# The Role of Phonetic Units in Speech Emotion Recognition

###### Abstract

We propose a method for emotion recognition through emotion-dependent speech
recognition using Wav2vec 2.0. Our method achieved a significant improvement
over most previously reported results on IEMOCAP, a benchmark emotion dataset.
Different types of phonetic units are employed and compared in terms of
accuracy and robustness of emotion recognition within and across datasets and
languages. Models of phonemes, broad phonetic classes, and syllables all
significantly outperform the utterance model, demonstrating that phonetic
units are helpful and should be incorporated in speech emotion recognition.
The best performance is from using broad phonetic classes. Further research is
needed to investigate the optimal set of broad phonetic classes for the task
of emotion recognition. Finally, we found that Wav2vec 2.0 can be fine-tuned
to recognize coarser-grained or larger phonetic units than phonemes, such as
broad phonetic classes and syllables.

Index Terms: speech emotion recognition, Wav2vec 2.0, fine-tuning, broad
phonetic classes

##  1 Introduction

Speech emotion recognition is essentially a sequence-to-one classification
problem whereas speech recognition is a sequence-to-sequence problem. In this
paper we attempt to bridge these two problems. More specifically, we propose a
method for emotion recognition through emotion-dependent speech recognition.

Speech emotion recognition has witnessed a stable advancement over the last
two decades [1, 2, 3, 4, 5]. Much of the earlier effort was concentrated on
feature engineering. The emobase feature set in the widely used OpenSMILE
toolkit [6], for example, consists of 998 acoustic features for emotion
recognition, including prosodic, spectral, as well as voice quality features.
In recent years, more effort has been devoted to improving deep learning model
architectures. Many studies in the literature are based on IEMOCAP [7], a
benchmark emotion dataset. At ICASSP 2020, half a dozen papers reported a
cross-validation accuracy of 70% or better on this dataset [8, 9, 10, 11, 12],
based on acoustic data only. [8] achieved the highest accuracy of 73% from
employing a dual-sequence LSTM model.

A confounding factor for emotion recognition is the acoustic variability of
different phonetic units such as phonemes. Figure 1 compares the spectra of
two vowels (from the same speaker) in the same emotion. Clearly, their spectra
are different in terms of the location of spectral peaks, which determines
vowel quality. There are two approaches to overcome this problem. The
predominant one is to make emotion features and models independent and more
robust to phonetic variability. The other approach is to take consideration of
phonetic variability by developing phonetically-aware features and models.
This study adopts the second approach.

A relatively small number of studies of speech emotion recognition have
investigated the effect of phonetic variability. [13] incorporated
articulatory information in emotion recognition. They observed that for the
vowel /AE/ anger forces a larger opening of the jaw as opposed to sadness,
while for the vowel /IY/ anger makes lips more protruded towards the outside.
[14] computed statistics of Mel-Frequency Cepstral Coefficients over three
phonetic classes (stressed vowels, unstressed vowels, and consonants)
respectively for emotion recognition. They found that spectral features
computed from consonant regions of the utterance contain more information
about emotion than either stressed or unstressed vowel features. [15]
investigated whether acoustic emotion recognition strongly depends on phonetic
content. They demonstrated that phoneme-specific emotion models can lead to
higher accuracies. [16] used a self-attention based emotion classification
model to understand the phonetic bases of emotions by discovering the most
“attended” phonemes for each emotion. They found that the distribution of
these attended-phonemes varies significantly between natural versus acted
emotions.

To make use of the effect of phonetic variability for emotion recognition, we
train emotion-dependent speech recognition models. For example, the vowel /AA/
in “happy” and “sad” are different phonemes and have different acoustic
models. With emotion-dependent speech recognition models, emotion recognition
can be done as a by-product of speech recognition. Similar efforts have been
made in the literature. [17] trained two sets of HMM/GMM acoustic models of
phonemes for high-arousal and low-arousal/neutral emotions respectively, and
determined the emotion of each phoneme in an utterance (generated from speech
recognition) by applying and comparing the two models of the phoneme. Compared
to [17], our approach is simpler. We combine speech recognition and emotion
classification into one step of recognizing emotion-dependent phonetic units
such as phonemes.

Emotion-dependent models may require more training data than emotion-
independent models because the number of units/models is increased by multiple
times (n times where n is the number of emotions). However, with the recent
advancement of pretrained acoustic models, the amount of data needed for
training speech recognition models is greatly reduced. For example, Wav2vec
2.0 [18] outperforms the previous state of the art on the Librispeech [19]
test set in terms of word error rate while using 100 times less labeled data.
By fine-tuning Wav2vec 2.0, we can train well-performing emotion-dependent
models even with a small amount of data.

We can also reduce the size of phonetic units in emotion-dependent models by
using coarser-grained (e.g., broad phonetic classes) or larger (e.g.,
syllables) units than phonemes. Besides to increase the training data for each
unit, we explore this idea in the study from three perspectives: 1. Can
pretrained acoustic representations be fine-tuned to recognize coarser or
larger phonetic units than phonemes? 2. What are the best phonetic units for
emotion recognition? 3. Given that phonemes are language-specific while broad
phonetic classes and syllables are language-general, are broad-phonetic-class
or syllable models more robust than phoneme models for cross-lingual emotion
recognition?

In the following sections we will first introduce the method of fine-tuning
Wav2vec 2.0 for emotion recognition, followed by experiments on the English
IEMOCAP dataset and three other datasets, in German, Arabic, and Mandarin
Chinese respectively. Conclusions and discussion are made in the last section.

Figure 1: Spectra of happy /AE/ and /IY/.

##  2 Fine-tuning Wav2vec2.0 for emotion recognition

###  2.1 The procedure

Wav2vec 2.0 is a framework for self-supervised learning of speech
representations. It consists of multiple convolution layers and self-attention
layers, and pretrains on audio alone by masking the speech input in the latent
space and solving a contrastive task defined over a quantization of the latent
representations. Pre-trained wav2vec models can be fine-tuned for speech
recognition with labeled data.

The procedure of fine-tuning Wav2vec 2.0 for emotion recognition is
illustrated and described in Figure 2. The core idea is to use emotion-
dependent units as labeled targets. A randomly initialized linear projection
is added on top of the contextual representations of Wav2vec 2.0 to map the
representations into emotion-dependent units (i.e., classes), and the entire
model is optimized by minimizing the CTC loss [20] through fine-tuning.

The fine-tuned model can be used to transcribe speech into emotion-dependent
units. For the purpose of speech emotion recognition, the recognized emotion-
dependent units of an utterance are mapped to an emotion category by majority
vote. For example, if most of the recognized units are of “happy”, then the
utterance will be classified as “happy”.

Figure 2: Fine-tuning wav2vec 2.0 for emotion recognition.

###  2.2 Phonetic units

We tried four types of phonetic units for recognition: phonemes, broad
phonetic classes, syllables, and the entire utterance. They are summarized in
Figure 4 with examples.

Figure 3: Phonetic units for speech emotion recognition.

##  3 Experiments on IEMOCAP

###  3.1 Data

IEMOCAP is a benchmark emotion dataset in English. It consists of 12 hours of
speech from 10 professional actors. Following the literature, we extracted
5531 utterances of four emotion types from the dataset: 1708 neutral, 1636
happy (also including excited), 1103 angry, and 1084 sad.

###  3.2 Fine-tuning

The wav2vec 2.0 large model pre-trained on 960 hours of Librispeech audio,
libri960_big.pt, was used for fine-tuning. The model was fine-tuned with 15k
updates in all our experiments. For the first 10k updates only the output
classifier is trained, after which the Transformer is also updated. The
max_tokens was set to 1m (which is equivalent to 62.5-second audio with
sampling rate of 16 kHz), the learning rate was 5e-5.

Figure 4 shows a typical training loss curve observed in fine-tuning.

Figure 4: A typical training loss curve observed in fine-tuning.

To be consistent with the literature, we conducted 10-fold cross validation on
the dataset. In each fold, the utterances of one speaker were used for
testing, and the other nine speakers for fine-tuning.

###  3.3 Results

####  3.3.1 Classification of emotion

The performance of emotion recognition has been evaluated in terms of both
weighted and unweighted accuracy. Weighted accuracy (WA) is the overall
accuracy on the entire dataset while unweighted accuracy (UA) is the average
of accuracies (recalls) of the emotion categories. The accuracies of models
using different phonetic units are reported in Table 1.

Table 1: Emotion recognition accuracy from using different phonetic units. Phonetic units | WA | UA
---|---|---
Phonemes | 75.5% | 75.6%
Broad phonetic classes | 76.8% | 77.2%
Syllables | 75.4% | 75.4%
Utterance | 62.6% | 62.5%

From Table 1 we can see that all models perform significantly better than most
previously reported results, except the utterance model. In the utterance
model, the entire utterance is a unit therefore there are only four emotion
dependent units (one for each of the four emotions) for recognition. This is
essentially a classification on the utterance without considering its phonetic
content. The results demonstrate that phonetic units are helpful and should be
incorporated in speech emotion recognition.

The best performance is from broad phonetic classes, and phonemes and
syllables perform equally well. Table 2 is the confusion matrix of the emotion
recognition results from fine-tuning broad phonetic class models.

Table 2: Confusion matrix of emotion recognition using broad phonetic classes. Target emotion | Predicted emotion
---|---
| Neutral | Happy | Angry | Sad
Neutral | 1189 | 298 | 100 | 121
Happy | 222 | 1328 | 64 | 22
Angry | 67 | 69 | 958 | 9
Sad | 195 | 93 | 25 | 771

####  3.3.2 Recognition of phonetic units

Although Wav2vec 2.0 has achieved great success on speech recognition in terms
of word error rate and phoneme error rate, it remains to be investigated
whether the model can be fine-tuned to recognize other phonetic unites such as
broad phonetic classes and syllables.

We evaluated the recognition results of phonemes, broad phonetic classes, and
syllables using the NIST scoring toolkit SCTK [21], both including and
excluding emotions. The error rates are listed in Table 3.

Table 3: Recognition error rates of different phonetic units counting or not counting emotion. The percentages in brackets are substitution, deletion, and insertion error rates respectively. Phonetic units | with emotion | without emotion
---|---|---
Phonemes | 31.9% | 14.9%
(N=194,473) | (24.2%,5.0%,2.8%) | (6.6%,5.2%,3.0%)
Broad phon. cl. | 28.2% | 11.5%
(N=194,473) | (20.3%,5.2%,2.8%)% | (3.0%,5.4%,3.0%)
Syllables | 32.4% | 16.3%
(N=78,943) | (21.4%,8.8%,2.2%) | (4.9%,9.0%,2.4%)

If we don’t count the emotion type of each recognized phonetic unit, the
recognition error rates for phonemes, broad phonetic classes, and syllables
are 14.9%, 11.5%, and 16.3% respectively. The results show that Wav2vec 2.0
can be fine-tuned to recognize not only phonemes, but also coarser-grained or
larger phonetic units such as broad phonetic classes and syllables, without a
language model.

##  4 Experiments on datasets in other languages

###  4.1 Data

We re-trained models of phonemes, broad phonetic classes, and syllables on the
entire dataset of IEMOCAP, with the same hyperparameters as used in the
experiments above. We then tested these models on three other emotion
datasets, EmoDB [22], KSUEmotions [23], and Mandarin Affective Speech [24].

EmoDB is a German dataset of emotional speech, containing utterances performed
in seven target emotions (including neutral) by ten actors. KSUEmotions
contains emotional Modern Standard Arabic (MSA) speech from 23 subjects, in
six emotions. Mandarin Affective Speech contains recordings in five emotions
by 68 Mandarin speakers. The datasets all contain neutral, happy, angry and
sad emotions. The utterances in these emotions were extracted from these
datasets for our experiments. The total number of utterances in each emotion
and each dataset is listed in Table 4.

Table 4: The number of utterances in each emotion and each dataset. Datasets | Neutral | Happy | Angry | Sad
---|---|---|---|---
EmoDB (German) | 79 | 71 | 127 | 62
KSU (Arabic) | 280 | 280 | 280 | 280
Affective (Mandarin) | 4079 | 4080 | 4080 | 4080

###  4.2 Results

Table 5 lists the recognition results on the three datasets.

