# Does Microservices Adoption Impact the Development Velocity? A Cohort Study

(Registered Report)

Nyyti Saarimäki  Tampere University
TampereFinland<EMAIL_ADDRESS>,  Mikel Robredo  University of Oulu
OuluFinland<EMAIL_ADDRESS>,  Sira Vegas  Universidad Politécnica
de Madrid
MadridSpain<EMAIL_ADDRESS>,  Natalia Juristo  Universidad Politécnica de
Madrid
MadridSpain<EMAIL_ADDRESS>,  Davide Taibi  University of Oulu
OuluFinland<EMAIL_ADDRESS>and  Valentina Lenarduzzi  University of Oulu
OuluFinland<EMAIL_ADDRESS>
###### Abstract.

[Context] Microservices enable the decomposition of applications into small
and independent services connected together. The independence between services
could positively affect the development velocity of a project, which is
considered an important metric measuring the time taken to implement features
and fix bugs. However, no studies have investigated the connection between
microservices and development velocity.

[Objective and Method] The goal of this study plan is to investigate the
effect microservices have on development velocity. The study compares GitHub
projects adopting microservices from the beginning and similar projects using
monolithic architectures. We designed this study using a cohort study method,
to enable obtaining a high level of evidence.

[Results] The result of this work enables the confirmation of the effective
improvement of the development velocity of microservices. Moreover, this study
will contribute to the body of knowledge of empirical methods being among the
first works adopting the cohort study methodology.

Empirical Software Engineering, Cohort Study, Microservices, Development
velocity

##  1\. Introduction

Microservices have become popular over the last years as they are considered
to have several benefits over the traditional monolithic structure. They
divide a project into smaller autonomous services each having a clear purpose
(Fowler and Lewis, 2014). Therefore, microservice architecture has a
significant effect on the overall development of a project and it allows
independent and concurrent development and deployment (Waseem et al., 2020),
reusability (Fowler and Lewis, 2014), and enhanced scalability (Dragoni et
al., 2018). In theory, this should make also the development process faster.
Development velocity is an important measure in current software projects and
a higher velocity implies faster implementation of features and quicker
resolution of issues. However, there are no studies investigating the
connection of microservices and development velocity.

The goal of this study is to investigate the effect microservices have on
development velocity. We hypothesize that microservice projects are faster to
develop than monolithic ones. To investigate this, we have planned a cohort
study comparing actively developed projects adopting microservices from the
beginning and similar projects using a monolithic structure.

The proposed study plan covers all aspects required for conducting the study,
including data sources, variable definitions, and data analysis. The study is
designed as a cohort study because of the causal nature of the goal. As a
result, the outcomes of the study are expected to provide high-level evidence
on the topic.

Paper structure: Section 2 describes the background and the related work.
Section 3 describes our study design and the execution. Section 4 discusses
the possible threats. Section 5 draws conclusions and future work.

##  2\. Background and Related Work

###  2.1. Microservices

The Microservice style is derived from Service-Oriented Architecture (SOA),
where services have dedicated responsibilities but are not independent (Fowler
and Lewis, 2014; Dragoni et al., 2017). In SOA, the individual services are
neither full-stack (e.g., the same database is shared among multiple services)
nor fully autonomous (e.g., service A depends on service B). On the contrary,
microservices systems are decentralized systems composed of a large number of
small independent services that communicate through different lightweight
mechanisms (Pahl et al., 2018).

Microservices are relatively small and autonomous services deployed
independently, with a single and clearly defined purpose (Fowler and Lewis,
2014). Microservices enable vertically decomposing applications into a subset
of business-driven independent services. Each service can be developed,
deployed, and tested independently by different development teams using
different technology stacks. Microservices have a variety of advantages
(Jamshidi et al., 2018). They can be developed in different programming
languages, can scale independently from other services, and can be deployed on
the hardware that best suits their needs. Moreover, because of their size,
they are easier to maintain and more fault-tolerant since the failure of one
service will not disrupt the whole system, which could happen in a monolithic
system (Fowler and Lewis, 2014).

###  2.2. Docker

Docker111Docker: https://www.docker.com/ is a platform for developing,
shipping and running applications in loosely isolated environments called
containers. A Docker container includes everything needed to run an
application and, therefore, it enables the developer to develop on their
platform on choice without having to worry about where the program is deployed
(Jangla, 2018a). This is one of the several reasons why Docker is a popular
tool among developers, and it was voted as the most important tool in
StackOverflow’s developer survey in
2022222https://survey.stackoverflow.co/2022/#most-popular-technologies-tools-
tech-prof. The tool is commercial but is has a free version for open-source
communities and individual developers.

A dockerized project can consist of one or several containers. The containers
can communicate with each other but the implementation of each container is
otherwise independent of other containers. Therefore, Docker is one way of
creating microservices. In practice, each container has a file called
`Dockerfile`. It is a ”is text document that contains all the commands a user
could call on the command line to assemble an image.” (Inc., 2023).

###  2.3. Development velocity

Development velocity estimates the amount of productive work a developer team
can complete in a given time frame.

The influence of microservices on velocity can vary depending on different
factors. Here are some ways in which the adoption of microservices can affect
velocity:

Independent Development and Deployment: Microservices facilitate the
autonomous development and deployment of individual services. This enables
teams to concurrently work on different services, reducing dependencies and
bottlenecks. Consequently, development cycles can be expedited, allowing for
quick iteration and delivery of new features or improvements (Soldani et al.,
2018)(Fowler and Lewis, 2014).

Enhanced Scalability: Microservices architecture empowers the independent
scaling of individual services based on their specific requirements. This
flexibility enables teams to optimize performance and responsiveness, ensuring
efficient utilization of resources. Fine-grained scaling aligned with demand
can bolster velocity by effectively handling the increased workload (Dragoni
et al., 2018).

Concurrent Development and Testing: Microservices architecture facilitates
parallel development and testing as services can be developed and tested in
isolation. Teams can independently work on different services, enabling
concurrent progress. This significantly speeds up the development lifecycle,
as changes and updates can be implemented in parallel, reducing overall
development time (Waseem et al., 2020).

Reusability and Modularity: Microservices encourage the creation of small,
reusable components that can be shared across services. This promotes
reusability and modularity, accelerating development by leveraging existing
services, libraries, and frameworks. Developers can build upon existing
functionalities, reducing redundant efforts and hastening the development
process (Fowler and Lewis, 2014).

Ultimately, the influence of microservices on velocity depends on the
effectiveness of the architecture design, the maturity level of microservices
adoption, the skills and experience of the development teams, and the
efficiency of supporting processes and tools (Soldani et al., 2018)(Taibi et
al., 2017).

To the best of our knowledge, this is the first study empirically
investigating the impact of microservices on velocity.

##  3\. The Empirical Study: Design and Execution Plan

In this Section, we describe our empirical study focusing on the design and
the execution plan.

###  3.1. Goal and Research Questions

The goal of the study is to understand the effect of Docker-based microservice
architecture on the development velocity during the early stages of the
evolution of a software project. Our goal is answered by the following
research question:

RQ. _Do projects adopting microservices from the beginning have higher
development velocity than monolithic projects?_

The hypothesis is that using Docker-based microservice architecture may help
to have a higher development velocity.

Decomposing software projects into independent dockerized microservices are
expected to accelerate the development process as each service can be
independently deployed, and optimized according to their needs (Jangla,
2018b). Despite Docker being a popular tool among developers, there are no
studies on the topic exploring the tool’s effect on velocity. Therefore, this
is the first study that empirically investigates the impact of microservices
on velocity.

###  3.2. Study Design

The research question of this paper is causal in nature and the most suitable
methodology for studying causality is a controlled experiment. However, the
data is observational and historical which prevents us from conducting one.
Thus, the study was designed as a retrospective cohort study which is an
analytical observational study methodology capable of obtaining high-level
evidence from such data. The overall design of our study is presented in
Figure 1.

Figure 1. The design of the observational study.

A cohort study (Grimes and Schulz, 2002b; Gordis, 2014) investigates whether
an exposure (independent variable) causes an outcome (dependent variable) by
comparing the outcomes of two or more groups of study subjects with different
levels of exposure. It carefully selects a study population and follows it
over a defined time period to see what naturally happens. The subjects are
selected from the source population (source of data) using unambiguous
eligibility criteria. They are needed to ensure the study population is
capable to answer the research questions in the context of the population of
interest.

As the methodology is observational, the results can be influenced also by
other factors than the exposure and outcome, such as project age or size.
Therefore, in cohort studies, it is crucial to identify these factors from
literature or using domain knowledge, measure them, and control for their
effect.

The exposure, confounders, and outcome are measured for all study subjects at
the start of the follow-up. The outcome is measured at the end of the follow-
up period. The gathered data is then used to analyze whether there is a
relationship between the exposure and the outcome. Note that the outcome is
measured also at the start of the study to control for the starting point.
This provides a temporal framework that makes it possible to assess causality.

###  3.3. Setting

The study investigates open-source projects from GitHub which are created
between 2020-2021. The cases of the study are projects which have adopted a
microservice architecture within six months of their creation while the
controls of the study have a monolithic structure.

The projects are tracked for 18 months starting from when they are half a year
old until they reach two years of age. Therefore, the study subjects have
different data measurement dates, but the same fixed length of follow-up time
(Figure 1).

###  3.4. Subjects

The subjects of the study are open-source software projects. To ensure the
study subjects can be used to answer the study goals, eligibility criteria are
needed.

The inclusion criteria determines what is required from each subject in order
to be considered as a part of the study.

  * •

Open-source project created in GitHub between 2020-2021

  * •

Uses GitHub to track issues

Exclusion criteria defines subjects which are excluded from the study at the
start of the follow-up period. We applied the following criteria:

  * •

Project has one or two Dockerfiles. Projects having only one or two
Dockerfiles are purely neither controls nor cases. We consider projects having
less than three to not have fully adopted the microservices, or not being
large enough to be considered in this study. We do not consider files like
dockerized databases or volumes in this count.

In addition to selection criteria, we define a loss to follow-up criteria. The
criteria define the subjects which are excluded from the study based on their
activity during the follow-up period. Subjects not meeting the criteria at the
end of the follow-up period are considered drop-outs and excluded from the
final study subjects.

  * •

The overall monthly trend of commits is decreasing during the follow-up.
Ensures the project is developed during the follow-up which is required to be
able to detect the outcome. Inspecting the trend includes projects with
different resources in the data set.

  * •

The overall monthly trend of issues is decreasing during the follow-up.
Ensures tracking issues using a well-integrated part of the development
process.

