#  $\mathrm{C}_{60}$+$\mathrm{C}_{60}$ molecular bonding revisited and
expanded

Jorge Laranjeira<EMAIL_ADDRESS>Departamento de Física and CICECO,
Universidade de Aveiro, 3810-193 Aveiro, Portugal  Karol Strutyński
Departamento de Química and CICECO, Universidade de Aveiro, 3810-193 Aveiro,
Portugal  Leonel Marques  Departamento de Física and CICECO, Universidade de
Aveiro, 3810-193 Aveiro, Portugal  Emilio Martínez-Núñez  Departamento de
Química Física, Universidade de Santiago de Compostela, 15782, Santiago de
Compostela, Spain  Manuel Melle-Franco<EMAIL_ADDRESS>Departamento de
Química and CICECO, Universidade de Aveiro, 3810-193 Aveiro, Portugal

###### Abstract

Several dimerization products of fullerene $\mathrm{C}_{60}$ are presented and
thoroughly characterized with a quantum chemical DFT model augmented by
dispersion. We reanalyze and expand significantly the number of known dimers
from 12 to 41. Many of the novel bonding schemes were found by analyzing more
than 2 nanoseconds of high energy molecular dynamics semiempirical
trajectories with AutoMeKin, a methodology previously used to compute the
reactivity of much smaller molecules. For completeness, this was supplemented
by structures built by different geometric considerations. Also, spin-
polarization was explicitly considered yielding 12 new bonding schemes with
magnetic ground states. The results are comprehensively analyzed and discussed
in the context of yet to be explained 3D fullerene structures and recent
fullerene 2D systems.

Fullerene Dimers, DFT calculations, AutoMeKin, HOMO-LUMO Gap

††preprint: APS/123-QED

## Introduction

At room temperature and atmospheric pressure, $\mathrm{C}_{60}$ is a van der
Waals solid with a face-centered cubic (fcc) structure where the molecules are
rotating freely [1]. When subjected to visible or ultraviolet light, or high-
pressures high-temperatures (HPHT) treatments the $\mathrm{C}_{60}$ molecules
bond to each other via 66/66 2+2 cycloaddition forming aggregates and dimers
[2, 3, 4], see figure 1.

Figure 1: Two views of the most stable and well studied $\mathrm{C}_{60}$
dimer with molecules bonded via 66/66 2+2 cycloaddition.

Increasing the pressure and temperature of the $\mathrm{C}_{60}$ HPHT
treatment leads to the formation of extended crystalline networks [5, 6, 7, 8,
9, 10, 11]. At pressures below 8 GPa the formation of low-dimensionality
polymers occurs yielding 1D orthorhombic, 2D tetragonal and 2D rhombohedral
phases. In all these low-dimensionality polymers, the molecules arrange
themselves in covalently bonded chains (1D) or sheets (2D) formed exclusively
by 66/66 2+2 cycloadditions.

It is above 8 GPa that fully 3D polymerized crystals are formed [8, 9, 10,
11]. Here other bonding schemes, besides the 66/66 2+2 cycloaddition, start to
play an important role, in fact, 3D polymers proposed to date have different
bonding schemes such as the 56/56 2+2 cycloaddition [7, 8], the 6/6 3+3
cycloaddition and the double 66/66 4+4 cycloaddition [9], the 56/65 2+2
cycloaddition and the 5/5 3+3 cycloaddition [10] or even the double 5/5 2+3
cycloaddition [11].

Recently, other $\mathrm{C}_{60}$ polymers have been produced from a different
route [12, 13]. This has yielded a pure carbon bulk material based on
covalently linked fullerenes forming 2D sheets with hexagonal symmetry dubbed
graphurelene [13]. These layers are bonded via single-bond and 56/65 2+2
cycloaddition, unlike any 2D polymers synthesized to date by HPHT treatment,
showcasing the potential of alternative synthetic routes to produce new
structures.