Table 5: The (weighted) accuracies of emotion recognition on three other datasets using models trained on IEMOCAP (English). Datasets | IEMOCAP models
---|---
| Phonemes | Broad | Syllables
EmoDB (German) | 60.4% | 66.7% | 68.4%
KSU (Arabic) | 54.2% | 60.6% | 55.0%
Affective (Mandarin) | 40.7% | 39.9% | 41.0%

The Wav2vec 2.0 pre-trained models were trained on audio with sampling rate of
16 kHz. Recordings in the IEMOCAP, EmoDB, and KSUEmotions are also sampled at
16 kHz. The Mandarin Affective Speech dataset, however, uses 8 kHz sampling
rate. We upsampled the audio to 16 kHz for testing this dataset. Although the
sampling rate is the same, the audio upsampled from 8 kHz to 16 kHz contains
no components in frequencies between 4 and 8 kHz, which is dramatically
different from the training data. This may explain why the IEMOCAP models
performed poorly on the Mandarin dataset. We also directly tested on 8k Hz
audio with 16 kHz models. However, the results were even worse.

We conducted a 7-fold cross validation experiment on the Mandarin Affective
Speech dataset, by partitioning the dataset into training and test sets. We
used initials and finals as the phonetic units in the experiment. The overall
(weighted) accuracy was 75.2%. This result shows that the poor performance of
IEMOCAP models on this dataset is not due to the nature of the dataset. The
mismatch of sampling rate is probably the main factor responsible for the poor
performance, together with other possible factors such as linguistic and
cultural differences.

The models performed reasonably well on the German and Arabic datasets,
suggesting that our method has the potential for cross-lingual emotion
recognition. The models of broad phonetic classes and syllables seem to work
better than the phoneme models. This may be because broad phonetic classes and
syllables are language-general whereas phonemes are language-specific. It may
also be because there are more training data for broad phonetic classes and
syllables than phonemes, therefore the models of broad phonetic classes and
syllables are more robust. Further studies are needed to test these
hytpotheses.

##  5 Conclusions and discussion

We propose a method for emotion recognition through fine-tuning wav2vec 2.0
for recognition of emotion-dependent phonetic units, including phonemes, broad
phonetic classes, syllables, as well as the entire utterance. The models of
phonemes, broad phonetic classes, and syllables all significantly outperform
the utterance model, demonstrating that phonetic units are helpful and should
be incorporated in speech emotion recognition.

The best performance is from broad phonetic classes. The advantages of using
broad phonetic classes for other tasks such as speaking rate estimation [25]
and speech enhancement [26] have been reported in the literature. Using broad
phonetic classes may force the model to disregard differences of phonemes
within a broad phonetic class, but also encourage the model to pay attention
to the phonetic variability among broad phonetic classes in emotion
recognition. It will be interesting to further investigate the optimal set of
broad phonetic classes for the task of emotion recognition.

Models of broad phonetic classes and syllables outperform phonemes in the
setting of cross-lingual emotion recognition. This may be because broad
phonetic classes and syllables are language-general whereas phonemes are
language-specific, or because more training data are available for broad
phonetic classes and syllables than phonemes. Further research is needed to
test these hypotheses.

Our study also shows that Wav2vec 2.0 can be fine-tuned to recognize not only
phonemes, but also coarser-grained or larger phonetic units such as broad
phonetic classes and syllables, without a language model.

The proposed method of fine-tuning Wav2vec 2.0 for emotion recognition
achieved a significant improvement over most previously reported results on
IEMOCAP. In the future we will investigate how to make fine-tuned models more
robust across datasets where there are mismatches in recording conditions as
well as sampling rate.

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# A physical noise model for quantum measurements

Faedi Loulidi<EMAIL_ADDRESS>Laboratoire de Physique Théorique,
Université de Toulouse, CNRS, UPS, France  ,  Ion Nechita  ion.nechita@univ-
tlse3.fr Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS,
France  and  Clément Pellegrini<EMAIL_ADDRESS>Institut de Mathématiques, Université de Toulouse, CNRS, UPS, France

###### Abstract.

In this paper we introduce a novel noise model for quantum measurements
motivated by an indirect measurement scheme with faulty preparation. Averaging
over random dynamics governing the interaction between the quantum system and
a probe, a natural, physical noise model emerges. We compare it to existing
noise models (uniform and depolarizing) in the framework of incompatibility
robustness. We observe that our model allows for larger compatibility regions
for specific classes of measurements.

###### Contents

  1. 1 Introduction
  2. 2 Quantum measurements
  3. 3 Effective noise model via indirect measurements: the two outcomes case.
    1. 3.1 The two outcomes POVM induced by a perfect indirect measurement scheme
    2. 3.2 The noisy two outcomes POVM induced by an imperfect indirect measurement scheme
  4. 4 The case of arbitrary POVMs
    1. 4.1 General induced POVM
    2. 4.2 Physical noise model
  5. 5 Applications to compatibility

##  1\. Introduction

In quantum information theory, measurements are modelled by POVMs (_Positive
Operator Valued Measure_). POVMs are tuples of positive semidefinite operators
that sum up the identity. The POVM formalism allows one to describe the
outcome probabilities of the measurement process without knowing the
microscopic details of the measurement device. However, there exists another
way of extracting the probability of obtaining some outcome without destroying
the quantum system. This procedure is based on _indirect measurement process_.
For a given quantum state $\rho$ in a Hilbert space
$\mathcal{H}_{S}\cong\mathbb{C}^{d}$, the indirect measurement process
consists in performing a measurement of the probe on the evolved quantum state
coupled to a general probe in a Hilbert space
$\mathcal{H}_{P}\cong\mathbb{C}^{n}$. Initially, the quantum state is coupled
to a probe, and the evolution of the total system is given by a unitary
operator $U\in\mathcal{U}(d\,n)$. More precisely the evolution of the total
system is given by:

$\displaystyle U:\mathcal{H}_{S}\otimes\mathcal{H}_{P}$
$\displaystyle\to\mathcal{H}_{S}\otimes\mathcal{H}_{P}$
$\displaystyle\rho\otimes\beta$ $\displaystyle\to
U\big{(}\rho\otimes\beta\big{)}U^{*}.$

Now consider an observable $A$ on the probe system with spectral decomposition
$A=\sum\lambda_{i}P_{i}$. In the indirect measurement process, one oserves the
eigenvalue $\lambda_{i}$ of A with probability

$\mathbb{P}(i):=\operatorname{Tr}\Big{[}U(\rho\otimes\beta)U^{*}(I\otimes
P_{i})\Big{]}.$

Models of indirect measurement have been studied in the physical and
mathematical community. Such models are at the cornerstone of the
understanding of major experiments in quantum optics [GBD+07a, GBD+07b] see
also [Car93, HR06, WM10]. For a mathematical approach where such model are
precisely studied, one can consult [BJPP18, BFPP19, BB11]. In what follows, we
shall distinguish two different types of indirect measurement, a _perfect
measurement_ and an _imperfect measurement_ which is interpreted as noisy
measurement [APR13].

More precisely, we say that we obtain a perfect measurement if the probe is
_well prepared_ initially in a prescribed pure state. As an illustration,
without lost of generality, consider the probe $\beta=|0\rangle\langle 0|$.
The probability of obtaining an outcome $\lambda_{i}$ is given by:

$\mathbb{P}(i)=\operatorname{Tr}\Big{[}U(\rho\otimes|0\rangle\langle
0|)U^{*}(I\otimes P_{i})\Big{]}.$

An imperfect measurement can be understood physically as _noise_ that occurs
in a quantum measurement device. To give a description of an imperfect
measurement, we shall consider that, instead of preparing the probe in the
reference state $|0\rangle\langle 0|$, preparation errors will result in the
probe being in a state $\beta$ close to the desired state $|0\rangle\langle
0|$. For example, $\beta$ could be a _convex combination of two projectors_
with a parameter $t\in[0,1]$: $\beta_{t}:=(1-t)|0\rangle\langle
0|+t|1\rangle\langle 1|$; here, $t$ plays the role of the noise parameter. In
the general case, the probability of obtaining the outcome $\lambda_{i}$ is
given by :

$\mathbb{P}(i)=\operatorname{Tr}\Big{[}U(\rho\otimes\beta)U^{*}(I\otimes
P_{i})\Big{]}.$

In both situations described above, the indirect measurement process induces
an effective POVM on the system of interest

$\mathbb{P}(i)=\operatorname{Tr}[\rho\,A_{i}].$

In this article, we shall describe _noisy POVMs_ as _emergent_ from a
_physical motivated model_ based on an _indirect measurement process_ with an
application to the _compatibility of quantum measurements_. The effect of
noise has recently received attention for practical reasons, in noisy quantum
computers or what is known as NISQ devices see [BCLK+22, LLSK22], in quantum
metrology [LGRK22], and in the compatibility of quantum measurements [DFK19].

Noisy POVM $\mathcal{A}^{\alpha}:=(A_{1}^{\alpha},\cdots,A_{n}^{\alpha})$ in
the literature are described, by a convex combination of an initial POVM
$\mathcal{A}=(A_{1},\cdots,A_{n})$ and a trivial POVM
$\mathcal{T}:=(t_{1}I,\cdots,t_{n}I)$ with some parameter $\alpha\in[0,1]$:

$A_{i}^{\alpha}:=\alpha\,A_{i}+(1-\alpha)t_{i}I,$ (1)

where we shall distinguish two main types of noise that were previously
considered:

  * •

The _uniform noise_ where, for all $i\in[n]$ we have $t_{i}=1/n$.

  * •

The _depolarizing noise_ where, for all $i\in[n]$ we have
$t_{i}=\operatorname{Tr}A_{i}/d$.

The different types of noise models above were introduced to study the effect
of noise on the compatibility of quantum measurements. The notion of
compatibility is highly explored in different contexts, such as the
compatibility of quantum channels [HM17], the compatibility dimension [LN21,
UKD+21, HMT21], and the link between incompatibility and nonlocality [WPGF09,
LN22].

In the following, we will give a brief overview of the different results in
this work.

In the case of POVMs with two outcomes, we shall assume that the system
$\rho\in\mathbb{C}^{d}$ is coupled to a probe given by a two-level system
$\beta\in\mathbb{C}^{2}$, that we fix initially in $|0\rangle\langle 0|$ (in
the perfect measurement context). Considering a probe observable
$A=\lambda_{0}|0\rangle\langle 0|+\lambda_{1}|1\rangle\langle 1|$, the
probability of obtaining the outcome $i\in\\{0,1\\}$ system will induce a POVM
with two outcomes $\mathcal{A}$. This POVM depends on the structure of the
interaction between the system and the probe modelled by a bipartite unitary
operator $U\in\mathcal{U}(d\cdot 2)$. Next, to introduce our main _physical
noise model_ based on the _indirect measurement process_ , we shall assume
that initially the probe is given by a general two-level state
$\beta\in\mathbb{C}^{2}$ which is interpreted as a noisy perturbation of
$|0\rangle\langle 0|$ (the probe is not perfectly prepared or is perturbed
before interacting with the system). In this context we still consider the
same probe observable $A=\lambda_{0}|0\rangle\langle
0|+\lambda_{1}|1\rangle\langle 1|$. This time the probability of obtaining the
outcome $i\in\\{0,1\\}$ will induce the POVM $\mathcal{A}^{\beta}$ that will
depend on $\beta$ and the block elements of $U$ (as in the perfect case). Our
noise model _emerges naturally_ , by considering a random model for $U$ and
taking the average of $\mathcal{A}^{\beta}$ with respect to the distribution
of the interaction unitary $U$. This procedure gives rise to the _noisy two
outcome POVM_
$\mathbb{E}[\mathcal{A}^{\beta}]=(\mathbb{E}[A_{0}^{\beta}],\mathbb{E}[A_{1}^{\beta}])$,
where its elements are given by:

$\mathbb{E}[A_{0}^{\beta}]=\beta_{00}\,A_{0}+\beta_{11}\,\frac{\operatorname{Tr}[A_{1}]}{d}I_{d}\quad\text{and}\quad\mathbb{E}[A_{1}^{\beta}]=\beta_{00}\,A_{1}+\beta_{11}\,\frac{\operatorname{Tr}[A_{0}]}{d}I_{d}.$

Naturally, we obtain a noisy POVM of the form (1). The particularity is that
it is close to the depolarizing noise model where the indices of the effect
operators inside the trace are switched. While usual models such as uniform or
depolarizing are usually introduced as ad-hoc noisy POVMs, our model is
physically motivated through indirect measurement. Interestingly, we end up
with a "model close to" the depolarising model where the usual noise
parameters are switched.