  * •

Use of Docker is interrupted during the follow-up period. Projects which at
any point before the end of the follow-up period introduced Docker and then
removed it. This ensures the projects maintain the exposure through the
follow-up period and the projects are free of any potential effects of
previous Docker usage.

###  3.5. Variables

The variables included in the study are described below. The hypothesized
relationships between the variables which are included in the study are
visualized in Figure 3.

The independent variable is adoption of microservice architecture. It is a
boolean variable indicating if a project uses the microservice architecture or
not. We focus on projects implementing the microservices using Docker and
define a project to have adopted microservice architecture if the repository
contains at least three docker files at the age of six months. Three docker
files are considered to be the minimum threshold for a system to be considered
distributed.

We consider the dependent variable as the development velocity of a project at
the end of the follow-up period. The velocity of a project is the average time
taken to close issues within a three month period. Therefore, lower velocity
indicates faster issue fixing time. Figure 2 presents an example of the
calculation while the measurement period is visualized in Figure 1.

Figure 2. Velocity calculation. The issues closed in the selected time window
(green) are included in the velocity calculation. The velocity in the figure
is (150 + 95) days / 2 = 122.5 days.

Based on our domain knowledge, the following preliminary set of confounding
variables is considered for the study. Even though collecting the planned
variables should be possible, the set of considered confounders might be
altered during data collection. All confounding variables are measured at the
start of the follow-up period.

  * •

Start velocity. The velocity of the project before the follow-up period.

  * •

Size of the project: The number of lines of code in the repository.

  * •

Development language: The main development language reported by GitHub.
Different languages might have a different impact on developers’ productivity,
and therefore on velocity. High-level languages might enable faster
development (i.e. higher velocity) while lower-level languages might require a
longer time between deployments (e.g. C or C++).

  * •

Number of programming languages: a higher number of different programming
languages might fragment the development community, and therefore reduce
velocity in case no developers are able anymore to modify a service written in
a specific language.

  * •

Age: The number of days from the first commit to the start of the follow-up.

  * •

Number of commits: Number of commits to the main branch of the repository by
the start of the follow-up reported by GitHub. Each commit might create or fix
an issue, and therefore, it could affect the velocity.

  * •

Number of issues: The number of created issues (open and closed) at the start
of the follow-up. The number of issues reported for the

  * •

Number of developers: The number of persons contributing to the project
according to GitHub. In theory, more developers should mean higher development
velocity. projects affect its velocity.

In addition to confounders, we have identified the following potential
relationships between them which have potential effects on the outcome
variable. These relationships will not be included in the study as additional
variables.

  * •

Size $/$ #Developers: A proxy for the amount of code a single developer
produces.

Figure 3. Graph visualizing the included variables and their (hypothesized)
connections.

###  3.6. Data sources and measurement

The source population (list of potential projects) will be extracted from the
World of Code (WoC) version U (updated 10/2021) or the following ones, based
on their availability when conducting the study. It is a Free/Libre Open
Source Software (FLOSS) ecosystem and a computational and statistical
infrastructure that aims to provide an operational, research-ready, updatable,
and expandable dataset (Mockus et al., 2023). The huge dataset is curated by
fully collecting and cross-referencing project objects (e.g., authors,
projects, commits, and blobs) mainly from public version control systems. WoC
enables getting information on project commits, blogs, and files.

The value of the independent variable (microservice adoption) is determined
using data from version control history. We crawl the commit history of the
repositories to determine the dates on which dockerfiles were added or
modified. From that information, we are able to determine the number of
dockerfiles in the project at any given date.

The data for determining the development velocity (dependent variable) is
gathered from GitHub using its REST API. We will crawl information about all
issues of the included repositories. Specifically, we will gather the
creation, update, and closing date for each issue.

The confounding variables are collected also from the projects’ GitHub
repositories. As GitHub API does not provide the size of the projects, we will
clone all included projects. If needed, we will analyze the repository using a
tool to determine the value of a confounder.

###  3.7. Study size

Being able to detect a difference between two groups requires a sufficient
number of subjects. Despite the data being mined from GitHub, which has a
large set of projects, the strict eligibility criteria of the study might
result in a small set of suitable projects. To understand and mitigate this
risk, we adopt power analysis which is a statistical calculation conducted to
determine the minimum required sample size for detecting significant effects
from the data. It ensures the study results are valid with a given level of
confidence and can detect meaningful effects.

Its calculation requires defining the power, significance level, and estimated
effect size for the study. Power is the probability of rejecting the null-
hypothesis when it is false while significance level determines the
probability of rejecting the null-hypothesis when it is true. We used value
80% for the power and 5% for the significance level, both commonly adopted
values. The size of the relevant effect in this study was determined as small
(Cohen’s d $\geq$ 0.2 (Cohen, 1988)). Even Docker is adopted mainly for other
reasons than enhancing development velocity., it is expected to have an effect
also on the velocity. Therefore, we considered negligible changes irrelevant
to this study.

Due to the strict eligibility criteria, the number of cases in the obtained
data set could be lower than expected for a 1:1 case-to-control ratio. Having
up to four controls for each case can increase the statistical power of a
study (Song and Chung, 2010). Therefore, we calculated the required sample
sizes for several case-to-control ratios. The analysis was conducted using
SPSS’s (version 29) power analysis for ”Independent-samples T Test”
functionality and t results from the analysis are presented in Table 1.

| Case to control ratio
---|---
| 1:1 | 1:2 | 1:3 | 1:4
#Cases | 394 | 295 | 263 | 246
#Controls | 394 | 590 | 787 | 983
Total | 788 | 885 | 1,050 | 1,229
Power=0.8, p=0.05, Effect size=0.2
Table 1. Results from the power calculation for the different case-to-control
ratios.

As the potential controls for this study are projects in GitHub using a
monolithic structure, the number of potential controls is expected to be more
than quadruple compared to the potential cases. In such case, we will consider
performing random sampling for the selection of the control projects.

###  3.8. Statistical analysis

The analysis will be conducted using SPSS (version 29) and R (version 4.1.1).

The connections between exposure, outcome, and confounder are visualized in
Figure 4. The data analysis of the study investigates several aspects of the
figure.

Figure 4. The relationships between exposure, outcome, and confounder.

####  3.8.1. Crude analysis

First, we determine the unadjusted relationship between the cases and controls
(arrow 2). The analysis does not include any external variables and it
compares the velocity between cases and controls. The results serve as base
knowledge and justification for further analysis including the external
variables. The groups are compared using independent samples t-test and
Cohen’s d effect size which both require (approximate) normality and
homogeneity of variances. If these requirements are not fulfilled, data
transformation is performed in order to fulfill them. In case the data
transformation does not provide approximated normality, the comparison is
conducted using the non-parametric Mann-Whitney test and Cliff’s delta effect
size.

####  3.8.2. Treating confounders

We investigate if the included variables are actually confounders. A
confounder needs to have a relationship to both exposure and outcome. First,
we analyze if there is a relationship between the confounders and the exposure
(arrow 1) by comparing the values of potential confounders between the case
and control groups.

The treatment method for the included confounders is chosen individually. The
choices are made based on the characteristics of the collected data. Below we
present the common techniques for treating confounding (Grimes and Schulz,
2002a), however, as the methods cannot be used in all circumstances, the used
techniques depend on the observed scenario.

  * •

Restriction removes the effect of a confounding variable by ensuring all
subjects are exposed to the same level of confounding. It is the simplest of
the methods as in practice this is done by adding exclusion criteria and no
further analysis is needed. However, after restricting a variable, its effect
cannot be assessed.

  * •

Matching ensures the study population contains $n$ similar controls for each
case and is done in the design phase of the study. This ensures the groups are
exposed to the effect confounders similarly. The matching criteria can consist
of one or a combination of several variables and the matching can be conducted
using several different algorithms (Rosenbaum, 2010). However, matching
prevents studying the variables used in the process.

  * •

Stratification ensures in the analysis phase of the study that subjects with
different levels of confounder have similar effects between the cases and
controls. The study subjects are divided into groups (or strata) based on the
level of confounding they are exposed to. The analysis is conducted separately
for each group and if the results differ, an adjusted value is calculated, for
example, using meta-analysis (Borenstein et al., 2021).

  * •

Statistical adjustment uses mathematical models to determine the relationship
on interest while controlling for the effect confounders. The confounders are
added to the considered models as additional independent covariates. To
determine the covariates’ impact on the results for the outcome variable, we
will consider potential methodologies of regression analysis based on the
distributional assumptions derived from the data. Similarly, based on the
assumed distribution, we will analyze the variance of the dependent variable
subject to the independent variable while accounting for the rest of
covariates.

To assess the significance of the considered models, we utilize existing
information criteria methods such as Akaike Information Criterion (AIC) or
Bayesian Information Criterion. Moreover, we consider performing Backward
Selection (BS) procedure based on the results from the mentioned information
criteria to achieve the model that better describes the relationship between
the dependent and independent variables.

##  4\. Threats to Validity

Construct Validity. The measurement of microservice adoption is a threat in
the study as version control does not provide data for validating its actual
usage. Thus, we will rely on the number of dockerfiles in the repository. We
try to mitigate this threat by requiring at least three dockerfiles and
excluding volumes etc from the count. However, this does not guarantee active
microservice adoption.

Internal Validity. The projects are collected from a version control system
that automatically collects and logs the data. This ensures the similarity of
the data collection between cases and controls. Additionally, GitHub has a
large number of projects and we should be able to gather a data set that
matches the minimum sample size determined in Section 3.7. However, the
exclusion and loss to follow-up criteria we set for this study might lead to a
sharp reduction of analyzable projects. To mitigate this, we would consider
collecting projects from additional data sources, such as GitLab, as well as
Jira or Bugzilla for issue tracking.

Cohort studies are sensitive to case and control groups not being comparable.
We plan to ensure their similarity by suitable eligibility criteria and
controlling for confounders using suitable methods such as restriction or
matching.

The projects are followed from their creation until they are two years old
and, therefore, they are in the same phase of their evolution. This could make
the data collection process difficult as data measurement dates vary between
the subjects. However, this is considered only a minor threat as GitHub
automatically records the data.

The velocity of the project might differ between projects for several reasons.
This is addressed by including external variables we have identified as
potentially having an effect on it. To further mitigate this, we measure the
development velocity also at the start of the follow-up and include the value
in the analysis. The velocity of the project is calculated using a three month
time window which might be inadequate for some projects, especially if the
window is during holidays. The projects can also use some other issue tracking
system than GitHub. However, we mitigate this threat by requiring a constant
or growing trend of commits and issues during the follow-up.