3D $\mathrm{C}_{60}$ polymers show quite remarkable physical properties for
pure carbon phases, such as metallicity and low-compressibility which fosters
interest in these materials [8, 9, 10, 11]. Nevertheless, diffraction patterns
from these phases lack the resolution needed to solve their crystalline
structure. Thus, it is necessary to build complementary computational models
fitting the overall symmetry and lattice parameters to the diffraction pattern
and study their stability recurring to quantum simulation tools like Density
Functional Theory (DFT) [7].

The distances between the two $\mathrm{C}_{60}$ covalently bonded molecules,
i.e. a $\mathrm{C}_{60}$ dimer, computed from their molecular centers are
similar to the corresponding distance in a bulk crystalline phase. For
instance, the $\mathrm{C}_{60}$ molecular distance, we compute for the 66/66
2+2 cycloaddition dimer, is $\sim$9.1 Å while in the low-dimensionality
polymers it is found to be between 9.02 and 9.18 Å [5, 6]. Also, for the
recent graphulerene [12, 13], the computer dimer distances are 9.23 and 9.09 Å
while the experimental bonding distance of 9.23 and 9.16 Å respectively.
Hence, relating bonding distances to bonding schemes constitutes a simple yet
accurate interpretation tool for the $\mathrm{C}_{60}$ crystal structure
assignment, rendering covalent $\mathrm{C}_{60}$ dimers specifically relevant
to compute and understand.

In the 90’s and beginning of the 2000’s, several studies regarding dimers were
presented, nevertheless, to the best of our knowledge, no more than 12 dimers
have been described altogether. Here, We consider, unless noted,
$\mathrm{C}_{60}$ covalent dimers, that is dimers where the forming fullerene
molecules are joined by intermolecular bonds. Strout et al. [4] performed
probably the most extensive $\mathrm{C}_{60}$ dimerization study which
addressed the 2+2 cycloadditions and a 2+4 cycloaddition. This last bonding
scheme, following our notation, corresponds to a double 6/56 4+2
cycloaddition, since two 2+4 cycloaddition reactions, involving one hexagon
(6) and one intramolecular single bond (56) each, are present.

Later, in 1994, Adams et al. [14] proposed three other bonding schemes, two
bonding the molecules via their hexagons and one between their pentagons. In
the bonding between hexagons, the six atoms from one hexagon bond to the six
atoms of the other hexagon, which can be obtained in the following two ways:
1) the hexagons of the two molecules align with each other (as well as the
pentagons); 2) the hexagons from one molecule align with the pentagons of the
other. When possible, we will keep the cycloaddition notation for the formed
dimers nomenclature thus, in this case the dimers will be named, for instance,
”triple 66/66 2+2 cycloaddition” (or ”triple 56/56 2+2 cycloaddition”) and
”triple 56/66 2+2 cycloaddition” respectively. In the bonding between
pentagons the five atoms from one pentagon bond to the five atoms of the
other, the dimer will be referred to as ”double 56/56 2+2 cycloaddition plus
single bond”. More details about this bonding nomenclature are available in
the supporting information (SI) section A. In 2007 Liu et al. [15] studied
again the two dimers bonded via hexagons confirming that they are
thermodynamically stable and thus, could be observed experimentally.

Osawa et al. [16] reported the two possible 6/6 4+4 cycloaddition bonded
dimers, while Matsuzawa et al. [17] also considered the 6/6 4+4 cycloaddition
dimers and added one of the two 66/6 2+4 cycloaddition dimers to their
calculations. Besides the mentioned dimers, the single bond dimer (SB) was
also considered [18]. In all of the works mentioned so far, the lowest energy
dimer was consistently found to be the 66/66 2+2 cycloaddition [4, 18, 15,
14].