In the rest of the article, we give a complete generalization of our noise
model for any number of outcomes $i\in[0,N]$ for arbitrary $N\in\mathbb{N}$.
In this general context, we describe physically motivated effective POVM
$\mathcal{A}^{\beta}$ with $N+1$ outcomes. Again, we concentrate on averaged
POVM $\mathbb{E}[\mathcal{A}^{\beta}]$ by considering random models for the
involved unitary operator.

Finally, we shall exploit our noise model by providing some applications for
the compatibility of quantum measurements. In particular, we compare it with
the usual noise models considered in the literature.

The paper is organized as follows. In Section 2, we recall the notion of
compatibility and the different type of noise models known in the literature.
In Section 3, we introduce our main _physically noise model based on an
indirect measurement process_ for POVMs with two outcomes. In Section 4, we
will generalize the results for POVMs with more than two outcomes. In Section
5, we will give examples of the application of our noise model to the
compatibility of quantum measurements.

##  2\. Quantum measurements

In this section, we will recall basic notions from quantum information theory.
We shall recall the (in-)compatibility of quantum measurement and the
different types of noise models established in the literature. In Quantum
Information theory, a quantum state is described by the framework of _density
matrices_ in a finite-dimensional Hilbert space
$\mathcal{H}\cong\mathbb{C}^{d}$. Formally we shall denote by the set of
quantum states by $\mathcal{M}_{d}^{1,+}$ defined as

$\mathcal{M}^{1,+}_{d}:=\\{\rho\in\mathcal{M}_{d}\,:\,\rho\geq 0\text{ and
}\operatorname{Tr}\rho=1\\},$

The set of density matrices encodes all the information about the physical
system. One of the main differences between classical mechanics and quantum
theory, the measurement outcomes of a given experiment are intrinsically
probabilistic. The celebrated _Born rule_ allows us to obtain the probability
of a given outcome. In general, the measurement process is characterized by
Positive Operator Valued Measures (POVMs)[NC00]. Formally a _positive operator
valued measure_ on $\mathcal{M}_{d}$ with $N$ outcomes is a $N$-tuple
$\mathcal{A}=(A_{1},\ldots,A_{N})$ of self-adjoint operators from
$\mathcal{M}_{d}$ which are positive semidefinite and sum up to the identity:

$\forall i\in[N],\quad A_{i}\geq 0\qquad\text{ and
}\qquad\sum_{i=1}^{N}A_{i}=I_{d}.$

When measuring a quantum state $\rho$ with the apparatus described by
$\mathcal{A}$, we obtain a random outcome from the set $[N]$:

$\forall i\in[N],\qquad\mathbb{P}(\text{outcome}=i)=\operatorname{Tr}[\rho
A_{i}].$

We shall write $[0,N]:=\\{0,1,\ldots,N\\}$ for the set of the $N+1$ positive
integers and the set for $N$ positive elements are given by
$[N]:=\\{1,\cdots,N\\}$. In particular, if the POVM framework reduces to the
_projective measurements_ (or _von Neumann measurements_) the measurement are
projectors of the form $A_{i}=|a_{i}\rangle\langle a_{i}|$. The POVM
framework, allows us to understand only the measurement outcome which is
relevant from given experiments without knowing the microscopic details of the
measurement device see figure 1 for an illustration. Another main difference
between classical physics and quantum mechanics is the notion of
_compatibility_. In classical mechanics, the measurement does not affect the
physical system. However, it is well known since the discovery of the quantum
theory the existence of measurements that cannot be measured at the same time
we called incompatible, and others that are compatible. We say that two POVMs
$\mathcal{A}=(A_{1},\cdots,A_{N})$ and $\mathcal{B}=(B_{1},\cdots,B_{M})$ are
compatible if there exists a third POVM $\mathcal{C}=(C_{11},\cdots,C_{NM})$
from where they are marginals, see figure 2 for an illustration. Formally, two
POVMs $\mathcal{A}$ and $\mathcal{B}$ are compatible if there exists a joint
POVM $\mathcal{C}=(C_{11},\cdots,C_{NM})$ such that:

$\forall i\in[N],\,A_{i}=\sum_{j=1}^{M}C_{ij},\quad\forall
j\in[M],\,B_{j}=\sum_{i=1}^{N}C_{ij}.$

In particular, the definition of compatibility reduces to the commutativity
for Projective measurements. Moreover, the definition of compatibility can be
extended naturally to _tuples of POVMs_ see [HMZ16] and the references therein
for more details.

Figure 1. Diagrammatic representation of a quantum measurement apparatus. The
device has an input canal and a set of $5$ LEDs which will turn on when the
corresponding outcome is achieved. After the measurement is performed, the
particle is destroyed, and the apparatus displays the classical outcome (here,
$3$).

In the following, we shall recall different types of _noise models_ considered
in the literature. A noise model is a POVM constructed from a convex
combination with a parameter $\alpha\in[0,1]$ of an original POVM
$\mathcal{A}$ and a _trivial_ POVM $\mathcal{T}$. We shall focus on two
different noise models well studied in the literature: the uniform noise, and
the depolarizing noise.

Figure 2. The joint measurement of $A$ and $B$ is simulated by by a third
measurement $C$, followed by classical post-processing.

  * •

The uniform noise model for $\alpha\in[0,1]$ is defined by:

$\mathcal{A}\to\mathcal{A}^{\alpha}:=\alpha\mathcal{A}+(1-\alpha)\mathcal{T},$

where $\mathcal{T}:=(t_{1}I,\cdots,t_{N}I)$, with $t_{i}:=1/N$ for all
$i\in[N]$.

  * •

The depolarizing noise for $\alpha\in[0,1]$ is defined by:

$\mathcal{A}\to\mathcal{A}^{\alpha}:=\alpha\mathcal{A}+(1-\alpha)\mathcal{T},$

where $\mathcal{T}:=(t_{1}I,\cdots,t_{N}I)$, with
$t_{i}:=\operatorname{Tr}A_{i}/d$ for all $i\in[N]$.

Noise models play several important roles in quantum information and more
specifically in the compatibility of quantum measurements. One should only
mention that for a given incompatible measurement, one can ask how much noise
one can add to the original POVMs in order to make them compatible. This
question will also be addressed when we will provide applications of our
physical noise model to the compatibility question and compare it with the
standard noise model in the last section of this paper.

##  3\. Effective noise model via indirect measurements: the two outcomes
case.

In this section, we will describe our _effective noise model_ obtained from an
_indirect measurement process_ in the case of POVMs with two outcomes. As
outlined in the Introduction, POVMs can be induced by indirect measurement
(see figure 3 for a pictorial representation). Here we shall consider a probe
of dimension $2$ and an observable with two outcomes. This will indeed give
rise to two POVMS with two outcomes. Then introducing noise in the probe
preparation will induce noisy POVMs that we can compare with the usual noise
models in the literature.

In the sequel, we shall describe the induced POVMs in terms of the unitary
interaction between the system and the probe. Briefly speaking the description
of the interaction will impose the form of the POVMs. In the case of dimension
$2$ (for the probe), we are able to describe an equivalence between the form
of the induced POVMs and the structure of the unitary interaction. This
structure combined with noise in the probe allows us to introduce random
models which allow us to derive effective POVMs that enter into the form (1).
Up to our knowledge, the main difference is that naturally, we obtain noises
that have never been studied in the literature and which are physically
motivated.

Subsection 3.1 concerns the model of indirect measurement in the perfect
situation and Subsection 3.2 concerns the models where we introduce noise in
the probe preparation.

Figure 3. The measurement of a quantum state on a POVM is equivalent to an
indirect measurement process, where one measures the probe on an evolved
quantum state coupled to a prepared probe on $|0\rangle\langle 0|$. By
destroying the probe in this process, we recover the initial POVM.

###  3.1. The two outcomes POVM induced by a perfect indirect measurement
scheme

Here we consider $\mathcal{H}_{P}=\mathbb{C}^{2}$. In the following, we will
assume that initially the probe is perfectly prepared in the state
$|0\rangle\langle 0|$. The evolution of the initial state $\rho$ of the system
coupled to the prepared probe on $|0\rangle\langle 0|$, where we have:

$\rho\otimes|0\rangle\langle 0|\to U\Big{(}\rho\otimes|0\rangle\langle
0|\Big{)}\,U^{*},$ (2)

where $U\in\mathcal{U}(2d)\subseteq\mathcal{M}_{d}\otimes\mathcal{M}_{2}$
given by:

$U=\sum_{i,j=0}^{1}U_{ij}\otimes|i\rangle\langle
j|=\begin{bmatrix}U_{00}&U_{01}\\\ U_{10}&U_{11}\end{bmatrix},$ (3)

with $U_{ij}\in\mathcal{M}_{d}(\mathbb{C})$ are the blocks of $U$.

We consider an observale $A=\lambda_{0}|0\rangle\langle
0|+\lambda_{1}|1\rangle\langle 1|$ on $\mathcal{H}_{P}$. We denote by
$\mathbb{P}(i)$ the probability of obtaining the outcome $i\in\\{0,1\\}$ by
measuring the probe in the evolved total system, which is given by:

$\mathbb{P}(i):=\operatorname{Tr}\Big{[}U\,\rho\otimes|0\rangle\langle
0|\,U^{*}\Big{(}I_{d}\otimes|i\rangle\langle i|\Big{)}\Big{]}.$ (4)

In the following proposition, we will show the probe measurement on the total
evolved system will induce an effective POVM $\mathcal{A}$.

######  Proposition 3.1.

The probability of obtaining the outcome $i\in\\{0,1\\}$ induces a POVM with
two outcomes $\mathcal{A}=(A_{0},A_{1})$ given by:

$\mathcal{A}:=(A_{0},A_{1})\quad\text{with}\quad
A_{0}:=U^{*}_{00}\,U_{00}\quad\text{and}\quad A_{1}:=U^{*}_{10}\,U_{10}.$ (5)

###### Proof.

Let $\mathbb{P}(0)$ and $\mathbb{P}(1)$ the probability of obtaining the
outcome $0$ and $1$, an explicit computation shows that the probability of
obtaining the outcome $i=0$ is :

$\displaystyle\mathbb{P}(0)$
$\displaystyle=\operatorname{Tr}\Big{[}U\,\Big{(}\rho\otimes|0\rangle\langle
0|\Big{)}\,U^{*}\,\Big{(}I_{d}\otimes|0\rangle\langle 0|\Big{)}\Big{]}$
$\displaystyle=\operatorname{Tr}\Big{[}\sum_{i,j=0}^{1}\,U_{ij}\otimes|i\rangle\langle
j|\Big{(}\rho\otimes|0\rangle\langle
0|\Big{)}\sum_{a,b=0}^{1}U^{*}_{ab}\otimes|b\rangle\langle
a|\Big{(}I_{d}\otimes|0\rangle\langle 0|\Big{)}\Big{]}$
$\displaystyle=\sum_{i,j=0}^{1}\,\sum_{a,b=0}^{1}\operatorname{Tr}\Big{[}U_{ij}\,\rho\,U^{*}_{ab}\otimes\ket{i}\braket{j}{0}\braket{0}{b}\braket{a}{0}\bra{0}\Big{]}$
$\displaystyle=\operatorname{Tr}\Big{[}U^{*}_{00}\,U_{00}\,\rho\Big{]}=\operatorname{Tr}\Big{[}A_{0}\rho\Big{]},$

where we have defined $A_{0}:=U^{*}_{00}U_{00}$.

A simple computation of $\mathbb{P}(1)$ as the one before, shows that:

$\mathbb{P}(1)=\operatorname{Tr}\Big{[}U^{*}_{10}\,U_{10}\,\rho\Big{]}=\operatorname{Tr}\Big{[}A_{1}\rho\Big{]},$

where we have defined $A_{1}:=U^{*}_{10}U_{10}.$

Using that $U\in\mathcal{U}(2d)$ is a unitary matrix, one can check easily
that $U^{*}_{00}\,U_{00}\,+\,U^{*}_{10}\,U_{10}=I_{d}$ which implies
$A_{0}+A_{1}=I_{d}$ which is the requirement for $A:=(A_{0},A_{1})$ being a
POVM.