External Validity. The included projects are open-source projects from GitHub
which reach a certain level of maturity within two years of their creation.
Therefore, the results are generalizable to young and active open-source
projects, that is, this study is not applicable to older projects with greater
maturity. However, the real generalizability of the results will depend on the
collected data.

Conclusion Validity. The planned data analysis follows the structure generally
adopted in cohort studies. As we do not have the actual data yet, we do not
know what are the most suitable data analysis methods. However, we have
presented the general directions and alternatives for the planned analysis.

##  5\. Conclusion

The objective of this study plan is to examine the impact of microservices on
development velocity. The study will involve a comparison between GitHub
projects that initially implemented microservices and similar projects that
utilized monolithic architectures. We have structured this study using a
cohort study methodology to ensure a robust level of evidence.

The outcome of this research will validate the potential enhancement in
development velocity achieved through the utilization of microservices.
Additionally, this study will make a valuable contribution to the existing
body of knowledge on empirical methods, as it will be one of the pioneering
works adopting the cohort study methodology.

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# Wigner functions for the pair angle and orbital angular momentum:
Possible applications in quantum information

H. A. Kastrup<EMAIL_ADDRESS>DESY Hamburg, Theory Group, Notkestrasse
85, D-22607 Hamburg, Germany

###### Abstract

The framework of Wigner functions for the canonical pair angle and orbital
angular momentum, derived and analyzed in 2 recent papers [H. A. Kastrup,
Phys. Rev. A 94, 062113(2016) and Phys. Rev. A 95, 052111(2017)], is applied
to elementary concepts of quantum information like qubits and 2-qubits, e.g.,
entangled EPR/Bell states etc.. Properties of the associated Wigner functions
of such superposed states (pure and mixed) are discussed and illustrated. The
Wigner functions of EPR/Bell states are distinguished by their topologically
“twisted” domain on the configuration subspace
$\mathbb{S}^{1}\times\mathbb{S}^{1}$, a torus, of the total phase space. Like
the applications of Wigner functions in quantum optics the results presented
in the present paper may be useful for the description and analysis of quantum
information experiments with orbital angular momenta of light beams or
electron beams, respectively.

##  I Introduction

In two recent papers Kastrup (2016, 2017) basic properties of Wigner functions
on cylindrical phase spaces $\mathbb{S}^{1}\times\mathbb{R}$ (angle and
orbital angular momentum, denoted by “A-OAM” in the following) were derived
and discussed. The possible usefulness of that concept has, of course, to be
demonstrated by its applications to special systems and associated
experiments.
A few simple typical example were discussed in Ch. IV C of Ref. Kastrup
(2016). The present paper suggests possible applications to such elementary
concepts as “qubits” and “2-qubits” of quantum information, see, e.g., Refs.
Nielsen and Chuang (2010); Barnett (2009).

The quantized canonical system of the pair angle and orbital angular momentum
is of special theoretical interest for quantum information because it provides
as a basic framework an infinite dimensional Hilbert space
$L^{2}(\mathbb{S}^{1},d\varphi/2\pi;\delta)$, with orthonormal basis

$e_{m,\delta}(\varphi)=e^{i(m+\delta)\varphi},\;\;m\in\mathbb{Z},\;\delta\in[0,1),$
(1)

with scalar product

$(\psi^{[\delta]}_{2},\psi^{[\delta]}_{1})=\int_{-\pi}^{\pi}\frac{d\varphi}{2\pi}\psi^{[\delta]\ast}_{2}(\varphi)\psi^{[\delta]}_{1}(\varphi),\;\;(e_{m,\delta},e_{n,\delta})=\delta_{mn}$
(2)

($\delta_{mn}$: Kronecker symbol) and the expansions

$\displaystyle\psi^{[\delta]}(\varphi)=$
$\displaystyle\sum_{m\in\mathbb{Z}}c_{m}\,e_{m,\delta]}(\varphi),\;\;c_{m}=(e_{m,\delta}\psi),$
(3)
$\displaystyle\psi^{[\delta]}(\varphi+2\pi)=e^{i2\pi\delta}\psi^{[\delta]}(\varphi).$

The number $\delta\in[0,1)$ mathematically characterizes a covering of the
group $U(1)$ and physically a fractional OAM Kastrup (2006). The following
discussions assume $\delta=0$ and denote
$L^{2}(\mathbb{S}^{1},d\varphi/2\pi)\equiv
L^{2}(\mathbb{S}^{1},d\varphi/2\pi;\delta=0),\,e_{m}\equiv e_{m,\delta=0}$
etc.. The essential generalizations of the main results obtained for
$\delta=0$ for the case $\delta\neq 0$ are discussed in Appendix A. Note that
the coefficients $c_{m}$ in Eq. (3) are independent of $\delta$.

One of the advantages of A-OAM systems for quantum information theories is
that one can select finite dimensional subspaces of any dimension $d$: $d=2$:
qubits, $d=3$: qutrits, $\ldots,~{}d$: “qudits”, like, e.g.,

$(e_{m_{0}}+e_{m_{1}}+\ldots+e_{m_{d-1}})/\sqrt{d}$ (4)

and associated tensor product spaces which then contain entangled states. (For
a recent general discussion of entanglement in quantum theory see Ref.Witten
(2018).)

In those $d$-dimensional subspaces $\varphi$-independent (“global”) unitary
transformations $U(d)$ and other linear mappings (called “gates” in quantum
information) may act.

Especially one can incorporate the usual elementary qubits from 2-dimensional
spaces bar (a) , e.g.,

$(|0\rangle\pm|1\rangle)/\sqrt{2},$ (5)

and associated entangled EPR/Bell product states bar (a)

$\displaystyle(|00\rangle\pm|11\rangle)/\sqrt{2}\equiv$
$\displaystyle(|0\rangle\otimes|0\rangle\pm|1\rangle\otimes|1\rangle)/\sqrt{2},$
(6) $\displaystyle(|01\rangle\pm|10\rangle)/\sqrt{2}\equiv$
$\displaystyle(|0\rangle\otimes|1\rangle\pm|1\rangle\otimes|0\rangle)/\sqrt{2}$
(7)

etc.

In the following the A-OAM Wigner functions of the most general qubits and
2-qubits will be derived and some special cases discussed and illustrated in
more detail. The discussion is mainly restricted to pure states. The
generalization to mixed states is indicated by Eqs. (30) – (33) below.

### Experiments

Experimentally A-OAM systems have been investigated particularly with
(Laguerre - Gauss) laser light beams (see, e.g., the reviews Allen  _et al._
(2003); Yao and Padgett (2011); Andrews and Babiker (2013); opt (2017)) and
with electron beams Lloyd  _et al._ (2017). For recent related experiments
see, e.g., Refs. Rego  _et al._ (2019); Chen  _et al._ (2019).

The crucial property of such beams is that they carry OAM $\vec{p}$ along
their directions, i.e., those beams “rotate” around their “axis”! For
experimental investigations of entangled OAM states see, e.g., the articles
Mair  _et al._ (2001); Exter  _et al._ ; Krenn  _et al._ (2017) and references
therein.

The use of associated A-OAM Wigner functions may be helpful for descriptions
and analyses of those experiments! Recall that, in principle, all statistical
properties of a quantum state of a system can be derived from its Wigner
function on the associated classical phase space! In possible applications one
should integrate over the radial coordinate of the cylindrical system Kastrup
(2016) before employing the A-OAM Wigner function framework.

##  II Wigner functions for qubits

###  II.1 Pure states

The most general qubit of a A-OAM system is given by

$\displaystyle\chi_{m_{0},m_{1}}^{(\alpha,\beta)}(\varphi)=\cos\beta\,e_{m_{0}}(\varphi)+\sin\beta\,e^{i\alpha}\,e_{m_{1}}(\varphi),$
(8) $\displaystyle m_{0},m_{1}\in\mathbb{Z},\;m_{1}\neq
m_{0};\,\beta\in(0,\frac{\pi}{2});\,\alpha\in[0,2\pi),$
$\displaystyle|\chi_{m_{0},m_{1}}^{(\alpha,\beta)}(\varphi)|^{2}=1+\sin
2\beta\cos[(m_{0}-m_{1})\varphi-\alpha].$ (9)

The states (8) are elements of the 2-dimensional subspace

$Q^{2}_{m_{0},m_{1}}=\\{\chi_{m_{0},m_{1}}=c_{0}\,e_{m_{0}}+c_{1}\,e_{m_{1}},\;|c_{0}|^{2}+|c_{1}|^{2}=1\\}$
(10)

of the overall Hilbert space $L^{2}(\mathbb{S}^{1},d\varphi/2\pi)$. Here we
assume the correspondences

$|0\rangle\leftrightarrow e_{m_{0}},\,\,|1\rangle\leftrightarrow e_{m_{1}}$
(11)

for the bases of the 2-dimensional space of qubits and the space
$Q^{2}_{m_{0},m_{1}}$.

The angular momentum operator ($\hbar=1$ in the following)

$L=(1/i)\partial_{\varphi}$ (12)

has the - obvious - expectation value

$(\chi_{m_{0},m_{1}}^{(\alpha,\beta)},L\chi_{m_{0},m_{1}}^{(\alpha,\beta)})=m_{0}\cos^{2}\beta+m_{1}\sin^{2}\beta,$
(13)

which for $m_{1}=-m_{0}$ becomes

$(\chi_{m_{0},-m_{0}}^{(\alpha,\beta)},L\chi_{m_{0},-m_{0}}^{(\alpha,\beta)})=m_{0}\,\cos
2\beta\,.$ (14)

This vanishes for $\beta=\pi/4$.

If

$\chi_{m_{0},m_{1}}^{(\hat{\alpha},\hat{\beta})}(\varphi)=\cos\hat{\beta}\,e_{m_{0}}(\varphi)+\sin\hat{\beta}\,e^{i\hat{\alpha}}\,e_{m_{1}}(\varphi)$
(15)

is another qubit of the type (8) in the same 2-dimensional space, then the
scalar product of both is given by

$\displaystyle(\chi_{m_{0},m_{1}}^{(\alpha,\beta)},\chi_{m_{0},m_{1}}^{(\hat{\alpha},\hat{\beta})})=$
(16)
$\displaystyle\cos\beta\,\cos\hat{\beta}+e^{-i(\alpha-\hat{\alpha})}\sin\beta\sin\hat{\beta},$

with the associated transition probability

$\displaystyle|(\chi_{m_{0},m_{1}}^{(\alpha,\beta)},\chi_{m_{0},m_{1}}^{(\hat{\alpha},\hat{\beta})})|^{2}=$
(17)
$\displaystyle\cos^{2}\beta\cos^{2}\hat{\beta}+\sin^{2}\beta\sin^{2}\hat{\beta}+\frac{1}{2}\sin
2\beta\sin 2\hat{\beta}\cos(\alpha-\hat{\alpha}),$

which equals $\cos^{2}(\beta-\hat{\beta)}$ for $\hat{\alpha}=\alpha$. Eq. (17)
is of interest in a discussion below (see Eq. (28)).

