# Silicon Pixel Sensor R&D for the CLIC Tracking Detector

J. Kröger,11footnotetext: Corresponding author.

###### Abstract

The physics aims at the proposed high-energy $e^{+}e^{-}$ collider CLIC pose
challenging demands on the performance of the detector system. Precise hit-
time tagging, an excellent spatial resolutions, and a low mass are required
for the vertex and tracking detectors. To meet these requirements, an all-
silicon vertex and tracking detector system is foreseen, for which a broad R&D
programme on a variety of novel silicon detector technologies is being
pursued. For the ultra-low mass vertex detector, different hybrid technologies
with innovative sensor concepts and interconnection techniques are explored.
For the large-scale tracking detector, the focus of the R&D lies on monolithic
HV-MAPS and HR-CMOS technologies. This contribution gives an overview of the
ongoing activities with a focus on monolithic technologies for the CLIC
tracking detector. Recent results from laboratory and test-beam measurement
campaigns of the ATLASpix_Simple and the CLICTD sensor prototypes are
presented.

##  1 Introduction

The Compact Linear Collider (CLIC) [1, 2] is a proposed high-energy electron-
positron collider based at CERN in Geneva, Switzerland, for the era beyond the
High-Luminosity Large Hadron Collider (HL-LHC) [3]. Its physics goals comprise
top quark and Higgs boson measurements with unprecedented precision, as well
as searches for Physics Beyond Standard Model (BSM). It is proposed to be
constructed in three energy stages reaching a centre-of-mass energy of up to
$3\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ and a total length of
$50\text{\,}\mathrm{k}\mathrm{m}$ at the final stage.

An innovate two-beam acceleration scheme is being developed for the CLIC
accelerator [1]. A low-energy high-current drive beam transfers its energy to
the main beam with high energy and low current in normal-conducting cavities,
the so-called two-beam modules. This acceleration technique allows for
gradients of more than $100\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{m}$.

The proposed physics aims for CLIC [4] pose challenging demands on the
performance of the detector system [5]. For pile-up rejection and to mitigate
the impact of beam-induced background, a precise hit-time tagging with a
resolution of $\sim 5\text{\,}\mathrm{n}\mathrm{s}$ is required for the vertex
and tracking detectors. In addition, an ultra-low material budget of $\sim
0.2\text{\,}\mathrm{\char 37\relax}$ of a radiation length ($X_{0}$) per layer
for the vertex detector and $\sim 1\text{\,}\mathrm{\char 37\relax}$ $X_{0}$
per layer for the tracking detector are required while achieving a single-
plane spatial resolution of a few micrometers. The detector requirements are
summarised in Table 1.

Table 1: Summary of the requirements for the CLIC vertex and tracking detectors [5]. | Vertex Detector | Tracking Detector
---|---|---
detector area | $0.84\text{\,}\mathrm{m}\mathrm{{}^{2}}$ | $137\text{\,}\mathrm{m}\mathrm{{}^{2}}$
timing resolution | $\sim 5\text{\,}\mathrm{n}\mathrm{s}$ | $\sim 5\text{\,}\mathrm{n}\mathrm{s}$
hit detection efficiency |  $99.7-99.9$ $\text{\,}\mathrm{\char 37\relax}$ |  $99.7-99.9$ $\text{\,}\mathrm{\char 37\relax}$
maximum pixel size | $(25\times 25)~{}$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{{}^{2}}$$ | $30-50~{}$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$\times 1-10~{}$\text{\,}\mathrm{m}\mathrm{m}$$
material budget per layer |  $\sim 0.2\text{\,}\mathrm{\char 37\relax}$ $X_{0}$ |  $1-1.5$ $\text{\,}\mathrm{\char 37\relax}$ $X_{0}$
average power dissipation | $<50\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$ | $<150\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{2}}$

To meet the stringent detector requirements, an all-silicon vertex and
tracking detector system is foreseen as the central part of the CLIC detector
model (CLICdet) [6]. In order to identify suitable technologies, a broad R&D
programme on a variety of novel silicon detector technologies is being pursued
[5].

For the ultra-low mass vertex detector, different small pitch
($25\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) hybrid technologies with
innovative sensor concepts are explored [7]. A dedicated readout chip called
CLICpix2 [8] has been developed in $65\text{\,}\mathrm{n}\mathrm{m}$ CMOS
technology and bump-bonded to thin planar active-edge sensors [9]. To overcome
the challenges of the fine-pitch bump-bonding, alternative interconnection
techniques such as capacitive coupling and anisotropic conductive films are
explored [10]. In addition, Silicon-On-Insulator (SOI) test chips are also
under investigation [11].

Monolithic CMOS technologies are promising for the large-area tracking
detector due to their cost effectiveness and large-scale production
capabilities. Different sensors with large and small collection electrodes are
under investigation and recent results from the ATLASpix_Simple and the CLICTD
test chips are presented in Section 2 of this paper.

In order to predict and further optimise the performance of the various
prototype technologies, a fast and versatile simulation tool, Allpix Squared
[12], has been developed. It allows for high statistics Geant4-based [13]
Monte Carlo simulations, for which detailed electric field maps from TCAD
finite-element simulations can be imported for a highly accurate description
of the sensor properties.

To allow fast prototyping cycles, a versatile data acquisition system called
Caribou [14], has been developed to be used in both laboratory measurements
and test-beam campaigns. It reduces the effort of taking new pixel sensor
prototypes into operation by maximizing the fraction of common hardware and
software blocks of the system while keeping chip-specific components to a
minimum.

In addition, the flexible and modular test-beam data reconstruction framework
Corryvreckan [15] has been developed. It fulfils the requirements of a complex
offline event building in data taking environments combining detector
subsystems with different readout architectures.

##  2 Depleted monolithic CMOS Sensors for the CLIC Tracking Detector

For the tracking detector with its large area of $\sim
140\text{\,}\mathrm{m}\mathrm{{}^{2}}$ fully monolithic CMOS technologies are
considered the best choice due to their cost efficiency and large-scale
production capabilities. In addition, they allow for a reduced material budget
compared to hybrid technologies.

A high voltage CMOS sensor with a large collection diode, the ATLASpix_Simple
[16], has been characterised both in laboratory and test-beam measurements and
initial results have been presented in [5]. Recent results focusing on a
comparison of different substrate resistivities are presented in Section 2.1.

3D TCAD simulations and previous test results have led to the development of
innovative design concepts for CMOS sensors with a small collection electrode
[17]. These have been implemented in various prototype chips targeting both
CLIC and other future projects. The CLICTD chip [18] has recently been
produced using a modified $180\text{\,}\mathrm{n}\mathrm{m}$ CMOS imaging
process implemented on a high-resistivity epitaxial layer. The design includes
an innovative sub-pixel segmentation scheme, and first results are presented
in Section 2.2.

###  2.1 The ATLASpix_Simple HV-MAPS Prototype

The ATLASpix_Simple sensor is a High-Voltage Monolithic Active Pixel Sensor
(HV-MAPS), which was designed as a candidate for the ATLAS ITk upgrade [16]
but also targets the requirements of the CLIC tracking detector.

Figure 1: Photograph of an ATLASpix sensor with its three submatrices glued
and wire-bonded to a printed circuit board.

Figure 2: Schematic drawing of the HV-MAPS concept [19] (modified).

It was produced in a commercial $180\text{\,}\mathrm{n}\mathrm{m}$ HV-CMOS
process on wafers with substrate resistivities between 20 and
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ and features an
active matrix consisting of 25 columns and 400 rows of pixels with a pitch of
$130\times 40$
$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{{}^{2}}$. A photograph
is shown in Figure 2.

As depicted in Figure 2, each pixel consists of a large collection electrode,
which is placed in a deep N-well on a p-substrate and houses the in-pixel
electronics comprising charge-sensitive amplifier and comparator. After
production, the sensors can be thinned down to
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ by removing undepleted bulk
material from the backside. A high bias voltage of up to
$\mathcal{O}$($100\text{\,}\mathrm{V}$) leads to a large depleted volume with
a high electric field resulting in a fast charge collection via drift.

For each hit, the time-of-arrival (ToA) is recorded with
$10\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$ a precision and a binning of
$16\text{\,}\mathrm{n}\mathrm{s}$. In addition, the signal charge is
determined with a time-over-threshold (ToT) measurement with a
$6\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$ precision. The pixel matrix is read
out in a data-driven column-drain scheme.

It has previously been shown that the measured spatial resolution of the
ATLASpix_Simple in column and row direction is approximately the pixel
pitch$/\sqrt{12}$. This corresponds to a binary resolution and is in agreement
with the fact that almost no charge is shared with neighbouring pixels [5].

####  2.1.1 Hit Detection Efficiency for Different Substrate resistivities

(a) Threshold dependence at a bias voltage of $-50\text{\,}\mathrm{V}$.

(b) Bias voltage dependence at a detection threshold of
$650\text{\,}\mathrm{e}^{-}$.

Figure 3:  Hit detection efficiency of ATLASpix_Simple samples with different
substrate resistivities and thicknesses.

The hit detection efficiency and its dependence on the detection threshold as
well as the bias voltage have also been studied in DESY-II test-beam
campaigns. The results are shown in Figure 3.

As can be seen in Figure 3(a), the efficiency drops with an increasing
detection threshold. Comparing the different substrate resistivities, it drops
significantly slower for the
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ sample than for
the lower substrate resistivities. As a consequence, the
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ sample can be
operated for a much larger range of the threshold while maintaining a high
detection efficiency.

This behaviour is expected because a higher substrate resistivity leads to a
larger depleted volume for a given bias voltage as described in Section 2.1.2.
Consequently, the signal induced by the generated charge carriers of an
incident particle is larger so that it can exceed the detection threshold for
higher thresholds before becoming inefficient.

Figure 3(b) shows how the hit detection efficiency changes with the applied
bias voltage. It can be observed that the efficiency rises with increasing
bias voltage. For the lower resistivities, the efficiency saturates only
slowly whereas it already reaches a stable plateau at a bias voltage around
$10\text{\,}\mathrm{V}$ for the
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$. Therefore, the
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ sample becomes
fully efficient at a much lower bias voltage.

Also this observation meets the expectation since a higher substrate
resistivity leads to a larger depleted volume for a given bias voltage. In
return, the $200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ sample
reaches a given depleted volume, and therefore a given signal size, for a
smaller applied bias voltage. Consequently, it becomes fully efficient at a
lower bias voltage.

It should be noted that the sensor thickness is not expected to have an
influence on the hit detection efficiency because by the thinning only
undepleted bulk material is removed from the back of the sensor and the
contribution to the charge collection from the undepleted bulk is
insignificant.

####  2.1.2 Timing Performance for Different Substrate resistivities

The timing performance for the different substrate resistivities of the
ATLASpix_Simple has been determined in measurement campaigns at the DESY-II
test-beam facility [20] using an EUDET-type beam telescope [21] and an
additional Timepix3 plane [22] as a timing reference.

(a) $20\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$,
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

(b) $80\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$,
$62\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

(c) $200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$,
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

Figure 4:  Track-cluster time residual versus the seed pixel ToT of ATLASpix
samples for different resistivities and thicknesses at a detection threshold
of $650\text{\,}\mathrm{e}^{-}$ and a bias voltage of $-50\text{\,}\mathrm{V}$
before applying any correction.

