}}\partial_{x}\hat{\varphi}+\frac{1}{\sqrt{\pi}}\hat{\Pi}$ |  Linear
Free Dirac |  $\hat{\psi}^{\dagger}(x)\left(J_{1}i\nu_{F}\partial_{x}\right)\hat{\psi}(x)$ |  $\hat{\Pi}^{2}+(\partial_{x}\hat{\varphi})^{2}$ |  Quadratic
Backscattering |  $\hat{\psi}_{+}^{\dagger}\hat{\psi}_{-}(e^{2ik_{F}x})-\hat{\psi}_{-}^{\dagger}\hat{\psi}_{+}(e^{-2ik_{F}x})$ |  $\frac{1}{2\pi}\cos{\sqrt{4\pi}(\hat{\varphi})}$ |  Nonlinear
Umklapp |  $\sum_{si\in\pm}\hat{\psi}^{\dagger}_{s1}\hat{\psi}^{\dagger}_{s2}\hat{\psi}_{s3}\hat{\psi}_{s4}$ |  Cosines and Sines |  Nonlinear
Table 3.1: Different Blocks of the Tomonaga Luttinger liquid Hamiltonian and
thier respective bosonized form.

###  3.2 Charge and Spin Bosonization

The spinless bosonization treatment does not allows for the proper
anticommutation relations. Namely for the terms
$\hat{\psi}_{+}^{\dagger}\hat{\psi}_{-}(e^{2ik_{F}x})$ and
$\hat{\psi}_{-}^{\dagger}\hat{\psi}_{+}(e^{-2ik_{F}x})$. Interactions that
allow backscattering and also maintain proper anticommutation relations, we
introduce two spin bosons $\hat{\varphi}_{\uparrow}$ and
$\hat{\varphi}_{\downarrow}$, that are related by the charged and spin bosons,

$\displaystyle\hat{\varphi}_{c}$
$\displaystyle=\frac{1}{\sqrt{2}}(\hat{\varphi}_{\uparrow}+\hat{\varphi}_{\downarrow})$
$\displaystyle\hat{\varphi}_{s}$
$\displaystyle=\frac{1}{\sqrt{2}}(\hat{\varphi}_{\uparrow}-\hat{\varphi}_{\downarrow})$
(3.18)

as well as chiral fields $\hat{\phi}_{c,s}$ and $\hat{\bar{\phi}}_{c,s}$.
Furthermore, with the extra species of fermions, we utilize the Klein Factors
$\eta_{\mu}$ which follow anticommutation relations,

$\displaystyle\\{\eta_{\mu},\eta_{\nu}\\}$
$\displaystyle=\\{\bar{\eta}_{\mu},\bar{\eta}_{\nu}\\}=2\delta_{\mu,\nu}$
$\displaystyle\\{\eta_{\mu},\bar{\eta}_{\nu}\\}$ $\displaystyle=0$ (3.19)

and preserve the fermionic anticommutation relations discussed above. We now
have a new form of bosonized fermion fields given as,

$\displaystyle\hat{\psi}_{+}(z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\eta_{\mu}e^{-i\sqrt{4\pi}\hat{\phi}_{\mu}(z)},$
$\displaystyle\hat{\psi}_{+}^{\dagger}(z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\eta_{\mu}e^{i\sqrt{4\pi}\hat{\phi}_{\mu}(z)},$
(3.20) $\displaystyle\hat{\psi}_{-}(\bar{z})$
$\displaystyle=\bar{\eta}_{\mu}\frac{1}{\sqrt{2\pi}}e^{i\sqrt{4\pi}\hat{\bar{\phi}}_{\mu}(\bar{z})},$
$\displaystyle\hat{\psi}_{-}^{\dagger}(\bar{z})$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\bar{\eta}_{\mu}e^{-i\sqrt{4\pi}\hat{\bar{\phi}}_{\mu}(\bar{z})}.$

Using these definitions and recognizing that for this system we set
$\eta_{\mu}\bar{\eta}_{\nu}=1$ [85, 86], we can see now that our bosonized
Hamiltonian can be split into two sections impurity scattering (the terms
without the factors of $(e^{\pm 2ik_{F}x})$) which result follows from the
pointscattering methods in Appendix A, and our impurity backscattering terms.
The impurity scattering bosonized Hamiltonian is given by,

$\hat{H}^{F}_{\mathrm{int}}=J\hat{\mu}(t)(\frac{i}{\sqrt{\pi}}(2\hat{\Pi}_{c}+2\hat{\varpi}_{s}))$
(3.21)

where $\hat{\varpi}\equiv\frac{1}{v}\partial_{t}\hat{\vartheta}$ is the
conjugate momentum of the dual field. Notice here if we suppress the spin
terms (make the system spinless) we retrieve the same Hamiltonian in Eq. 3.13.
If we were able to suppress the $(e^{\pm 2ik_{F}x})$ terms as mentioned before
but consider both spin and charge we have two noncommuting observables that
could be used to create our non-zero capacity quantum channel.

The back-scattering Hamiltonian follows from eqs. 3.20,

$\hat{H}^{B}_{\mathrm{int}}=J\hat{\mu}(t)(\frac{1}{2\pi}\cos{\sqrt{2\pi(\hat{\varphi}_{c}+\hat{\vartheta}_{s})}})$
(3.22)

This could be used as well to create a different arrangement of operators
necessary to test for a non-zero capacity quantum channel. As we mentioned in
the previous chapter, to evaluate these systems we turn to the computational
basis states of a quantum field, coherent states.

###  3.3 Coherent States and Their Problems.

One recurring theme in this work, is the need to tackle the nonorthogonality
of coherent states. In Chapter 2, we showed that while you can never totally
eliminate the inner product of different coherent states, you can greatly
reduce it’s value through parameter tuning. While this is true for all
bosonized form of table 3.1, other problems begin to develop stemming from the
inner product of coherent states. To demonstrate this we look at the bosonized
form of the backscattering block.

Starting with the standard definitions of coherent states in field theory,

$\ket{\pm\alpha}\equiv\exp{(\pm
i\varphi)}\ket{0}=\hat{D}(\alpha)\ket{0}=\exp{\left(\int
d^{n}\textbf{k}\left[\alpha(k)\hat{a}_{\textbf{K}}^{\dagger}-\alpha(k)^{*}\hat{a}_{\textbf{K}}\right]\right)}.$
(3.23)

We want to be able to calculate

$e^{i\cos{\sqrt{4\pi}\varphi}}\ket{0},$ (3.24)

but to do this we first evaluate

$e^{\hat{D}(\alpha)}\ket{0}$ (3.25)

in the Fock Basis. This will allow for a straightforward Taylor expansion of
Eq. 3.24. Coherent states in the Fock basis are defined by

$\ket{\alpha}\equiv
e^{\frac{-|\alpha|^{2}}{2}}\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}.$
(3.26)

Using the Eq. 3.26, Baker–Campbell–Hausdorff (BCH), and the following
relation,

$\hat{D}(\alpha)\hat{D}(\beta)=e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}\hat{D}(\alpha+\beta),$
(3.27)

we can expand Eq. 3.25 to

$\displaystyle e^{\hat{D}(\alpha)}\ket{0}$
$\displaystyle=\ket{0}+\hat{D}(\alpha)\ket{0}+\frac{1}{2!}(\hat{D}(\alpha))^{2}\ket{0}+\frac{1}{3!}(\hat{D}(\alpha))^{3}\ket{0}...$
(3.28)
$\displaystyle=\ket{0}+\ket{\alpha}+\frac{1}{2!}\ket{2\alpha}+\frac{1}{3!}\ket{3\alpha}...$
(3.29)
$\displaystyle=\ket{0}+e^{\frac{-|\alpha|^{2}}{2}}\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}+\frac{1}{2!}e^{\frac{-|(2\alpha)|^{2}}{2}}\sum^{\infty}_{n=0}\frac{(2\alpha)^{n}}{\sqrt{n!}}\ket{n}$
$\displaystyle+\frac{1}{3!}e^{\frac{-|3\alpha|^{2}}{2}}\sum^{\infty}_{n=0}\frac{(3\alpha)^{n}}{\sqrt{n!}}\ket{n}+...$
(3.30) $\displaystyle=\sum_{m=0}^{\infty}\frac{1}{m!}\ket{m\alpha}$ (3.31)
$\displaystyle=\sum^{\infty}_{m=0}\sum^{\infty}_{n=0}e^{-\frac{|m\alpha|^{2}}{2}}\frac{(m\alpha)^{n}}{m!\sqrt{n!}}\ket{n}.$
(3.32)

While coherent states are not orthogonal, fock states are and therefore can be
normalized using the scheme;

$\displaystyle 1$
$\displaystyle=\sum_{n}\braket{\beta}{n}\\!\\!\braket{n}{\beta}$ (3.33)
$\displaystyle=N^{2}\sum^{\infty}_{m=0}\sum^{\infty}_{n^{\prime}=0}e^{-|m\alpha|^{2}}\frac{(m\alpha)^{2n^{\prime}}}{(m!)^{2}{n^{\prime}!}}$
(3.34)

where we have set $\ket{\beta}\equiv e^{\hat{D}(\alpha)}\ket{0}$. Using the
same line of reasoning above and remaining in the coherent state basis we find