According to Ch. IV of Ref. Kastrup (2016) the A-OAM Wigner function
$V_{\psi}(\theta,p)$ for a wave function $\psi(\varphi)$ from Eq. (3) is given
by (with $\hbar$ explicit in the following two Eqs.)

$\displaystyle V_{\psi}(\theta,p)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{d\vartheta}{2\pi}e^{-i(p/\hbar)\vartheta}\psi^{\ast}(\theta-\vartheta/2)\,\psi(\theta+\vartheta/2)$
$\displaystyle=\sum_{m,n\in\mathbb{Z}}c_{m}^{\ast}V_{mn}(\theta,p)c_{n}\,,$
(18) $\displaystyle V_{mn}(\theta,p)$
$\displaystyle=\frac{1}{2\pi}e^{i(n-m)\theta}\int_{-\pi}^{\pi}\frac{d\vartheta}{2\pi}e^{i[(m+n)/2-p/\hbar]\vartheta}$
(19)
$\displaystyle=\frac{1}{2\pi}e^{i(n-m)\theta}\operatorname{sinc}\pi[p/\hbar-(m+n)/2],$
$\displaystyle\operatorname{sinc}\pi x=\frac{\sin\pi x}{\pi
x}=\frac{1}{2\pi}\int_{-\pi}^{\pi}d\vartheta\,e^{ix\vartheta}.$ (20)

Here $\theta$ describes (by means of the pair $(\cos\theta,\sin\theta)$) the
points on the (configuration) circle $\mathbb{S}^{1}$ of the classical phase
space
$\mathcal{P}^{2}(\theta,p)=\\{(\theta,p)\in\mathbb{S}^{1}\times\mathbb{R}\\}$,
with $p\in\mathbb{R}$ the classical canonically conjugate OAM. Integration
measure on $\mathcal{P}^{2}$ is $d\theta dp\;(\hbar=1)$. The letter $"V"$,
denoting the Wigner function (II.1), stands for “Vortex”.

The remarkable significance of the sinc-function in the context of the A-OAM
Wigner function is elaborately discussed in Ref. Kastrup (2016).

Taking for $\psi(\varphi)$ the wave function (8) gives

$\displaystyle 2\pi\,V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)$ (21)
$\displaystyle=\cos^{2}\beta\operatorname{sinc}\pi(p-m_{0})+\sin^{2}\beta\operatorname{sinc}\pi(p-m_{1})$
$\displaystyle+\sin
2\beta\cos[(m_{0}-m_{1})\theta-\alpha]\operatorname{sinc}\pi[p-(m_{0}+m_{1})/2].$

The last line in Eq. (21) represents the probability interference term in
$(\chi_{m_{0},m_{1}}^{(\alpha,\beta)},\chi_{m_{0},m_{1}}^{(\alpha,\beta)})$ –
see Eq. (9) – and describes, therefore, essential quantum mechanical
properties of the state!

That means the $\theta$-dependent part of the A-OAM Wigner function
$V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)$ of the qubit (8) is solely
determined by its probability interference term!

That part vanishes for $p-(m_{0}+m_{1})/2=k,\;k\in\\{\mathbb{Z}-\\{0\\}\\}$
and/or if $\alpha$ and $\theta$ are such that
$\cos[(m_{0}-m_{1})\theta-\alpha]=0$.

On the other hand, the factors of $\cos[(m_{0}-m_{1})\theta-\alpha]$ in Eq.
(21) are maximal $(=1)$ for $\beta=\pi/4$ and $p=(m_{0}+m_{1})/2$!

The angles $\alpha$ and $\beta$ may depend on other parameters, e.g., time
$t$, space coordinates, external fields etc., and may, therefore, be
manipulated from outside. Their values can be represented by points on the
2-dimensional surface of a “Bloch” sphere bar (b).

### Examples

It is instructive to look at a few special examples:

1\. $m_{1}=-m_{0}$:

$\displaystyle 2\pi\,V_{m_{0},-m_{0}}^{(\alpha,\beta)}(\theta,p)$ (22)
$\displaystyle=\cos^{2}\beta\operatorname{sinc}\pi(p-m_{0})+\sin^{2}\beta\operatorname{sinc}\pi(p+m_{0})$
$\displaystyle+\sin 2\beta\cos[2m_{0}\theta-\alpha]\operatorname{sinc}\pi\,p.$

The special case $\alpha=0,\beta=\pi/4,m_{0}=1$ of Eq. (22) is illustrated in
Fig. 1 (see also Fig. 2 in Ref. Kastrup (2016)):

Figure 1: A-OAM Wigner function
$2\pi\,V_{1,-1}^{(\alpha=0,\beta=\pi/4)}(\theta,p)=\frac{1}{2}[\operatorname{sinc}\pi(p-1)+\operatorname{sinc}\pi(p+1)]+\cos
2\theta\,\operatorname{sinc}\pi p$ of the qubit $(e_{+1}+e_{-1})/\sqrt{2}$.

2\. $m_{1}=0$ (ground state of $H\propto L^{2}$):

$\displaystyle 2\pi\,V_{m_{0},0}^{(\alpha,\beta)}(\theta,p)$ (23)
$\displaystyle=\cos^{2}\beta\operatorname{sinc}\pi(p-m_{0})+\sin^{2}\beta\operatorname{sinc}\pi\,p$
$\displaystyle+\sin
2\beta\cos[m_{0}\theta-\alpha]\operatorname{sinc}\pi(p-m_{0}/2),$

Here the interference term vanishes for
$p=m_{0}/2+k,\;k\in\\{\mathbb{Z}-\\{0\\}\\}$. (Recall that $m_{0}\neq 0$
because $m_{1}=0$.)

The Wigner function (23) for the special values $\alpha=0,\beta=\pi/3,m_{0}=1$
is shown in Fig. 2:

Figure 2:  A-OAM Wigner function
$2\pi\,V_{1,0}^{(\alpha=0,\beta=\pi/3)}(\theta,p)=\frac{1}{4}[\operatorname{sinc}\pi(p-1)+3\operatorname{sinc}\pi
p]+\frac{\sqrt{3}}{2}\cos\theta\,\operatorname{sinc}\pi(p-1/2)$ for the qubit
$(e_{+1}+\sqrt{3}\,e_{0})/2$.

###  II.2 Marginal probability distributions

The quantum mechanical marginal probability distributions
$|\chi_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta)|^{2}$ (angular distribution
density) and $\\{\cos^{2}\beta,\sin^{2}\beta\\}$ (OAM distribution) of the
state (8) can be obtained, according to Ref. Kastrup (2016), from the A-OAM
Wigner function (21) as follows:

$\displaystyle\int_{-\infty}^{\infty}dp\,V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)$
(24) $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\\{1+\sin
2\beta\,\cos[(m_{0}-m_{1})\theta-\alpha]\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}|\chi_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta)|^{2}$

– compare Eq. (9) –, where the relation

$\int_{-\infty}^{\infty}dp\,\operatorname{sinc}\pi(p+a)=1,\;a\in\mathbb{R},$
(25)

has been used. Comparing Eqs. (24) and (9) one has to take into account that
in the present case the integration measure for the phase space is
$d\theta\,dp$ (with $\hbar=1$), whereas we have $d\varphi/(2\pi)$ in the
integral (2) Kastrup (2016, 2017).

Integration over $\theta$ gives Whittaker’s cardinal function Kastrup (2016)

$\displaystyle\int_{-\pi}^{\pi}d\theta\,V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)=\omega_{m_{0},m_{1}}^{(\alpha,\beta)}(p)$
(26)
$\displaystyle=\cos^{2}\beta\,\operatorname{sinc}\pi(p-m_{0})+\sin^{2}\beta\,\operatorname{sinc}\pi(p-m_{1}),$

from which the quantum mechanical OAM probabilities $\cos^{2}\beta$ and
$\sin^{2}\beta$ can be extracted immediately with the help of the
orthonormality relations Kastrup (2016)

$\int_{-\infty}^{\infty}dp\,\operatorname{sinc}\pi(p-m)\,\operatorname{sinc}\pi(p-n)=\delta_{mn}.$
(27)

If $V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)$ and
$V_{m_{0},m_{1}}^{(\hat{\alpha},\hat{\beta)}}(\theta,p)$ are A-OAM Wigner
functions of the qubits (8) and (15), then the transition probability (17) is
now given Kastrup (2016) by the integral

$\displaystyle 2\pi\int_{-\infty}^{\infty}dp\int_{-\pi}^{\pi}d\theta
V_{m_{0},m_{1}}^{(\alpha,\beta)}(\theta,p)V_{m_{0},m_{1}}^{(\hat{\alpha},\hat{\beta)}}(\theta,p)=$
(28)
$\displaystyle\cos^{2}\beta\cos^{2}\hat{\beta}+\sin^{2}\beta\sin^{2}\hat{\beta}+\frac{1}{2}\sin
2\beta\sin 2\hat{\beta}\cos(\alpha-\hat{\alpha}),$

where the relations (27), (25) and

$\int_{-\infty}^{\infty}dp\,\operatorname{sinc}^{2}\pi(p+a)=1,\;a\in\mathbb{R},$
(29)

have been used ka (5).

###  II.3 Mixed states

I briefly indicate the changes if the pure state $\chi_{m_{0},m_{1}}$ of the
subspace (10) is replaced by a mixed state represented by a $2\times 2$
density matrix $\rho^{[m_{0},m_{1}]}$ with elements
$\rho_{m_{j},m_{k}},\,j,k=0,1$: Now $V_{\psi}(\theta,p)$ of Eq. (II.1) is
replaced by Kastrup (2016, 2017)

$\displaystyle V_{\rho^{[m_{0},m_{1}]}}(\theta,p)=$
$\displaystyle\operatorname{tr}\\{\rho^{[m_{0},m_{1}]}\cdot
V^{[m_{o},m_{1}]}(\theta,p)\\},$ (30) $\displaystyle
V^{[m_{0},m_{1}]}(\theta,p)=(V_{m_{j}m_{k}}(\theta,p)),$

with the matrix elements $V_{m_{j}m_{k}}(\theta,p),\,j,k=0,1$ as in Eq. (19).