Apart from dimers formed just with pristine, i.e. unfunctionalized,
$\mathrm{C}_{60}$, there have also been numerous studies of dimers formed with
other intermediates as well as coalescence products [19, 4, 20]. In addition,
there have been studies on the dimerization and coalescence of
$\mathrm{C}_{60}$ inside carbon nanotubes [5, 21, 22]. Note that although some
of these systems also did appear in our calculations, for simplicity, they
were not considered in our analysis and will not be discussed. Nevertheless,
coalescence products might improve the understanding of amorphous carbon phase
formation from $\mathrm{C}_{60}$ [23] or other phases were the
$\mathrm{C}_{60}$ cage is broken [24] and might be the object of future
studies.

The new systems reported here have been obtained fundamentally with AutoMeKin
[25, 26, 27], an automatic computational procedure to find and characterize
chemical reaction paths, and extended by manual construction following
geometric considerations. We report a total of 41 stable dimers computed at
DFT level. From these, 25 were obtained with AutoMeKin, the others were built
from geometric considerations or obtained from the literature. The molecular
structure and relative stability of all dimers were computed at the TPSS-
Def2-TZVPP/B3LYP-6-31G(d,p) theory level augmented by D3 dispersion with
Becke–Johnson damping. All the dimers studied present $\mathrm{C}_{60}$
distances between 8.04 Å and 9.23 Å which renders some of these bridging
patterns suitable candidates to be present in 3D $\mathrm{C}_{60}$ polymers.

## Methods

First, we performed a benchmark calculation of the dimer bonding energies for
four different Hamiltonians, namely: TPSS-
Def2-TZVPP-D3BJ/TPSS-6-31G(d,p)-D3BJ, TPSS-
Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ, TPSS-6-31G(d,p)-D3BJ and
B3LYP-6-31G(d,p)-D3BJ, on the lowest five energy $\mathrm{C}_{60}$ dimers,
table LABEL:tcomp_tpssb3lyp. The TPSS-Def2-TZVPP-D3BJ Hamiltonian was chosen
after a recent benchmark study comparing the $\mathrm{C}_{60}$ isomerization
energies with 115 methodologies which recommended this method based on its
accuracy [28]. The bonding energy was computed by subtracting the energy of
two isolated $\mathrm{C}_{60}$ to the energy of each dimer: $\Delta
E_{Bond}=E_{dimer}-2E_{\mathrm{C}_{60}}$.

Table 1: Bonding energy of the five lowest energy $\mathrm{C}_{60}$+$\mathrm{C}_{60}$ dimers optimized with different levels of theory given in eV/dimer. Theory level | Min1 | Min2 | Min3 | Min4 | Min5
---|---|---|---|---|---
B3LYP-6-31G(d,p)-D3BJ | -0.6541 | 0.2055 | 0.5839 | 1.0571 | 1.0934
TPSS-6-31G(d,p)-D3BJ | -0.5415 | 0.2125 | 0.5304 | 0.9635 | 0.9904
TPSS-Def2-TZVPP-D3BJ/TPSS-6-31G(d,p)-D3BJ | -0.2512 | 0.4931 | 0.7398 | 1.2428 | 1.2676
TPSS-Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ | -0.2494 | 0.4989 | 0.7440 | 1.2501 | 1.2771
Table 2: Bonding energy of the non-covalent $\mathrm{C}_{60}$+$\mathrm{C}_{60}$ dimer optimized with different levels of theory and corrected by the basis set superposition error (BSSE), given in eV/dimer. Theory level |  $\Delta\mathrm{E}_{Bond}$ (eV/dimer) |  $\Delta\mathrm{E}_{Bond}$\+ BSSE (eV/dimer)
---|---|---
B3LYP-6-31G(d,p)-D3BJ | -0.4569 | -0.3512
TPSS-Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ | -0.3604 | -0.3460