∎

In the proposition above, we have seen that every unitary
$U\in\mathcal{U}(2d)$, induces by an indirect measurement scheme, an effective
POVM $\mathcal{A}=(A_{0},A_{1})$ on the target system $\mathbb{C}^{d}$. We now
ask the reverse question: _given a fixed POVM $\mathcal{A}$ on
$\mathbb{C}^{d}$,what is the set of interaction unitary operators $U$ which
yield $\mathcal{A}$ as an effective POVM?_ This question motivates the
following definition, where we will fix the POVM elements and search for
unitaries where their first column is given by the POVM elements.

######  Definition 3.2.

Given a fixed two outcomes POVM $\mathcal{A}=(A_{0},A_{1})$, we define a
subset of unitary matrices $\mathbb{U}(A_{0},A_{1})$ as:

$\mathbb{U}(A_{0},A_{1}):=\Big{\\{}U\in\mathcal{U}(2d)\Big{|}\,A_{0}=U^{*}_{00}\,U_{00},\,A_{1}=U^{*}_{10}\,U_{10}\Big{\\}}\subseteq\mathcal{U}(2d).$
(6)

In the following proposition, we will show that the set
$\mathbb{U}(A_{0},A_{1})$ can be completely characterized with the help of
three unitary matrices $V,W,Z\in\mathcal{U}(d).$

######  Proposition 3.3.

The set $\mathbb{U}(A_{0},A_{1})$ is completely characterized as follows:

$\displaystyle\mathbb{U}(A_{0},A_{1})=\Big{\\{}V,W,Z\in\mathcal{U}(d)$
$\displaystyle\Big{|}\,U_{00}=V\,\sqrt{A_{0}},U_{10}=W\,\sqrt{A_{1}},$
$\displaystyle
U_{01}=V\,\sqrt{A_{1}}\,Z^{*}\,,\,U_{11}=-W\,\sqrt{A_{0}}\,Z^{*}\Big{\\}}.$

In other words $U\in\mathbb{U}(A_{0},A_{1})$ if and only if there exists
$V,W,Z\in\mathcal{U}(d)$ such that

$U=\begin{bmatrix}V\sqrt{A_{0}}&V\,\sqrt{A_{1}}\,Z^{*}\\\
W\sqrt{A_{1}}&-W\,\sqrt{A_{0}}\,Z^{*}\end{bmatrix}.$ (7)

###### Proof.

Let $\mathbb{W}(A_{0},A_{1})$ be the set from the right-hand side in the
statement. Straightforward matrix block computations easily yield the first
inclusion

$\mathbb{W}(A_{0},A_{1})\subseteq\mathbb{U}(A_{0},A_{1})$ (8)

Let us concentrate on the reverse inclusion:

$\mathbb{U}(A_{0},A_{1})\subseteq\mathbb{W}(A_{0},A_{1}).$ (9)

To this end let $U\in\mathbb{U}(A_{0},A_{1})$, that is

$A_{0}=U^{*}_{00}U_{00}\quad\text{and}\quad A_{1}=U^{*}_{10}U_{10}.$ (10)

Let the polar decomposition of $U_{00}$ given by:

$U_{00}=V\,P_{0},$ (11)

where $V\in\mathcal{U}(d)$ and $P_{0}$ a positive semidefinte matrix.

By combining $A_{0}$ from equations (10) and (11):

$A_{0}=U^{*}_{00}U_{00}=(V\,P_{0})^{*}V\,P_{0}=P_{0}^{2}\implies
P_{0}=\sqrt{A_{0}},$ (12)

we deduce that $U_{00}$ is given by:

$U_{00}=V\sqrt{A_{0}}.$ (13)

Similarly, let the polar decomposition of $U_{10}$ given by:

$U_{10}=W\,P_{1},$ (14)

where $W\in\mathcal{U}(d)$ and $P_{1}$ a positive semidefinite matrix.

We have that:

$A_{1}=U^{*}_{10}U_{10}=(W\,P_{1})^{*}W\,P_{1}=P_{1}^{2}\implies
P_{1}=\sqrt{A_{1}},$ (15)

where we can deduce the form of $U_{10}$, which is given by:

$U_{10}=W\sqrt{A_{1}}.$ (16)

From the unitary property of $U\in\mathcal{U}(2d)$, we shall determine the
other blocks of $U$.

$U^{*}U=I_{2d}\implies(\text{S}_{1}):\begin{cases}U^{*}_{00}\,U_{00}\,+\,U^{*}_{10}\,U_{10}=I_{d}&\\\
U^{*}_{00}\,U_{01}\,+\,U^{*}_{10}\,U_{11}=0&\\\
U^{*}_{01}\,U_{00}\,+\,U^{*}_{11}\,U_{10}=0&\\\
U^{*}_{01}\,U_{01}\,+\,U^{*}_{11}\,U_{11}=I_{d}&\end{cases}$ (17)

and

$UU^{*}=I_{2d}\implies(\text{S}_{2}):\begin{cases}U_{00}\,U^{*}_{00}\,+\,U_{01}\,U^{*}_{01}=I_{d}&\\\
U_{00}\,U^{*}_{10}\,+\,U_{01}\,U^{*}_{11}=0&\\\
U_{10}\,U^{*}_{00}\,+\,U_{11}\,U^{*}_{01}=0&\\\
U_{10}\,U^{*}_{10}\,+\,U_{11}\,U^{*}_{11}=I_{d}&\end{cases}$ (18)

By using the fourth equation from the system $(\text{S}_{2})$ (18) and the
equation (16):

$WA_{1}W^{*}\,+\,U_{11}\,U^{*}_{11}=I_{d},$ (19)

where from the fact that $A_{0}+A_{1}=I_{d}$, we deduce that:

$(U_{11}^{*}\,W)^{*}\,U_{11}^{*}\,W=A_{0}.$ (20)

Let the polar decomposition of $U_{11}^{*}\,W$ given by:

$U_{11}^{*}W=-Z\,Q,$ (21)

where $Z\in\mathcal{U}(d)$ and positive semidefinite $Q$.

By combining the equations (20) and (21), we have:

$(U_{11}^{*}\,W)^{*}\,U_{11}^{*}\,W=Q^{2}=A_{0}\implies Q=\sqrt{A_{0}},$ (22)

where we can deduce the form of $U_{11}$, which is given by :

$U_{11}=-W\,\sqrt{A_{0}}\,Z^{*}.$ (23)

The only remaining element we need to determine is $U_{01}$, for that we shall
use the first equation of the system $(\text{S}_{2})$ (18) and from
$A_{0}+A_{1}=I_{d}$, we have:

$U_{00}\,U^{*}_{00}\,+\,U_{01}\,U^{*}_{01}=I_{d}\iff(U_{01}^{*}\,V)^{*}U_{01}^{*}\,V=A_{1}.$
(24)

Let the polar decomposition of $U_{01}^{*}\,V$ given by:

$U_{01}^{*}\,V=\Gamma\,\tilde{Q},$ (25)

with $\Gamma\in\mathcal{U}(d)$ and $\tilde{Q}$ positive matrix.

By combining the equations (25) and (24), we have:

$(U_{01}^{*}\,V)^{*}U_{01}^{*}\,V=\tilde{Q}^{2}=A_{1}\implies\tilde{Q}=\sqrt{A_{1}},$
(26)

where we can deduce the form of $U_{01}$, which is given by:

$U_{01}=V\,\sqrt{A_{1}}\Gamma^{*}.$ (27)

The only remaining matrix to determine is $\Gamma$, for that we shall use the
second equation of $(\text{S}_{2})$ (18) and analyze the invertibility of the
effect operators $A_{0}$ and $A_{1}$.

From the second equation of $(\text{S}_{2})$ (18) we have:

$U_{00}\,U^{*}_{10}\,+\,U_{01}\,U^{*}_{11}=0\iff\sqrt{A_{1}}\,\sqrt{A_{0}}=\sqrt{A_{1}}\,\Gamma^{*}\,Z\,\sqrt{A_{0}}$
(28)

Remark that $\mathrm{Ran}(A_{0})+\mathrm{Ran}(A_{1})=\mathbb{C}^{d}$, to
discuss the inversibility $A_{0}$ and $A_{1}$, we shall distinguish two cases:

  * •

Assuming $A_{0}$ invertible and $A_{1}$ invertible only on its range:

Let $\mathbb{C}^{d}\ni x=x_{0}+x_{1}$ with $x_{i}\in\mathrm{Ran}(A_{i})$, and
$y_{1}^{T}:=x_{1}^{T}(\sqrt{A_{1}})^{-1}$, we have:

$y_{1}^{T}\sqrt{A_{1}}=x_{1}^{T}\,I_{d}\,|_{\mathrm{Ran}(A_{1})}=x_{1}^{T}\,I_{d}|_{\mathrm{Ran}(A_{1})}\Gamma^{*}\,Z.$
(29)

where $I_{d}|_{\mathrm{Ran}(A_{1})}$ is the identity on the range of $A_{1}$.

Hence, we have that:

$\Gamma^{*}Z\,|_{\mathrm{Ran}(A_{1})}=I_{d}\,|_{\mathrm{Ran}(A_{1})}\implies\Gamma|_{\mathrm{Ran}(A_{1})}=Z|_{\mathrm{Ran}(A_{1})}$
(30)

  * •

Assuming $A_{1}$ invertible and $A_{0}$ invertible only on its range:

As before $\mathbb{C}^{d}\ni x=x_{0}+x_{1}$ with
$x_{i}\in\mathrm{Ran}(A_{i})$, and $y_{0}:=(\sqrt{A_{0}})^{-1}x_{0}$ we have:

$\sqrt{A_{0}}\,y_{0}=\,I_{d}\,|_{\mathrm{Ran}(A_{0})}x_{0}=\Gamma^{*}\,Z\,I_{d}|_{\mathrm{Ran}(A_{1})}\,x_{0},$
(31)

where $I_{d}|_{\mathrm{Ran}(A_{0})}$ is the identity on the range of $A_{0}$.

We obtain that:

$\Gamma^{*}Z\,|_{\mathrm{Ran}(A_{0})}=I_{d}\,|_{\mathrm{Ran}(A_{0})}\implies\Gamma|_{\mathrm{Ran}(A_{0})}=Z|_{\mathrm{Ran}(A_{0})}.$
(32)

By combining the two consequences above we obtain:

$\Gamma=\Gamma|_{\mathrm{Ran}(A_{0})}+\Gamma|_{\mathrm{Ran}(A_{1})}=Z|_{\mathrm{Ran}(A_{0})}+Z|_{\mathrm{Ran}(A_{0})}=Z$
(33)

Therefore, we can deduce in both cases where $A_{0}$ and $A_{1}$ are both
invertible and only one of the two is invertible, that $U_{01}$ is given by:

$U_{01}=V\,\sqrt{A_{1}}\,Z^{*}.$ (34)

This ends the proof of the second inclusion, therefore the end of the proof of
the proposition.

∎

######  Remark 3.4.

The set $\mathbb{U}(A_{0},A_{1})$ is not a subgroup of $\mathcal{U}(2d)$, one
need to check that $I\notin\mathbb{U}(A_{0},A_{1})$. From the definition given
above we have $U_{00}=V\sqrt{A_{0}}$ and $U_{10}=W\sqrt{A_{1}}$ for
$V,W\in\mathcal{U}(d)$. Let $I_{2d}\in\mathcal{U}(2d)$ we have
$U_{00}=V\,\sqrt{A_{0}}=I_{d}$ and $U_{10}=W\,\sqrt{A_{1}}=0_{d}$, hence one
can check that $A_{0}+A_{1}\neq I_{d}$, therefore $\mathbb{U}(A_{0},A_{1})$ is
not a subgroup of $\mathcal{U}(2d)$.