For a general $2\times 2$ density matrix

$\rho^{[m_{0},m_{1}]}=\frac{1}{2}(I_{2}+\vec{a}\cdot\vec{\sigma}],$ (31)

with $I_{2}$ the unit matrix in 2 dimensions,
$\vec{a}\in\mathbb{R}^{3},\,\vec{a}^{2}\leq 1$ and $\sigma_{j},\,j=1,2,3,$
Pauli’s matrices, we get

$\displaystyle 2\pi\operatorname{tr}[\rho^{[m_{0},m_{1}]}\cdot
V^{[m_{0},m_{1}]}(\theta,p)]$ (32)
$\displaystyle=\frac{1+a_{3}}{2}\operatorname{sinc}\pi(p-m_{0})+\frac{1-a_{3}}{2}\operatorname{sinc}\pi(p-m_{1})$
$\displaystyle+\\{a_{1}\cos[(m_{0}-m_{1})\theta]+a_{2}\sin[(m_{0}-m_{1})\theta]\\}$
$\displaystyle\times\operatorname{sinc}\pi[p-(m_{0}+m_{1})/2].$

Note that here, too, (cf. Eq. (21)) the $\theta$-dependent term of
$V_{\rho^{[m_{0},m_{1}]}}(\theta,p)$ is solely determined by the interference
part of $\rho^{[m_{0},m_{1}]}$ The essential changes of the main results
obtained for $\delta=0$ for the case $\delta\neq 0$ are discussed in
$\rho^{[m_{0},m_{1}]}$ in Eq. (31): the numbers $a_{1}\pm i\,a_{2}$ form its
anti-diagonal. For the pure case $\vec{a}^{2}=1$ the expression (32) becomes
equivalent to the function (21).

Important for applications is the overlap formula of two density matrices

$\displaystyle\operatorname{tr}(\rho^{[m_{0},m_{1}]}_{1}\cdot\rho^{[m_{0},m_{1}]}_{2})$
(33)
$\displaystyle=2\pi\int_{-\infty}^{\infty}d(p/\hbar)\int_{-\pi}^{\pi}d\theta\,{V_{\rho^{[m_{0},m_{1}]}_{1}}}(\theta,p)\,{V_{\rho^{[m_{0},m_{1}]}_{2}}}(\theta,p),$

derived in Refs.Kastrup (2016, 2017).

##  III Wigner functions for 2-qubits, especially EPR/Bell states

###  III.1 The quantum state space of 2-qubits

For the discussion of the tensor product of the Hilbert space
$L^{2}(\mathbb{S}^{1},d\varphi/2\pi)$ from above – characterized by the Eqs.
(1)–(3) – with itself we have to go slightly beyond the A-OAM framework
discussed in Refs. Kastrup (2016, 2017):

There we had a phase space
$\mathcal{P}^{2}(\theta,p)=\\{(\theta,p)\in\mathbb{S}^{1}\times\mathbb{R}\\}$
with the circle $\mathbb{S}^{1}$ as configuration space and the real line
$\mathbb{R}$ as cotangent (canonical momentum) space. Coordinates for the
former are provided by the pair $(\cos\theta,\sin\theta)$ and the angular
momentum $p\in\mathbb{R}$ for the latter. By doubling the system we get the
phase space

$\displaystyle\mathcal{P}^{4}(\tilde{\theta},\tilde{p})=$
$\displaystyle\\{(\theta_{1},\theta_{2};p_{1},p_{2})\in\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{R}\times\mathbb{R})\\},$
(34)
$\displaystyle\tilde{\theta}\equiv\\{\theta_{1},\theta_{2}\\},\;\;\tilde{p}\equiv\\{p_{1},p_{2}\\}.$

Configuration space is now the torus $\mathbb{S}^{1}\times\mathbb{S}^{1}$.

A crucial tool for the derivation of the Wigner function (II.1) in Ref.
Kastrup (2016) are the unitary representations of the Euclidean group $E(2)$
of the plane ka (6). In our case, $\mathcal{P}^{4}(\tilde{\theta},\tilde{p})$,
we have to employ the direct product $E(2)\times E(2)$ and the associated
unitary representations. The procedure for deriving the A-OAM Wigner function
in question is then stricly analogue to that of Ch. II in Ref. Kastrup (2016)
for the expression (II.1) above and the result is as expected:

We have the product Hilbert space

$L^{2}(\mathbb{S}^{1}\times\mathbb{S}^{1},d\varphi_{1}d\varphi_{2}/(2\pi)^{2}),$
(35)

with basis

$e_{mn}(\tilde{\varphi})=e_{m}(\varphi_{1})\,e_{n}(\varphi_{2})=e^{im\varphi_{1}+in\varphi_{2}},\,m,n\in\mathbb{Z},$
(36)

scalar product

$\displaystyle(\psi_{2},\psi_{1})=$
$\displaystyle\int_{-\pi}^{\pi}\frac{d^{2}\tilde{\varphi}}{(2\pi)^{2}}\psi^{\ast}_{2}(\tilde{\varphi})\psi_{1}(\tilde{\varphi}),$
(37) $\displaystyle(e_{km},e_{ln})=\delta_{kl}\,\delta_{mn},$

and expansions

$\psi(\tilde{\varphi})=\sum_{m,n\in\mathbb{Z}}c_{mn}\,e_{mn}(\tilde{\varphi}),\,\,c_{mn}=(e_{mn},\psi).$
(38)

The functions (36) are eigenfunctions of the total OAM operator:

$L=\frac{1}{i}\partial_{\varphi_{1}}+\frac{1}{i}\partial_{\varphi_{2}},\;\;Le_{mn}=(m+n)e_{mn}.$
(39)

We here assume for both Hilbert space factors of their tensor product that
$\delta_{1}=0=\delta_{2}$. See the discussion after Eq. (3). The cases
$\delta_{1},\delta_{2}\neq 0$ will be discussed in Appendix A.

Comparison of the basis (36) with the correspondences (11) implies the
following correspondences for a basis of a 4-dimensional subspace
$Q^{4}_{m_{0}m_{1},n_{0}n_{1}}$ considered as the tensor product of the space
(10) with itself (without the normalization $|c_{1}|^{2}+|c_{2}|^{2}=1$):

$\displaystyle
e_{m_{0}n_{0}}\leftrightarrow|00\rangle,\;e_{m_{1}n_{1}}\leftrightarrow|11\rangle,$
(40) $\displaystyle
e_{m_{0}n_{1}}\leftrightarrow|01\rangle,\;e_{m_{1}n_{0}}\leftrightarrow|10\rangle.$

The elements of the 4-dimensional space $Q^{4}_{m_{0}m_{1},n_{0}n_{1}}$,
generated by the tensor product of qubits, are called “2-bits”. Another
possible basis for $Q^{4}_{m_{0}m_{1},n_{0}n_{1}}$ constitute the four
EPR/Bell states (6) and (7).

###  III.2 Wigner functions for general 2-qubits

Applying the same arguments of Ch. II in Ref. Kastrup (2016) – which lead to
the Wigner function (II.1) above – now to the products
$\mathcal{P}^{4}(\tilde{\theta},\tilde{p})$ and $E(2)\times E(2)$ we then get,
on the phase space $\mathcal{P}^{4}$ for the wave function $\psi$ of Eq. (38),
the A-OAM Wigner function

$\displaystyle V_{\psi}(\tilde{\theta},\tilde{p})=$ (41)
$\displaystyle\frac{1}{(2\pi)^{2}}\int_{-\pi}^{\pi}\frac{d^{2}\tilde{\vartheta}}{(2\pi)^{2}}e^{-i(p_{1}\vartheta_{1}+p_{2}\vartheta_{2})}\,\psi^{\ast}(\tilde{\theta}-\tilde{\vartheta}/2)\,\psi(\tilde{\theta}+\tilde{\vartheta}/2),$

which is the obvious generalization of the expression (II.1). We next
determine the A-OAM Wigner functions for general 2-qubit elements

$\displaystyle\psi_{2\\!-\\!qb}(\tilde{\varphi})$
$\displaystyle=c_{00}\,e_{m_{0}n_{0}}(\tilde{\varphi})+c_{10}\,e_{m_{1}n_{0}}(\tilde{\varphi})$
(42)
$\displaystyle+c_{01}\,e_{m_{0}n_{1}}(\tilde{\varphi})+c_{11}\,e_{m_{1}n_{1}}(\tilde{\varphi}),$
$\displaystyle|c_{00}|^{2}+|c_{01}|^{2}+|c_{10}|^{2}+|c_{11}|^{2}=1.$

of the 4-dimensional tensor product space The four complex coefficients
$c_{jk}$ may be parametrized by real numbers as follows:

$\displaystyle
c_{00}=b_{00},\;c_{10}=e^{i\alpha_{10}}b_{10},\;c_{01}=e^{i\alpha_{01}}b_{01},\;c_{11}=e^{i\alpha_{11}}b_{11},$
$\displaystyle\alpha_{jk}\in[0,2\pi),\;b_{jk}\in\mathbb{R},\;b_{00}^{2}+b_{10}^{2}+b_{01}^{2}+b_{11}^{2}=1.$
(43)

Finally, a convenient parametrization for the real $b_{jk}$ is

$\displaystyle
b_{00}=\cos\beta,\;b_{10}=\sin\beta\cos\gamma,\;\beta,\gamma\in[0,\pi)$ (44)
$\displaystyle
b_{01}=\sin\beta\sin\gamma\cos\phi,\;b_{11}=\sin\beta\sin\gamma\sin\phi,\;\phi\in[0,2\pi).$

Inserting the wave function (42) into the expression (41) yields the most
general 2-qubit A-OAM Wigner function

$\displaystyle(2\pi)^{2}V_{\psi_{2\\!-\\!qb}}(\tilde{\theta},\tilde{p})$ (45)
$\displaystyle=b_{00}^{2}\,\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-n_{0})$
$\displaystyle+b_{10}^{2}\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi(p_{2}-n_{0})$
$\displaystyle+b_{01}^{2}\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-n_{1})$
$\displaystyle+b_{11}^{2}\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi(p_{2}-n_{1})$
$\displaystyle+2\,b_{00}b_{10}\cos[(m_{1}-m_{0})\theta_{1}+\alpha_{10}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\operatorname{sinc}\pi(p_{2}-n_{0})$
$\displaystyle+2\,b_{00}b_{01}\cos[(n_{1}-n_{0})\theta_{2}+\alpha_{01}]$
$\displaystyle\times\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1})/2]$
$\displaystyle+2\,b_{00}b_{11}\cos[(m_{1}-m_{0})\theta_{1}+(n_{1}-n_{0})\theta_{2}+\alpha_{11}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1})/2]$
$\displaystyle+2\,b_{01}b_{10}\cos[(m_{1}-m_{0})\theta_{1}-(n_{1}-n_{0})\theta_{2}+\alpha_{10}-\alpha_{01}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1})/2]$
$\displaystyle+2\,b_{01}b_{11}\cos[(m_{1}-m_{0})\theta_{1}+\alpha_{11}-\alpha_{01}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2)]\operatorname{sinc}\pi(p_{2}-n_{1})$
$\displaystyle+2\,b_{10}b_{11}\cos[(n_{1}-n_{0})\theta_{2}+\alpha_{11}-\alpha_{10}]$
$\displaystyle\times\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi[p_{2}-(n_{1}+n_{0})/2].$

Essential properties of this general expression can be seen from the
discussion of the following special examples:

###  III.3 Wigner functions for special 2-qubits

####  III.3.1 Wigner function for the basis vector $e_{m_{0}n_{0}}$

The A-OAM Wigner function for the basis vector
$e_{m_{0}n_{0}}(\tilde{\varphi})$ is given by

$V_{m_{0}n_{0}}(\tilde{\theta},\tilde{p})=\frac{1}{(2\pi)^{2}}[\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-n_{0})].$
(46)

This follows from Eq. (45) with $b_{00}=1,b_{01}=b_{10}=b_{11}=0$.