Figure 4 shows 2D histograms of the seed pixel ToT of a cluster on the
ATLASpix_Simple plotted against the time residual of the reference track and
the associated cluster on the ATLASpix_Simple. The reference track timestamp
has been assigned by a Timepix3 plane [22] with a precision of
$1.56\text{\,}\mathrm{n}\mathrm{s}$ [23]. The seed pixel of a cluster is the
one with the earliest timestamp, which is used to define the timestamp of the
whole cluster. The ToT is a measure of the signal size, i.e. a large ToT
corresponds to a large signal size. A clear dependence of the peak position of
the time residual on the seed pixel ToT can be observed for the low substrate
restivity of $\sim 20\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$
(see Figure 4(a)). This effect, referred to as timewalk, is strongly reduced
with increasing substrate resistivity (see Figures 4(b) and 4(c)).

The dependence on the substrate resistivity can be explained as follows. A
higher substrate resistivity leads to a larger depleted volume for a given
bias voltage. As a consequence, more charge carriers can move freely and the
induced signal is higher. This means that the crossing of the detection
threshold is steeper and less affected by jitter. In addition, a larger
electric field leads to a faster charge collection due to a shorter drift time
of the charge carriers. Hence, the timewalk effect is expected to be less
pronounced.

Figure 5:  Comparison of the track-cluster time residual for ATLASpix_Simple
sensors with different substrate resistivities and thicknesses after row-
dependent and timewalk correction.

A correction for a row-dependent signal delay as well as a timewalk correction
of the cluster timestamp have been applied offline. The resulting time
residuals are shown in Figure 5 for the three investigated substrate
resistivities at equal operating conditions with a threshold of around
$650\text{\,}\mathrm{e}^{-}$ and a bias voltage of $-50\text{\,}\mathrm{V}$.
Gaussian fits have been performed to determine the time resolution as the
standard deviation $\sigma$ of the fit function. The results are summarized in
Table 2.

As in Figure 4, a clear dependence on the substrate resistivity is observed
and can be explained in the same way. Hence, the timewalk effect is expected
to be less pronounced and the timing resolution is better for a higher
substrate resistivity. The overall best result of $\sigma\sim 6.8$
$\text{\,}\mathrm{n}\mathrm{s}$ was obtained with a
$200\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$ sample at a
threshold of $480\text{\,}\mathrm{e}^{-}$ and a bias voltage of
$-50\text{\,}\mathrm{V}$.

It should be noted that all measured time residuals contain a contribution of
about $1.56\text{\,}\mathrm{n}\mathrm{s}$ from the time resolution of the
reference track timestamp.

Table 2: Comparison of the time resolutions for ATLASpix_Simple samples with different substrate resistivities at a threshold of $\sim 650\text{\,}\mathrm{e}^{-}$ and a bias voltage of $-50\text{\,}\mathrm{V}$. The quoted uncertainties correspond to the errors on the fit. substrate resistivity [$\text{\,}\mathrm{\SIUnitSymbolOhm}\mathrm{c}\mathrm{m}$] | thickness [$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$] |  $\sigma$ [$\text{\,}\mathrm{n}\mathrm{s}$]
---|---|---
20 | 100 | $13.3\pm 0.04$
80 | 62 | $9.4\pm 0.03$
200 | 100 | $8.3\pm 0.02$

###  2.2 The CLICTD HR-CMOS Prototype

Figure 6: Photograph of a CLICTD sensor glued and wire-bonded to a printed
circuit board.

The CLICTD sensor has been designed to meet the requirements of the CLIC
tracking detector. It is fabricated in a modified CMOS imaging process, which
has been optimised with the help of 3D TCAD studies [17]. The chip features a
matrix of $128\times 128$ pixels with a pitch of $37.5\times 30$
$\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{{}^{2}}$. A photograph
is shown in Figure 6. In an innovative readout segmentation scheme, $8\times
1$ pixels each are combined into a single readout channel with a combined
8-bit ToA measurement with $10\text{\,}\mathrm{n}\mathrm{s}$ binning and a
5-bit ToT measurement. This scheme allows to save space for digital circuitry
while maintaining the small-collection electrode design at a low pixel pitch
[18].

The sensor has been characterised both in laboratory and test-beam measurement
campaigns.

####  2.2.1 Hit Detection Efficiency

(a) Hit detection efficiency for a threshold range from 150 to
$2500\text{\,}\mathrm{e}^{-}$.

(b) Hit detection efficiency for a threshold range from 150 to
$700\text{\,}\mathrm{e}^{-}$.

Figure 7:  Hit detection efficiency measurements for a CLICTD sample. Figure
8: In-pixel hit detection efficiency of a CLICTD sample for a threshold of
$1500\text{\,}\mathrm{e}^{-}$.

Measurements at the DESY-II test-beam facility [20] show that the CLICTD
sensor can be operated fully efficiently, i.e. with a hit detection efficiency
above $99.8\text{\,}\mathrm{\char 37\relax}$ for threshold values below
$400\text{\,}\mathrm{e}^{-}$, as can be seen in Figures 7(a) and 7(b). It has
been measured down to a threshold of $150\text{\,}\mathrm{e}^{-}$, where the
sensor can be operated without noise hits. At thresholds above
$400\text{\,}\mathrm{e}^{-}$, the hit detection efficiency drops because the
induced signal of the collected charge is less likely to cross the detection
threshold.

As the in-pixel distribution of the hit detection efficiency in Figure 8
reveals, the efficiency remains high in the centre of a pixel and drops
towards edges and especially the corners. This is explained by an increased
amount of charge sharing into the neighbouring cells for particles incident
close to the edges and corners of the sensor. As a consequence, the collected
charge per pixel is smaller and the induced signal is less likely to cross the
detection threshold.

####  2.2.2 Timing Performance

The time resolution of the CLICTD sensor can be deduced from the width of the
time residual between the track timestamp and the CLICTD cluster timestamp,
which is defined as the earliest pixel timestamp within the cluster.

Figure 9(a), which shows the track-cluster time residual plotted against the
seed pixel ToT of the cluster, reveals a clear timewalk effect. A timewalk
correction can be applied offline to improve the time resolution. The result
is shown in Figure 9(b). A Gaussian fit to the time residuals results in a
time resolution of $\sigma\sim 6.3$ $\text{\,}\mathrm{n}\mathrm{s}$.

(a) Track-cluster time residual vs. seed pixel ToT of the CLICTD.

(b) Track-cluster time residual after timewalk correction.

Figure 9:  Timing performance measurements for the CLICTD.

##  3 Summary and Outlook

The experimental conditions at the proposed linear $e^{+}e^{-}$ collider CLIC
pose stringent requirements on the detector performance. To meet these
requirements, an all-silicon vertex and tracking detector system is foreseen,
for which a diverse R&D programme is being pursued in order to identify
suitable technologies. As part of these efforts, hardware and software tools
for simulation, data acquisition, as well as test-beam analysis have been
developed within the CLICdp collaboration.

For the ultra-low mass vertex detector, different small pitch hybrid
technologies with innovative interconnection techniques are under
investigation. For the large-scale tracking detector, the focus of the R&D
lies on depleted monolithic CMOS technologies. Various sensor prototypes, such
as the ATLASpix_Simple and the CLICTD, are being characterized in laboratory
and test-beam measurement campaigns.

Samples of the ATLASpix_Simple HV-MAPS test chip with different substrate
resistivies have been compared in view of the timing performance as well as
the hit detection efficiency. The ATLASpix_Simple reaches hit detection
efficiencies above $99.8\text{\,}\mathrm{\char 37\relax}$ and a timing
resolution of $\sim 6.8\text{\,}\mathrm{n}\mathrm{s}$. A high substrate
resistivity is essential both for an excellent timing behaviour as well as a
larger operating window maintaining a high hit detection efficiency.

The CLICTD HR-CMOS sensor prototype has been designed to explore a novel
readout segmentation scheme. A hit detection efficiency above
$99.8\text{\,}\mathrm{\char 37\relax}$ and a time resolution as good as $\sim
6.3\text{\,}\mathrm{n}\mathrm{s}$ have been measured.

In summary, the presented results show that HV-MAPS and HR-CMOS are promising
technologies and have a large potential to be used in the CLIC tracking
detector. Most of the detector requirements are fulfilled by the test sensors
and further prototypes with newer process technologies and improved designs
are planned to be fabricated and tested in the future.

## Acknowledgments

This work has been sponsored by the Wolfgang Gentner Programme of the German
Federal Ministry of Education and Research (grant no. 05E15CHA).

The measurements leading to these results have been performed at the Test Beam
Facility at DESY Hamburg (Germany), a member of the Helmholtz Association
(HGF).

We would like to gratefully acknowledge CERN and their accelerator staff for
the reliable test-beam operation. This project has received funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 654168. It benefited from services provided by the ILC Virtual
Organisation, supported by the national resource providers of the EGI
Federation and was done using resources provided by the Open Science Grid,
which is supported by the National Science Foundation and the U.S. Department
of Energy’s Office of Science.

## References

  * [1] M. Aicheler, P. Burrows, M. Draper et al., eds., _A Multi-TeV Linear Collider Based on CLIC Technology: CLIC Conceptual Design Report_.  CERN Yellow Reports: Monographs. CERN, 2012, 10.5170/CERN-2012-007.
  * [2] P. Burrows, N. Lasheras, L. Linssen et al., eds., _The Compact Linear Collider (CLIC) - 2018 Summary Report_.  CERN Yellow Reports: Monographs. CERN, 2018, 10.23731/CYRM-2018-002.
  * [3] G. Apollinari, I. Béjar Alonso, O. Brüning et al., _High-Luminosity Large Hadron Collider (HL-LHC): Technical Design Report V. 0.1_.  CERN Yellow Reports: Monographs. CERN, 2017, 10.23731/CYRM-2017-004.
  * [4] P. Roloff, R. Franceschini, U. Schnoor and A. Wulzer, _The Compact Linear e +e- Collider (CLIC): Physics Potential_, _2018_ , [1812.07986].
  * [5] D. Dannheim, K. Krüger, A. Levy et al., eds., _Detector technologies for CLIC_.  CERN Yellow Reports: Monographs. CERN, 2019, 10.23731/CYRM-2019-001.
  * [6] D. Arominski, J.-J. Blaising, E. Brondolin et al., _A detector for CLIC: main parameters and performance_ , _2018_ , [1812.07337].
  * [7] A. Velyka and H. Jansen, _Development and simulations of Enhanced Lateral Drift Sensors_ , _JINST_ 14 (2019) C05005, [1905.03119].
  * [8] E. Santin, P. Valerio and A. Fiergolski, _CLICpix2 User’s Manual_ , _https://edms.cern.ch/document/1800546/1_ (2017) .
  * [9] M. Williams, _R &D for the CLIC vertex and tracking detectors_, _JINST_ 15 (2020) C03045–C03045.
  * [10] S. Spannagel, _Silicon Vertex & Tracking Detectors for the Compact Linear Collider_, _Pos_ Vertex2019 (2019) 044, [2002.10169].
  * [11] R. Bugiel et al., _Test-beam results of a SOI pixel-detector prototype_ , _Nucl. Instrum. Meth._ A901 (2018) 173–179.
  * [12] S. Spannagel et al., _Allpix squared: A modular simulation framework for silicon detectors_ , _Nucl. Instrum. Meth._ A901 (2018) 164 – 172, [1806.05813].
  * [13] S. Agostinelli et al., _Geant4 – a simulation toolkit_ , _Nucl. Instrum. Meth._ A506 (2003) 250 – 303.
  * [14] T. Vanat, _Caribou – A versatile data acquisition system_ , _PoS_ TWEPP2019 (2020) 100.
  * [15] J. Kröger, S. Spannagel and M. Williams, _User Manual for the Corryvreckan Test Beam Data Reconstruction Framework, Version 1.0_ , _2019_ , [1912.00856].
  * [16] I. Perić et al., _A high-voltage pixel sensor for the ATLAS upgrade_ , _Nucl. Instrum. Meth._ A924 (2019) 99–103.
  * [17] M. Munker et al., _Simulations of CMOS pixel sensors with a small collection electrode, improved for a faster charge collection and increased radiation tolerance_ , _JINST_ 14 (2019) C05013.
  * [18] I. Kremastiotis, R. Ballabriga and N. Egidos, _Design of a monolithic HR-CMOS sensor chip for the CLIC silicon tracker_ , _PoS_ TWEPP2018 (2019) 072.
  * [19] I. Perić, _A novel monolithic pixelated particle detector implemented in high-voltage CMOS technology_ , _Nucl. Instrum. Meth._ A582 (2007) 876–885.
  * [20] R. Diener et al., _The DESY II test beam facility_ , _Nucl. Instrum. Meth_ A922 (2019) 265–286.
  * [21] H. Jansen et al., _Performance of the EUDET-type beam telescopes_ , _EPJ Tech. Instrum._ 3 (2016) 7, [1603.09669].
  * [22] T. Poikela et al., _Timepix3: a 65K channel hybrid pixel readout chip with simultaneous ToA/ToT and sparse readout_ , _JINST_ 9 (2014) C05013.
  * [23] F. Pitters et al., _Time resolution studies of Timepix3 assemblies with thin silicon pixel sensors_ , _JINST_ 14 (2019) P05022, [1901.07007].