######  Remark 3.5.

Let the action $\circ^{\prime}:\,\mathcal{U}(d)^{\times
3}\curvearrowright\mathbb{U}(A_{0},A_{1})$ defined by:

$\displaystyle\circ^{\prime}:\,\mathcal{U}(d)^{\times
3}\times\mathbb{U}(A_{0},A_{1})$ $\displaystyle\to\mathbb{U}(A_{0},A_{1}),$
$\displaystyle\Big{(}(V,W,Z),U\Big{)}$
$\displaystyle\to(V,W,Z)\circ^{\prime}U.$

For $U\in\mathbb{U}(A_{0},A_{1})$, the action of $\circ^{\prime}$ is defined
by:

$\displaystyle(V,W,Z)\circ U:$ $\displaystyle=\begin{bmatrix}V&0\\\
0&W\end{bmatrix}\begin{bmatrix}V^{\prime}\sqrt{A_{0}}&V^{\prime}\,\sqrt{A_{1}}\,Z^{\prime*}\\\
W^{\prime}\sqrt{A_{1}}&-W^{\prime}\,\sqrt{A_{0}}\,Z^{\prime*}\end{bmatrix}\begin{bmatrix}I&0\\\
0&Z^{*}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\tilde{V}\sqrt{A_{0}}&\tilde{V}\,\sqrt{A_{1}}\,\tilde{Z}^{\prime*}\\\
\tilde{W}^{\prime}\sqrt{A_{1}}&-\tilde{W}^{\prime}\,\sqrt{A_{0}}\,\tilde{Z}^{\prime*}\end{bmatrix}\in\mathbb{U}(A_{0},A_{1}),$

where $\tilde{V}=VV^{\prime}$,$\tilde{W}=WW^{\prime}$ and
$\tilde{Z}=ZZ^{\prime}$ are in $\mathcal{U}(d)$.

With this at hand, one can understand the set $\mathbb{U}(A_{0},A_{1})$ in the
proposition 3.3 as the orbit of the simple matrix $\tilde{U}$ given by:

$\tilde{U}:=\begin{bmatrix}\sqrt{A_{0}}&\sqrt{A_{1}}\\\
\sqrt{A_{1}}&-\sqrt{A_{0}}\end{bmatrix}.$

We can also check that the following property holds:

$\displaystyle\Big{(}(V,W,Z)\circ^{\prime}(V^{\prime},W^{\prime},Z^{\prime})\Big{)}\circ\tilde{U}=(VV^{\prime},WW^{\prime},ZZ^{\prime})\circ\tilde{U}.$

where $\circ^{\prime}$ is the group action on $\mathcal{U}(d)^{\times 3}$.

###  3.2. The noisy two outcomes POVM induced by an imperfect indirect
measurement scheme

To introduce our main _physical noise model_ , we shall assume that initially
the quantum state $\rho$ is coupled to a general two-level probe $\beta$, and
the unitaries are random. The imperfect measurement of the probe will induce a
general POVM $\mathcal{A}^{\beta}$. The average over the randomness of the
POVM $\mathcal{A}^{\beta}$ will induce our physical noise model.

In the basis $\\{\ket{0},\ket{1}\\}$ we write $\beta\in\mathcal{M}_{2}^{1,+}$
as:

$\beta=\sum_{i,j=0}^{1}\beta_{ij}|i\rangle\langle j|\in\mathcal{M}_{2}^{1,+}.$

Again, we consider the evolution of the form

$\rho\otimes\beta\to U\Big{(}\rho\otimes\beta\Big{)}U^{*},$

with $U\in\mathbb{U}(A_{0},A_{1})$.

We shall denote by $\mathbb{P}_{\beta}(i)$ the probability of obtaining the
outcome $i\in\\{0,1\\}$ on the evolved quantum state $\rho$ coupled to the
probe $\beta$. In the following Proposition, we shall show that the outcome
probability of obtaining $i\in\\{0,1\\}$ will induce an effective POVM
$\mathcal{A}^{\beta}$ that will depend on the probe.

######  Proposition 3.6.

Let the evolution of two-level probe $\beta$ coupled to a quantum state $\rho$
governed by $U\in\mathbb{U}(A_{0},A_{1})$ with associated unitary operators
$V,W,Z$. The probability of obtaining the outcome $i\in\\{0,1\\}$ given by
$\mathbb{P}_{\beta}(i)$ induces an effective POVM with two outcomes
$\mathcal{A}^{\beta}$ given by:

$\mathcal{A}^{\beta}=(A_{0}^{\beta},A_{1}^{\beta}),$

where $A_{0}^{\beta}$ and $A_{1}^{\beta}$ are explicitly given by:

$\displaystyle A_{0}^{\beta}$
$\displaystyle=\beta_{00}\,A_{0}+\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}+\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{1}\,Z,$
$\displaystyle A_{1}^{\beta}$
$\displaystyle=\beta_{00}\,A_{1}-\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}-\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{0}\,Z.$

###### Proof.

An explicit computation of $\mathbb{P}_{\beta}(i)$ shows that:

$\displaystyle\mathbb{P}_{\beta}(i)$
$\displaystyle:=\operatorname{Tr}\Big{[}U\,(\rho\otimes\beta)\,U^{*}\,(I_{d}\otimes|i\rangle\langle
i|)\Big{]}$
$\displaystyle=\sum_{c,d=0}^{1}\beta_{cd}\,\operatorname{Tr}\Big{[}U^{*}_{id}\,U_{ic}\,\rho\Big{]}$
$\displaystyle=\operatorname{Tr}[A_{i}^{\beta}\rho],$

where we have defined
$A_{i}^{\beta}:=\sum_{c,d=0}^{1}\beta_{cd}\,U^{*}_{id}\,U_{ic}.$

By using Proposition 3.3, the effects of the POVM $\mathcal{A}^{\beta}$ are
given explicitly by:

$\displaystyle
A_{0}^{\beta}:=\sum_{c,d=0}^{1}\beta_{cd}\,U^{*}_{0d}\,U_{0c}=\beta_{00}\,A_{0}+\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}+\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{1}\,Z$
$\displaystyle
A_{1}^{\beta}:=\sum_{c,d=0}^{1}\beta_{cd}\,U^{*}_{1d}\,U_{1c}=\beta_{00}\,A_{1}-\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}-\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{0}\,Z$

∎

To introduce our noise model from an indirect measurement process, we shall
define a _centered 1-design_ probability measure on $\mathcal{U}(d)$.

######  Definition 3.7.

A probability distribution $\mu$ on $\mathcal{U}(d)$ is called a _centered
1-design_ if $\forall X\in\mathcal{M}_{d}$

$\mathbb{E}_{Z\sim\mu}[Z\,X]=\mathbb{E}_{Z\sim\mu}[Z^{*}\,X]:=0$ (35)

and

$\mathbb{E}_{Z\sim\mu}[Z\,X\,Z^{*}]:=\frac{\operatorname{Tr}[X]}{d}I_{d}.$
(36)

######  Remark 3.8.

The _Haar probability distribution_ is an example of a centered $1$-design.

As we have described in the introduction of this subsection, our physical
noise model will emerge naturally by taking the average of the effective POVM
$\mathcal{A}^{\beta}$. For that, we shall assume that $Z\in\mathcal{U}(d)$ are
sampled from a centered 1-design distribution.

######  Theorem 3.9.

Assume $U\in\mathbb{U}(A_{0},A_{1})$ and assume that the corresponding
$Z\in\mathcal{U}(d)$ is sampled from a centered 1-design distribution $\mu$.
The expectation value of the induced noisy effective POVM
$\mathcal{A}^{\beta}$ will induce a general noise model described by the two
outcomes POVM $\mathbb{E}_{Z\sim\mu}[\mathcal{A}^{\beta}]$ given by:

$\mathbb{E}_{Z\sim\mu}[\mathcal{A}^{\beta}]=(\mathbb{E}_{Z\sim\mu}[A_{0}^{\beta}],\mathbb{E}_{Z\sim\mu}[A_{1}^{\beta}]),$

where $\mathbb{E}_{Z\sim\mu}[A_{0}^{\beta}]$ and
$\mathbb{E}_{Z\sim\mu}[A_{1}^{\beta}]$ are given by:

$\mathbb{E}_{Z\sim\mu}[A_{0}^{\beta}]=\beta_{00}\,A_{0}+\beta_{11}\,\frac{\operatorname{Tr}[A_{1}]}{d}I_{d}\quad\text{and}\quad\mathbb{E}_{Z\sim\mu}[A_{1}^{\beta}]=\beta_{00}\,A_{1}+\beta_{11}\,\frac{\operatorname{Tr}[A_{0}]}{d}I_{d}.$

In other words, we have:

$(\mathbb{E}_{Z\sim\mu}[A_{0}^{\beta}]\,,\,\mathbb{E}_{Z\sim\mu}[A_{1}^{\beta}])=\beta_{00}(A_{0}\,,\,A_{1})+\beta_{11}\left(\frac{\operatorname{Tr}[A_{1}]}{d}I_{d}\,,\,\frac{\operatorname{Tr}[A_{0}]}{d}I_{d}\right).$

###### Proof.

From Proposition 3.6 we have shown that:

$\displaystyle A_{0}^{\beta}$
$\displaystyle=\beta_{00}\,A_{0}+\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}+\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{1}\,Z.$
(37) $\displaystyle A_{1}^{\beta}$
$\displaystyle=\beta_{00}\,A_{1}-\beta_{01}\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}-\beta_{10}\,\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}+\beta_{11}\,Z^{*}\,A_{0}\,Z.$
(38)

By taking the average on $Z$ and by linearity we have:

$\displaystyle\mathbb{E}_{Z\sim\mu}[A_{0}^{\beta}]$
$\displaystyle=\beta_{00}\,A_{0}+\beta_{01}\,\mathbb{E}_{Z\sim\mu}[\,Z\,\sqrt{A_{1}}\,\sqrt{A_{0}}]+\beta_{10}\,\mathbb{E}_{Z\sim\mu}[\sqrt{A_{1}}\,\sqrt{A_{0}}\,Z^{*}]+\beta_{11}\,\mathbb{E}_{Z\sim\mu}[Z^{*}\,A_{1}\,Z].$
$\displaystyle=\beta_{00}\,A_{0}+\beta_{11}\,\frac{\operatorname{Tr}[A_{1}]}{d}I_{d}.$

With the same computation as the previous one, we obtain
$\mathbb{E}_{Z\sim\mu}[A_{1}^{\beta}]$ as announced in the statement of the
Theorem. ∎

Having established the general description of noise models from a two-level
probe model $\beta$, we shall specify two different probes: the _probabilistic
probe_ $\sigma_{t}$ and the _cat probe_ $\gamma_{t}$. We will show that
starting from these two different types of probes when one takes the average
on $Z$ they give the _same type of noisy effective POVM_ , hence the same type
of noise model.

######  Definition 3.10.

We shall define the probabilistic and the cat probes are given respectively by
$\sigma_{t}$ and $\gamma_{t}$ for $t\in[0,1]$ as:

$\sigma_{t}:=(1-t)|0\rangle\langle 0|+t|1\rangle\langle
1|\quad\text{and}\quad\gamma_{t}:=|\lambda_{t}\rangle\langle\lambda_{t}|.$
(39)

where $\ket{\lambda_{t}}$ is defined as

$\ket{\lambda_{t}}:=\sqrt{1-t}\ket{0}+\sqrt{t}\ket{1}.$ (40)

As a direct consequence of Theorem 3.9, we will see in the following Corollary
that the resulting noise model is independent of using a probabilistic or a
cat probe.

######  Corollary 3.11.

Assuming the evolution $U\in\mathcal{U}(2d)$ of $\rho$ coupled either to the
probabilistic or the cat probe given respectively by $\sigma_{t}$ and
$\gamma_{t}$. The expectation value of $\mathcal{A}^{\sigma_{t}}$ and
$\mathcal{A}^{\gamma_{t}}$ are equal, and we have
$\mathbb{E}_{Z\sim\mu}[\mathcal{A}^{\sigma_{t}}]=\mathbb{E}_{Z\sim\mu}[\mathcal{A}^{\gamma_{t}}]$.

###### Proof.

The proof of the Corollary is basically that the probabilistic and the cat
probe have the same diagonal elements, moreover, the off-diagonal elements
will be all canceled when we take the expectation value.