$e^{i\cos{\sqrt{4\pi}\varphi}}\ket{0}=e^{\hat{D}(-\alpha)}e^{\hat{D}(\alpha)}\ket{0}=N\sum_{m=0}^{\infty}\sum_{m^{\prime}=0}^{\infty}\frac{1}{m!}\frac{1}{(m+m^{\prime})!}(\ket{m\alpha}).$
(3.35)

The first equality follows from Eulers formula and BCH and the phase factors
have been absorbed into the displacement operator $\hat{D}(\pm\alpha)$. From
here we can calculate the correlator,

$\displaystyle\braket{0}{e^{i\cos{\sqrt{4\pi}\varphi}}}{0}$
$\displaystyle=\braket{0}{e^{\hat{D}(-\alpha)}e^{\hat{D}(\alpha)}}{0}$ (3.36)
$\displaystyle=\sum_{m,m^{\prime}=0}^{\infty}\sum_{s,s^{\prime}\in{\pm}}\frac{1}{m!}\frac{1}{m^{\prime}!}\braket{s^{\prime}m^{\prime}\alpha}{sm\alpha}$
(3.37)
$\displaystyle=N^{\prime}\sum_{m=0}^{\infty}\sum_{m^{\prime}=0}^{\infty}\frac{1}{m!}\frac{1}{m^{\prime}!}(e^{-\frac{1}{2}|\alpha|^{2}(m-m^{\prime})^{2}}+e^{-\frac{1}{2}|\alpha|^{2}(m+m^{\prime})^{2}})$
(3.38)

where the last equality follows from the identity

$\braket{\beta}{\alpha}=e^{-\frac{1}{2}(|\beta|^{2}+|\alpha|^{2}-2\beta^{*}\alpha)}.$
(3.39)

As one can see, calculating these correlators is highly nontrivial.
Numerically it appears these series converge, but without analytic proofs,
it’s challenging to pin down the normalization factors. Moreover, implementing
these series into the numerical simulations of coherent information for the
electron density and current blocks of the LL Hamiltonian is extremely taxing
computationally.

In the next chapter, we introduce the formalism of quantum Shannon theory
which focuses on evaluating quantum information channels. To understand the
advances we have made to RQI using quantum Shannon theory, we demonstrate the
background of quantum channels from a first principles formalism. Readers who
are not familiar with quantum Shannon theory will hopefully find this chapter
very readable.

##  Chapter 4 Quantum Information and Quantum Computing

###  4.1 Introduction to Information Channels

Those who study information sciences, have developed a series of mathematical
tools to help understand the inter-workings behind how information transfers
from inputs to outputs. These transfers, are aptly named information
channels[4, 87].

Claude Shannon’s remarkable works in the 1940s [88] paved the way for a deeper
understanding of how to treat classical information. Shannon theory has since
been developed and blossomed into a vibrant field to study for mathematicians
and scientists alike. The community of QI scientists have subsequently coined
quantum communication theory as “Quantum Shannon Theory”, as an homage to
Shannon and the works that he produced [4].

Just as with other fields of physics, classical and quantum information
channels follow separate rules. The quantum revolution of the 1920s brought
with it a confusing perspective of how we are to view the governing mechanics
of the micro-world. Up until that moment, nature could be described by a set
of deterministic rules. Only limited by the complexity of each system and the
ignorance of the observer. The quantum founding fathers showed that at the
smallest of scales, nature is fundamentally determined by probabilities[26,
89].

The probabilistic nature of quantum mechanics would thus play a lofty role in
how we understand information. In this chapter, we introduce a guide to
quantum information channels and do this through the guise of an
“interpretation free” version of quantum mechanics. Measurement theory is a
popular topic in quantum communication theory, as the physical interpretation
of the measurement has consequences regarding the information that is being
treated. Despite this, physical interpretations of the evolution of QI do not
impact the results experienced in a laboratory. For this reason we provide a
simple and clear, interpretation free, definition to quantum information.

######  Definition 1.

Quantum Information is the information contained in the entanglement between
states and therefore cannot exist in systems whose states are “separable”.

This definition offers a straight-forward test of the “quantum-ness” of a
channel. Namely, at any given point in the processing of the information, do
the states become separable. At the end of this chapter we outline the
mathematics that determines the separability of states and demonstrate how one
can distinguish quantum information from classical information channels.

Despite the differences in the types of information, certain properties of
these information channels will be analogous to each other. Therefore, we dive
first into a more familiar domain of classical information channels while
preparing to understand the more complicated but coinciding quantum
information channels. The aim of this chapter, is to introduce some of the
tools necessary to understand the properties of information channels. While
the mathematics is deep and complex, I do my best to keep it self-contained as
much as possible, with the intent to offer a broader audience the ideas behind
information theory.

###  4.2 Classical Information Channels

As mentioned previously, complexity of a classical system can shroud its
deterministic properties. Probabilities are often deployed as a simplification
to these complex systems. While there is nothing quantum about flipping a fair
coin, the fact that quantum theory is probabilistic, allows us to draw similar
allusions by following such a system. To unpack the information hidden in this
random process of coin flipping, we need a theory to measure information as it
changes through these classical processes. For that, we turn to Shannon’s
information theory.

Shannon’s theory is best understood with the usage of bits. A bit is a single
piece of information that describes the state of an object. Consider the above
example of a fair coin. The state of the coin can be described by assigning
the value of zero to heads and one to tails. Therefore, reading out the bit is
equivalent to measuring the state of the coin. To measure the information of a
system111We will be using $\log_{2}$ for the entirety of this chapter for
consistency., we first consider the Shannon entropy $H(X)$ given by,

$H(X)=\sum_{x}-p_{X}(x)\log p_{X}(x)$ (4.1)

with probability distribution $p_{X}(x)$. $H(x)$ will measure the randomness
or surprise in the output of the system. If we consider our above example of
the fair coin we expect a Shannon Entropy of one, which is the maximum amount
of uncertainty for this system. This can easily be seen by replacing the fair
coin with a weighted coin. A weighted coin with two-thirds probability of
heads and one-third probability of tails, has a Shannon Entropy of 0.918. A
weighted coin with nine-tenths probability of heads and one-tenth tails will
yields 0.469. As the coin gets more weighted, the uncertainty in the result of
the coin flip goes down, and subsequently the measure of Shannon Entropy does
as well. Generally, we can say if $x$ is the probability of obtaining heads,
the probability of obtaining tails is $1-x$ and Eq. 4.1 becomes,

$H(X)=x\log x+(1-x)\log(1-x).$ (4.2)

We can see in figure 4.1, that the information content of $X$ is maximized
when both the events are equally likely. This makes sense because having equal
probability outcomes, makes the task of guessing the hardest. So in essence,
the entropy is closely related to the contents of the information channel.

Shannon entropy specifically measures information content of the random
variable $X$. If we refer to this “surprise” or “randomness” of an outcome $x$
as the information content $i(x)$, and define it as
$i(x)=\log(\frac{1}{p(x)})=-\log(p(x))$. This measurement has the property
that it is high for low probability outcomes and low for high probability
outcomes. Moreover, this also has another desirable property of “additivity”
i.e. the information content of two outcomes $x$ and $y$ get added since the
joint probability $p(x,y)=p(x).p(y)\implies
i(x,y)=-\log(p(x)p(y))=-\log(p(x))-\log(p(y))=i(x)+i(y)$. Thus, the total
information content of the random variable $X$ can be written as the weighted
average of the information of possible outcomes.

Figures/coin_toss_entropy

Figure 4.1: Graphical representation of H(x) for our fair-coin flipping
system. Notice the maximum entropy occurs when our probability $x=1/2$.

For both the classical and the quantum case of information channels, we will
model a very simple operation that has purpose both in classical and quantum
computing. For the classical system we will use a flip operation. The flip
operation will take our input bit and flip the value of it. For instance, our
fair coin modeled with heads as zero and tails as one, will do just as it
says. Flip the side of the coin from its initial configuration to the
opposite. Figure 4.2 gives a pictorial representation of this process.

Figures/bit_flip

Figure 4.2: A classical information channel that shows a flip operation acting
on a coin. The coin is initially in state of heads but the flip operation
changes this to tails.

A useful calculation in information theory is Channel Capacity, which is a
measure of information loss (or conversely propagated) through our circuit.
Information lost is often do to the noisiness of a channel. Noise is
attributed to unwanted interactions and will affect the input states by
altering them. For instance, a sender Alice and a receiver Bob may be
attempting to communicate classically. Choosing a noisy channel to save
resources could alter the inputs enough that Bob would be unable to decipher
the original message Alice has sent. If instead of a fair coin, suppose Alice
has access to a set of random variables $X$. She encodes a message onto the
elements of $X$ and sends it via some noisy channel to Bob who receives a set
of random variables $Y$ to decipher. Our aim now must be to mathematically
evaluate the channel that led from $X$ to $Y$. This will show if the channel
reliably transferred Alice’s message. Pictorially this is shown in figure 4.3.