$\tilde{\Phi}(A^{(N)}):=\Pi_{\rm
alg}^{(N+M)}\left(A^{(N)}\otimes\mathbf{1}^{(M)}\right).$ (73)

Then this map is not a valid embedding. Namely, it is not in general
multiplicative, i.e. there exist $A,B\in\mathcal{A}_{\rm alg}^{(N)}$ with
$\tilde{\Phi}(AB)\neq\tilde{\Phi}(A)\tilde{\Phi}(B)$.

###### Proof.

A tedious but straightforward calculation yields

$\displaystyle\tilde{\Phi}\left(|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|\right)$
$\displaystyle=$
$\displaystyle\frac{1}{|\mathcal{G}|}\sum_{g\in\mathcal{G}}\sum_{\mathbf{g}\in\mathcal{G}^{M}}|\mathbf{h},\mathbf{g};\mathbf{1}\rangle\langle\mathbf{j},g\mathbf{g};\mathbf{1}|$
(74)
$\displaystyle+\frac{1}{|\mathcal{G}|}\delta_{\mathbf{h},\mathbf{j}}\sum_{\mathbf{g}\in\mathcal{G}^{M}}\Pi_{\mathbf{h},\mathbf{g};\chi\neq\mathbf{1}}.$

But then, for $\mathbf{h}\neq\mathbf{j}$, we obtain

$\tilde{\Phi}(|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|)\tilde{\Phi}(|\mathbf{j};\mathbf{1}\rangle\langle\mathbf{h};\mathbf{1}|)\neq\tilde{\Phi}(|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{h};\mathbf{1}|).$

∎

A similar argument applies if we try to embed $\mathcal{A}_{\rm inv}^{(N)}$
into $\mathcal{A}_{\rm inv}^{(N+M)}$: the analog of the above construction,
with $\Pi_{\rm alg}^{(N+M)}$ replaced by $\Pi_{\rm inv}^{(N+M)}$, does not
yield a valid embedding.

###  4.2 A class of invariant traces

For every embedding $\Phi^{U}$ of Lemma 28, we obtain a corresponding
“invariant trace”:

######  Lemma 30.

For every
$U=\bigoplus_{\mathbf{h}\in\mathcal{G}^{N-1}}U_{g(\mathbf{h})}^{\otimes
N}\in\mathcal{U}_{\rm sym}$, define a corresponding “invariant trace” ${\rm
Trinv}^{U}:=\left(\Phi^{U}\right)^{\dagger}$. It is trace-preserving, and it
maps invariant operators to invariant operators. In particular, ${\rm
Trinv}^{U}_{(M)}\left(\mathcal{A}_{\rm alg}^{(N+M)}\right)=\mathcal{A}_{\rm
alg}^{(N)}$, and can be explicitly written in the following form:

${\rm
Trinv}^{U}_{(M)}\,\rho=\sum_{\mathbf{h},\mathbf{j}\in\mathcal{G}^{N-1}}|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|\sum_{\mathbf{g}\in\mathcal{G}^{M}}\langle\mathbf{h},g(\mathbf{h})^{-1}\mathbf{g};\mathbf{1}|\rho|\mathbf{j},g(\mathbf{j})^{-1}\mathbf{g};\mathbf{1}\rangle+\sum_{\mathbf{h}\in\mathcal{G}^{N-1}}\Pi_{\mathbf{h};\chi\neq\mathbf{1}}\sum_{\mathbf{g}\in\mathcal{G}^{M}}\frac{{\rm
tr}(\Pi_{\mathbf{h},\mathbf{g};\chi\neq\mathbf{1}}\rho)}{|\mathcal{G}|-1}.$

We omit the straightforward proof.

Since the different invariant traces formalize different ways to embed the
two-particle observables into the three-particle observables, it is clear that
the answer to the question raised at the beginning of this section will depend
on the embedding. In other words, the question of whether the phase $\theta$
is accessible on the first two particles depends on the operational definition
of how to access the first two particles within the _total_ three-particle
Hilbert space.

To illustrate this fact, let us apply two different invariant traces to the
paradox. In contrast to the usual partial trace, every invariant trace yields
identical results when applied to the equivalent states $\Psi$ and
$\Psi^{\prime}$ in Eqs. (56) and (57) (hence the name “invariant”). That is,

${\rm Trinv}_{3}^{U}|\Psi\rangle\langle\Psi|={\rm
Trinv}_{3}^{U}|\Psi^{\prime}\rangle\langle\Psi^{\prime}|$ (75)

for all $U\in\mathcal{U}_{\rm sym}$. However, different $U$ yield different
results. For example, consider the invariant trace ${\rm Trinv}^{\mathbf{1}}$
which is the adjoint of the embedding $\Phi^{(1)}$ that we have constructed in
Subsection 4.1. A straightforward calculation gives

$\displaystyle{\rm Trinv}^{\mathbf{1}}_{3}|\Psi\rangle\langle\Psi|$
$\displaystyle=$ $\displaystyle\frac{1}{2n}|h;\mathbf{1}\rangle\langle
h;\mathbf{1}|+\frac{1}{2n}|j;\mathbf{1}\rangle\langle j;\mathbf{1}|$
$\displaystyle+\frac{1}{2n}\Pi_{h;\chi\neq\mathbf{1}}+\frac{1}{2n}\Pi_{j;\chi\neq\mathbf{1}}$
$\displaystyle=$ $\displaystyle\Pi_{\rm inv}^{(2)}\left[|e\rangle\langle
e|_{1}\otimes\left(\frac{1}{2}|h\rangle\langle h|+\frac{1}{2}|j\rangle\langle
j|\right)_{2}\right],$

Up to observational equivalence, we hence get a mixed state

${\rm Trinv}^{\mathbf{1}}_{3}|\Psi\rangle\langle\Psi|\sim|e\rangle\langle
e|_{1}\otimes\left(\frac{1}{2}|h\rangle\langle h|+\frac{1}{2}|j\rangle\langle
j|\right)_{2}.$

In particular, the phase $\theta$ has disappeared. This is not surprising:
ultimately, ${\rm Trinv^{\mathbf{1}}}$ amounts to taking the usual partial
trace in the representation of the state relative to particle $1$, i.e. of the
state $|\Psi^{\prime}\rangle$ of Eq. (57).

On the other hand, consider the symmetry transformation $U$ from Example 20
which transforms to the center of mass. Like in Ref. [50], let us choose
$m_{1}$ and $m_{2}$ such that $m_{1}a=m_{2}b$. Using Lemma 30, It is clear
that

${\rm
Trinv}_{3}^{U}\left(\Pi_{\mathbf{h};\chi\neq\mathbf{1}}\right)=\Pi_{h_{1};\chi\neq\mathbf{1}}$
(76)

and similarly for $\mathbf{h}$ replaced by $\mathbf{j}$. Furthermore,

$\displaystyle{\rm
Trinv}_{3}^{U}(|\mathbf{l};\mathbf{1}\rangle\langle\mathbf{p};\mathbf{1}|)$
$\displaystyle=$
$\displaystyle\sum_{h,j\in\mathbb{Z}_{n}}|h;\mathbf{1}\rangle\langle
j;\mathbf{1}|\sum_{g\in\mathbb{Z}_{n}}\langle
h,g+\left\lfloor\frac{m_{2}}{m}h\right\rfloor;\mathbf{1}|\mathbf{l};\mathbf{1}\rangle\langle\mathbf{p};\mathbf{1}|j,g+\left\lfloor\frac{m_{2}}{m}j\right\rfloor;\mathbf{1}\rangle$
(77) $\displaystyle=$ $\displaystyle|l_{1};\mathbf{1}\rangle\langle
p_{1};\mathbf{1}|\delta_{l_{2}-\lfloor\frac{m_{2}}{m}l_{1}\rfloor,p_{2}-\lfloor\frac{m_{2}}{m}p_{1}\rfloor}$

for all $\mathbf{l},\mathbf{p}\in\mathbb{Z}_{n}^{2}$. Since ${\rm
Trinv}_{3}^{U}|\Psi\rangle\langle\Psi|={\rm Trinv}_{3}^{U}\left(\Pi_{\rm
alg}(|\Psi\rangle\langle\Psi|)\right)$, we can expand $\Pi_{\rm
alg}(|\Psi\rangle\langle\Psi|)$ into basis elements and apply the above
equations. As a result, this yields exactly Eq. (55) (since $a$ is an integer,
it turns out that we can ignore all $\lfloor\cdot\rfloor$). That is,

${\rm Trinv}_{3}^{U}\left(|\Psi\rangle\langle\Psi|\right)=\Pi_{\rm
inv}\left(|\psi\rangle\langle\psi|\right)\sim|\psi\rangle\langle\psi|.$ (78)

That is, up to observational equivalence, we obtain the original pure
alignable state $|\psi\rangle$ of Eq. (53). Moreover, in this observational
equivalence class, $|\psi\rangle$ is _unique up to symmetry equivalence_ :
namely, if there is another pure alignable state $|\psi^{\prime}\rangle$ with
${\rm
Trinv}_{3}^{U}(|\Psi\rangle\langle\Psi|)\sim|\psi^{\prime}\rangle\langle\psi^{\prime}|$,
then $\psi\sim\psi^{\prime}$. But due to Lemma 22, this implies that
$\psi\simeq\psi^{\prime}$. Thus, in particular, there is (up to a global
phase) a unique representation of this state relative to the $i$-th particle,
$|\psi\rangle\simeq|e\rangle_{i}\otimes|\varphi\rangle_{\bar{i}}$. According
to Eq. (54), it is

$|\varphi\rangle_{\bar{1}}=\frac{1}{\sqrt{2}}\left(|a+b\rangle_{2}+e^{i\theta}|-a-b\rangle_{2}\right)$
(79)

relative to the first particle. Thus, under the “center of mass” relational
trace ${\rm Trinv}^{U}$, the phase $\theta$ survives, in contrast to the
result for ${\rm Trinv}^{\mathbf{1}}$.