The Wigner function (46) is independent of $\theta_{1}$ and $\theta_{2}$!

The expression (46) is exactly the product
$V_{m_{o}}(\theta_{1},p_{1})\,V_{n_{0}}(\theta_{2},p_{2})$ of the Wigner
functions for the basis vectors $e_{m_{0}}$ and $e_{n_{0}}$
respectivelyKastrup (2016).

####  III.3.2 Wigner functions of factorizable 2-qubits

If $c_{00}=b_{00}=\cos\beta$ and $c_{10}=b_{10}e^{i\alpha_{10}}=\sin\beta
e^{i\alpha_{10}}$, $b_{01}=0=b_{11}$, then we have for the state (42)

$\displaystyle\psi^{(\alpha_{10},\beta)}_{00,10}(\tilde{\varphi})=$
$\displaystyle\cos\beta e_{m_{0}n_{0}}(\tilde{\varphi})+\sin\beta
e^{i\alpha_{10}}\,e_{m_{1}n_{0}}(\tilde{\varphi})$ (47) $\displaystyle=$
$\displaystyle[\cos\beta e_{m_{0}}(\varphi_{1})+\sin\beta
e^{i\alpha_{10}}\,e_{m_{1}}(\varphi_{1})]\,e_{n_{0}}(\varphi_{2}).$

According to Eq. (45) the associated Wigner function is given by

$\displaystyle V_{00,10}^{(\alpha_{10},\beta)}(\tilde{\theta},\tilde{p})$ (48)
$\displaystyle=\frac{1}{(2\pi)^{2}}\\{\cos^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{0})+\sin^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{1})$
$\displaystyle+\sin
2\beta\cos[(m_{1}-m_{0})\theta_{1}+\alpha_{10}]\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\\}$
$\displaystyle\times\operatorname{sinc}\pi(p_{2}-n_{0})=V_{m_{0},m_{1}}^{(\alpha_{10},\beta)}(\theta_{1},p_{1})\,V_{n_{0}}(\theta_{2},p_{2}).$

Thus, the Wigner function of the product state (47) is a product of the qubit
Wigner function (21) with $\theta=\theta_{1}$ and $p=p_{1}$ times the Wigner
functions
$V_{n_{0}}(\theta_{2},p_{2})=\operatorname{sinc}\pi(p_{2}-n_{0})/(2\pi)$ of
the basis vector $e_{n_{0}}$ Kastrup (2016). Corresponding arguments hold for
the special cases $(b_{00},b_{10},b_{01},b_{11})$ =
$(\cos\beta,0,\sin\beta,0)$, $(0,0,\cos\phi,\sin\phi)$ or
$(0,\cos\gamma,0,\sin\gamma)$. Note that all four cases of the type (48) do
depend on $\theta_{1}\text{ or }\theta_{2}$ only!

####  III.3.3 A simplified general case

Considerable simplifications are obtained for the terms in expression (45) if
$m_{0}+m_{1}=0,\,n_{0}+n_{1}=0,\;m_{0}\neq 0\neq n_{0}$, and additionally
$n_{0}=m_{0}$ and $\alpha_{10}=0$ as well. This corresponds to a physical
situation where a system with total angular momentum zero is decomposed or
decays into two subsystems which move in opposite directions, one with angular
momentum $m_{0}$ and the other with $m_{1}=-m_{0}$. Example is a neutral
particle with spin zero (e.g., a neutral pion or a Higgs particle) which - in
its rest frame - decays into 2 photons with opposite spins one.

With the simplifications mentioned we get for the expression (45):

$\displaystyle(2\pi)^{2}V_{\psi_{2\\!-\\!qb}}(\tilde{\theta},\tilde{p})$ (49)
$\displaystyle=b_{00}^{2}\,\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-m_{0})$
$\displaystyle+b_{10}^{2}\operatorname{sinc}\pi(p_{1}+m_{0})\operatorname{sinc}\pi(p_{2}-m_{0})$
$\displaystyle+b_{01}^{2}\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}+m_{0})$
$\displaystyle+b_{11}^{2}\operatorname{sinc}\pi(p_{1}+m_{0})\operatorname{sinc}\pi(p_{2}+m_{0})$
$\displaystyle+2\,b_{00}b_{10}\cos(2m_{0}\theta_{1})\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi(p_{2}-m_{0})$
$\displaystyle+2\,b_{00}b_{01}\cos(2m_{0}\theta_{2}+\alpha_{01})\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi
p_{2}$
$\displaystyle+2\,b_{00}b_{11}\cos[2m_{0}(\theta_{1}+\theta_{2})+\alpha_{11}]\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi p_{2}$
$\displaystyle+2\,b_{01}b_{10}\cos[2m_{0}(\theta_{1}-\theta_{2})-\alpha_{01}]\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi p_{2}$
$\displaystyle+2\,b_{01}b_{11}\cos(2m_{0}\theta_{1}+\alpha_{11}-\alpha_{01})\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi(p_{2}+m_{0})$
$\displaystyle+2\,b_{10}b_{11}\cos(2m_{0}\theta_{2}+\alpha_{11})\operatorname{sinc}\pi(p_{1}+m_{0})\operatorname{sinc}\pi
p_{2}.$

This expression will be useful below (see example 6).

####  III.3.4 Wigner functions for non-factorizable 2-qubits: EPR/Bell states

The non-factorizable EPR/Bell states (6) or (7) are special cases of the
states (42) with either $b_{00}b_{11}\neq 0,\,b_{10}=0=b_{01}$ or
$b_{10}b_{01}\neq 0,\,b_{00}=0=b_{11}$, i.e.

$\psi^{(\alpha_{11},\beta)}_{00,11}(\tilde{\varphi})=\cos\beta
e_{m_{0}n_{0}}(\tilde{\varphi})+\sin\beta\,e^{i\alpha_{11}}\,e_{m_{1}n_{1}}(\tilde{\varphi})$
(50)

and

$\psi^{(\alpha_{01},\gamma)}_{10,01}(\tilde{\varphi})=\cos\gamma\,\
e_{m_{1}n_{0}}(\tilde{\varphi})+\sin\gamma\,e^{i\alpha_{01}}\,e_{m_{0}n_{1}}(\tilde{\varphi}).$
(51)

According to Eq. (45) the Wigner function of the state (50) is given by

$\displaystyle V_{00,11}^{(\alpha_{11},\beta)}(\tilde{\theta},\tilde{p})$ (52)
$\displaystyle=\frac{1}{(2\pi)^{2}}\\{\cos^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-n_{0})$
$\displaystyle+\sin^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi(p_{2}-n_{1})$
$\displaystyle+\sin
2\beta\cos[(m_{1}-m_{0})\theta_{1}+(n_{1}-n_{0})\theta_{2}+\alpha_{11}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1})/2]\\}.$

with a corresponding expression for the Wigner function of the state (51):

$\displaystyle V_{10,01}^{(\alpha_{01},\gamma)}(\tilde{\theta},\tilde{p})$
(53)
$\displaystyle=\frac{1}{(2\pi)^{2}}\\{\cos^{2}\gamma\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-n_{0})$
$\displaystyle+\sin^{2}\gamma\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi(p_{2}-n_{1})$
$\displaystyle+\sin
2\gamma\cos[(m_{1}-m_{0})\theta_{1}-(n_{1}-n_{0})\theta_{2}-\alpha_{01}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1})/2]\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1})/2]\\}.$

The Wigner functions (52) and (53) have the following essential properties:

1\. They are not factorizable into two factors each of which depends on
$\theta_{1}$ or $\theta_{2}$ only.

2\. The $\theta_{1}$ \- and $\theta_{2}$ \- dependences of the expression (52)
are solely given by the argument

$\vartheta_{+}(\tilde{\theta})=(m_{1}-m_{0})\theta_{1}+(n_{1}-n_{0})\theta_{2}+\alpha_{11}$
(54)

of the non-classical $\cos$-interference term and that of the function (53)
correspondingly by

$\vartheta_{-}(\tilde{\theta})=(m_{1}-m_{0})\theta_{1}-(n_{1}-n_{0})\theta_{2}-\alpha_{01};$
(55)

with the inversions

$\displaystyle\theta_{1}$
$\displaystyle=\frac{1}{2(m_{1}-m_{0})}[\vartheta_{+}+\vartheta_{-}+\alpha_{01}-\alpha_{11}],$
(56) $\displaystyle\theta_{2}$
$\displaystyle=\frac{1}{2(n_{1}-n_{0})}[\vartheta_{+}-\vartheta_{-}+\alpha_{01}+\alpha_{11}].$
(57)

Note that here $m_{1}-m_{0}\neq 0\neq n_{1}-n_{0}$.

3\. As the function (52) does not depend on $\vartheta_{-}$ it is instructive
to consider the configuration space curve

$\displaystyle\mathcal{C}_{+}=$
$\displaystyle\\{(\theta_{1}=\theta_{1}(\vartheta_{+}),\theta_{2}=\theta_{2}(\vartheta_{+})\,)\in\mathbb{S}^{1}\times\mathbb{S}^{1},$
(58)
$\displaystyle\vartheta_{+}\in\mathbb{R};\,\,\vartheta_{-},\,\alpha_{01},\,\alpha_{11}=\text{const.}\\}.$

Thus, the domain of the curve (58) is a spiral around a torus with “slopes”
$\partial\theta_{1}/\partial\vartheta_{+}=1/[2(m_{1}-m_{0})]$ and
$\partial\theta_{2}/\partial\vartheta_{+}=1/[2(n_{1}-n_{0})]$.