Figures/classical_circuit

Figure 4.3: Alice’s message is encoded into the elements $x_{i}$ where it is
passed through some noisy channel $\mathcal{N}$ and later deciphered by Bob.

After the information transfer, Bob has access to $Y$, and the amount of his
randomness/surprise about $X$ is quantified by the Conditional Entropy.
Mathematically this has the shape,

$H(Y|X)=-\sum_{x,y}p_{Y|X}(y|x)\log\left(\frac{p_{Y|X}(y|x)}{p_{Y}(y)}\right)$
(4.3)

where $p_{X,Y}(x,y)$ is an expression of the multiplicity of probability
namely, $p_{X,Y}(x,y)=p_{X}(x)p_{Y}(y)$. Notice that $H(X)\geq H(X|Y)$, and
subsequently $I(X;Y)\geq 0$. Here these probabilities are a result of the
interactions within the noisy quantum channels. The measure of how much
information Bob recovers, can be expressed as the Mutual Information $I(X;Y)$
where,

$I(X;Y)=H(Y)-H(Y|X).$ (4.4)

The mutual information in essence quantifies the similarity or correlations
between the random variables of Alice and Bob. This is because $H(Y|X)$
quantifies the amount of error due to noise. In a perfect channel this value
is zero because $Y$ is fully aware of how $X$ will change.

Up until this point, we have very purposefully not discussed the contents of
the noisy information channels. We want to have a means to describe an
information channel either classically or quantum mechanically, but the
contents of the channels may be messy and hard to understand. This is where
channel capacity comes into play. Channel capacity uses information to
quantifiably measure the reliability of an information channel $\mathcal{N}$.
Note: We have reserved the notation for $\mathcal{N}$ to describe non-
relativistic information channels and $\Xi$ we have utilized for quantum
information channels that utilize qubit-field interactions. The means as to
which the information passes through that channel is equivalent to knowledge
of the information given and received. After all, the channel is a
probabilistic mapping of the set $X$ onto the set $Y$,

$\mathcal{N}:p_{Y|X}(y|x).$ (4.5)

Under this construction, we define the channel capacity as,

$\mathcal{C}(\mathcal{N})=\max_{p_{X}(x)}I(X;Y)$ (4.6)

which has a physical interpretation such that, if Alice can arrange her
information in a way that allows a maximal amount of correlation with Bob’s,
than this is the upper bound to the capacity of the channel. Using what we
learned about information, we can quantify the capacity of a channel which
will help identify if these channels are efficient uses of resources.

###  4.3 Quantum Information Channels

A quantum channel is a linear, completely positive, and trace preserving
mapping corresponding to a physical evolution [4, 90]. That is to say, that it
too, is a probabilistic mapping from initial states to final states. Yet
something deeper is going on here. The probabilities involved the information
transfer is not necessarily due to our inability to evaluate a complex system.
Instead, the mechanics are fundamentally probabilistic. This leads to a
relationship deeper than correlation, entanglement.

Entanglement, while mysterious in many ways, is a fundamental ingredient in
quantum mechanics and is necessary for successful quantum computing. Similar
to our classical bit, we introduce a quantum bit or qubit, that plays the same
role as arbiter of information, but for quantum systems. If we assign the spin
degrees of freedom of an electron to a value of our qubit we can create a
similar arrangement to the classical coin flip.

For our purposes, lets assign spin up to our qubit $\ket{0}$, and spin-down to
$\ket{1}$ and construct the state of our entangled pair (a Bell State),

$\ket{\psi}=\frac{\ket{0}\ket{0}+\ket{1}\ket{1}}{\sqrt{2}}$ (4.7)

where the coefficient of $\frac{1}{\sqrt{2}}$ guarantees that $50\%$ of the
time we will measure both spin-up and $50\%$ of the time we will measure both
spin down.

Now that our descriptive states are set up as qubits, we can begin to evaluate
the information in our states. Lets, first establish a parallel to our Shannon
Entropy. It may surprise you that quantum mechanical entropy came first, this
was accomplished by John von Neumann in 1932. Nevertheless, our von Neumann
entropy has a similar form to the Shannon Entropy,

$S(\hat{\rho})=-\operatorname{Tr}{(\hat{\rho}\log\hat{\rho})}$ (4.8)

where $\hat{\rho}$ is a density matrix whose purpose is to describe, in
entirety, the probabilities and states of our system. If these states are
eigenstates (eigenstates are states we can measure eg. spin-up and spin-down)
then our density matrix takes the appearance of
$\hat{\rho}=\sum_{j}n_{j}\ket{j}\bra{j}$ with $\ket{j}$ denoting an
eigenstate, and we can simplify equation 4.8 to,

$S(\hat{\rho})=-\sum_{j}n_{j}\log n_{j}$ (4.9)

where $n_{j}$ is the probability of measuring an eigenstate. But what does
this quantify? In essence, this quantifies the purity of the state. While
purity has some analogous meaning to classical information, we should press on
with our development of quantum information and maybe a more intuitive
understanding will present itself. By now you have probably noticed that
equations 4.1 and 4.9 are mostly identical. That changes quickly when we
introduce Quantum Mutual Information.

To understand Quantum Mutual Information, lets give Alice one qubit and Bob
another. We can entangle them such that our input state matches that of
equation 4.7. We can then model the mutual information between Alice’s and
Bob’s qubits as,

$I(A:B)=S(\hat{\rho}_{A})+S(\hat{\rho}_{B})-S(\hat{\rho}_{AB})$ (4.10)

where $\hat{\rho}_{A}=\operatorname{Tr}_{B}(\hat{\rho}_{AB})$ and $\rho_{A}$
is the density matrix describing the state of the Alice’s qubit. Tracing over
part of the system, in essence ignores that part of the system. When we trace
over Bob’s information we are focused solely on the information of Alice. The
physical interpretation of mutual information is thus a quantifiable
measurement of the entanglement shared between Alice and Bob. For a highly
entangled state such as our bell state, we find a value of two which indicates
a maximum entanglement between the two qubits. While quantum mutual
information is insightful for understanding entanglement between states, these
states remain as initial states. We have made a modification to our system
between the classical and quantum cases. The modification is the usage of Bob.
In our classical noisy channel arrangement, Bob received some data $Y$ from
Alice whose input was $X$. The classical mutual information was a correlation
between input and output. Therefore, We need another tool that is analogous to
classical mutual information, that will evaluate how a quantum channel could
alter this entanglement. Luckily, we’ll find a familiar analogy to classical
mutual information.

The flow of quantum information is quantifiable by how much entanglement
passes from the initial state to the final state. Quantum Coherent information
is a quantitative measurement of this transfer. To understand it better, lets
bring in a third qubit belonging to Carla. Let’s suppose Alice and Carla
entangle their qubits together and create a Bell Pair like that of Eq. 4.7.
Bob’s qubit is initially in a $\ket{0}$ (spin-up) state with $100\%$
probability. We now define a quantum channel from Alice to Bob where we aim to
pass the entanglement. A popular quantum channel that is well known to
preserve entanglement is done using a SWAP gate. The SWAP gate will do just as
it says; swap the values of Alice’s and Bob’s qubits and with it the
information associated. We will explore the mathematical structure of the SWAP
gate in Chapter 6. Figure 4.4 shows this quantum information channel,
outlining the SWAP of entanglement from Alice’s qubit to Bob’s.

Figure 4.4: SWAP channels exchange quantum information between qubits as
demonstrated with a reference qubit (Carla).

We can now define our quantum coherent information as,

$I_{C}(\hat{\rho}_{in,AB},\mathcal{N})=S(\hat{\rho}_{out,B})-S(\hat{\rho}_{out,BC}).$
(4.11)

where, $\hat{\rho}_{in,AB}$ is the input to the channel and $\mathcal{N}$ is
the channel itself. One should note here, that the Channel $\mathcal{N}$ is
equivalent to $\hat{\rho}_{out,B}$. The difference in notations is that
$\mathcal{N}$ expresses the operators that make up the channel, and their
acting on the initial states. Whereas, $\hat{\rho}_{out,B}$ is the output
state which is a result of this channel. This is because $\mathcal{N}$,
mathematically is a mapping from initial states to final states. The analogy
to the classical mutual information (Eq. 4.4) should be more evident now, and
as we did in the classical case, we elevate this to the Quantum Channel
Capacity by taking,

$Q(\mathcal{N})=\max_{\hat{\rho}_{in,AB}}I_{C}(\hat{\rho}_{in,AB},\mathcal{N}).$
(4.12)

Just as it was in classical information theory, the channel capacity is a
measurement of reliability in transferring quantum information (ie.
entanglement). Entanglement is one of the best metrics of a quantum computer,
measuring the propagation of this entanglement, remains to be an important
calculation as we further progress quantum computers.