The Basis Set Superposition Error (BSSE) was evaluated in a non-covalent
dimer, and found to be sizable, around 0.1 eV, for bonding energies with the
smaller, DZP, basis sets, table LABEL:bsse. After the BSSE correction, both
levels of theory yielded values differing less than 5 meV. Interestingly,
relative energies were very similar, and also similar results were found for
all DFT functionals. Considering this, we selected the TPSS-
Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ methodology as it yields a good
compromise between accuracy and computational cost. Namely, we performed the
geometry optimizations at the B3LYP-6-31G(d,p) [29, 30] level augmented by the
D3 van der Waals correction [31, 32] with Becke–Johnson damping [33]
(B3LYP-6-31G(d,p)-D3BJ) followed by a single point calculation with TPSS-
Def2-TZVPP Hamiltonian [34, 35] also augmented by the D3 van der Waals
correction with Becke–Johnson damping (TPSS-Def2-TZVPP-D3BJ) for the energies.
Hessian calculations were performed in all presented geometries at the
B3LYP-6-31G(d,p)-D3BJ level to explicitly check the absence of negative
frequencies. All calculations were performed with Gaussian09 [36].

We explicitly analyzed the dispersion contribution to the bonding energy, as
this contribution is missing in most early DFT and semiempirical studies. The
dispersion correction shifts bonding energies to more binding values, for the
B3LYP-6-31G(d,p) functional the maximum shift is 1.34 eV and the average one
is 0.78 eV. For the TPSS-Def2-TZVPP functional the stabilization is lower, the
maximum shift is 1.15 eV and the average 0.68 eV. This indicates that there is
a consistent attractive dispersion contribution to the bonding interaction in
all covalently bonded $\mathrm{C}_{60}$ dimers. In fact, without van der Waals
correction, all dimers are metastable compared to the isolated
$\mathrm{C}_{\mathrm{60}}$ molecule, see SI table S1 and figures S1 and S2.

To systematically explore possible minima we employed AutoMeKin (AMK) [25, 26,
27], a software designed to find reaction mechanisms. AMK analyzes high-energy
molecular dynamics trajectories via graph theory to search for structural
transformations. When a structural transformation, such as bonds breaking or
forming, is detected it finds and characterizes the transition state via its
second-order derivatives. From the transition state, the intrinsic reaction
coordinate (IRC) method is then used to obtain reactants and products. For
this, 2.25 ns were run altogether via 4500, 500 fs long, independent molecular
dynamics of two $\mathrm{C}_{60}$ molecules in a high energy bonding
configuration. The dynamics were run with a specially modified version of
MOPAC2016 [37] using the PM7 [38] Hamiltonian with a time step of 0.5 fs. Note
that AMK so far has been used in molecules $\leq 30$ atoms [39], so this
study, on a system with 120 atoms, showcases how the AMK methodology can be
successfully extended to other problems.

In addition, for completeness, we have comprehensively searched for reported
dimers in literature and also constructed other bonding schemes considering
the geometric possibilities allowed by the molecule. In the tables, the
following labels reveal the procedure followed for each minimum: L,
literature, G, geometric considerations and AMK. To build dimers from
geometric considerations, we went through the possibilities of bonding between
each molecular element of geometry avoiding large deformations. Three geometry
elements: chemical bonds, pentagons, or hexagons were considered with a
maximum of two of these elements being combined at a time. For instance, the
two ”5/6 3+3” bonds may be constructed taking into consideration the bridging
of one pentagon to one hexagon while the ”double 66/66 4+4” takes into
consideration the bridging of two hexagons from each molecule. For simplicity,
we only considered systems with an even number of intermolecular bonds in this
procedure.

We present the HOMO-LUMO gaps at the B3LYP-6-31G(d,p) level as this
Hamiltonian is expected to yield closer values to experiment [40]. Note that
all calculations were first run without spin polarization, and, in the cases
where structures were found to be unstable under these conditions, they were
then run considering triplet or quintuplet ground states. This procedure
yielded 12 dimers with spin-polarized ground states. In all cases, no initial
symmetry constraints were used in the geometry optimizations.