###  4.3 Definition of the relational trace

Recalling the invariant algebra inclusions $\mathcal{A}_{\rm
phys}\subset\mathcal{A}_{\rm alg}\subset\mathcal{A}_{\rm inv}$, we have thus
far focused on constructing an invariant trace for $\mathcal{A}_{\rm alg}$,
since by Lemma 26, it is the smallest algebra containing the invariant part of
all alignable states and observables. However, in the previous subsection, we
have seen that there does not exist a unique invariant trace on it, and the
same conclusion applies to $\mathcal{A}_{\rm inv}$. We will now show that, by
contrast, there _does_ exists a natural embedding of the algebra
$\mathcal{A}_{\rm phys}^{(N)}$ of relational $N$-particle operators into the
algebra $\mathcal{A}_{\rm phys}^{(N+M)}$ of relational $(N+M)$-particle
operators. This will also lead to a natural definition of an invariant trace
in terms of relational states.

This trace, which we hence call the _relational trace_ , has a natural and
simple definition that is manifestly invariant under relative translations
between the two particle groups. It is therefore independent of the various
physically distinct conventions discussed in the previous subsections. On the
relational subspace $\mathcal{H}_{\rm phys}$, the paradox is therefore
unambiguously resolved. It is sufficient to focus on this subspace because
Lemma 22 tells us that the relational observables are tomographically complete
for the invariant information in all alignable states. Thus, $\mathcal{H}_{\rm
phys}$ contains all the information we are interested in.

######  Lemma 31.

The map $\tilde{\Phi}_{\rm phys}:\mathcal{A}_{\rm
phys}^{(N)}\rightarrow\mathcal{A}_{\rm phys}^{(N+M)}$, defined by

$\displaystyle\tilde{\Phi}_{\rm phys}(A_{\rm phys}^{(N)}):=\hat{\Pi}_{\rm
phys}^{(N+M)}\left(A_{\rm phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)$ (80)

is an embedding. It simplifies to

$\displaystyle\tilde{\Phi}_{\rm phys}(A_{\rm phys}^{(N)})=A_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)},$ (81)

but it is _not_ unital.

Recall that this construction does not yield an embedding of all invariant
observables $\mathcal{A}^{(N)}_{\rm inv}$, or even of $\mathcal{A}_{\rm
alg}^{(N)}$, as we have seen in Lemma 29. Thus, it is remarkable that it works
for the subalgebra of relational observables.

###### Proof.

Noting that every $A_{\rm phys}^{(N)}\in\mathcal{A}_{\rm phys}^{(N)}$
satisfies $A_{\rm phys}^{(N)}=\hat{\Pi}_{\rm phys}^{(N)}(A_{\rm phys}^{(N)})$,
we can recast $\tilde{\Phi}_{\rm phys}$ in the form

$\displaystyle\tilde{\Phi}_{\rm phys}(A_{\rm phys}^{(N)})$ $\displaystyle=$
$\displaystyle\Pi_{\rm phys}^{(N+M)}\left(\Pi_{\rm
phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)\left(A_{\rm
phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)$
$\displaystyle{\quad}{\quad}{\quad}{\quad}{\quad}\times\left(\Pi_{\rm
phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)\Pi_{\rm phys}^{(N+M)}.$

By considering the action on an arbitrary basis state
$\ket{g_{1},\ldots,g_{N+M}}$, it is easy to verify that

$\displaystyle\left(\Pi_{\rm phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)\Pi_{\rm
phys}^{(N+M)}=\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}.$ (82)

Hence, using idempotence of the projector,

$\displaystyle\tilde{\Phi}_{\rm phys}(A_{\rm phys}^{(N)})=A_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)},$ (83)

and the image commutes with $\Pi_{\rm phys}^{(N+M)}$. Checking the embedding
properties is now trivial. Furthermore, note that the unit element of
$\mathcal{A}_{\rm phys}^{(N)}$ is $\Pi_{\rm phys}^{(N)}$, while the unit
element of $\mathcal{A}_{\rm phys}^{(N+M)}$ is $\Pi_{\rm phys}^{(N+M)}$.
However, $\tilde{\Phi}_{\rm phys}(\Pi_{\rm phys}^{(N)})=\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}$. To see that $\tilde{\Phi}_{\rm
phys}(\Pi_{\rm phys}^{(N)})$ does not act as the identity on $\mathcal{H}_{\rm
phys}^{(N+M)}$, observe that ${\rm tr}\left(\Pi_{\rm
phys}^{(N)}\right)=\dim\mathcal{H}_{\rm phys}^{(N)}=|\mathcal{G}|^{N-1}$,
hence ${\rm tr}\left(\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm
phys}^{(M)}\right)=|\mathcal{G}|^{N+M-2}$, but ${\rm tr}\left(\Pi_{\rm
phys}^{(N+M)}\right)=|\mathcal{G}|^{N+M-1}$. ∎

Returning to the discussion of how to define the relations of the additional
$M$ particles to the first group of $N$ particles (cf. Fig. 6), we now have a
unique answer: the projector $\Pi_{\rm phys}^{(M)}$ in $\tilde{\Phi}_{\rm
phys}$ takes the _coherent average_ over all possible such relations. This can
also be seen by inspecting the action of $\tilde{\Phi}_{\rm phys}$ on the
basis elements of $\mathcal{A}_{\rm phys}$:

$\tilde{\Phi}_{\rm
phys}\left(|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|\right)=\frac{1}{|\mathcal{G}|}\sum_{g\in\mathcal{G},\mathbf{g}\in\mathcal{G}^{M}}|\mathbf{h},\mathbf{g};\mathbf{1}\rangle\langle\mathbf{j},g\mathbf{g};\mathbf{1}|.$

In contrast to Eq. (72) and the embeddings of Subsection 4.1, this embedding
does _not_ assign to every $M$-particle configuration $\mathbf{g}$ another
one, $\mathbf{g}^{\prime}$, which has “the same relation” to the first $N$
particles in branches $\mathbf{h}$ and $\mathbf{j}$. Instead, it generates the
_uniform superposition_ of all the possibilities.

This averaging is also the reason for the failure of the unitality property.
However, as we will see shortly, the absence of unitality is precisely the
reason why the relational trace defined below maps relational $(N+M)$-particle
states into relational $N$-particle states. It will thus be rather a feature
than a failure.

While $\tilde{\Phi}_{\rm phys}$ is not unital, note that the embedding of the
relational $N$-particle unit $\tilde{\Phi}_{\rm phys}(\Pi_{\rm
phys}^{(N)})=\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}$ certainly _does_
act as the identity on its image,

$\displaystyle\mathcal{H}_{\rm phys}^{(N\otimes M)}:=\Big{\\{}\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm
phys}^{(M)}\ket{\psi}\,\Big{|}\,\ket{\psi}\in\mathcal{H}^{\otimes(N+M)}\Big{\\}}.$

This subspace of the space of relational $(N+M)$-particle states
$\mathcal{H}_{\rm phys}^{(N+M)}$ will be essential below when resolving the
paradox of the third particle.

The natural embedding induces a natural trace.

######  Definition 32 (Relational trace).

The relational trace is defined to be the Hilbert-Schmidt adjoint ${\rm
Trel}:=\Phi_{\rm phys}^{\dagger}$ of the extended embedding map $\Phi_{\rm
phys}:\mathcal{L}(\mathcal{H}^{\otimes
N})\rightarrow\mathcal{L}(\mathcal{H}^{\otimes(N+M)})$ defined by $\Phi_{\rm
phys}:=\tilde{\Phi}_{\rm phys}\circ\hat{\Pi}_{\rm phys}^{(N)}$. That is, the
relational trace is the unique map with the property

$\displaystyle{\rm tr}\left(\Phi_{\rm phys}(A^{(N)})\,\rho\right)={\rm
tr}\left(A^{(N)}\,{\rm Trel}_{(M)}\rho\right)$ (84)

for all $A^{(N)}\in\mathcal{L}(\mathcal{H}^{\otimes N})$ and all
$\rho\in\mathcal{L}(\mathcal{H}^{\otimes(N+M)})$ (in particular for all
states).

Specifically, note that

$\displaystyle{\rm tr}\left(\Phi_{\rm phys}(A^{(N)})\,\rho\right)$
$\displaystyle=$ $\displaystyle{\rm tr}\left(\Phi_{\rm
phys}(A^{(N)})\,\hat{\Pi}_{\rm phys}^{(N+M)}(\rho)\right)$ (85)
$\displaystyle=$ $\displaystyle{\rm tr}\left(\Phi_{\rm
phys}(A^{(N)})\,\rho_{\rm phys}\right).$

This is precisely the expectation value of the relational observable
$\Phi_{\rm phys}(A^{(N)})\in\mathcal{A}_{\rm phys}^{(N+M)}$ in the relational
state $\rho_{\rm phys}\in\mathcal{S}(\mathcal{H}_{\rm phys}^{(N+M)})$,
evaluated in the manifestly invariant inner product on
$\mathcal{S}(\mathcal{H}_{\rm phys}^{(N+M)})$. The relational trace is thus
unambiguously defined in terms of the so-called physical inner product of
constraint quantization [61, 62, 30], i.e. the inner product on
$\mathcal{H}_{\rm phys}$. We will analyze this in more detail in our upcoming
work [72].

######  Theorem 33.

The relational trace takes the explicit form

$\displaystyle{\rm Trel}_{(M)}\rho={\rm Tr}_{(M)}\left[\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\,\rho\,\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\right],$

where ${\rm Tr}_{(M)}$ is the standard partial trace over particles
$N+1,\ldots,N+M$. It maps relational operators onto relational operators, i.e.
${\rm Trel}_{(M)}\left(\mathcal{A}_{\rm phys}^{(N+M)}\right)=\mathcal{A}_{\rm
phys}^{(N)}$, and is trace-preserving for states in
$\mathcal{S}(\mathcal{H}_{\rm phys}^{(N\otimes M)})$, but trace-decreasing
outside of it. Furthermore, it preserves observational equivalence, i.e.
$\rho\sim\sigma$ implies ${\rm Trel}_{(M)}\rho={\rm Trel}_{(M)}\sigma$.

###### Proof.

The first statement follows from

$\displaystyle{\rm tr}\left(\Phi_{\rm phys}(A^{(N)})\,\rho\right)={\rm
tr}\left[\left(\Pi_{\rm phys}^{(N)}A^{(N)}\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm
phys}^{(M)}\right)\rho\right]$ $\displaystyle={\rm
tr}\left[\left(A^{(N)}\otimes\mathbf{1}^{(M)}\right)\left(\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\,\rho\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\right)\right],$

which holds for any $A^{(N)}\in\mathcal{L}(\mathcal{H}^{\otimes N})$ and any
$\rho\in\mathcal{L}(\mathcal{H}^{\otimes(N+M)})$.

Given the conjugation of its input with the projector $\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}$, it is clear that ${\rm Trel}_{(M)}$
is trace-preserving for states in $\mathcal{S}(\mathcal{H}_{\rm
phys}^{(N\otimes M)})$, but not for states outside of it. It is also clear
that ${\rm Trel}_{(M)}$ maps operators from $\mathcal{A}_{\rm phys}^{(N+M)}$
into operators in $\mathcal{A}_{\rm phys}^{(N)}$, since the projectors
$\Pi_{\rm phys}^{(N)}$ can be taken outside of the trace over particles
$N+1,\ldots,N+M$. To see that it is surjective, it is straightforward to check
that $|\mathcal{G}|^{1-M}{\rm Trel}_{(M)}\left(A_{\rm
phys}^{(N)}\otimes\mathbf{1}^{(M)}\right)=A_{\rm phys}^{(N)}$ for all $A_{\rm
phys}^{(N)}\in\mathcal{A}_{\rm phys}^{(N)}$.