We have spoken in detail about two channels of information, yet there remains
two more. Channels that change classical information to quantum information
(the process that creates entanglement) and vice versa (measurement). One
thing that is a consequence of our Def. 1, is that when states are separable,
the entanglement is gone and what is left is a tensor product of states. The
procedure for verifying this is well established but will be fruitful for us
to see written out.

###  4.4 Determining if a Quantum Channel is Entanglement Breaking

####  4.4.1 The Quantum Channel

Up until this point we have spoken about the quantum channel at surface level.
For the remainder of the chapter we unpack some of the mathematics and show
how these aforementioned ideas are shaped into understanding quantum
information in quantum circuit.

Suppose there exists a quantum system A and an environment system $\phi$. We
define a quantum channel $\Xi_{A\phi}$ as a linear, completely positive, and
trace preserving mapping corresponding physical evolution from a state
$\hat{\rho}_{A}\in\mathcal{H}_{A}$ to an environment (in our case the field
$\phi$) $\hat{\rho}_{\phi}\in\mathcal{H}_{\phi}$ which is arranged as a pure
state (eg. ${\ket{0}\\!\\!\bra{0}}_{\phi}$). The initial state of the whole
system is
$\hat{\rho}_{A\phi}=\hat{\rho}_{A}\otimes{\ket{0}\\!\\!\bra{0}}_{\phi}$ where
$\hat{\rho}_{A\phi}\in(\mathcal{H}_{A}\otimes\mathcal{H}_{\phi})$ and there
exists some interaction between the systems with Unitary Operator
$\hat{U}_{A\phi}$ such that after the interaction, we can find a state
$\Xi_{A\phi}(\hat{\rho}_{A})$ from the channel

$\Xi_{A\phi}(\hat{\rho}_{A})=\operatorname{Tr}_{A}\\{\hat{U}_{A\phi}(\hat{\rho}_{A}\otimes\ket{0}_{\phi}\bra{0})\hat{U}_{A\phi}^{\dagger}\\}$
(4.13)

it is helpful here to rewrite our initial state as

$\hat{\rho}_{A\phi}=\hat{\rho}_{A}\otimes\ket{0}_{\phi}\bra{0}=(\mathbb{1}_{A}\otimes\ket{0}_{\phi})(\hat{\rho}_{A})(\mathbb{1}_{A}\otimes\bra{0}_{\phi})$
(4.14)

as well as understand a partial trace to be defined as

$\operatorname{Tr}_{B}\\{X_{AB}\\}\equiv\sum_{l}\bra{l}_{B}X_{AB}\ket{l}_{B}$
(4.15)

plugging these two identities into Eq. 4.13 we find,

$\displaystyle\Xi_{A\phi}(\hat{\rho}_{A})$
$\displaystyle=\sum_{l}(\mathbb{1}_{A}\otimes\bra{l}_{\phi})\hat{U}_{A\phi}(\mathbb{1}_{A}\otimes\ket{0}_{\phi})(\hat{\rho}_{A})(\mathbb{1}_{A}\otimes\ket{0}_{\phi})\hat{U}_{A\phi}^{\dagger}(\mathbb{1}_{A}\otimes\ket{1}_{\phi})$
(4.16) $\displaystyle=\sum_{l}\hat{M}_{l}\hat{\rho}_{A}\hat{M}_{l}^{\dagger}$
(4.17)

where we have defined
$\hat{M}_{l}\equiv(\mathbb{1}_{A}\otimes\bra{l}_{\phi})\hat{U}_{A\phi}(\mathbb{1}_{A}\otimes\ket{0}_{\phi})$.
This operator will become important when finding our if our channel is
entanglement breaking.

####  4.4.2 Separable States

Before, we get into entanglement breaking we need to identify separability of
states for our system. To do this we describe a joint ensemble of our systems
A and $\phi$

$\\{p_{l}(l),\ket{A}_{l}\otimes\ket{\phi}_{l}\\}$ (4.18)

where $p_{l}(l)$ is some probability distribution and the set
$\\{\ket{A_{l}}\\}$ and $\\{\ket{\phi_{l}}\\}$ are sets of pure states on
their respective system. We define the density operators of our system A and
environment $\phi$ separately (ignoring the other exists) to be,

$\displaystyle\hat{\rho}_{A}$
$\displaystyle=\sum_{l}p_{l}(l){\ket{A_{l}}\\!\\!\bra{A_{l}}}$
$\displaystyle\hat{\rho}_{\phi}$
$\displaystyle=\sum_{l}p_{l}(l){\ket{\phi_{l}}\\!\\!\bra{\phi_{l}}}$ (4.19)

so the density operator of this ensemble is given as,

$\hat{\rho}_{A\phi}=\sum_{l}p_{l}(l)({\ket{A_{l}}\\!\\!\bra{A_{l}}}\otimes{\ket{\phi_{l}}\\!\\!\bra{\phi_{l}}})$
(4.20)

which classifies the density operator $\hat{\rho}_{A\phi}$ as “separable” by
definition. It can be noted here that a density operator that is not separable
is classified as “entangled”.

####  4.4.3 Breaking Entanglement

For a deeper dive into this material, please see Refs. [4, 2]. Now, we revisit
our operators $\hat{M}_{l}$ in Eq. 4.17 and $\hat{U}_{A\phi}$ that exists in
$\hat{M}_{l}$. We wish to show that if $\hat{U}_{A\phi}$ is a Schmidt rank-one
unitary like that of Eq. 2.20, than the density operator $\hat{\rho}_{A\phi}$
is separable222we show this for a single qubit in system A but it remains true
for any number of qubits in A.. Let us suppose we entangle our qubit In system
A with some ancillary qubit in another system C. We define the bipartite
density operator $\hat{\rho}_{AC}\in\mathcal{H}_{A}\otimes\mathcal{H}_{C}$ and
since $\hat{\rho}_{AC}$ is entangled than it is not separable.

We then define a Channel from A and C to the environment $\phi$

$(\mathbb{1}_{C}\otimes\Xi_{A\phi})(\hat{\rho}_{CA})=\operatorname{Tr}_{A}\\{\hat{U}_{A\phi}(\hat{\rho}_{AC}\otimes\ket{0}_{\phi}\bra{0})\hat{U}_{A\phi}^{\dagger}\\}$
(4.21)

The observable defined by our two-state qubit can be rewritten as a projection
onto spin up or spin down,

$\hat{\mu}_{A}=\sum_{s\in\\{\pm\\}}s\hat{P}_{s}$ (4.22)

With this decomposition of $\hat{\mu}_{A}$ we can express our single rank one
unitary $\hat{U}_{A\phi}$ as,

$\hat{U}_{A\phi}=\sum_{s\in\\{\pm\\}}\hat{P}_{s}\otimes\hat{U}_{s}$ (4.23)

Where our operator $\hat{U}_{s}$ follows from Eq. 2.20, and is therefore
$\hat{U}_{s}=\exp{(is\lambda\hat{\mathcal{O}}})$. We can then rewrite Eq. 4.21
as,

$(\mathbb{1}_{C}\otimes\Xi_{A\phi})(\hat{\rho}_{CA})=\sum_{s}\operatorname{Tr}_{A}\\{\hat{P}_{s}(\hat{\rho}_{AC}\otimes\hat{U}_{s}\ket{0}_{\phi}\bra{0}\hat{U}_{s}^{\dagger}).$
(4.24)

We then recognize the following;

$\displaystyle p_{s}(s)$
$\displaystyle=\operatorname{Tr}\\{\hat{P}_{s}\hat{\rho}_{AC}\\}$ (4.25)
$\displaystyle\operatorname{Tr}_{A}\\{\hat{P}_{s}\hat{\rho}_{AC}\\}$
$\displaystyle=\sum_{s}\bra{s}_{A}\hat{P}_{s}\hat{\rho}_{AC}\ket{s}_{A}=p_{s}(s)\hat{\rho}_{AC}(s)$
$\displaystyle\hat{\rho}_{\phi}(s)$
$\displaystyle=\hat{U}_{s}\ket{0}_{\phi}\bra{0}\hat{U}_{s}^{\dagger}$

plugging these quantities into Eq. 4.24 we get our final form of the channel,

$(\mathbb{1}_{C}\otimes\Xi_{A\phi})(\hat{\rho}_{CA})=\sum_{s}p(s)\hat{\rho}_{AC}(s)\otimes\hat{\rho}_{\phi}(s).$
(4.26)

which we recognize to be separable.

##  Chapter 5 Applications of Shannon Theory to UDWQCs

###  5.1 The Skeleton of a Quantum Computer

So far we have introduced the UDW detector formalism as well as scalar fields
$\hat{\varphi}$ and $\hat{\Pi}$ and have seen some overlap between the
formalism of UDW detectors and HLLs. In Chapter 4, quantum communication
theory was introduced as the means to find quantum channels that can process
and transmit quantum information. In this chapter, we demonstrate the quantum
Shannon properties that we can develop using the UDWQC model.