By Theorem 3.9 we obtain:

$\mathbb{E}_{Z\sim\mu}[A_{0}^{\sigma_{t}}]=t\,A_{0}+(1-t)\,\frac{\operatorname{Tr}[A_{1}]}{d}I_{d}=\mathbb{E}_{Z\sim\mu}[A_{0}^{\gamma_{t}}]\,,\,\mathbb{E}_{Z\sim\mu}[A_{1}^{\sigma_{t}}]=t\,A_{1}+(1-t)\,\frac{\operatorname{Tr}[A_{0}]}{d}I_{d}=\mathbb{E}_{Z\sim\mu}[A_{1}^{\gamma_{t}}].$

∎

##  4\. The case of arbitrary POVMs

In this section, we will generalize our _physical noise model_ description
from an _indirect measurement process_ for the case of POVMs _having more than
two outcomes_. Here we shall consider a probe of dimension $N+1$. As in the
previous section, the POVMs can be induced by indirect measurement, where the
resulting POVMs are of $N+1$ outcomes. The noise in the probe preparation will
induce noisy POVMs of $N+1$ outcomes.

The unitary interaction between the system and the probe will induce and give
the form of the POVMs. This will allow us, to introduce the set of the
unitaries that give rise to the POVMs. With the noisy probe and set of all the
unitaries that we shall consider as random, a natural noise model emerges by
taking the average over the unitaries.

In Subsection 4.1, we will give the induced POVMs in the perfect situation. In
Subsection 4.1, we will give our physical noise model description.

###  4.1. General induced POVM

In the following subsection, we will give a general model where we will
describe an emergent POVM $\mathcal{A}$ with $N+1$ outcomes. For that, as in
Section 3, we shall assume the probe is initially prepared in
$|0\rangle\langle 0|$ and the total system is given by
$\rho\otimes|0\rangle\langle 0|$. The dynamics are given by unitary matrices
$U\in\mathcal{U}((N+1)d)$. Explicitly the evolution of the total system, as
usual, is given by:

$\rho\otimes|0\rangle\langle 0|\to U\Big{(}\rho\otimes|0\rangle\langle
0|\Big{)}\,U^{*},$

with
$U\in\mathcal{U}((N+1)d)\subseteq\mathcal{M}_{d}\otimes\mathcal{M}_{N+1}.$

We denote by $\mathbb{P}(i)$ the probability of obtaining the outcome
$i\in[0,N]$ by measuring the probe on the evolved total system. In the
following proposition, we will show that the measurement over the probe will
induce an effective emergent POVM $\mathcal{A}$ with $N+1$ outcomes.

######  Proposition 4.1.

The probability of obtaining the outcome $i\in[0,N]$ induce an effective POVM
$\mathcal{A}$ with $N+1$ outcomes given by:

$\mathcal{A}:=(A_{0},\cdots,A_{N})\quad\text{with}\quad
A_{i}:=U^{*}_{i0}\,U_{i0},\quad\forall i\in[0,N].$ (41)

###### Proof.

An explicit computation of $\mathbb{P}(i)$ shows that:

$\displaystyle\mathbb{P}(i)$
$\displaystyle=\operatorname{Tr}\Big{[}U\,\Big{(}\rho\otimes|0\rangle\langle
0|\Big{)}\,U^{*}\,\Big{(}I_{d}\otimes|i\rangle\langle i|\Big{)}\Big{]}$
$\displaystyle=\operatorname{Tr}\Big{[}U^{*}_{i0}\,U_{i0}\,\rho\Big{]}=\operatorname{Tr}\Big{[}A_{i}\rho\Big{]}.$

Where we have defined $A_{i}:=U^{*}_{i0}\,U_{i0}\geq 0$.

Therefore $A_{i}\geq 0$ for all $i\in[0,N]$ and by using the unitarity of $U$
one can check easily that $\sum_{i=0}^{N}\,A_{i}=I_{d}$ holds, hence
$\mathcal{A}:=(A_{0},\cdots,A_{N})$ defines a POVM. ∎

The proposition above motivates the following definition, where we will define
a set of unitaries $U\in\mathcal{U}((N+1)d)$ where for fixed $U_{i0}$ for all
$i\in[0,N]$ is given by $U^{*}_{i0}\,U_{i0}=:A_{i}$.

######  Definition 4.2.

Consider a POVM $\mathcal{A}=(A_{0},\cdots,A_{N})$. We define a subset of
unitary matrices $\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ as:

$\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}:=\Big{\\{}U\in\mathcal{U}((N+1)d)\Big{|}\,U^{*}_{i0}\,U_{i0}=A_{i}\Big{\\}}.$
(42)

######  Remark 4.3.

It is a very difficult task to explicitly characterize the set of unitaries
given above, in comparison with the set given in Definition 3.2 which was
fully characterized in Proposition 3.3.

######  Remark 4.4.

The set $\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ is non-empty. To
show this, we shall show it is possible to complete the matrix $U$ having as
it first block column $U_{i0}=\sqrt{A_{i}}$. This is possible as long as the
first $d$ columns of $U$ are orthonormal. For that, let two different columns
$c_{1}$, $c_{2}$ of $U$, where $c_{j}\in[d]$ and $j\in\\{1,2\\}$. The scalar
product of two columns is given by:

$\sum_{l=1}^{d}\sum_{i=0}^{N}\Big{(}\sqrt{A_{i}}\,(l,c_{1})\Big{)}^{*}\,\sqrt{A_{i}}(l,c_{2})=\sum_{l=1}^{d}\sum_{i=0}^{N}A_{i}(c_{1},c_{2})=\delta_{c_{1},c_{2}},$

where in the equality above we have used that $\sum_{i=0}^{N}A_{i}=I_{d}$.

Therefore, the first $d$ columns of $U$ are orthonormal, hence we can always
complete a unitary matrix
$U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$.

######  Remark 4.5.

The set $\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ is invariant
invariant under the action of element in $\mathcal{U}^{\times 2N+1}(d)$
defined as below.

Let the action $\circ^{\prime}:\mathcal{U}(d)^{\times
2N+1}\curvearrowright\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ defined
by:

$\displaystyle\circ^{\prime}:\mathcal{U}(d)^{\times
2N+1}\times\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$
$\displaystyle\to\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)},$
$\displaystyle\Big{(}(V_{0},\cdots,V_{N},Z_{1},\cdots,Z_{N}),U\Big{)}$
$\displaystyle\to\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}.$

For $U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ the action of
$\circ^{\prime}$ is defined by:

$U^{\prime}=(V_{0},\cdots,V_{N},Z_{1},\cdots,Z_{N})\circ^{\prime}U:=\begin{bmatrix}V_{0}&&\\\
&V_{1}\\\ &&\ddots&\\\ &&&V_{N}\end{bmatrix}\,U\,\begin{bmatrix}I_{d}&&\\\
&Z_{1}^{*}\\\ &&\ddots&\\\ &&&Z_{N}^{*}\end{bmatrix}.$

Remark from a simple computation that:

$U_{i0}^{\prime*}U_{i0}^{\prime}=U_{i0}^{*}V_{i}^{*}V_{i}U_{i0}=A_{i},$

therefore we have
$U^{\prime}\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$.

The Definition 4.2 will play a crucial role in the rest of this section to
introduce our physical noise model.

###  4.2. Physical noise model

In the following subsection, we will introduce our main general _physical
noise_ model. For that, we will assume initially that the quantum state $\rho$
is coupled to a probe given by an $(N+1)$-level quantum state
$\beta\in\mathcal{M}_{N+1}^{1,+}$.

######  Definition 4.6.

Let $\beta\in\mathcal{M}_{N+1}^{1,+}$ the $N+1$ level quantum state describing
the probe:

$\beta=\sum_{i,j=0}^{N}\beta_{i,j}|i\rangle\langle
j|\in\mathcal{M}_{N+1}^{1,+}.$

Assuming the evolution of the total system, quantum state $\rho$ coupled to a
probe $\beta$ is given by:

$\rho\otimes\beta\to U\Big{(}\rho\otimes\beta\Big{)}U^{*},$

with $U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$. We denote by
$\mathbb{P}_{\beta}(i)$ the probability of obtaining an outcome $i\in[0,N]$ by
measuring the probe on the evolved total system.

In the following we will show the outcome probability of obtaining $i\in[0,N]$
will induce an effective noisy POVM $\mathcal{A}^{\beta}$.

######  Proposition 4.7.

The probability of obtaining the outcome $i\in[0,N]$ by measuring the probe on
the evolved total system, will induce an effective POVM $\mathcal{A}^{\beta}$
where:

$\mathcal{A}^{\beta}=(A_{0}^{\beta},\cdots,A_{N}^{\beta}),\quad\text{and}\quad
A_{i}^{\beta}=\sum_{c,k=0}^{N}\,\beta_{ck}\,U^{*}_{ik}\,U_{ic},\quad\forall
i\in[0,N].$

###### Proof.

An explicit computation of $\mathbb{P}_{\beta}(i)$ shows that:

$\displaystyle\mathbb{P}_{\beta}(i)$
$\displaystyle=\operatorname{Tr}\Big{[}U\,(\rho\otimes\beta)\,U^{*}\,(I_{d}\otimes|i\rangle\langle
i|)\Big{]}$
$\displaystyle=\sum_{c,k=0}^{N}\beta_{ck}\,\operatorname{Tr}\Big{[}U^{*}_{ik}\,U_{ic}\,\rho\Big{]}$
$\displaystyle=\operatorname{Tr}[A_{i}\rho],$

where we have defined
$A_{i}^{\beta}:=\sum_{c,k=0}^{N}\beta_{ck}\,U^{*}_{ik}\,U_{ic}\geq 0$ and
$\sum_{i=0}^{N}A_{i}^{\beta}=\operatorname{Tr}\beta=1$, therefore
$\mathcal{A}^{\beta}:=(A_{0}^{\beta},\cdots,A_{N}^{\beta})$ defines a POVM. ∎

To introduce our physical noise model in this general setting, we shall assume
that the unitaries $U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ are
randomly sampled from a _nice probability_ measure.

######  Definition 4.8.

A probability distribution $\mu$ on
$\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$ is a _nice probability
measure_ if it is invariant under right multiplication with a unitary
$Z\in\mathcal{U}((N+1)d)$ of the form
$Z:=\sum_{i=0}^{N}\,Z_{i}\otimes|i\rangle\langle i|\quad\text{with}\quad
Z_{0}=I_{d}\quad\text{and}\quad Z_{i}\in\mathcal{U}(d)$ for all $i\neq 0.$
More precisely, we have for a random unitary
$U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$:

$\mathbb{E}_{U\sim\mu}[f(U)]=\mathbb{E}_{U\sim\mu}[f(U\cdot Z)].$

######  Remark 4.9.

To construct an example of a nice probability measure, consider a fixed
element $U_{0}\in\mathbb{U}(\\{A_{i}\\}_{i\in[0,N]})$. Let also
$V_{1},\ldots,V_{N}\in\mathcal{U}(d)$ be independent Haar-distributed random
unitary matrices. Define the random variable

$U:=U_{0}(I_{d}\oplus V_{1}\oplus\cdots\oplus V_{N}).$

We claim that $U$ has a nice distribution. Indeed, if
$Z_{1},\ldots,Z_{N}\in\mathcal{U}(d)$ are arbitrary unitary matrices,

$\mathbb{E}[f(U\cdot
Z)=\mathbb{E}[f(U_{0}VZ)]=\mathbb{E}[f(U_{0}\tilde{V})]=\mathbb{E}[f(U)],$

where we have used the fact that $V$ and $\tilde{V}=VZ$ have the same (block-
Haar) distribution.

######  Proposition 4.10.

Let $\mu$ be a nice probability measure on
$\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$. We have:

$U\in\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)},\quad\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}]=0.$
(43)

###### Proof.

By using the right invariance property of the nice measure $\mu$ from
Definition 4.8 we have:

$\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}]=Z^{*}_{k}\,\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}].$

Since this last equation should hold for all $Z_{k}\in U(d)$, the proposition
holds. ∎

The physical motivation of introducing the definition above, all the
microscopical degrees of freedom generating the dynamics of the total system
are unknown. Therefore, defining them as random becomes relevant. However,
assuming they are sampled from a nice probability measure will become clear in
the following theorem. In the following, we will show that when one averages
over the unitaries an effective noise model emerges.