The above properties 1.-3. show that the non-factorizàbility of EPR/Bell
states is associated with the non-factorizability of the corresponding Wigner
function.

The Wigner functions (52) and (53) do depend on the configuration space
variables $\theta_{1}$ and $\theta_{2}$ only through the non-classical
interference terms $\cos\vartheta_{+}(\theta_{1},\theta_{2})$ or
$\cos\vartheta_{-}(\theta_{1},\theta_{2})$, in such a way that their domain is
a curve which spirals around a torus. Thus, characteristic features of A-OAM
Wigner functions for EPR/Bell states are related to the non-trivial
topological properties of a torus, the configuration subspace of the total
phase space.

####  III.3.5 Marginal distributions

The marginal (separate) probability distribuions for the angles
$\tilde{\theta}$ and the momenta $\tilde{p}$, respectively, can - like in
subsection II.B. - be obtained by integrating, e.g., the Wigner function (52)
over the complementary canonical variables:

$\displaystyle\int_{-\infty}^{\infty}dp_{1}dp_{2}V_{00,11}^{(\alpha_{11},\beta)}(\tilde{\theta},\tilde{p})$
(59) $\displaystyle=\frac{1}{(2\pi)^{2}}[1+\sin
2\beta\cos[\vartheta_{+}(\tilde{\theta})],$

where the relation (25) has been used. The expression (59) equals
$1/(2\pi)^{2}|\psi^{(\alpha_{11},\beta)}_{00,11}(\tilde{\varphi})|^{2}$ with
$\psi$ of Eq. (50) and $\tilde{\varphi}$ replaced by $\tilde{\theta}$. The
factor $1/(2\pi)^{2}$ corresponds to the same factor in the integral (37).

Integrating the function (52) over $\tilde{\theta}$ yields the Whittaker
cardinal functionKastrup (2016)

$\displaystyle\int_{-\pi}^{\pi}d\theta_{1}d\theta_{2}V_{00,11}^{(\alpha_{11},\beta)}(\tilde{\theta},\tilde{p})$
(60)
$\displaystyle=\cos^{2}\beta[\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}+n_{0})]$
$\displaystyle+\sin^{2}\beta[\operatorname{sinc}\pi(p_{1}-m_{1})\operatorname{sinc}\pi(p_{2}-n_{1})]$
$\displaystyle=\omega_{00,11}(\tilde{p}).$

Again, using the orthonormality relations (27) the probabilities
$|c_{00}|^{2}=\cos^{2}\beta$ and $|c_{11}|^{2}=\sin^{2}\beta$ for finding the
OAM pairs $(m_{0},m_{0})$ or $(m_{1},n_{1})$, respectively, can be obtained
from the Whittaker cardinal function $\omega_{00,11}(\tilde{p})$ of Eq. (60).

The Wigner function (53) can be treated accordingly.

####  III.3.6 Wigner functions of the EPR/Bell states basis

According to the correspondences (40) we get for the basic EPR/Bell states (6)
and (7) the functions (with
$n_{0}=m_{0}\in\mathbb{Z}-\\{0\\},\,n_{1}=m_{1}=-m_{0}$)

$\displaystyle\psi_{00,11;\pm}(\tilde{\varphi})$ (61)
$\displaystyle=\frac{1}{\sqrt{2}}[e_{m_{0}}(\varphi_{1})e_{m_{0}}(\varphi_{2})\pm
e_{-m_{0}}(\varphi_{1})e_{-m_{0}}(\varphi_{2})],$
$\displaystyle~{}~{}~{}~{}~{}L\,\psi_{00,11;\pm}(\tilde{\varphi})=2(m_{0}\mp
m_{0})\psi_{00,11;\pm}(\tilde{\varphi});$ (62)
$\displaystyle\psi_{01,10;\pm}(\tilde{\varphi})$ (63)
$\displaystyle=\frac{1}{\sqrt{2}}[e_{m_{0}}(\varphi_{1})e_{-m_{0}}(\varphi_{2})\pm
e_{-m_{0}}(\varphi_{1})e_{m_{0}}(\varphi_{2})],$
$\displaystyle~{}~{}~{}~{}~{}L\,\psi_{01,10;\pm}(\tilde{\varphi})=0.$ (64)

Taking $\beta=\pi/4,\,\alpha_{11}=0,\,\pi$ in Eq. (52) and
$\gamma=\pi/4,\,\alpha_{01}=0,\,\pi$ in Eq. (53) gives the associated Wigner
functions (see also the expression (49))

$\displaystyle(2\pi)^{2}$ $\displaystyle
V_{00,11;\pm}(\tilde{\theta},\tilde{p})$ (65)
$\displaystyle=\frac{1}{2}[\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}-m_{0})]$
$\displaystyle+\frac{1}{2}[\operatorname{sinc}\pi(p_{1}+m_{0})\operatorname{sinc}\pi(p_{2}+m_{0})]$
$\displaystyle\pm\cos[2m_{0}(\theta_{1}+\theta_{2})]\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi p_{2};$ $\displaystyle(2\pi)^{2}$ $\displaystyle
V_{01,10;\pm}(\tilde{\theta},\tilde{p})$ (66)
$\displaystyle=\frac{1}{2}[\operatorname{sinc}\pi(p_{1}-m_{0})\operatorname{sinc}\pi(p_{2}+m_{0})]$
$\displaystyle+\frac{1}{2}[\operatorname{sinc}\pi(p_{1}+m_{0})\operatorname{sinc}\pi(p_{2}-m_{0})]$
$\displaystyle\pm\cos[2m_{0}(\theta_{1}-\theta_{2})]\operatorname{sinc}\pi
p_{1}\operatorname{sinc}\pi p_{2}.$

Let us look at special cases of the expression (66), with the minus sign in
Eq. (63) and the related one in Eq. (66) as well:

An example of such functions (66) with $m_{0}=1$ and $p_{2}=1/2$ is shown in
Fig. 3.

Figure 3:  Wigner function
$(2\pi)^{2}V_{01,10;-}(\theta_{1},\theta_{2},p_{1},p_{2}=1/2)=\\{-\frac{1}{3}\operatorname{sinc}\pi(p_{1}-1)+\operatorname{sinc}\pi(p_{1}+1)-2\cos[2(\theta_{1}-\theta_{2})]\operatorname{sinc}\pi
p_{1}\\}/\pi$ of the EPR/Bell state $\psi_{01,10;-}(\tilde{\varphi})$ from Eq.
(66).

Other examples are

$V_{01,10;-}(\tilde{\theta},p_{1},p_{2})=0\text{ for
}p_{1},p_{2}\in\\{\mathbb{Z}-\\{0\\}\\\ \\},$ (67)

and

$\displaystyle(2\pi)^{2}V_{01,10;-}$
$\displaystyle(\tilde{\theta},p_{1},p_{2}=0)$ (68)
$\displaystyle=-\cos[2m_{0}(\theta_{1}-\theta_{2})]\operatorname{sinc}\pi
p_{1},$

with the corresponding relation for $p_{1}=0$. A graphical illustration of the
function (68) with $m_{0}=1$ is given in Fig. 4.

Figure 4:  Wigner function
$(2\pi)^{2}V_{01,10;-}(\theta_{1},\theta_{2},p_{1},p_{2}=0)=-\cos[2(\theta_{1}-\theta_{2})]\operatorname{sinc}\pi
p_{1}$, according to Eq. (68) with $m_{0}=1$, of the EPR/Bell state
$\psi_{01,10;-}(\tilde{\varphi})$ from Eq. (63).

Furthermore,

$V_{01,10;-}(\theta_{1},\theta_{2},p_{1}=0,p_{2}=0)=-\frac{1}{4\pi^{2}}\cos[2m_{0}(\theta_{1}-\theta_{2})],$
(69)

showing explicitely that the Wigner function is negative on certain subsets of
the phase space.

###### Acknowledgements.

I very much thank the DESY Theory Group for its sustained and very kind
hospitality after my retirement from the Institute for Theoretical Physics of
the RWTH Aachen. I am grateful to Hartmann Römer for a fruitful discussion and
to David Kastrup for providing the figures. I thank Jakub Rembieliński and
Krzystof Kowalski for the invitation to their Theoretical Physics Institute at
the University of Łódz (May 2017) and for many stimulating discussions with
them. Equally I thank Gerd Leuchs and Luis Sanchez-Soto for their invitation
to the MPI for the Science of Light, Erlangen, and for many fruitful
discussions (February 2018). Finally I am obliged to family and friends for
help and encouragements during a longer period of illnesses.

##  Appendix A Fractional OAM with $\delta\neq 0$

###  A.1 Generalities

In the main text above we have discussed A-OAM Wigner functions in the Hilbert
space framework for the special case $\delta=0$ of the more general frameworks
as described by the Eqs. (1)-(3). The mathematical and physical meanings of
the “fractional” case $\delta\in(0,1)$ were discussed in Ref.Kastrup (2006).
In the present appendix the generalization of the main results for Wigner
functions with $\delta=0$ to the more general case $\delta\neq 0$ will be
sketched (for the same topic see also Subsection V.A. of Ref. Kastrup (2016)):

The essential recipe for doing so are the replacements

$m_{0},\,m_{1}\rightarrow m_{0}+\delta,\,m_{1}+\delta$ (70)

in Section II and

$m_{0},m_{1}\rightarrow
m_{0}+\delta_{1},m_{1}+\delta_{1},\,\,n_{0},n_{1}\rightarrow
n_{0}+\delta_{2},n_{1}+\delta_{2}$ (71)

in Section III, where the two Hilbert spaces of the tensor product may be
different.

Instead of the state (8) we now have

$\displaystyle\chi_{m_{0},m_{1}}^{[\delta](\alpha,\beta)}(\varphi)=$
$\displaystyle\cos\beta\,e_{m_{0},\delta}(\varphi)+\sin\beta\,e^{i\alpha}\,e_{m_{1},\delta}(\varphi),$
(72) $\displaystyle m_{0},m_{1}\in\mathbb{Z},\;m_{1}\neq m_{0}.$

Its absolute value squared is the same as in Eq. (9).

In Eqs. (10) and (11) the $e_{m_{j}}$ have to be replaced by
$e_{m_{j},\delta}$.