One might now understand the figure 1.1 better. In our model, if we implant a
two-state Qdot (Alice) into a HLL, we can detect if information has been
embedded into the field111This is the fundamental underpinning detector
models. See [91]. The information will then be attached to right- and left-
moving fields which can be collected at another Qdot (Bob). Entanglement of an
ancilliary qubit with Alice can be evaluated to understand the effectiveness
of the entanglement propagation of the channel, otherwise known as the quantum
channel’s capacity.

###  5.2 A Brief Review of the System

We have shown that our “naive” Hamiltonian Eq. 2.5 was constructed
specifically to demonstrate the connection of UDW detectors to a CFT (namely
HLLs) to realize qubit-field transduction. Given that we wish to take our
system and extend it into general quantum Shannon theory, we redefine our Eq.
2.5 as

$\displaystyle\hat{H}_{\mathrm{int}}(t)$
$\displaystyle=\lambda\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\hat{\psi}^{\dagger}_{+}\hat{\psi}_{+}-\hat{\psi}^{\dagger}_{-}\hat{\psi}_{-})$
(5.1)

so that we can begin to implement constraints and simulate numerically,
quantum Shannon theory properties. Notice in Eq. 5.1 we have used a more
generic form of coupling $\lambda$ which is common in QIS. We will remain with
this coupling notation for this chapter. As we did in Chapter 2, we have
suppressed the terms containing the $e^{\pm 2ik_{F}x}$ through a Gaussian
smearing function.

An interpretation of this quantum computer thus far, may be considered as
follows through the methodology of UDW detectors: Consider the spin-state of
our Qdot Alice at some initial time $t_{A}$. At this time we allow an
interaction between Alice and the HLL. The information is then smeared onto
the left- and right-moving components of the field through the smearing
function $p(x(t_{A}),y)$222Smearing and collecting information will also be
referred to as encoding and decoding the information respectively.. It is then
collected by Qdot Bob at some time $t_{B}$. Given the time evolution of the
field one could expect that the the function required for decoding the
information will be different than that of the encoding function. At least at
the level of the correlator that is surely true and a calculable quantity. At
most, for field-mediated processes, this extra value will end up being an
overall phase. For now, we presume that in one dimension, it remains to be the
same shape as the smearing function at Alice.

Without strong coupling, our channel from Alice to Bob has a perturbatively
limited quantum capacity. Whereas, the goal of quantum computing is to reach
maximum quantum capacity of one. With a perturbative approach we can find an
array of literature describing the interaction Hamiltonians to analyze.
However, the literature on strong coupling is severely more limited. We are
subject to study a case that is well understood with the aim to uncover more
systems in future work. What is understood in earnest are interactions between
our two state Qdot system and a scalar bosonic field. Implementing what was
shown in Chapter 3 we can express Eq. 5.1 as,

$\displaystyle\hat{H}_{\mathrm{int}}(t)$
$\displaystyle=\lambda\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(\partial_{z}\hat{\phi}+\partial_{\bar{z}}\hat{\bar{\phi}}))$
(5.2) $\displaystyle=\lambda\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(2\hat{\Pi}))$ (5.3)

These two equations provide us with our first to versions of quantum gates to
explore numerically. Unfortunately, Eq. 5.3 constructs a rank-one unitary
operator which behaves as a controlled unitary operator for qubit-field
interactions. Controlled-NOT (CNOT) gates, also controlled unitaries, are
known to be destructive to QI and as was shown at the end of Chapter 4, so are
the channels made up of this controlled unitary. However two CNOT gates, in
series, is known as a quantum state transfer and acts as the ideal comparison
for our UDWQC when considering the propagation of QI. This comparison will be
demonstrated in Chapter 6 via the diamond norm.

###  5.3 UDWQC Quantum Channels

Recall, an entanglement breaking channel has the necessity that the
information at some point throughout the process becomes classical. This is an
indication that whether some observer recognizes it or not, a measurement
takes place throughout the evolution processes. Initializing our field in a
pure state $\ket{0}\\!\\!\bra{0}_{\phi}$, we define our mapping
$\Xi_{A\rightarrow\phi}$ as the mapping from initial states of system A and
field $\hat{\phi}$ to final states of $\hat{\phi}$ which is given by the
equation,

$\Xi_{A\rightarrow\phi}(\hat{\rho}_{A})=\operatorname{Tr}_{A}\\{\hat{U}_{A\phi}(\rho_{A}\otimes\ket{0}_{\phi}\\!\\!\bra{0})\hat{U}_{A\phi}^{\dagger}\\}$
(5.4)

where $\Xi_{A\rightarrow\phi}(\hat{\rho}_{A})$ is some state of the field
$\hat{\phi}$ after the interaction between $A$ and $\hat{\phi}$ and
$\hat{U}_{A\phi}$ is a Unitary operator. We will discuss in Chapter 7, the
relevance that this system is relativistic. While we remain in euclidean space
(through Wick rotation) through this thesis, we can see that these circuits
fit naturally onto any given manifold.

Now suppose we entangle our system A with some external system C. Our new
quantum channel takes the shape,

$(\mathbb{1}_{C}\otimes\Xi_{A\phi})(\hat{\rho}_{CA})=\operatorname{Tr}_{A}\\{\hat{U}_{A\phi}(\hat{\rho}_{AC}\otimes\ket{0}_{\phi}\\!\\!\bra{0})\hat{U}_{A\phi}^{\dagger}\\}$
(5.5)

where our $\hat{\rho}_{AC}$ is an entangled bipartite state between System $A$
and $C$. If at some point we find that the system C and the field states in
$\hat{\rho}_{A\phi}$ are separable ie.
$\hat{\rho}_{AC}\otimes\hat{\rho}_{\phi}$ than our entanglement has been
broken. As shown in Chapter 4, and consistent with the literature [47, 2, 92]
this channel is broken as it can be written in the form,

$(\mathbb{1}_{C}\otimes\Xi_{A\rightarrow\phi})(\sigma_{CA})=\sum_{s}p(s)\rho_{AC}(s)\otimes\rho_{\phi}(s).$
(5.6)

The entire quantum channel in question here, is from Alice to Bob (our two
Qdots). This will be represented by a longer quantum channel,

$\Xi_{A\rightarrow
B}(\hat{\rho}_{A})=\operatorname{Tr}_{A\phi}\\{\hat{U}_{\phi
B}\hat{U}_{A\phi}(\hat{\rho}_{A\phi}\ket{0}_{\phi}\\!\\!\bra{0}\hat{\rho}_{\phi
B})\hat{U}_{A\phi}^{\dagger}\hat{U}_{\phi B}^{\dagger}\\}$ (5.7)

and visualized in Fig. 5.1. We now aim to address the types of unitaries that
preserve entanglement and propagate it through this channel from Alice to Bob.

Figure 5.1: Encoding and Decoding quantum information onto and off of the
field $\hat{\varphi}$.

###  5.4 Possible Gates For Our Quantum Computer

####  5.4.1 Gate Formations of Interest

We have found that our system (UDWQC) is adequate for transferring information
onto and off of our fields that have the same form as the fields of Eq. 2.1.
As discussed in Chapter 2, it is known in the literature that the minimum
necessity for an entanglement safe channel is two Schmidt rank-one unitaries
that have the shape,

$\hat{U}_{\nu}=\exp{(i\lambda_{\nu 2}\hat{\mu}_{\nu
2}\otimes\hat{\mathcal{O}}_{\nu 2})}\exp{(i\lambda_{\nu 1}\hat{\mu}_{\nu
1}\otimes\hat{\mathcal{O}}_{\nu 1})}.$ (5.8)

We now explore regimes that have potential to realize this type of unitary in
our LL scheme.

####  5.4.2 Left- and Right-Moving Gates

If we rewrite Eq. 5.1 as the following,

$\hat{H}_{\mathrm{int}}(t)=\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\lambda_{\alpha}\hat{\psi}^{\dagger}_{\alpha,+}\hat{\psi}_{\alpha,+}-\lambda_{\beta}\hat{\psi}^{\dagger}_{\beta,-}\hat{\psi}_{\beta,-})$
(5.9)

where the coupling constants are now defined with separate coupling to left-
and right-movers. Following the formalism in Chapter 3 we can find a bosonized
form of Eq. 5.2

$\displaystyle\hat{H}_{\mathrm{int}}(t)$
$\displaystyle=\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(\lambda_{\alpha}\partial_{z}\hat{\phi}+\lambda_{\beta}\partial_{\bar{z}}\hat{\bar{\phi}})).$
(5.10)

Decomposing this Hamiltonian into
$\hat{H}_{\mathrm{int}}(t)=\hat{H}_{\alpha}(t)+\hat{H}_{\beta}(t)$ we can
evaluate a single left- or right-moving interaction Hamiltonian and we express
the $\partial_{z}\hat{\phi}$ terms as field and momentum observables such
that,

$\displaystyle\hat{H}_{\alpha}(t)$ $\displaystyle=\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\frac{1}{\sqrt{\pi}}(\lambda_{\alpha\phi}\partial_{x}\hat{\phi}(t,x)-\lambda_{\alpha\pi}\hat{\Pi}(t,x))).$
(5.11)

We also include the switching function into the coupling as the expectation
here is that we time order the couplings with delta-like functions. Doing
this, we outline two observables that are able to satisfy Eq. 5.8. Namely,

$\hat{U}_{\nu}=\exp{(i\lambda_{\alpha\phi}\hat{\mu}_{\phi}\otimes\partial_{x}\hat{\phi})}\exp{(i\lambda_{\alpha\pi}\hat{\mu}_{\pi}\otimes\hat{\Pi})}.$
(5.12)

The change in this system is that we only work with left- and right-movers
separately. This systems exists as a Chiral Luttinger Liquid (CLL) [93, 94]
and as such, could offer a more complex quantum computer set up.