Finally, note that the image of $\Phi_{\rm phys}$ is contained in
$\mathcal{A}_{\rm phys}^{(N+M)}$, thus $\Phi_{\rm phys}=\hat{\Pi}_{\rm
phys}^{(N+M)}\circ\Phi_{\rm phys}$. Taking the Hilbert-Schmidt adjoint of this
equation yields ${\rm Trel}_{(M)}\circ\hat{\Pi}_{\rm phys}^{(N+M)}={\rm
Trel}_{(M)}$. Now suppose we have $\rho\sim\sigma$, then Lemma 14 and
$\hat{\Pi}_{\rm phys}^{(N+M)}\circ\Pi_{\rm inv}^{(N+M)}=\hat{\Pi}_{\rm
phys}^{(N+M)}$ imply $\hat{\Pi}_{\rm phys}^{(N+M)}(\rho)=\hat{\Pi}_{\rm
phys}^{(N+M)}(\sigma)$. Altogether this implies that ${\rm Trel}_{(M)}$
preserves observational equivalence. ∎

We can write

${\rm Trel}_{(M)}=\hat{\Pi}_{\rm phys}^{(N)}\circ{\rm
Tr}_{(M)}\circ\hat{\Pi}_{\rm phys}^{(N+M)}.$ (86)

We can thus view the relational partial trace ${\rm Trel}_{(M)}$ as an
invariant extension of the standard partial trace ${\rm Tr}_{(M)}$. The non-
unitality of $\tilde{\Phi}_{\rm phys}$ is reflected in the final application
of $\hat{\Pi}_{\rm phys}^{(N)}$. Without this projection, the image of ${\rm
Trel}_{(M)}$ would not in general be contained in $\mathcal{A}_{\rm
phys}^{(N)}$. For example, the uniform mixture $\rho^{(N+M)}:=\Pi_{\rm
phys}^{(N+M)}/|\mathcal{G}|^{N+M-1}$ on the physical subspace of $N+M$
particles yields ${\rm
Tr}_{(M)}\rho^{(N+M)}=\mathbf{1}^{(N)}/|\mathcal{G}|^{N}$, whose decomposition
according to Theorem 12 contains operators outside of $\mathcal{A}_{\rm
phys}^{(N)}$.888More generally, the image of $\mathcal{H}_{\rm
phys}^{(N+M)}\setminus\mathcal{H}_{\rm phys}^{(N\otimes M)}$ under
$\rm{Tr}_{(M)}$ does not lie in the $N$-particle relational subspace
$\mathcal{H}_{\rm phys}^{(N)}$. _Thus, non-unitality is the price to pay for
remaining relational._

This leads to ${\rm Trel}_{(M)}$ being trace-_decreasing_ , unless the initial
state is fully supported on the subspace $\mathcal{H}_{\rm phys}^{(N\otimes
M)}$. Should we be worried about this fact — shouldn’t marginals of normalized
quantum states be normalized? Not in this case. In contrast to the standard
partial trace, the relational trace is _not_ supposed to tell us what the
reduced quantum state on a subsystem is. Instead, it is constructed to tell us
precisely the following:

######  Theorem 34.

Given some $(N+M)$-particle state
$\rho^{(N+M)}\in\mathcal{S}(\mathcal{H}^{\otimes(N+M)})$, the following
_conditional state of $N$ particles_ is normalized or subnormalized:

$\rho^{(N)}:=\frac{{\rm Trel}_{(M)}\rho^{(N+M)}}{{\rm
tr}\left(\rho^{(N+M)}\Pi_{\rm phys}^{(N+M)}\right)}.$ (87)

Consider any relational projector $0\leq E^{(N)}_{\rm phys}\leq\Pi^{(N)}_{\rm
phys}$ which we interpret as a “relational event”. Then the state $\rho^{(N)}$
tells us the probabilities of this $N$-particle event, _conditioned on the
$(N+M)$-particle system being relational_:

${\rm tr}\left(E^{(N)}_{\rm phys}\rho^{(N)}\right)={\rm Prob}\left(E_{\rm
phys}^{(N)}|\,\,\Pi_{\rm phys}^{(N+M)}\right).$ (88)

That is, the renormalized result $\rho^{(N)}$ of the relational trace gives us
the expectation values of all $N$-particle relational observables, conditioned
on the global $(N+M)$-particle state being fully relational.

###### Proof.

It follows from Theorem 33 and Eq. (82) that

$\displaystyle{\rm tr}\left({\rm Trel}_{(M)}\rho^{(N+M)}\right)$
$\displaystyle=$ $\displaystyle{\rm tr}\left(\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\rho^{(N+M)}\right)$ (89)
$\displaystyle\leq$ $\displaystyle{\rm tr}\left(\Pi_{\rm
phys}^{(N+M)}\rho^{(N+M)}\right),$

hence $\rho^{(N)}$ is not supernormalized. Now, the probability that an
initial global measurement of the projector $\Pi_{\rm phys}^{(N+M)}$ yields
outcome “yes”, and then a subsequent local measurement of $E_{\rm phys}^{(N)}$
yields “yes” too, is

$\displaystyle{\rm Prob}\left(E_{\rm phys}^{(N)},\Pi_{\rm
phys}^{(N+M)}\right)$ $\displaystyle{\quad}{\quad}{\quad}={\rm
tr}\left(\Phi_{\rm phys}(E_{\rm phys}^{(N)})\hat{\Pi}_{\rm
phys}^{(N+M)}(\rho^{(N+M)})\right)$ $\displaystyle{\quad}{\quad}{\quad}={\rm
tr}\left(E_{\rm phys}^{(N)}{\rm Trel}_{(M)}\circ\hat{\Pi}_{\rm
phys}^{(N+M)}(\rho^{(N+M)})\right)$ $\displaystyle{\quad}{\quad}{\quad}={\rm
tr}\left(E_{\rm phys}^{(N)}{\rm Trel}_{(M)}\rho^{(N+M)}\right),$ (90)

where we have used Definition 32 and Eq. (86). The rest of the claim follows
from the definition of conditional probability. ∎

Recall that in Lemma 22, we have seen that the projection $\rho_{\rm
phys}:=\hat{\Pi}_{\rm phys}^{(N+M)}(\rho)$ for alignable states $\rho$ is
subnormalized, but is sufficient to determine the expectation values of all
invariant observables. Thus, $\rho_{\rm phys}$ should not be seen as the
marginal of $\rho$ on some subsystem, but as the “relational part” of $\rho$.
The “relational weight” ${\rm tr}\,\rho_{\rm phys}$ is in general less than
one,999This assumes that $\rho\in\mathcal{L}(\mathcal{H}^{\otimes(N+M)})$ in
$\rho_{\rm phys}=\hat{\Pi}^{(N+M)}_{\rm phys}(\rho)$ is normalized, as
appropriate in the context of our manuscript where there is an external
observer who could measure also non-invariant observables in the presence of
an external frame. By contrast, in the perspective-neutral approach [39, 41,
42, 43, 44, 45] (and more generally in constraint quantization), one
disregards any external structure and works directly with normalized
relational states. That is, one would _define_ the normalization of the
$(N+M)$-particle relational state as $\rm{tr}\rho_{\rm phys}=1$. Indeed, note
that the _full_ standard trace $\rm{tr}\rho_{\rm phys}=\rm{tr}\left(\Pi_{\rm
phys}^{(N+M)}\rho\right)$ is precisely the extension of the so-called physical
inner product [61, 62, 30] to density matrices. However, note also from the
previous footnote that the standard _partial_ trace $\rm{Tr}_{(M)}$ is not in
general appropriate for relational states $\rho_{\rm phys}$. The additional
projection with $\hat{\Pi}_{\rm phys}^{(N)}$ in the relational trace fixes
this issue, but reduces the norm of states with support outside of
$\mathcal{H}_{\rm phys}^{(N\otimes M)}$. This has a transparent physical
interpretation which is best seen through the norm reduction of invariant
basis states:
$\rm{Trel}_{(M)}(\ket{\mathbf{h}_{N},\mathbf{h}_{M};\mathbf{1}}\\!\bra{\mathbf{j}_{N},\mathbf{j}_{M};\mathbf{1}})$,
where $\mathbf{h}_{N},\mathbf{j}_{N}\in\mathcal{G}^{N-1}$ and
$\mathbf{h}_{M},\mathbf{j}_{M}\in\mathcal{G}^{M}$, is equal to
$\frac{1}{|\mathcal{G}|}\,\ket{\mathbf{h}_{N};\mathbf{1}}\\!\bra{\mathbf{j}_{N};\mathbf{1}}$
if $\mathbf{h}_{M}=g\mathbf{j}_{M}$ for some $g\in\mathcal{G}$, and is zero
otherwise. The variables $\mathbf{h}_{M},\mathbf{j}_{M}$ lying in
$\mathcal{G}^{M}$ rather than $\mathcal{G}^{M-1}$ reflects the fact that they
encode not only the $M-1$ internal relations of the $M$ particles, but also a
definite relation between the two particle groups, which is a property of
_both_ groups together. The normalization reduction factor $1/|\mathcal{G}|$
comes from the coherent averaging over the relations between the two particle
groups (which is partially a property of the $N$ particles), and quantifies
the corresponding ignorance of the relational $N$-particle state obtained via
$\rm{Trel}_{(M)}$. By construction, the latter contains all information about
the relational $N$-particle observables which are independent of the relation
between the particle groups. However, the ‘ignorance factor’ $1/|\mathcal{G}|$
has to be taken into account. and it can decrease when disregarding some of
the particles.

###  4.4 Relational resolution of the paradox

In order to apply these insights to the paradox of the third particle, let us
first discuss some further aspects of the interplay of composition and
relational trace, for the generic case of $M+N$ particles. Suppose
$\rho^{(N)}$ is a state of the $N$ particles before taking the additional $M$
particles into account, prepared by an observer with access to the external
reference frame. The corresponding relational state is $\rho_{\rm
phys}^{(N)}=\hat{\Pi}_{\rm phys}^{(N)}(\rho^{(N)})$. Next, suppose the $M$
additional particles are prepared in a normalized state $\rho^{(M)}$, and the
composite state of all particles is of the product form
$\rho^{(N+M)}=\rho^{(N)}\otimes\rho^{(M)}$. We can then construct the
relational state corresponding to this composition. Here it is important to
note that $\rho_{\rm phys}^{(N+M)}=\hat{\Pi}_{\rm
phys}^{(N+M)}\left(\rho^{(N+M)}\right)=\hat{\Pi}_{\rm
phys}^{(N+M)}\left(\tilde{\rho}^{(N+M)}\right)$, for any
$\tilde{\rho}^{(N+M)}\simeq\rho^{(N+M)}$. That is, all members of the symmetry
equivalence class of $\rho^{(N+M)}$ (incl. states featuring entanglement
between the two particle groups) yield the same relational $(N+M)$-particle
state. However, due to Theorem 33, it is only the projection into
$\mathcal{H}_{\rm phys}^{(N\otimes M)}$ that matters for the relational trace.
But Eq. (82) implies that

$\displaystyle\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}\,\rho_{\rm
phys}^{(N+M)}\,\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}$
$\displaystyle{\quad}{\quad}{\quad}=\Pi_{\rm phys}^{(N)}\otimes\Pi_{\rm
phys}^{(M)}\left(\rho^{(N)}\otimes\rho^{(M)}\right)\Pi_{\rm
phys}^{(N)}\otimes\Pi_{\rm phys}^{(M)}$
$\displaystyle{\quad}{\quad}{\quad}=\rho_{\rm phys}^{(N)}\otimes\rho_{\rm
phys}^{(M)},$ (91)

and so we have ${\rm Trel}_{(M)}\rho^{(N+M)}_{\rm phys}=\rho_{\rm
phys}^{(N)}\cdot{\rm tr}_{(M)}\rho_{\rm phys}^{(M)}$. Up to a constant factor,
this is precisely the initial relational $N$-particle state that we had before
taking the additional $M$ particles into account.