The expressions (II.1) and (19) become now

$\displaystyle V_{\psi}^{[\delta]}(\theta,p)$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{d\vartheta}{2\pi}e^{-i(p/\hbar)\vartheta}\psi^{[\delta]\ast}(\theta\\!-\\!\frac{\vartheta}{2})\psi^{[\delta]}(\theta\\!+\\!\frac{\vartheta}{2})$
$\displaystyle=\sum_{m,n\in\mathbb{Z}}c_{m}^{\ast}V_{mn}^{[\delta]}(\theta,p)c_{n}\,,$
(73) $\displaystyle V_{mn}^{[\delta]}(\theta,p)$
$\displaystyle=\frac{1}{2\pi}e^{i(n-m)\theta}\int_{-\pi}^{\pi}\frac{d\vartheta}{2\pi}e^{i[(m+n+2\delta)/2-p/\hbar]\vartheta}$
(74)
$\displaystyle=\frac{1}{2\pi}e^{i(n-m)\theta}\operatorname{sinc}\pi[p/\hbar-(m+n+2\delta)/2],$

which implies for the Wigner function (21) the generalization

$\displaystyle 2\pi\,V_{m_{0},m_{1}}^{[\delta](\alpha,\beta)}(\theta,p)$ (75)
$\displaystyle=\cos^{2}\beta\operatorname{sinc}\pi(p-m_{0}-\delta)+\sin^{2}\beta\operatorname{sinc}\pi(p-m_{1}-\delta)$
$\displaystyle+\sin 2\beta\cos[(m_{0}-m_{1})\theta-\alpha]$
$\displaystyle\times\operatorname{sinc}\pi[p-(m_{0}+m_{1}+2\delta)/2].$

Note that the $\theta$-dependent $\cos$-interference term remains independent
of $\delta$, whereas the different $\operatorname{sinc}$-functions have their
maximum $1$ now at $p=m_{0}+\delta,\;p=m_{1}+\delta$ and
$p=(m_{0}+m_{1}+2\delta)/2$.

The additional resulting changes for the formulae in Subsections II.A. and
II.B. due to the replacements (70) are obvious. The relation (27) also remains
valid under those substitutions.

For the mixed states of Subsection II.C. we obtain instead of Eq. (30)

$\displaystyle V_{\rho^{[m_{0},m_{1}]}}^{[\delta]}(\theta,p)=$
$\displaystyle\operatorname{tr}\\{\rho^{[m_{0},m_{1}]}\cdot
V^{[\delta;m_{o},m_{1}]}(\theta,p)\\},$ (76) $\displaystyle
V^{[\delta;m_{0},m_{1}]}(\theta,p)=(V_{m_{j}m_{k}}^{[\delta]}(\theta,p)),$

with the matrix elements $V_{m_{j}m_{k}}^{[\delta]}(\theta,p),\,j,k=0,1$ from
Eq. (74). The main result (32) is replaced by

$\displaystyle 2\pi\operatorname{tr}[\rho^{[m_{0},m_{1}]}\cdot
V^{[\delta;m_{0},m_{1}]}(\theta,p)]$ (77)
$\displaystyle=\frac{1+a_{3}}{2}\operatorname{sinc}\pi(p-m_{0}-\delta)+\frac{1-a_{3}}{2}\operatorname{sinc}\pi(p-m_{1}-\delta)$
$\displaystyle+\\{a_{1}\cos[(m_{0}-m_{1})\theta]+a_{2}\sin[(m_{0}-m_{1})\theta]\\}$
$\displaystyle\times\operatorname{sinc}\pi[p-(m_{0}+m_{1}+2\delta)/2].$

The generalizations of the formulae in Section III are slightly more
complicated because the Hilbert space factors of Eq. (35) may have different
$\delta_{1}$ and $\delta_{2}$:

$L^{2}(\mathbb{S}^{1}\times\mathbb{S}^{1},d\varphi_{1}d\varphi_{2}/(2\pi)^{2};\delta_{1},\delta_{2}),$
(78)

with basis

$\displaystyle e_{mn;\delta_{1},\delta_{2}}(\tilde{\varphi})=$ $\displaystyle
e_{m,\delta_{1}}(\varphi_{1})\,e_{n,\delta_{2}}(\varphi_{2})$ (79)
$\displaystyle=$ $\displaystyle
e^{i(m+\delta_{1})\varphi_{1}+i(n+\delta_{2})\varphi_{2}},\,m,n\in\mathbb{Z},$

scalar product

$\displaystyle(\psi_{2}^{[\delta_{1},\delta_{2}]},\psi_{1}^{[\delta_{1},\delta_{2}]})=$
$\displaystyle\int_{-\pi}^{\pi}\frac{d^{2}\tilde{\varphi}}{(2\pi)^{2}}\psi^{[\delta_{1},\delta_{2}]\ast}_{2}(\tilde{\varphi})\psi_{1}^{[\delta_{1},\delta_{2}]}(\tilde{\varphi}),$
(80)
$\displaystyle(e_{km;\delta_{1},\sl_{2}},e_{ln;\delta_{1},\delta_{2}})=\delta_{kl}\,\delta_{mn},$

and expansions

$\displaystyle\psi^{[\delta_{1},\delta_{2}]}(\tilde{\varphi})=$
$\displaystyle\sum_{m,n\in\mathbb{Z}}c_{mn}\,e_{mn;\delta_{1},\delta_{2}}(\tilde{\varphi}),$
(81) $\displaystyle
c_{mn}=(e_{mn;\delta_{1},\delta_{2}},\psi^{[\delta_{1},\delta_{2}]}).$

The functions (79) are eigenfunctions of the total OAM operator:

$L=\frac{1}{i}\partial_{\varphi_{1}}+\frac{1}{i}\partial_{\varphi_{2}},\;\;Le_{mn;\delta_{1},\delta_{2}}=(m+n+\delta_{1}+\delta_{2})e_{mn;\delta_{1},\delta_{2}}.$
(82)

Instead of the correspondences (40) we now have

$\displaystyle
e_{m_{0}n_{0};\delta_{1},\delta_{2}}\leftrightarrow|00\rangle,\;e_{m_{1}n_{1};\delta_{1},\delta_{2}}\leftrightarrow|11\rangle,$
(83) $\displaystyle
e_{m_{0}n_{1};\delta_{1},\delta_{2}}\leftrightarrow|01\rangle,\;e_{m_{1}n_{0};\delta_{1},\delta_{2}}\leftrightarrow|10\rangle.$

In the Wigner function (41) the wave function $\psi$ has to be replaced by
$\psi^{[\delta_{1},\delta_{2}]}$ from Eq. (81):

$\displaystyle(2\pi)^{2}V_{\psi}^{[\delta_{1},\delta_{2}]}(\tilde{\theta},\tilde{p})=$
(84)
$\displaystyle\int_{-\pi}^{\pi}\frac{d^{2}\tilde{\vartheta}}{(2\pi)^{2}}e^{\\!-\\!i(p_{1}\vartheta_{1}+p_{2}\vartheta_{2})}\,\psi^{[\delta_{1},\delta_{2}]\ast}(\tilde{\theta}\\!-\\!\tilde{\vartheta}/2)\psi^{[\delta_{1},\delta_{2}]}(\tilde{\theta}\\!+\\!\tilde{\vartheta}/2).$

The general 2-qubit elements (42) are replaced by

$\displaystyle\psi_{2\\!-\\!qb}^{[\delta_{1},\delta_{2}]}(\tilde{\varphi})=$
(85) $\displaystyle
c_{00}\,e_{m_{0}n_{0};\delta_{1},\delta_{2}}(\tilde{\varphi})+c_{10}\,e_{m_{1}n_{0};\delta_{1},\delta_{2}}(\tilde{\varphi})$
$\displaystyle+c_{01}\,e_{m_{0}n_{1};\delta_{1},\delta_{2}}(\tilde{\varphi})+c_{11}\,e_{m_{1}n_{1};\delta_{1},\delta_{2}}(\tilde{\varphi}),$
$\displaystyle|c_{00}|^{2}+|c_{01}|^{2}+|c_{10}|^{2}+|c_{11}|^{2}=1.$

Inserting this state into the expression (84) gives the obvious generalization
of the 2-qubit Wigner function (45).

It is remarkable that the $\theta$-dependent $\cos$-factors in all
interference terms of the general expression (45) remain unchanged under the
substitutions (71). See also the corresponding invariances of the Wigner
functions (52) and (53) and those of the arguments (54) and (55) of their
$\cos$-factors.

For, e.g., the Wigner function (52) we have explicitly

$\displaystyle(2\pi)^{2}V_{00,11}^{[\delta_{1},\delta_{2}](\alpha_{11},\beta)}(\tilde{\theta},\tilde{p})$
(86)
$\displaystyle=\cos^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{0}-\delta_{1})\operatorname{sinc}\pi(p_{2}-n_{0}-\delta_{2})$
$\displaystyle+\sin^{2}\beta\operatorname{sinc}\pi(p_{1}-m_{1}-\delta_{1})\operatorname{sinc}\pi(p_{2}-n_{1}-\delta_{2})$
$\displaystyle+\sin
2\beta\cos[(m_{1}-m_{0})\theta_{1}+(n_{1}-n_{0})\theta_{2}+\alpha_{11}]$
$\displaystyle\times\operatorname{sinc}\pi[p_{1}-(m_{0}+m_{1}+2\delta_{1})/2]$
$\displaystyle\times\operatorname{sinc}\pi[p_{2}-(n_{0}+n_{1}+2\delta_{2})/2]\\}.$

Thus, the $\theta$-dependence of the Wigner functions as defined on classical
phase space is not affected by the substitutions (70) and (71). The changes of
the momenta $p$ and $\tilde{p}$ are intuitively obvious.

###  A.2 An important application

An essential case of a $\delta\neq 0$ Hilbert space was already discussed in
the context of example 3 in Subsection IV.C. of Ref. Kastrup (2016): Suppose
$\psi_{b}(\varphi)$ is a function of $\varphi$ with the property

$\psi_{b}(\varphi+2\pi)=e^{i2\pi
b\varphi}\,\psi_{b}(\varphi),\,b\in\mathbb{R}.$ (87)

The real number $b$ can be decomposed uniquely into an integer plus a $\delta$
of Eq. (1):

$b=n_{b}+\delta,\;\delta\in[0,1),\;n_{b}\in\mathbb{Z}.$ (88)

If the integral

$\int_{-\pi}^{\pi}\frac{d\varphi}{2\pi}\psi^{\ast}_{b}(\varphi)\psi_{b}(\varphi)$
(89)

exists then $\psi_{b}(\varphi)$ can be considered as an element of
$L^{2}(\mathbb{S}^{1},d\varphi/2\pi;\delta)$. $\psi_{b}(\varphi)$ is a kind of
Bloch waveKastrup (2006).

## References

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