####  5.4.3 Dirac-type Current Hamiltonians

Another gate that will be examined thoroughly throughout this project is a
dirac style Hamiltonian that considers the current
$\hat{\psi}_{\pm}^{\dagger}\partial_{x}\hat{\psi}_{\pm}$. This hamiltonian
applied to our UDW+HLL system takes the following shape

$\displaystyle\hat{H}^{D}_{\mathrm{int}}(t)$
$\displaystyle=\lambda\chi(t)\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\hat{\psi}^{\dagger}_{+}\partial_{x}\hat{\psi}_{+}-\hat{\psi}^{\dagger}_{-}\partial_{x}\hat{\psi}_{-})$
(5.13)

we can follow the point-splitting procedure from Appendix B to find a
different form of this Dirac Hamiltonian,

$\hat{H}^{D}_{\mathrm{int}}(t)=\int_{\mathbb{R}}dy\
p(x(t),y)\hat{\mu}(t)(\lambda_{\Pi}\hat{\Pi}^{2}+\lambda_{\varphi}(\partial_{x}\hat{\varphi})^{2})$
(5.14)

which brings back our original fields from Eq. 2.1. This however is a
quadratic coupling to our Qdot and has a Unitary gate of the form,

$\hat{U}_{\nu}=\exp{(i\lambda_{\nu 2}\hat{\mu}_{\nu 2}\otimes\hat{O}^{2}_{\nu
2})}\exp{(i\lambda_{\nu 1}\hat{\mu}_{\nu 1}\otimes\hat{O}^{2}_{\nu 1})}.$
(5.15)

This regime of strong coupling in the UDW detector literature has not been
explored but perturbatively it has been [79]. Further investigations into
strong coupling of this interaction would provide unique quantum channels to
explore numerically.

Presented here are two versions of our LL model that can provide quantum
channels to numerically simulate quantum Shannon properties. There are of
course other blocks of the Tomonaga-Luttinger liquid Hamiltonian to explore as
well, but as demonstrated in the end of Chapter 3, there are other problems
with the coherent states of those unique field operators to tackle first. With
the concrete (and not-so-concrete) channels in place we move into the
numerical simulations and introduce new formalism that decreases our
computational workload.

###  5.5 Unruh–DeWitt Bosonic Dephasing-like Channels

####  5.5.1 The UDW Channel as a Bosonic Dephasing Channel

In quantum Shannon theory it is often useful to discuss entanglement in
multipartite channels that experience decoherence as dephasing channels,
usually under the guise of interactions with the environment[95, 96]. To that
regard, we may see a similarly behaving channel in the Fock basis as

$\mathcal{N}_{\gamma}(\hat{\rho})=\sum_{m,n=0}^{\infty}e^{-\frac{\gamma}{2}(n-m)}(c_{n}c^{*}_{m}){\ket{n}\\!\\!\bra{m}}$
(5.16)

where $\gamma$ is the factor that determines the rate of decoherence. For
example as $\gamma\rightarrow\infty$ the diagonal terms vanish. We revisit the
concept of dephasing due to the environment in Sec. 5.6.1.

In the case of the standard UDW channel, we aim to show that the dephasing is
a necessity following the non-orthogonality of the coherent states. Generating
the off-diagonals of the coherent state density matrix, scrambles the encoded
information from $\hat{U}_{A\varphi}$ as the controlled structure of our
unitaries encodes (decodes) QI onto (off of) the diagonals of the coherent
state density matrix. Utilizing the dephasing perspective, along with our free
parameters to send these states to zero, can accomplish near perfect channel
capacity.

To show the relation between the current formalism and the proposed dephasing
perspective, we can reformulate the state transfer channel of Ref. [2] to

$\hat{U}_{\nu\varphi}=\sum_{z,x\in\pm}\hat{P}_{x}\hat{P}_{z}\otimes\exp\left[ix\int
dk\,\sqrt{\gamma_{\Pi}(k)}\hat{\Pi}_{\nu}(k)\right]\\\ \times\exp\left[iz\int
dk\,\sqrt{\gamma_{\varphi}(k)}\hat{\varphi}_{\nu}(k)\right]$ (5.17)

where we have set

$\displaystyle\hat{\varphi}(\tilde{\gamma})_{\nu}$
$\displaystyle\coloneqq\lambda_{\varphi}\int
dk\tilde{F}_{\nu}(k)\hat{\varphi}(k,t_{\nu})=\int
dk\sqrt{\gamma_{\varphi}(k)}\hat{\varphi}(k,t_{\nu})$ (5.18)
$\displaystyle\hat{\Pi}(\tilde{\gamma})_{\nu}$
$\displaystyle\coloneqq\lambda_{\Pi}\int
dk\tilde{F}_{\nu}(k)\hat{\Pi}(k,t_{\nu})=\int
dk\sqrt{\gamma_{\Pi}(k)}\hat{\Pi}(k,t_{\nu}).$ (5.19)

Instead of redefining the coherent amplitudes to include our free parameters
as we did in Sec. 2.3.3, we have wrapped the free parameters up in some
function $\gamma_{O}(k)$ and leave the coherent amplitudes as they exist in
the field observables $\hat{O}$. This simplification allows for general
treatments of the channel regardless of the linearity of the ladder operators
in the field observables.

Now carrying out the inner product of Eq. 2.30 yields

$|\braket{+\sqrt{\gamma_{\varphi}(k)}\alpha(k)}{-\sqrt{\gamma_{\varphi}(k)}\alpha(k)}|=e^{-2\gamma_{\varphi}(k)|\alpha(k)|^{2}}$
(5.20)

where we have utilized the non-orthogonality of coherent states identity

$\braket{\beta(k)}{\alpha(k)}=\exp\left(-\frac{1}{2}|\alpha(k)|^{2}-\frac{1}{2}|\beta(k)|^{2}+\beta^{*}(k)\alpha(k)\right).$
(5.21)

It is obvious from Eq. 5.20 that the coherent states are part of a bosonic
dephasing channel $\Xi_{\varphi\rightarrow\varphi^{\prime}}$ with dephasing
constant $\sqrt{\gamma_{O}(k)}$. However, unlike canonical dephasing channels
the coupling, smearing, and switching are the mechanisms of dephasing. Sending
our dephasing function $\gamma_{\varphi}(k)\rightarrow\infty$ for any value
$k$, removes off diagonal elements from this channel. This should be expected
following the constraints in Sec. 2.3.3.

###  5.6 Applications of a UDW Channel From a Bosonic Dephasing perspective.

The main result of the previous section is that in the RQI channel outlined in
Fig. 2.1, we can model the coupling constant and smearing function as a
dephasing function (similar to the dephasing parameter). For the remainder of
this chapter, we demonstrate the results of this dephasing-like model using
numerical simulations. For this purpose, we utilize a Gaussian smearing
function and subsequently treat the dephasing function as a constant
$\gamma_{\hat{O}}$ with a single mode $k$. Since we aim to take advantage of
the new perspective, let’s first consider the canonical dephasing channel.

####  5.6.1 Canonical Bosonic Dephasing Channel

The canonical dephasing channel is one where the system interacts with the
environment, and the QI signal depreciates. Therefore, given the paradoxical
result in our UDW system, outlining such a dephasing channel (one where the
field is allowed to interact with the environment), may provide such a boost
in QI through our channel. However, given that dephasing channels acts on the
diagonal components of the density matrix we may find that the unwanted states
remain undeterred.