Let us see what this implies for the paradox of the third particle. Thus, let
us concretely compute the relational trace ${\rm Trel}_{3}$ of the state
$|\Psi\rangle$ of Eq. (56). Due to invariance, the result will be identical if
we apply it to the state $|\Psi^{\prime}\rangle$ of Eq. (57). As a first step,
we find

$\displaystyle\Pi_{\rm phys}^{(3)}(|\Psi\rangle\langle\Psi|)\Pi_{\rm
phys}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2n}|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{h};\mathbf{1}|+\frac{e^{-i\theta}}{2n}|\mathbf{h};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|$
$\displaystyle+\frac{e^{i\theta}}{2n}|\mathbf{j};\mathbf{1}\rangle\langle\mathbf{h};\mathbf{1}|+\frac{1}{2n}|\mathbf{j};\mathbf{1}\rangle\langle\mathbf{j};\mathbf{1}|.$

Using ${}_{3}\langle
g_{3}|\mathbf{h};\mathbf{1}\rangle=\frac{1}{\sqrt{|\mathcal{G}|}}\,|g_{3}h_{2}^{-1},g_{3}h_{2}^{-1}h_{1}\rangle$
yields

$\displaystyle{\rm Trel}_{3}(|\Psi\rangle\langle\Psi|)$ $\displaystyle=$
$\displaystyle\Pi_{\rm phys}^{(2)}{\rm Tr}_{3}\left[\Pi_{\rm
phys}^{(3)}(|\Psi\rangle\langle\Psi|)\Pi_{\rm phys}^{(3)}\right]\Pi_{\rm
phys}^{(2)}$ $\displaystyle=$ $\displaystyle\Pi_{\rm
phys}^{(2)}|\psi\rangle\langle\psi|\,\Pi_{\rm phys}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2n^{2}}|h;\mathbf{1}\rangle\langle
h;\mathbf{1}|+\frac{e^{-i\theta}}{2n^{2}}|h;\mathbf{1}\rangle\langle\
j;\mathbf{1}|$
$\displaystyle+\frac{e^{i\theta}}{2n^{2}}|j;\mathbf{1}\rangle\langle
h;\mathbf{1}|+\frac{1}{2n^{2}}|j;\mathbf{1}\rangle\langle j;\mathbf{1}|.$

Since $|\Psi\rangle$ is alignable, Lemma 22 tells us that
$\langle\Psi|\Pi_{\rm phys}^{(N+M)}|\Psi\rangle=1/n$. Thus, computing the
conditional state of Theorem 34, we obtain the projection of the state in Eq.
(55) into the relational subalgebra. Hence, we recover exactly the relational
state of the first two particles which we had before adding the third. In
particular, the phase $\theta$ is preserved: as expected, it remains
accessible on the first two particles.

The algebra $\mathcal{A}_{\rm phys}$ generated by relational observables and
the subspace $\mathcal{H}_{\rm phys}$ of relational states is the arena of the
perspective-neutral approach to QRFs [39, 41, 42, 43, 44, 45]. As such, there
is no paradox of the third particle in this approach. Furthermore, it provides
a compelling conceptual interpretation of this resolution: the relational
states are the perspective-neutral states, i.e. they correspond to a
description of the composite particle system prior to choosing an internal
reference relative to which the state is described. The perspective-neutral
states contain the entire information about all internal QRF perspectives at
once. The relational trace is performed at the perspective-neutral level, and
consistency at that level implies consistency in all internal perspectives. We
will further elaborate on this in Ref. [72].

###  4.5 Comparison to the resolution by Angelo et al.

Angelo et al. [50] also propose a resolution to the apparent paradox that they
have raised in their paper. Let us recapitulate their resolution in our
terminology and compare the two approaches. First, they introduce an operator
$T:=e^{-2i(a+b)\hat{p}_{r_{2}}}$ which in our notation simply implements a
translation of the two particles, given by
$T|g_{1},g_{2}\rangle=|g_{1}-2a,g_{2}+2b\rangle$. Computing the expectation
value in the two-particle state of Eq. (53) yields
$\langle\psi|T|\psi\rangle=\frac{1}{2}e^{i\theta}$. Thus, this operator (or
rather its real and imaginary parts) admit the measurement of the phase
$\theta$.

Since we are in the framework of Example 8, we can use the explicit form of
the characters to see that

$T|h;\chi_{k}\rangle=\chi_{k}(-2a)|h+2a+2b;\chi_{k}\rangle.$ (92)

In particular, the invariant part of this translation can be expressed in the
form

$T|h;\mathbf{1}\rangle=|h+2a+2b;\mathbf{1}\rangle.$ (93)

That is, $T$ increases the relative distance of the particles by $2a+2b$.
Indeed, if we define

$T_{\rm inv}:=\Pi_{\rm inv}(T)=\Pi_{\rm alg}(T)=\hat{\Pi}_{\rm phys}(T),$ (94)

then $T_{\rm inv}\in\mathcal{A}_{\rm phys}$ satisfies Eq. (93). Note, however,
that $T$ contains strictly more information than $T_{\rm inv}$: not only does
it tell us that the relative distance of the particles increases by $2a+2b$,
but it also tells us what happens to their _absolute_ positions.

Angelo et al. write: _“The crucial (and surprising) observation is that $T$
actually shifts the relative coordinate of particle $3$ as well as that of
particle $2$.”_ Strictly speaking, this is not a claim about $T$, but about
the embedding $T^{(3)}$ of $T$ into the three-particle observables. Using the
obvious embedding $T^{(3)}=T\otimes\mathbf{1}$, we obtain
$T^{(3)}|g_{1},g_{2},g_{3}\rangle=|g_{1}-2a,g_{2}+2b,g_{3}\rangle$, and thus

$T^{(3)}|h_{1},h_{2};\mathbf{1}\rangle=|h_{1}+2a+2b,h_{2}+2a;\mathbf{1}\rangle.$
(95)

That is, the relative coordinate of particle $3$ is also shifted by $2a$. This
reproduces Angelo et al.’s claim, but it is important to understand where the
shift of $h_{2}\mapsto h_{2}+2a$ comes from. It is certainly not possible to
deduce this shift from $T_{\rm inv}$ alone. Instead, it comes form _the
specific choice_ of implementing $T_{\rm inv}$ (a relative shift of $2a+2b$)
via $T$ (absolutely shifting particle $1$ by $-2a$ and particle $2$ by $2b$).
And the latter choice comes from Angelo et al.’s decision of _preserving the
center of mass_ , which fixes the non-invariant action of $T$.

In summary: Angelo et al.’s proposed resolution of the paradox comes from
_deciding to embed the two-particle observables via the center-of-mass
embedding_ that we have described in Subsection 4.2. As shown there, this
leads to a preservation of the phase $\theta$. While the center-of-mass may be
the most democratic choice among the particles, as also discussed in
Subsection 4.2, there exist other well-motivated, but physically inequivalent
choices of embedding, e.g. relative to one of the particles, which may or may
not preserve $\theta$. Basing a resolution of the paradox in the context of
frame covariance on a specific choice of frame seems unsatisfactory.
Furthermore, even if one wanted to argue that the center-of-mass is a
privileged choice in mechanics, this would have to rely on its _dynamical_
properties — in fact, postulating certain particle masses in the first place
is nothing but a claim about how the particles move. However, the paradox of
the third particle is a completely _kinematical_ thought experiment: it can be
formulated without any reference to the particle masses or time evolution
whatsoever, as explained at the beginning of Section 4. Thus, for the paradox,
there is in any case no reason to privilege the center of mass embedding over
any other choice of invariant embedding, and hence this type of argumentation
is insufficient to conclude that the phase $\theta$ is preserved.

In contrast, our resolution amounts to the construction of a _relational
embedding_ for which no such choice has to be made in the first place.
Nonetheless, Angelo et al’s insight is still important: embedding fewer into
more particles will in general “do something” to the additional particles, and
care has to be taken of how the embedding is accomplished.

##  5 Conclusions

The aim of this article is to elucidate the operational essence and
interpretation of the recent structural approach to QRFs [38, 39, 40], and to
also clarify the meaning of its QRF transformations as symmetry
transformations. These insights have then been exploited to illuminate the
physics behind the apparent ’paradox of the third particle’ of Ref. [50] and
to resolve it at a formal level through relational observables.

We began by providing a careful conceptual comparison of the quantum
information [12, 14, 16, 17, 13, 15, 18, 19] and structural approaches to
QRFs, illustrating the difference in their operational essence in terms of two
communication scenarios. While both approaches focus on an external relatum
independent description of physical observables and quantum states, they do so
in different manners and with different goals. As we have seen, technically a
distinction can be drawn between the two in terms of how they describe
external relatum independent states fundamentally: they are the _incoherently_
and _coherently_ group-averaged states in the QI and structural approach,
respectively.

A key ambition of the QI approach is to elucidate how to perform communication
protocols between different parties in the absence of a shared external
laboratory frame. To this end, it suffices to focus on physical properties of
the communicated quantum system that are meaningful relative to an arbitrary
choice of external frame. For example, this can be achieved by restricting to
speakable information that is encoded in decoherence-free subspaces or by
communicating an _additional_ reference quantum system that serves as a token
for the sender’s reference frame. Either way, the QI approach maintains the
reference frame external to the system of interest. For successfully carrying
out such operational protocols it is also not necessary to take an extra step
and choose an internal reference frame within the system of interest, and to
ask how the quantum system is described relative to one of its subsystems.

However, this additional step is precisely what the structural approach aims
for. Its primary goal is not the implementation of protocols for communicating
physical information. It has rather a more fundamental ambition: to dissolve
the distinction between quantum systems and reference frames and thereby to
extend the set of available reference frame choices to include subsystems of
the physical system of interest. Its focal point are thus not only external
relatum independent state descriptions, but _internal_ state descriptions. The
operational essence of the structural approach can be illustrated in a
scenario in which different agents agree on a (redundancy-free) _description_
of physical quantum states _without_ adhering to an external relatum. They can
always achieve this task by invoking certain “canonical choices” in the
representation of quantum states that exploit the _internal_ structure of the
quantum system to be described. In particular, these canonical choices of
representation are related by transformations that coincide with the QRF
transformations in the structural approach.

To show this explicitly, we have then formalized these conceptual observations
in the context of an $N$-particle quantum system (“$\mathcal{G}$-system”)
where the configuration space of each particle is a finite Abelian group
$\mathcal{G}$. We chose this simple setting in order to avoid technicalities
and to render all appearing structures completely transparent. But we
emphasize that our observations are of more general validity. They apply
directly to laboratory situations in which agents simply disregard, or do not
have access to, a relatum external to the system of interest, but in principle
also to the case that no external physical relatum exists in the first place
as, e.g., in quantum cosmology (see Refs. [76, 43] for a related discussion).