## Results and Discussion

The dimer structures found are comprehensively summarized in table LABEL:at1
by increasing energy with their symmetry, number of intermolecular bonds,
HOMO-LUMO gap, bonding energy, the distance between $\mathrm{C}_{60}$
molecules and a label indicating the minima origin. A graphical rendering for
all the bonding schemes found on these dimers together with some others that
were found by AMK but are not stable at the DFT level, can be found on the SI,
table S3. Relevantly, of the 12 dimers previously reported just the 6/66 2+4
cycloaddition reported by Matsuzawa et al. [17] was found not to be stable in
our study, although it was found by AMK (AMK minimum 3097) indicating that at
the semiempirical PM7 level, this dimer is stable.

Only one covalent dimer with a bonding energy of $-0.25$ eV was found to be
thermodynamically more stable, i.e. exothermic, than the isolated
$\mathrm{C}_{60}$ molecules. In comparison, the less stable dimer is
considerably more endothermic with a bonding energy of 7.95 eV. As a
reference, we also computed the energy of the non-covalent dimer, $-0.36$ eV,
which is more stable than the isolated molecule and any covalent dimer. These
results match experimental observations since the only, experimentally
observed, covalent dimer is fruit of the 66/66 2+2 cycloaddition [2, 3, 4] and
at ambient conditions, $\mathrm{C}_{60}$ molecules are not covalently bound in
the solid state. More exotic bonding configurations, for instance,
corresponding to the 56/56 2+2 cycloaddition [8], can be found in three-
dimensional crystals formed under high-pressure treatments, where the 66/66
2+2 cycloaddition bonding scheme alone cannot be used to bond all molecules in
a three-dimensional extended system.