######  Theorem 4.11.

Let $U$ be a random unitary operator sampled from a nice probability measure
$\mu$, and assume that the elements of the probe density matrix $\beta_{cc}$
are constant for all $c\neq 0$. The averaged POVM given by
$\mathbb{E}_{U\sim\mu}[\mathcal{A}^{\beta}]$ defines a noise model given by:

$\mathbb{E}_{U\sim\mu}[A^{\beta}_{i}]=\beta_{00}\,A_{i}+\frac{1-\beta_{00}}{N-1}\Big{(}1-\frac{1}{d}\operatorname{Tr}[A_{i}]\Big{)}\,I$

One can interpret the element $\beta_{00}$ of the probe
$\beta\in\mathcal{M}^{1,+}_{N+1}$ as a noise parameter.

###### Proof.

Let $\mu$ a nice probability measure on
$\mathbb{U}\Big{(}\\{A_{i}\\}_{i\in[0,N]}\Big{)}$, we recall from Proposition
4.7 that:

$A_{i}^{\beta}=\sum_{c,k=0}^{N}\beta_{ck}\,U^{*}_{ik}\,U_{ic}.$

An explicit computation of the expectation value of $\mathcal{A}^{\beta}$ on
$U\sim\mu$ yields to:

$\displaystyle\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]$
$\displaystyle=\sum_{c,k=0}^{N}\beta_{ck}\,\mathbb{E}_{U\sim\mu}[U^{*}_{id}\,U_{ic}]$
$\displaystyle=\beta_{00}\,A_{i}+\sum_{k=0}^{N}\beta_{0k}\,\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}]+\sum_{c=0}^{N}\beta_{c0}\,\mathbb{E}_{U\sim\mu}[U^{*}_{i0}\,U_{ic}]+\sum_{c,k\neq
0}^{N}\beta_{ck}\,\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{ic}]$
$\displaystyle=\beta_{00}\,A_{i}+\sum_{k=0}^{N}\beta_{0k}\,Z_{k}^{*}\,\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}]+\sum_{c=0}^{N}\beta_{c0}\,\mathbb{E}_{U\sim\mu}[U^{*}_{i0}\,U_{ic}]\,Z_{c}+\sum_{c,k\neq
0}^{N}\beta_{ck}\,Z^{*}_{k}\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{ic}]\,Z_{c}$
$\displaystyle=\beta_{00}\,A_{i}+\sum_{k=0}^{N}\beta_{0k}\,Z_{k}^{*}\,\underbrace{\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{i0}]}_{0}+\sum_{c=0}^{N}\beta_{c0}\,\underbrace{\mathbb{E}_{U\sim\mu}[U^{*}_{i0}\,U_{ic}]}_{0}\,Z_{c}+\sum_{c,k\neq
0}^{N}\beta_{ck}\,Z^{*}_{k}\underbrace{\mathbb{E}_{U\sim\mu}[U^{*}_{ik}\,U_{ic}]}_{\delta_{ck}\,\Gamma_{ic}\,I}\,Z_{c}.$

In the last line, the simplification is obtained by the use of Proposition
4.10, therefore we have the final expression given by:

$\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]=\beta_{00}\,A_{i}+\sum_{c\neq
0}^{N}\beta_{cc}\,\Gamma_{ic}\,I.$ (44)

where we have defined the constant $\Gamma_{ic}$ as:

$\Gamma_{ic}:=\frac{1}{d}\operatorname{Tr}\Big{[}\mathbb{E}_{U\sim\mu}[U^{*}_{ic}\,U_{ic}]\Big{]}.$
(45)

One can check with the definition of $\Gamma_{ic}$ that
$\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]=I$ is a POVM, from the expression of
$\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]$ the positivity condition holds, the
only remaining condition is to show that the
$\sum_{i}\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]$.

By using that $U^{*}U=I_{(N+1)d}$ we have:

$U^{*}U=I_{(N+1)d}\implies\sum_{i=0}^{N}U^{*}_{ic}\,U_{ic}=I_{d}\implies\sum_{i=0}^{N}\Gamma_{ic}=1.$
(46)

Then we have from equation (46) that:

$\sum_{i=0}^{N}\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]=\beta_{00}\,\sum_{i=0}^{N}\,A_{i}+\sum_{c\neq
0}^{N}\beta_{cc}\sum_{i=0}^{N}\,\Gamma_{ic}\,I=\operatorname{Tr}[\beta]=1.$
(47)

By assuming for all $\forall c\neq 0$, $\beta_{cc}$ are constants and do not
depend on $c$, we have that:

$\displaystyle\mathbb{E}_{U\sim\mu}[A_{i}^{\beta}]$
$\displaystyle=\beta_{00}\,A_{i}+\sum_{c\neq 0}^{N}\beta_{cc}\,\Gamma_{ic}\,I$
$\displaystyle=\beta_{00}\,A_{i}+\frac{(1-\beta_{00})}{N-1}\sum_{c\neq
0}^{N}\,\Gamma_{ic}\,I$
$\displaystyle=\beta_{00}\,A_{i}+\frac{(1-\beta_{00})}{N-1}\Big{(}1-\frac{1}{d}\operatorname{Tr}[A_{i}]\Big{)}I.$

To obtain the last equality we have used the unitary property of $U$ where we
have:

$UU^{*}=I_{(N+1)d}\implies\sum_{c=0}^{N}U_{ic}U^{*}_{ic}=I_{d}\implies\sum_{c=0}^{N}\Gamma_{ic}=1,$
(48)

and by combining the expression from equation (45),we deduce that:

$\sum_{c\neq
0}^{N}\Gamma_{ic}=1-\Gamma_{i0}=1-\frac{1}{d}\operatorname{Tr}\Big{[}\mathbb{E}_{U\sim\mu}[U^{*}_{i0}\,U_{i0}]\Big{]}=1-\frac{1}{d}\operatorname{Tr}[A_{i}].$
(49)

∎

######  Remark 4.12.

An interesting open question is to explore the possible noise models that can
be obtained by considering different probe states $\beta$ (with non-constant
diagonal) and / or different probability distributions for interaction unitary
$U$. Could the noise models from [DFK19] be obtained in this way?

##  5\. Applications to compatibility

In this section, we shall give an application of the physical noise model we
introduced in the setting of compatibility of quantum measurements. As we have
described in Section 2, one can always make incompatible measurements
compatible by adding noise. A natural notion that we shall encounter in this
section is the notion of _robustness_. Robustness captures the minimal amount
of noise needed to make incompatible measurements compatible. We shall give
the formulation of robustness in the case of three different noise models, the
uniform noise model, the depolarizing noise model, and our physical noise
model. We shall give the SDP and its dual formulation in the three types of
noise models mentioned above. We will end this section by giving an explicit
example of robustness in the case of a particular pair of measurements, using
the three noise models.

In general, deciding whether a set of quantum measurements is (in-)compatible
is a hard task, which can be formulated as a semidefinite program (SDP). These
are a type of convex optimization programs with positive semidefinite
constraints (see [BBV04] for a general introduction) that will allow us later
to give a formulation of robustness.

We recall that a noise model is given by a convex combination of a given POVM
$\mathcal{A}$ with a trivial POVM $\mathcal{T}$. We recall from Section 2 the
uniform noise and the depolarizing noise are given by:

  * •

The uniform noise model for $\alpha\in[0,1]$ is defined by:

$\mathcal{A}\to\mathcal{A}^{\alpha}:=\alpha\mathcal{A}+(1-\alpha)\mathcal{T},$

where $\mathcal{T}:=(t_{1}I,\cdots,t_{N}I)$, with $t_{i}:=1/N$ for all
$i\in[N]$.

  * •

The depolarizing noise for $\alpha\in[0,1]$ is defined by:

$\mathcal{A}\to\mathcal{A}^{\alpha}:=\alpha\mathcal{A}+(1-\alpha)\mathcal{T},$

where $\mathcal{T}:=(t_{1}I,\cdots,t_{N}I)$, with
$t_{i}:=\operatorname{Tr}A_{i}/d$ for all $i\in[N]$.

We shall also recall that our main noise model is based on an indirect
measurement process obtained in Theorem 4.11 where:

  * •

For a given probe $\beta$ where $\beta_{00}$ can be interpreted as a noise
parameter taking values in $[0,1]$,

$\mathcal{A}\to\mathcal{A}^{\beta}=\beta_{00}\mathcal{A}+(1-\beta_{00})\mathcal{T},$

and the trivial POVM is given by $\mathcal{T}=(t_{0}I,\cdots,t_{N}I)$, with
$t_{i}=\frac{1}{N-1}\Big{(}1-\frac{1}{d}\operatorname{Tr}[A_{i}]\Big{)}$ for
all $i\in[0,N]$.

Note that in the formulas above, it is the quantity $1-\alpha$, resp.
$1-\beta_{00}$, which measures the amount of noise present in the POVM
$\mathcal{A}^{\alpha}$. In the following, we shall give the definition of
_incompatibility robustness_ that computes the minimal amount of noise needed
to make two POVMs compatible, see [HMZ16, DFK19] and the references therein.

######  Definition 5.1.

For two POVMs $\mathcal{A}$ and $\mathcal{B}$, and a noise model
$\mathcal{T}$, the robustness of incompatibility of $\mathcal{A}$ and
$\mathcal{B}$ is defined by:

$\alpha^{*}:=\sup_{\alpha\in[0,1]}\\{\alpha\,:\,\mathcal{A}^{\alpha}\text{ and
}\mathcal{B}^{\alpha}\text{ are compatible}\\}.$ (50)

For a noise model given by a trivial operator $\mathcal{T}$, we can give the
SDP formulation of robustness. Therefore, we will recall the SDP and its dual
formulation for the uniform and depolarizing noise.

######  Definition 5.2.

The SDP of the robustness of two POVMs with $N$ outcomes $\mathcal{A}$ and
$\mathcal{B}$ with a uniform and depolarizing noise are given respectively by:

$\alpha^{*}_{u}:=\begin{cases}\max_{\alpha\in[0,1],\\{C_{ij}\\}}\alpha\quad\text{s.t}&\\\
\sum_{j=1}^{N}C_{ij}=\alpha A_{i}+(1-\alpha)I/N&\\\
\sum_{i=1}^{N}C_{ij}=\alpha
B_{j}+(1-\alpha)I/N&\end{cases}\quad\text{and}\quad\alpha^{*}_{d}:=\begin{cases}\max_{\alpha\in[0,1],\\{C_{ij}\\}}\alpha\quad\text{s.t}&\\\
\sum_{j=1}^{N}C_{ij}=\alpha A_{i}+(1-\alpha)\operatorname{Tr}A_{i}/d&\\\
\sum_{i=1}^{N}C_{ij}=\alpha
B_{j}+(1-\alpha)\operatorname{Tr}B_{j}/d&\end{cases},$

where the $\alpha^{*}_{u}$ and $\alpha^{*}_{d}$ stand respectively for the
case of the uniform and the depolarizing noise.