To evaluate this channel, we begin with a standard unitary that takes
advantage of the number operator

$\hat{U}_{\mathrm{E}}=e^{-i\sqrt{\gamma_{E}}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})}$
(5.22)

where $\gamma_{E}$ is the dephasing constant associated with the interaction
between the field and environment and $\hat{b}^{\dagger}(\hat{b})$ are the
creation (annihilation) operators of the environment. We can model the
channel’s effect on the field by writing out the composite channel

$\Xi_{\varphi\rightarrow\varphi^{\prime},E}=\operatorname{Tr}_{E}[\hat{U}_{\mathrm{E}}(\hat{\rho}_{1,\varphi}\otimes\ket{0}_{E}\\!\\!\bra{0})\hat{U}_{\mathrm{E}}^{\dagger}].$
(5.23)

Implanting this in the usual prescription of bosonic dephasing channels,
similar to that of Eq. 5.30, we get

$\Xi_{\varphi\rightarrow\varphi^{\prime},E}(\hat{\rho}_{\varphi})=\int_{-\infty}^{\infty}d\phi\,p(\phi)\,e^{-i\hat{a}^{\dagger}\hat{a}\phi}\hat{\rho}_{1,\varphi}e^{i\hat{a}^{\dagger}\hat{a}\phi},$
(5.24)

where $\Xi_{\varphi\rightarrow\varphi^{\prime},E}(\hat{\rho}_{\varphi})$
represents the dephasing channel and the effects on our field $\hat{\varphi}$,
and $p(\phi)$ is a Gaussian probability density given by

$p(\phi)=\sqrt{\frac{1}{2\pi\gamma_{E}}}e^{-\frac{1}{2}\frac{\phi^{2}}{\gamma_{E}}}.$
(5.25)

To evaluate this channel we convert to the Fock basis and continue by
substituting in Eq. 5.32 and simplifying Eq. 5.24 which then becomes

$\Xi_{\varphi\rightarrow\varphi^{\prime},E}(\hat{\rho}_{\varphi})=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)\times
e^{-i\hat{a}^{\dagger}\hat{a}\phi}{\ket{s\sqrt{\gamma_{\varphi}}\alpha}\\!\\!\bra{s^{\prime}\sqrt{\gamma_{\varphi}}\alpha}}e^{i\hat{a}^{\dagger}\hat{a}\phi}\\\
=\sum_{s,s^{\prime}\in\pm}\sum_{m,n}\int_{-\infty}^{\infty}d\phi\,p(\phi)e^{-\frac{1}{2}(\gamma_{\phi}|\alpha|^{2}(s^{2}+s^{\prime
2}))}\times\frac{(s\sqrt{\gamma_{\varphi}}\alpha)^{n}(s^{\prime}\sqrt{\gamma_{\varphi}}\alpha^{*})^{m}}{\sqrt{n!}\sqrt{m!}}e^{-i\hat{a}^{\dagger}\hat{a}\phi}{\ket{n}\\!\\!\bra{m}}e^{i\hat{a}^{\dagger}\hat{a}\phi}.$
(5.26)

Evaluating the operators we obtain

$\Xi_{\varphi\rightarrow\varphi^{\prime},E}(\hat{\rho}_{\varphi})=\sum_{s,s^{\prime}\in\pm}\sum_{m,n}\int_{-\infty}^{\infty}d\phi\,p(\phi)e^{-i\phi(m-n)}\\\
\times e^{-\frac{1}{2}(\gamma_{\phi}|\alpha|^{2}(s^{2}+s^{\prime
2}))}\frac{(s\sqrt{\gamma_{\varphi}}\alpha)^{n}(s^{\prime}\sqrt{\gamma_{\varphi}}\alpha^{*})^{m}}{\sqrt{n!}\sqrt{m!}}{\ket{n}\\!\\!\bra{m}}.$
(5.27)

Carrying out the integral yields

$\Xi_{\varphi\rightarrow\varphi^{\prime},E}(\hat{\rho}_{\varphi})=\sum_{s,s^{\prime}\in\pm}\sum_{m,n}e^{-\frac{1}{2}(m-n)^{2}\sqrt{\gamma_{E}}}\\\
\times e^{-\frac{1}{2}(\gamma_{\varphi}|\alpha|^{2}(s^{2}+s^{\prime
2}))}\frac{(s\sqrt{\gamma_{\varphi}}\alpha)^{n}(s^{\prime}\sqrt{\gamma_{\varphi}}\alpha^{*})^{m}}{\sqrt{n!}\sqrt{m!}}{\ket{n}\\!\\!\bra{m}}.$
(5.28)

It is evident that when $m=n$, we get no effect from the dephasing channel.
Moreover, when considering the change that this dephasing makes to our
original correlator in Eq. 2.25, we trace over the final state of the field
and assess the inner product of our new coherent states. What we find is that
non-zero results require $m=n$. Therefore, acting on the field with the number
operator results in an inner product identical to Eq. 5.20 and subsequently
will not change the coherent information of the UDW channel. This result
indicates that in this idealized dephasing channel, the environment will have
no effect on the quantum information through our channel. Coincidentally, the
condensed matter systems where these channels have been proposed, take
advantage of topologically protected edge states to transmit quantum
information [18]. Therefore, interactions with the environment were expected
to be minimal regardless.

####  5.6.2 Cross-Talk Noise

##### Setting up a UDW Noise Channel

One possible generation of unwanted disturbances of quantum information in
proposed condensed matter approaches to UDW channels, is unwanted interactions
with detectors, a type of Cross-Talk (CT) noise (see Fig. 5.2). Traditional
calculations of these additional interactions would increase the amount of
vertex operators in the correlator of Eq. 2.25, making computational times
significantly longer as well as more involved. However, considering the
dephasing perspective, we instead can incorporate additional detector effects
into the dephasing parameter.

Let’s assume that the new gate is defined as

$\hat{U}_{\mathrm{CT}}=\sum_{z\in\pm}\hat{P}_{z}\otimes\exp\left[iz\sqrt{\gamma_{N}}\hat{\varphi}_{\nu}\right],$
(5.29)

similar to that of Eq. 2.20. Following the prescription in Sec. 5.6.1 we model
the channel as

$\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})=\int_{-\infty}^{\infty}d\phi\,p(\phi)\,e^{i\hat{\varphi}\phi}\hat{\rho}_{1,\varphi}e^{-i\hat{\varphi}\phi}$
(5.30)

which expresses the channel
$\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})$, a
noisy composite channel describing the field evolution throughout the channel
$\Xi_{A\rightarrow B,\mathrm{N}}$. The channel indicates the probability
(which follows a random probably distribution $p(x)$ [95, 96]) that the CT
detector produces a different coherent state. For this calculation we will
utilize the probability distribution of Eq. 5.25 with the substitution of the
dephasing parameter $\gamma_{N}$. $\hat{\rho}_{1,\varphi}$ is the state after
the interaction between the field and qubit A and has the explicit density
form

$\displaystyle\hat{\rho}_{1,\varphi}$
$\displaystyle=\sum_{s,s^{\prime}\in\pm}{\ket{s\sqrt{\gamma_{\varphi}}\alpha}\\!\\!\bra{s^{\prime}\sqrt{\gamma_{\varphi}}\alpha}}$
(5.31)
$\displaystyle=\sum_{s,s^{\prime}\in\pm}e^{s\sqrt{\gamma_{\varphi}}\hat{\varphi}}{\ket{0}\\!\\!\bra{0}}e^{-s^{\prime}\sqrt{\gamma_{\varphi}}\hat{\varphi}}.$
(5.32)

Figure 5.2:  Introducing another UDW gate acts as additional information
entering the channel and subsequently will introduce cross-talk noise to the
original channel in Eq. 2.24.

Substituting Eq. 5.32 into Eq. 5.30 we get

$\displaystyle\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})$
$\displaystyle=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)$
$\displaystyle\times
e^{i(\phi+\sqrt{\gamma}_{\varphi})\hat{\varphi}}\hat{\rho}_{\varphi}e^{i(\phi+\sqrt{\gamma}_{\varphi})\hat{\varphi}}$
(5.33)
$\displaystyle=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)$
$\displaystyle\times{\ket{s(\phi+\sqrt{\gamma}_{\varphi})\alpha}\\!\\!\bra{s^{\prime}(\phi+\sqrt{\gamma}_{\varphi})\alpha}}.$
(5.34)

As was shown in Sec. 2.3.3, the unwanted states result from the correlator
formed by the partial trace over the field and the non-orthogonality of
coherent states. Tracing over
$\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})$ to
reproduce the effect of the correlator in Eq. 2.25 will demonstrate the
changes made to the inner product of the coherent states which we have denoted
by
$\braket{\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})}$
and defined as

$\operatorname{Tr}\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})\equiv\braket{\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})}=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)\times\braket{s^{\prime}(\phi+\sqrt{\gamma}_{\varphi})\alpha}{s(\phi+\sqrt{\gamma}_{\varphi})\alpha}\\\
=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)\,\exp\bigg{(}-\frac{1}{2}s^{\prime
2}(\phi+\sqrt{\gamma}_{\varphi})^{2}|\alpha|^{2}-\frac{1}{2}s^{2}(\phi+\sqrt{\gamma}_{\varphi})^{2}|\alpha|^{2}+s^{\prime}s(\phi+\sqrt{\gamma}_{\varphi})^{2}|\alpha|^{2}\bigg{)}\\\
=\sum_{s,s^{\prime}\in\pm}\int_{-\infty}^{\infty}d\phi\,p(\phi)\,\exp\bigg{(}-\frac{1}{2}(\phi+\sqrt{\gamma}_{\varphi})^{2}|\alpha|^{2}(s^{\prime}-s)^{2}\bigg{)}$
(5.35)

where we have utilized Eq. 5.21 as we did in Sec. 5.5.1. Notice Eq. 5.35 has
the same structure as Eq. 5.20 but allows new dephasing construction of the
parameters. As expected when $s^{\prime}=s$ the expectation value is trivially
one and will not effect the outcome of channel Eq. 2.24. However, when
$s^{\prime}\neq s$ we now have two parameters, $\gamma_{\varphi}$ and $\phi$,
to enforce dephasing of unwanted states.