We determined the symmetry group $\mathcal{G}_{\rm sym}$ associated with a
$\mathcal{G}$-system, which preserves all its external-relatum-independent
structure. We showed that states from the corresponding symmetry equivalence
class are alignable to a choice of reference system through a
$\mathcal{G}_{\rm sym}$ transformation: each equivalence class contains
“canonical choices” of state representations, and these correspond to
selecting any one of the $N$ particles as a reference system to define the
origin and to describing the remaining $N-1$ particles relative to it. These
canonical choices are the ’internal QRF perspectives’ on the $N$-particle
system of Refs. [38, 39, 40].

The symmetry group $\mathcal{G}_{\rm sym}$ contains the ’classical translation
group’ $\mathcal{G}$ as a strict _subgroup_ , but it also contains ‘relation-
conditional translations’ which turn out to include the QRF transformations of
Ref. [40], which are equivalent to those of Refs. [38, 39]. While it is
evident from these works that the QRF transformations are conditional
translations, the present article clarifies that they are _symmetry_
transformations with a precise and transparent physical interpretation.

Being translations conditional on the particle relations, the QRF
transformations make sense in a classical context when dealing with, for
example, statistical mixtures of particle positions rather than
superpositions, and indeed have classical analogs that have been exhibited in
Refs. [39, 41, 42, 43, 40]. Nevertheless, just like the CNOT gate has a
classical meaning, but can generate entanglement, the QRF transformations
similarly lead to interesting quantum effects such as a QRF dependence of,
e.g. entanglement and superpositions [38, 39, 40], classicality [39, 51, 52],
spin [46, 47], certain quantum resources [53], temporal locality [48, 44], and
of comparing quantum clock readings [42, 45].101010It would be interesting to
study the recent proposals [79, 80, 81] for quantum time dilation effects in
terms of the temporal QRF transformations as in Refs. [42, 43, 48, 44, 45].

Given the two groups $\mathcal{G}_{\rm sym}$ and $\mathcal{G}$ in the setup,
one has _a priori_ two distinct ways to construct invariant states and
observables. Interestingly, as we have shown, the invariant (pure) states and
thus the subspace $\mathcal{H}_{\rm phys}$ of relational states do _not_ in
fact depend on whether one requires invariance under the action of
$\mathcal{G}_{\rm sym}$ or its subgroup $\mathcal{G}$. By contrast, the set of
invariant observables _does_ depend on which group one works with: the
operator algebra invariant under $\mathcal{G}_{\rm sym}$ is a strict subset of
the operator algebra invariant under its subgroup $\mathcal{G}$. However, the
two invariant operator algebras coincide again in their restriction to the
space of relational states $\mathcal{H}_{\rm phys}$, which is the algebra
$\mathcal{A}_{\rm phys}$ generated by so-called relational observables [29,
30, 31, 32, 33, 34, 35, 36, 37, 42, 43, 44, 45]. The space of relational
states $\mathcal{H}_{\rm phys}$ and the relational operator algebra
$\mathcal{A}_{\rm phys}$ are key structures in constraint quantization [61,
62, 30] and the platform of the perspective-neutral approach to QRFs [39, 41,
42, 43, 44, 45] (part of the structural approach), which are thus independent
of the distinction between $\mathcal{G}_{\rm sym}$ and $\mathcal{G}$. The
difference between $\mathcal{G}_{\rm sym}$ and its subgroup $\mathcal{G}$ is,
however, crucial when aligning the _non-invariant_ description of quantum
states to a particle at the level of the full $N$-particle Hilbert space.

These observations also permitted us to first clarify the physics behind the
’paradox of the third particle’ discussed in Ref. [50], and subsequently to
resolve it at a formal level. First, we have illuminated why the usual partial
trace is _not_ suitable in the context of QRFs, because it ignores the
equivalence classes of states that are operationally indistinguishable in the
absence of an external relatum. Next, we have explained that, in order to take
the observational equivalence classes into account, one has to construct a
partial trace in terms of the invariant observables. However, even when
attempting to do so, we have seen that there does not exist a physically
distinguished choice for such an invariant partial trace _outside the space of
relational states_. The reason is that an invariant partial trace demands a
suitable embedding of the two-particle invariant observables into the three-
particle invariant observables. Yet such an embedding (while invariant under
symmetry transformations) depends on how one defines the relation of the third
to the first two particles, and there are multiple physically inequivalent
ways (e.g., distance to the first particle, center of mass, etc.). The two-
particle reduced state then depends on one’s convention of how to define the
relation between the third and the first two particles, despite restricting
attention to invariant observables.

However, when restricting attention further to the algebra $\mathcal{A}_{\rm
phys}$ generated by relational observables and the space of relational states
$\mathcal{H}_{\rm phys}$, we showed that there _does_ exist a physically
distinguished embedding of the relational two-particle observables and states
into the relational three-particle observables and states. Physically, this
embedding corresponds to _coherently averaging_ over all possible relations
between the third and the first two particles, and thereby defines an entirely
invariant embedding. This permitted us to define an unambiguous relational
partial trace that determines the expectation values of relational observables
on subsets of particles. In particular, this trace achieves for relational
states what a consistent partial trace should do: if a third particle is
independently prepared, then the two-particle reduced state, obtained from the
relational three-particle state, coincides with the relational two-particle
state prior to taking the third particle into account. At the level of
relational observables and relational states, the paradox of the third
particle of Ref. [50] is thus resolved; in this sense, the perspective-neutral
approach does not feature any paradox of additional particles. However, we
have not discussed what it would mean for an agent to operationally implement
this resolution in the lab and, specifically, how they may operationally
restrict to relational states and observables (although we believe this to be
possible). In this light, our resolution of the paradox is formal.

The paradox of the third particle and our resolution can be viewed as a
finite-dimensional analog of the problem of boundaries and edge modes in gauge
theory and gravity [64, 65, 66, 67, 68, 69, 70, 71]. Boundaries in space or
spacetime usually break gauge-invariance and constitute challenges for gauge-
invariant observables. The latter are typically non-local (such as Wilson
loops) and can thus have support in two neighbouring regions separated by a
boundary. Those gauge-invariant observables with support in both regions
determine the physical relation between the two and are accounted for in terms
of so-called edge modes when one of the regions is ignored. This is analogous
to the joining of two groups of $N$ and $M$ particles and asking for the
invariant relations between the two groups. The relative distances between the
two groups of particles are the finite-dimensional analog of the gauge-
invariant observables in gauge theories and gravity that have support in two
neighbouring regions. As we have seen, ignoring one group of particles by
simply taking the standard partial trace may indeed lead to an invariance
breaking in analogy to the field theory case, i.e. $N$-particle states that
are _not_ relational. This is because the set of relational observables for
the joint $(N+M)$-particle system is _not_ only the union of the sets of
relational $N$\- and $M$-particle observables, again in analogy to two
neighbouring subregions in spacetime. Our relational trace defines a purely
relational, i.e. invariant way of ‘ignoring’ a group of particles, and it
would be interesting to extend this tool to the study of edge modes in gauge
theories and gravity.

Lastly, we emphasize that our novel interpretation of the structural approach
applies in particular also to temporal quantum reference frames, i.e. quantum
clocks. For instance, the example of the cyclic group could model a set of
quantum clocks each with a finite set of readings. The external frame would
then be some laboratory clock that one external observer has access to, but
another may not. Nevertheless, the two observers can agree on the description
of the flow of time by focusing on a purely _internal_ choice of clock that
leads to a _relational notion of time_ entirely independent of any external
clock. It is in this sense that one can interpret the relational quantum
dynamics defined by temporal relational observables [29, 30, 31, 32, 33, 34,
35, 36, 42, 43, 44, 45, 82] or the Page-Wootters formalism [44, 45, 80, 48,
83, 84, 85, 86] (which recently have been shown to be equivalent [44, 45]) in
the context of laboratory situations.111111These frameworks, however, also
apply in the absence of external agents such as in quantum cosmology or
gravity. Indeed, this is precisely how the experimental illustration of the
Page-Wootters dynamics reported in [87] is to be understood.

In this manuscript, we focused purely on kinematical aspects of quantum
reference frame physics. In our companion article [72], we study in detail how
the insights gained here are affected when we take the dynamics of the
$N$-particle system into account. This question will link also with the
perspective-neutral approach to QRFs [39, 41, 42, 43, 44, 45], and we will
establish in detail the equivalence of its “quantum coordinate changes” with
the QRF transformations exhibited here.

## Acknowledgments

We thank Thomas Galley, Leon Loveridge and Alexander R. H. Smith for many
useful comments on an earlier draft version. MK is grateful to Časlav Brukner,
Esteban Castro-Ruiz, and David Trillo Fernandez for helpful discussions. MPM
would like to thank Anne-Catherine de la Hamette for inspiring discussions in
the initial phase of this project. MK acknowledges the support of the Vienna
Doctoral School in Physics (VDSP) and the support of the Austrian Science Fund
(FWF) through the Doctoral Programme CoQuS. PAH is grateful for support from
the Foundational Questions Institute under grant number FQXi-RFP-1801A. This
work was supported in part by funding from Okinawa Institute of Science and
Technology Graduate University. MK and MPM thank the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation, for support via grant number FQXi-RFP-1815. This
research was supported in part by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through the Department of Innovation, Science and Economic Development Canada
and by the Province of Ontario through the Ministry of Research, Innovation
and Science.