######  Proposition 5.3.

[DFK19] The dual SDP of the robustness of the case of the uniform and the
depolarising noise for two POVMs with $N$ outcomes are given respectively by:

$\displaystyle\alpha^{*}_{u}$
$\displaystyle=\begin{cases}\min_{\\{X_{i}\\}_{i\in[N]},\\{Y_{j}\\}_{j\in[N]}}1+\sum_{i=1}^{N}\operatorname{Tr}[X_{i}A_{i}]+\sum_{j=1}^{N}\operatorname{Tr}[Y_{j}B_{j}]\quad\text{s.t}&\\\
X_{i}+Y_{j}\geq 0\,\text{for all}\quad i,j\in[N]\quad\text{and}&\\\
1+\sum_{i=1}^{N}\operatorname{Tr}[X_{i}A_{i}]+\sum_{j=1}^{N}\operatorname{Tr}[Y_{j}B_{j}]\geq\sum_{i=1}^{N}\operatorname{Tr}X_{i}/N+\sum_{j=1}^{N}\operatorname{Tr}Y_{j}/N\end{cases}$
$\displaystyle\alpha^{*}_{d}$
$\displaystyle=\begin{cases}\min_{\\{X_{i}\\}_{i\in[N]},\\{Y_{j}\\}_{j\in[N]}}1+\sum_{i=1}^{N}\operatorname{Tr}[X_{i}A_{i}]+\sum_{j=1}^{N}\operatorname{Tr}[Y_{j}B_{j}]\quad\text{s.t}&\\\
X_{i}+Y_{j}\geq 0\,\text{for all}\quad i,j\in[N]\quad\text{and}&\\\
1+\sum_{i=1}^{N}\operatorname{Tr}[X_{i}A_{i}]+\sum_{j=1}^{N}\operatorname{Tr}[Y_{j}B_{j}]\geq\sum_{i=1}^{N}\operatorname{Tr}X_{i}\,\operatorname{Tr}X_{i}/d+\sum_{j=1}^{N}\operatorname{Tr}Y_{j}\,\operatorname{Tr}B_{j}/d\end{cases}$

###### Proof.

Let us consider the following Lagrangian $\mathcal{L}_{u}$ corresponding to
the optimization problem of robustness for the uniform noise given by the
primal SDP in Eq. (5.2):

$\mathcal{L}_{u}:=\alpha+\sum_{i,j=1}^{N}\Big{\langle}C_{ij},X_{ij}\Big{\rangle}-\sum_{i=1}^{N}\Big{\langle}\sum_{j=1}^{N}C_{ij}-\alpha
A_{i}-(1-\alpha)\frac{I}{N},X_{i}\Big{\rangle}-\sum_{j=1}^{N}\Big{\langle}\sum_{i=1}^{N}C_{ij}-\alpha
B_{j}-(1-\alpha)\frac{I}{N},Y_{j}\Big{\rangle}$

where $X_{ij}$, $X_{i}$ and $Y_{j}$ are positive semidefinite matrices for all
$i,j\in[N]$ encoding the different constrains of the primal optimisation
problem.

Since Slater’s condition is satisfied, the minimax duality holds, therefore we
have:

$\max_{\alpha,\\{C_{ij}\\}}\quad\inf_{\\{X_{ij}\\},\\{X_{i}\\},\\{Y_{j}\\}}\mathcal{L}_{u}=\inf_{\\{X_{ij}\\},\\{X_{i}\\},\\{Y_{j}\\}}\quad\max_{\alpha,\\{C_{ij}\\}}\mathcal{L}_{u},$
(51)

By expanding the expression of $\mathcal{L}_{u}$ and taking the maximum over
$\\{C_{ij}\\}$ and $\alpha$ we obtain:

$\max_{\\{C_{ij}\\},\alpha}\mathcal{L}_{u}=\sum_{i=1}^{N}\operatorname{Tr}X_{i}/N+\sum_{j=1}^{N}\operatorname{Tr}Y_{j}/N$

if the following constraints hold

$\displaystyle
1+\sum_{j=1}^{N}\Big{\langle}B_{j},Y_{j}\Big{\rangle}+\sum_{i=1}^{N}\Big{\langle}A_{i},X_{i}\Big{\rangle}$
$\displaystyle\geq\sum_{i=1}^{N}\operatorname{Tr}X_{i}/N+\sum_{j=1}^{N}\operatorname{Tr}Y_{j}/N\quad\text{and}$
$\displaystyle X_{ij}$ $\displaystyle=X_{i}+Y_{j}\geq 0,$

and $+\infty$ if they do not. Above we have used the Hilbert-Schmidt scalar
$\langle A,B\rangle_{\text{H.S}}=\operatorname{Tr}(A^{*}B)$. By plugging the
obtained value in the expression of
$\max_{\\{C_{ij}\\},\alpha}\mathcal{L}_{u}$ we obtain the result of the
statement.

The same computation can be explicitly done for the depolarising noise, where
it is easy to check by duality the results of the statement in the proposition
with the following Lagrangian:

$\displaystyle\mathcal{L}_{d}:=\alpha+\sum_{i,j=1}^{N}\Big{\langle}C_{ij},X_{ij}\Big{\rangle}$
$\displaystyle-\sum_{i=1}^{N}\Big{\langle}\sum_{j=1}^{N}C_{ij}-\alpha
A_{i}-(1-\alpha)\frac{\operatorname{Tr}A_{i}}{d},X_{i}\Big{\rangle}$
$\displaystyle-\sum_{j=1}^{N}\Big{\langle}\sum_{i=1}^{N}C_{ij}-\alpha
B_{j}-(1-\alpha)\frac{\operatorname{Tr}B_{j}}{N},Y_{j}\Big{\rangle}.$

∎

In the following, we shall give the SDP formulation of robustness in the case
of our physical noise model and its dual formulation.

######  Definition 5.4.

The SDP of the robustness of two POVMs with $N+1$ outcomes $\mathcal{A}$ and
$\mathcal{B}$ in the physical noise model is given by:

$\alpha^{*}_{p}:=\begin{cases}\max_{\beta_{00}\in[0,1],\\{C_{ij}\\}}\beta_{00}\quad\text{s.t}&\\\
\sum_{j=0}^{N}C_{ij}=\beta_{00}A_{i}+\frac{(1-\beta_{00})}{N-1}\Big{(}1-\frac{\operatorname{Tr}A_{i}}{d}\Big{)}I&\\\
\sum_{i=0}^{N}C_{ij}=\beta_{00}B_{j}+\frac{(1-\beta_{00})}{N-1}\Big{(}1-\frac{\operatorname{Tr}B_{j}}{d}\Big{)}I&\end{cases},$

where $\alpha^{*}_{p}$ stands for robustness in our physical model.

We give next the dual formulation.

######  Proposition 5.5.

The dual SDP of robustness in the case of our physical noise model is given
by:

$\alpha^{*}_{p}=\min_{\\{X_{i}\\}_{i\in[0,N]},\\{Y_{j}\\}_{j\in[0,N]}}1+\sum_{i=0}^{N}\operatorname{Tr}[X_{i}A_{i}]+\sum_{j=0}^{N}\operatorname{Tr}[Y_{j}B_{j}]$

with the following constraints:

$\displaystyle X_{i}+Y_{j}$ $\displaystyle\geq 0\,\text{for all}\quad
i,j\in[0,N]\,\text{and}$ $\displaystyle 1+\sum_{i=0}^{N}\langle
X_{i}A_{i}\rangle+\sum_{j=0}^{N}\operatorname{Tr}[Y_{j}B_{j}]$
$\displaystyle\geq\frac{1}{N-1}\Big{(}\sum_{i=0}^{N}\operatorname{Tr}X_{i}+\sum_{j=0}^{N}\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{j=0}^{N}\operatorname{Tr}B_{j}\,\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{i=0}^{N}\operatorname{Tr}X_{i}\,\operatorname{Tr}A_{i}\Big{)}.$

###### Proof.

Let us define the Lagrangian $\mathcal{L}_{p}$ corresponding to the
optimization problem of robustness in our main physical model given by:

$\displaystyle\mathcal{L}_{p}:=\beta_{00}+$
$\displaystyle\sum_{i,j=0}^{N}\Big{\langle}C_{ij},X_{ij}\Big{\rangle}-\sum_{i=0}^{N}\Big{\langle}\sum_{j=0}^{N}C_{ij}-\beta_{00}A_{i}-\frac{(1-\beta_{00})}{N-1}\Big{(}1-\frac{\operatorname{Tr}A_{i}}{d}\Big{)}I,X_{i}\Big{\rangle}$
$\displaystyle-\sum_{j=0}^{N}\Big{\langle}\sum_{i=0}^{N}C_{ij}-\beta_{00}B_{j}-\frac{(1-\beta_{00})}{N-1}\Big{(}1-\frac{\operatorname{Tr}B_{j}}{d}\Big{)}I,Y_{j}\Big{\rangle}.$

Due to Slater’s condition which is satisfied, the minimax duality holds,
therefore we have:

$\max_{\beta_{00},\\{C_{ij}\\}}\quad\inf_{\\{X_{ij}\\},\\{X_{i}\\},\\{Y_{j}\\}}\mathcal{L}_{p}=\inf_{\\{X_{ij}\\},\\{X_{i}\\},\\{Y_{j}\\}}\quad\max_{\beta_{00},\\{C_{ij}\\}}\mathcal{L}_{p}.$

By expanding the expression of $\mathcal{L}_{p}$ and taking the maximum of
$\\{C_{ij}\\}$ and $\beta_{00}$:

$\max_{\\{C_{ij}\\},\beta_{00}}\mathcal{L}_{p}=\frac{1}{N-1}\Big{(}\sum_{i=0}^{N}\operatorname{Tr}X_{i}+\sum_{j=0}^{N}\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{j=0}^{N}\operatorname{Tr}B_{j}\,\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{i=0}^{N}\operatorname{Tr}X_{i}\,\operatorname{Tr}A_{i}\Big{)},$

with the following constraints:

$1+\sum_{i=0}^{N}\langle
X_{i}A_{i}\rangle+\sum_{j=0}^{N}\operatorname{Tr}[Y_{j}B_{j}]\geq\frac{1}{N-1}\Big{(}\sum_{i=0}^{N}\operatorname{Tr}X_{i}+\sum_{j=0}^{N}\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{j=0}^{N}\operatorname{Tr}B_{j}\,\operatorname{Tr}Y_{j}-\frac{1}{d}\sum_{i=0}^{N}\operatorname{Tr}X_{i}\,\operatorname{Tr}A_{i}\Big{)},$

and

$X_{ij}=X_{i}+Y_{j}\geq 0\,\text{for all}\quad i,j\in[0,N],$

and $+\infty$ if the constraints above are not satisfied.

By plugging the obtained value in the expression of
$\max_{\\{C_{ij}\\},\alpha}\mathcal{L}_{p}$ we obtain the result of the
statement. ∎

Let us now compare the incompatibility robustness of a pair of measurements
using the three different noise models: uniform, depolarizing, and our
physical model. We consider an example of two POVMs on $\mathbb{C}^{3}$ with
two outcomes $\mathcal{A}=(A_{0},A_{1})$ and $\mathcal{B}=(B_{0},B_{1})$ given
by:

Figure 4. Compatibility region of the POVMs $\mathcal{A}$, $\mathcal{B}$ for
the different noise models: the uniform, the depolarizing, and our physical
noise model. On the right panel, we zoom in on the parameter region where the
noise models yield different results.

$A_{0}:=\begin{bmatrix}1/3&0&0\\\ 0&2/3&0\\\
0&0&0\end{bmatrix},\,A_{1}:=\begin{bmatrix}2/3&0&0\\\ 0&1/3&0\\\
0&0&1\end{bmatrix},$

and

$B_{0}=|f_{1}\rangle\langle f_{1}|+|f_{2}\rangle\langle
f_{2}|\quad\text{and}\quad B_{1}=|f_{3}\rangle\langle f_{3}|,$

where $\\{\ket{f_{i}}\\}$ for $i\in[3]$ are the columns of the Fourier matrix

$F_{3}=\frac{1}{\sqrt{3}}\begin{bmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\end{bmatrix},$

with $\omega=\exp(2\pi\mathrm{i}/3)$. We plot the _compatibility region_ of
the POVMs defined above in Figure 4. Recall that the compatibility region
[BN18] is a generalization of the incompatibility robustness from Eq. (50):

$\Gamma(\mathcal{A},\mathcal{B}):=\\{(p,q)\in[0,1]^{2}\,:\,\mathcal{A}^{p}\text{
and }\mathcal{B}^{q}\text{ are compatible}\\}.$

Notice in the formula above that the region $\Gamma(\mathcal{A},\mathcal{B})$
depends on the noise model used to define $\mathcal{A}^{p}$ and
$\mathcal{B}^{q}$. The compatibility regions corresponding to the different
noise models are approximately the same; small differences can be observed for
noise parameters in the interval $[0.7,1]$. One can notice that with our
physical noise model, we require _less_ noise in order to make the POVMs
$\mathcal{A}$ and $\mathcal{B}$ compatible with the other noise models
considered in the literature.

Acknowledgements. F.L. would like to thank Denis Rochette for help with
Mathematica code. The authors were supported by the ANR project ESQuisses,
grant number ANR-20-CE47-0014-01 as well as by the PHC program _Star_
(Applications of random matrix theory and abstract harmonic analysis to
quantum information theory). I.N. has also received support from the ANR
project STARS, grant number ANR-20-CE40-0008. C.P is also supported by the ANR
projects Q-COAST ANR- 19-CE48-0003, “Quantum Trajectories”
ANR-20-CE40-0024-01, and “Investissements d’Avenir” ANR-11-LABX-0040 of the
French National Research Agency.

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