Plugging the above probability distribution into Eq. 5.35 and evaluating the
integral for $s^{\prime}\neq s$ we get

$\braket{\Xi_{\varphi\rightarrow\varphi^{\prime},\mathrm{N}}(\hat{\rho}_{\varphi})}=\frac{\exp{\bigg{(}-\frac{2\gamma_{\varphi}|\alpha|^{2}}{1+4|\alpha|^{2}b^{2}\gamma_{\varphi}}\bigg{)}}}{\sqrt{1+4|\alpha|^{2}b^{2}\gamma_{\varphi}}}$
(5.36)

where we have redefined our CT dephasing parameter as a multiple of the
original $\gamma_{N}=b\gamma_{\varphi}$. Eq. 5.36 is directly comparable to
Eq. 5.20. It is straightforward to see in Fig. 5.3(a) as $b\rightarrow 0$ the
interaction with the noise is turned off, at $b=1$ the channel is the
noisiest, and as $b\rightarrow\infty$ $\gamma_{N}$ acts as the primary
dephasing factor for small values of $\gamma_{\varphi}$ which is demonstrated
in Fig. 5.3(b).

(a) As $b\rightarrow 0$ the coupling as well as the noise is turning off.

(b) As $b\rightarrow\infty$ it takes over the dephasing process for smaller
values of $\gamma_{\varphi}$

(c) As the noise is turning off the quantum capacity is increasing.

(d) For large $b$ and small values of $\lambda_{\varphi}$ the additional
interaction can result in better quantum capacity.

Figure 5.3: Comparing the coherent information of channels with and without
noise, it is clear to see that the noisy channel $\Xi_{A\rightarrow
B,N}(\lambda_{\varphi,b})$ reaches a channel capacity of one at a slower rate.
At $b=1$ the noisy signal is the highest, this is due to a signal boost in the
unwanted states. Regardless, we can see in (d) an increase in the lower bound
of coherent information for small values of $\gamma_{\varphi}$. Code for
simulations can be found at Ref. [1].

##### Noisy UDW Effects on Coherent Information

To understand the difference these new parameters make to the coherent
information we can look at the definition of $\gamma_{\varphi}$ in Eq. 5.18.
Since $\gamma_{\varphi}$ is a function of $\lambda_{\varphi}$ we can set up
the new inner product in terms of a new coupling constant
$\lambda_{\varphi,b}$ given by the relation

$\frac{\exp{\bigg{(}-\frac{2\gamma_{\varphi}|\alpha|^{2}}{1+4b^{2}\gamma_{\varphi}|\alpha|^{2}}\bigg{)}}}{\sqrt{1+4b^{2}\gamma_{\varphi}|\alpha|^{2}}}=\exp\bigg{[}-2(\lambda_{\varphi,b})^{2}\times\int
dk\,|\tilde{F}_{\nu}(k)|^{2}|\alpha|^{2}\bigg{]}.$ (5.37)

We then solve for $\lambda_{\varphi,b}$ in terms of $\lambda_{\varphi}$ and
ascertain

$\lambda_{\varphi,b}=\bigg{[}\frac{\lambda_{\varphi}^{2}}{1+\frac{4b^{2}|\alpha|^{2}\lambda_{\varphi}^{2}}{\sqrt{(2\pi)^{3}}\sigma}}-\frac{\sqrt{(2\pi)^{3}}\sigma}{2|\alpha|^{2}}\times\ln\left(1+\frac{4b^{2}|\alpha|^{2}\lambda_{\varphi}^{2}}{\sqrt{(2\pi)^{3}}\sigma}\right)\bigg{]}^{\frac{1}{2}}$
(5.38)

where the term $\sqrt{(2\pi)^{3}}\sigma$ is a consequence of Gaussian smearing
with width $\sigma$. Calculating channel capacity with this new value
$\lambda_{\varphi,b}$, while keeping the remaining parameters intact, still
reaches near perfect channel capacity but reaches it slower, when $b\leq 1$
demonstrated by Fig. 5.3(c). Furthermore, for high values of $b$ and low
values of $\lambda_{\varphi}$ one would expect $b$ to act as the primary
dephasing factor as shown in Fig. 5.3(d).

In this chapter, We have shown that the formalism of quantum channels produced
by UDW detectors provides a bosonic dephasing channel perspective. With this
perspective, we have demonstrated that the purpose of the dephasing is to
remove unwanted states that lead to the scrambling of the QI encoded on the
field. We have shown that the dephasing constant can be written in terms of
the strength of coupling between the qubit and field.

We aimed to demonstrate several applications of the dephasing perspective that
provide interpretations of unwanted interactions. Firstly, we applied the
canonical form of a bosonic dephasing channel, allowing the system to interact
with the environment. In this idealized dephasing channel, the prior
constraints on the off-diagonals were the only effects that remained when
tracing over the field. Subsequently, this indicated no additional noise
generated from interacting with the environment under this prescription.

Secondly, how does coherent information change given some noise due to
additional UDW detectors? To evaluate this numerically, we presented a unitary
operation that introduces CT noise to the system and calculated how that noise
affects the strength of coupling between the qubits and fields. Figure 5.3
demonstrates how the additional noise can affect the propagation of quantum
information through the channel given in Eq. 2.24. These values make intuitive
sense as one may expect an additional signal boost to non-orthogonal off-
diagonal elements of the density matrix in Eq. 5.34 which act to scramble the
QI.

An open problem remaining is the possibility of writing down a unitary
dephasing channel that increases coherent information overall. One might
notice that non-unitary interactions that increase the strength of the
diagonal elements of the coherent state density matrix while decreasing the
off-diagonal elements are possible. However, given the nature of dephasing
channels, accomplishing this with unitary operations requires much care.

##  Chapter 6 Universal UDWQC with Analogous Quantum Logic Gates

Elementary quantum logic gates, from which universal quantum computing is
established [87, 97], are a vital part of quantum information theory but are
absent in the developing theory of Relativistic Quantum Information (RQI).
Unruh–DeWitt (UDW) detectors have given RQI a formalism that allows for
applications in quantum Shannon theory, such as mutual information, coherent
information, and quantum capacity in field-mediated qubit interactions [2].
Experimental realizations of UDW-style qubits have been proposed during the
last few years[18], and a set of quantum logic gates will provide insights
into quantum communication and computational power in two-dimensional quantum
materials[24].

The introduction of elementary quantum logic gates, designed for UDW qubits to
carry out quantum processing tasks, allows for cross-discipline applications
for qubit-field quantum computing. To illustrate this, we introduce elementary
UDW quantum logic gates, a field-mediated equivalent to canonical qubit logic
gates, to demonstrate the diamond distance, a measure of similarity, between a
UDW state transfer channel and the canonical state transfer channel.
Measurements like diamond distance allow for a rigorous comparison between
field-mediated transduction through UDW detectors and local quantum mechanical
operations.

###  6.1 Canonical Quantum Logic Gates

####  6.1.1 Projectors and Quantum Logic Gates

Behind every quantum logic gate is a series of Pauli and projection operators
in computational basis states. A common example of this is the quantum “NOT”
gate. Commonly denoted as $\hat{X}$, the quantum NOT gate, also known as the
Pauli-X gate, has the mathematical forms

$\hat{X}=\hat{\sigma}^{x}=\sum_{\mu\in\pm}\mu\hat{P}^{\mu}_{X}={\ket{+}\\!\\!\bra{+}}-{\ket{-}\\!\\!\bra{-}}={\ket{1}\\!\\!\bra{0}}+{\ket{0}\\!\\!\bra{1}}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (6.1)

Here we have defined the projector
$\hat{P}^{\mu}_{X}\equiv\ket{\mu_{x}}\bra{\mu_{x}}$ with $\ket{\mu_{x}}$ being
the basis states in the x-basis. Similar projectors can be constructed for the
z- and y-basis.

A truth table is a stylistic representation of input and output states that
displays the changes that are undergone through a given process. We can
generate a truth table for the NOT gate that has the following form.

NOT Gate Input | Output
---|---
$\ket{0}$ | $\ket{1}$
$\ket{1}$ | $\ket{0}$
Table 6.1: A truth table is a simplified look at operating on an input state
with a computational, or in our case a quantum computational operation. In
this table we are acting on the input state with the NOT gate to achieve the
Output state.

The other Pauli matrices have a similar formalism