## References

  * [1] Y. Aharonov and L. Susskind, _Charge Superselection Rule_ , Phys. Rev. 155, 1428 (1967).
  * [2] Y. Aharonov and L. Susskind, _Observability of the Sign Change of Spinors under $2\pi$ Rotations_, Phys. Rev. 158, 1237 (1967).
  * [3] Y. Aharonov and T. Kaufherr, _Quantum frames of reference_ , Phys. Rev. D 30, 368 (1984).
  * [4] E. Wigner, _Die Messung quantenmechanischer Operatoren_ , Z. Physik 133, 101–108 (1952).
  * [5] H. Araki and M. M. Yanase, _Measurement of Quantum Mechanical Operators_ , Phys. Rev. 120, 622 (1960).
  * [6] M. M. Yanase, _Optimal Measuring Apparatus_ , Phys. Rev. 123, 666 (1961).
  * [7] L. Loveridge, B. Busch, and T. Miyadera, _Relativity of quantum states and observables_ , EPL 117, 40004 (2017).
  * [8] L. Loveridge, T. Miyadera, and P. Busch, _Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics_ , Found. Phys. 48, 135–198 (2018).
  * [9] T. Miyadera, L. Loveridge, and P. Busch, _Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off_ , J. Phys. A: Mathematical and Theoretical, 49(18), 185301 (2016).
  * [10] L. Loveridge, _A relational perspective on the Wigner-Araki-Yanase theorem_ , J. Phys.: Conf. Ser. 1638, 012009 (2020).
  * [11] P. A. Höhn and M. P. Müller, _An operational approach to spacetime symmetries: Lorentz transformations from quantum communication_ , New J. Phys. 18, 063026 (2016).
  * [12] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, _Reference frames, superselection rules, and quantum information_ , Rev. Mod. Phys. 79, 555 (2007).
  * [13] A. R. H. Smith, _Communicating without shared reference frames_ , Phys. Rev. A 99, 052315 (2019).
  * [14] I. Marvian, _Symmetry, Asymmetry and Quantum Information_ , PhD thesis, University of Waterloo, 2012.
  * [15] G. Gour and R. W. Spekkens, _The resource theory of quantum reference frames: manipulations and monotones_ , New J. Phys. 10, 033023 (2008).
  * [16] G. Gour, I. Marvian, and R. W. Spekkens, _Measuring the quality of a quantum reference frame: The relative entropy of frameness_ , Phys. Rev. A 80, 012307 (2009).
  * [17] I. Marvian and R. W. Spekkens, _Modes of asymmetry: The application of harmonic analysis to symmetric quantum dynamics and quantum reference frames_ , Phys. Rev. A 90, 062110 (2014).
  * [18] M. C. Palmer, F. Girelli, and S. D. Bartlett, _Changing quantum reference frames_ , Phys. Rev. A 89, 052121 (2014).
  * [19] A. R. H. Smith, M. Piani, and R. B. Mann, _Quantum reference frames associated with noncompact groups: the case of translations and boosts, and the role of mass_ , Phys. Rev. A 94, 012333 (2016).
  * [20] J. Åberg, _Catalytic Coherence_ , Phys. Rev. Lett. 113, 150402 (2014).
  * [21] M. Lostaglio, D. Jennings, and T. Rudolph, _Description of quantum coherence in thermodynamic processes requires constraints beyond free energy_ , Nat. Commun. 6, 6383 (2015).
  * [22] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, _Quantum Coherence, Time-Translation Symmetry, and Thermodynamics_ , Phys. Rev. X 5, 021001 (2015).
  * [23] M. Lostaglio and M. P. Müller, _Coherence and Asymmetry Cannot be Broadcast_ , Phys. Rev. Lett. 123, 020403 (2019).
  * [24] I. Marvian and R. W. Spekkens, _No-Broadcasting Theorem for Quantum Asymmetry and Coherence and a Trade-off Relation for Approximate Broadcasting_ , Phys. Rev. Lett. 123, 020404 (2019).
  * [25] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, _Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?_ , Phys. Rev. X 7, 031022 (2017).
  * [26] P. Ćwikliński, M. Studziński, M. Horodecki, and J. Oppenheim, _Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics_ , Phys. Rev. Lett. 115, 210403 (2015).
  * [27] M. P. Woods, R. Silva, and J. Oppenheim, _Autonomous Quantum Machines and Finite-Sized Clocks_ , Ann. Henri Poincaré 20, 125 (2019).
  * [28] M. P. Woods and M. Horodecki, _The resource theoretic paradigm of quantum thermodynamics with control_ , arXiv:1912.05562 [quant-ph].
  * [29] C. Rovelli, _Quantum gravity_ , Cambridge University Press, 2004.
  * [30] T. Thiemann, _Modern Canonical Quantum General Relativity_ , Cambridge University Press, 2007.
  * [31] J. Tambornino, _Relational Observables in Gravity: a Review_ , SIGMA 8, 017 (2012).
  * [32] C. Rovelli, _What is observable in classical and quantum gravity?_ , Class. Quant. Grav. 8, 297 (1991).
  * [33] C. Rovelli, _Quantum reference systems_ , Class. Quant. Grav. 8, 317 (1991).
  * [34] C. Rovelli, _Time in quantum gravity: An hypothesis_ , Phys. Rev. D 43, 442-456 (1991).
  * [35] B. Dittrich, _Partial and complete observables for Hamiltonian constrained systems_ , Gen. Rel. Grav. 39, 1891 (2007).
  * [36] B. Dittrich, _Partial and complete observables for canonical general relativity_ , Class. Quant. Grav. 23, 6155 (2006).
  * [37] L. Chataignier, _Construction of quantum Dirac observables and the emergence of WKB time_ , Phys. Rev. D 101, 086001 (2020).
  * [38] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, _Quantum mechanics and the covariance of physical laws in quantum reference frames_ , Nat. Commun. 10, 494 (2019).
  * [39] A. Vanrietvelde, P. A. Höhn, F. Giacomini, and E. Castro-Ruiz, _A change of perspective: switching quantum reference frames via a perspective-neutral framework_ , Quantum 4, 225 (2020).
  * [40] A. de la Hamette and T. Galley, _Quantum reference frames for general symmetry groups_ , Quantum 4, 367 (2020).
  * [41] A. Vanrietvelde, P. A. Höhn, and F. Giacomini, _Switching quantum reference frames in the N-body problem and the absence of global relational perspectives_ , arXiv:1809.05093 [quant-ph].
  * [42] P. A. Höhn and A. Vanrietvelde, _How to switch between relational quantum clocks_ , New J. Phys. 22, 123048 (2020).
  * [43] P. A. Höhn, _Switching Internal Times and a New Perspective on the ‘Wave Function of the Universe’_ , Universe 5, 116 (2019).
  * [44] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, _The Trinity of Relational Quantum Dynamics_ , Phys. Rev. D (in press), arXiv:1912.00033 [quant-ph].
  * [45] P. A. Höhn, A. R. H. Smith, and M. P. E. Lock, _Equivalence of approaches to relational quantum dynamics in relativistic settings_ , Front. Phys. 9, 587083 (2021).
  * [46] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, _Relativistic Quantum Reference Frames: The Operational Meaning of Spin_ , Phys. Rev. Lett. 123, 090404 (2019).
  * [47] L. F. Streiter, F. Giacomini, and Č. Brukner, _A Relativistic Bell Test within Quantum Reference Frames_ , Phys. Rev. Lett. 126, 230403 (2021).
  * [48] E. Castro-Ruiz, F. Giacomini, A. Belenchia, and Č. Brukner, _Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems_ , Nat. Commun. 11, 2672 (2020).
  * [49] J. M. Yang, _Switching Quantum Reference Frames for Quantum Measurement_ , Quantum 4, 283 (2020).
  * [50] R. M. Angelo, N. Brunner, S. Popescu, A. J. Short, and P. Skrzypczyk, _Physics within a quantum reference frame_ , J. Phys. A: Math. Theor. 44, 145304 (2011).
  * [51] T. P. Le, P. Mironowicz, and P. Horodecki, _Blurred quantum Darwinism across quantum reference frames_ , Phys. Rev. A 102, 062420 (2020).
  * [52] J. Tuziemski, _Decoherence and information encoding in quantum reference frames_ , arXiv:2006.07298 [quant-ph].
  * [53] M. F. Savi, and R. M. Angelo, _Quantum Resources Covariance_ , Phys. Rev. A 103, 022220 (2021).
  * [54] P. A. Guérin and Č. Brukner, _Observer-dependent locality of quantum events_ , New J. Phys. 20, 103031 (2018).
  * [55] L. Hardy, _Implementation of the Quantum Equivalence Principle_ , arXiv:1903.01289 [quant-ph].
  * [56] S. Gielen and L. Menéndez-Pidal, _Singularity resolution depends on the clock_ , Class. Quant. Grav. 37, 205018 (2020).
  * [57] K. Giesel, L. Herold, B. F. Li, and P. Singh, _Mukhanov-Sasaki equation in a manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields_ , Phys. Rev. D 102, 023524 (2020).
  * [58] K. Giesel, B. F. Li, and P. Singh, _Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential_ , Phys. Rev. D 102, 126024 (2020).
  * [59] P. A. Dirac, _Lectures on Quantum Mechanics_ , Yeshiva University Press, 1964.
  * [60] M. Henneaux and C. Teitelboim, _Quantization of Gauge Systems_ , Princeton University Press, 1992.
  * [61] D. Giulini and D. Marolf, _A Uniqueness theorem for constraint quantization_ , Class. Quant. Grav. 16, 2489 (1999).
  * [62] D. Marolf, _Group averaging and refined algebraic quantization: Where are we now?_ , arXiv:gr-qc/0011112 [gr-qc].
  * [63] C. Rovelli, _Why Gauge?_ , Found. Phys. 44, 91–104 (2014).
  * [64] W. Donnelly and A. C. Wall, _Entanglement Entropy of Electromagnetic Edge Modes_ , Phys. Rev. Lett. 114, 111603 (2015).
  * [65] W. Donnelly and L. Freidel, _Local subsystems in gauge theory and gravity_ , JHEP 09, 102 (2016).
  * [66] M. Geiller and P. Jai-akson, _Extended actions, dynamics of edge modes, and entanglement entropy_ , JHEP 20, 134 (2020).
  * [67] L. Freidel, M. Geiller and D. Pranzetti, _Edge modes of gravity. Part I. Corner potentials and charges_ , JHEP 2020, 26 (2020).
  * [68] H. Gomes and A. Riello, _Unified geometric framework for boundary charges and particle dressings_ , Phys. Rev. D 98, 025013 (2018).
  * [69] A. Riello, _Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back_ , SciPost Phys. 10, 125 (2021).
  * [70] W. Wieland, _New boundary variables for classical and quantum gravity on a null surface_ , Class. Quant. Grav. 34, no.21, 215008 (2017).
  * [71] W. Wieland, _Fock representation of gravitational boundary modes and the discreteness of the area spectrum_ , Annales Henri Poincare 18, no.11, 3695 (2017).
  * [72] P. A. Höhn, M. Krumm, and M. P. Müller, _Internal quantum reference frames for finite Abelian groups_ , arXiv:2107.07545 [quant-ph].
  * [73] B. Simon, _Representations of Finite and Compact Groups_ , American Mathematical Society, 1996.
  * [74] K. R. Davidson, _C ∗-Algebras by Example_, American Mathematical Society, 1996.
  * [75] A. Savage, _Modern Group Theory_ , lecture notes, University of Ottawa, 2017. Available at https://alistairsavage.ca/mat5145/notes/MAT5145-Modern_group_theory.pdf
  * [76] P. A. Höhn, _Reflections on the information paradigm in quantum and gravitational physics_ , J. Phys. Conf. Ser. 880, 012014 (2017).
  * [77] M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ , Cambridge University Press, New York, 2010.
  * [78] M. Tinkham, _Group Theory and Quantum Mechanics_ , Dover Publications, 1992.
  * [79] S. Khandelwal, M. P. E. Lock, and M. P. Woods, _Universal quantum modifications to general relativistic time dilation in delocalised clocks_ , Quantum 4, 309 (2020).
  * [80] A. R. H. Smith and M. Ahmadi, _Quantum clocks observe classical and quantum time dilation_ , Nat. Commun. 11, 5360 (2020).
  * [81] P. T. Grochowski, A. R. H. Smith, A. Dragan, and K. Debski, _Quantum time dilation in atomic spectra_ , Phys. Rev. Research 3, 023053 (2021).
  * [82] R. Gambini and J. Pullin, _The Montevideo Interpretation: How the inclusion of a Quantum Gravitational Notion of Time Solves the Measurement Problem_ , Universe 6, 236 (2020).
  * [83] D. N. Page, and W. K. Wootters, _Evolution without evolution: Dynamics described by stationary observables_ , Phys. Rev. D 27, 2885 (1983).
  * [84] W. K. Wootters, _“Time” replaced by quantum correlations_ , Int. J. Theor. Phys. 23, 701 (1984).
  * [85] V. Giovannetti, S. Lloyd, and L. Maccone, _Quantum time_ , Phys. Rev. D 92, 045033 (2015).
  * [86] A. R. H. Smith and M. Ahmadi, _Quantizing time: interacting clocks and systems_ , Quantum 3, 160 (2019).
  * [87] E. Moreva, G. Brida, M. Gramegna, V. Giovannetti, L. Maccone, and M. Genovese, _Time from quantum entanglement: An experimental illustration_ , Phys. Rev. A 89, 052122 (2014).