Table 3: Energetics and distances of $\mathrm{C}_{60}$+$\mathrm{C}_{60}$ dimers computed at TPSS-Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ level. The HOMO-LUMO gaps presented were computed with B3LYP-6-31G(d,p)-D3BJ with * indicating that the presented value is an average of the $\alpha$ and $\beta$ spin channels gaps (table S2 in SI has the used values). Letter L indicates a structure found in Literature while AMK and G indicate the structure was found using, respectively, AutoMeKin or geometric considerations. The number that follows AMK is an internal reference produced by AMK during the generation. Minima | Dimer | Sym. | interm. | HOMO-LUMO | $\Delta\mathrm{E}_{Bond}$ | Mol. | bond
---|---|---|---|---|---|---|---
order | name |  | bonds | gap (eV) | (eV/dimer) | dist. Å | origin
| $\mathrm{C}_{60}$ | $\mathrm{I}_{\mathrm{h}}$ | 0 | 2.8523 | - | - | -
| Non covalent | $\mathrm{C}_{\mathrm{1}}$ | 0 | 2.59561 | -0.3604 | 9.88 | -
Min1 | 66/66 2+2 | $\mathrm{D}_{2\mathrm{h}}$ | 2 | 2.50562 | -0.2494 | 9.05 | AMK66/L [4, 14, 18, 16, 17]
Min2 | 56/66 2+2 | $\mathrm{C}_{\mathrm{s}}$ | 2 | 1.81935 | 0.4989 | 9.07 | AMK203/L [4]
Min3 | SB | $\mathrm{C}_{2\mathrm{h}}$ | 1 | 1.9126* | 0.7440 | 9.23 | L [18]
Min4 | 56/65 2+2 | $\mathrm{C}_{2\mathrm{h}}$ | 2 | 1.69636 | 1.2501 | 9.09 | AMK1473/L [4]
Min5 | 56/56 2+2 | $\mathrm{C}_{2\mathrm{v}}$ | 2 | 1.74996 | 1.2771 | 9.10 | AMK1818/L [4]
Min6 | double 66/66 2+2 | $\mathrm{C}_{2\mathrm{v}}$ | 4 | 2.49311 | 1.4683 | 8.64 | G
Min7 | double 56/66 2+2 M2 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 2.45719 | 1.8367 | 8.65 | AMK1269
Min8 | 5/66 3+2 | $\mathrm{C}_{\mathrm{s}}$ | 2 | 1.8779* | 2.1629 | 8.91 | AMK3333
Min9 | double 56/56 2+2 | $\mathrm{C}_{2\mathrm{v}}$ | 4 | 2.34045 | 2.2527 | 8.66 | G
Min10 | AMK5678 | $\mathrm{C}_{\mathrm{1}}$ | 3 | 1.5557* | 2.3363 | 8.85 | AMK5678
Min11 | 6/66 3+2 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 0.74614 | 2.3867 | 8.87 | AMK3615
Min12 | 6/6 4+4 M1 | $\mathrm{C}_{2\mathrm{v}}$ | 2 | 1.97527 | 2.6221 | 8.81 | AMK2506/L [16, 17]
Min13 | triple 66/65 2+2 | $\mathrm{D}_{3\mathrm{d}}$ | 6 | 2.39487 | 2.6419 | 8.44 | L [14, 15]
Min14 | 6/6 4+4 M2 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 1.79731 | 2.6441 | 8.81 | AMK2507/L [16, 17]
Min15 | triple 66/66 2+2 | $\mathrm{D}_{3\mathrm{h}}$ | 6 | 2.4327 | 2.6783 | 8.44 | L [14, 15]
Min16 | 6/6 3+4 M2 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 1.8916* | 2.8918 | 8.80 | AMK4004
Min17 | 6/6 3+3 M3 | $\mathrm{C}_{\mathrm{2}}$ | 2 | 1.8632* | 3.1388 | 8.77 | AMK5325
Min18 | double 56/66 2+2 M1 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 1.56792 | 3.2546 | 8.69 | AMK2605
Min19 | 5/6 3+4 M2 | $\mathrm{C}_{\mathrm{s}}$ | 2 | 1.01063 | 3.3104 | 8.87 | AMK4307
Min20 | 5/6 3+4 M1 | $\mathrm{C}_{\mathrm{s}}$ | 2 | 1.01798 | 3.3326 | 8.87 | AMK4321
Min21 | 6/56 3+2 M2 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 1.6369* | 3.3799 | 8.87 | AMK4646
Min22 | 6/56 3+2 M1 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 1.6207* | 3.4201 | 8.87 | AMK4567
Min23 | double 56/65 2+2 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 1.52057 | 3.6534 | 8.71 | AMK3018
Min24 | 5/5 3+3 M1 | $\mathrm{C}_{2\mathrm{h}}$ | 2 | 1.6465* | 3.6760 | 8.89 | G
Min25 | 6/6 3+4 M1 | $\mathrm{C}_{\mathrm{1}}$ | 2 | 1.3896* | 3.9138 | 8.77 | AMK3893
Min26 | double 56/56 2+2 | $\mathrm{C}_{2\mathrm{v}}$ | 4 | 1.22506 | 4.4948 | 8.75 | G
Min27 | AMK6163 | $\mathrm{C}_{1}$ | 3 | 1.8342* | 4.6261 | 8.41 | AMK6163
Min28 | double 56/56 2+2 plus SB | $\mathrm{D}_{5\mathrm{h}}$ | 5 | 0.74260 | 4.8285 | 8.68 | L [14]
Min29 | double 6/56 4+2 | $\mathrm{C}_{2\mathrm{h}}$ | 4 | 2.34100 | 5.0253 | 8.62 | L [4]
Min30 | 5/56 3+2 plus 6/56 4+2 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 1.52955 | 5.1168 | 8.67 | G
Min31 | double 5/5 2+3 | $\mathrm{C}_{2\mathrm{h}}$ | 4 | 1.10995 | 5.2631 | 8.72 | G
Min32 | 5/6 3+3 plus 56/6 2+3 | $\mathrm{C}_{1}$ | 4 | 1.4211* | 5.5353 | 8.62 | AMK5775
Min33 | AMK7077 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 0.87974 | 6.3159 | 8.29 | AMK7077
Min34 | AMK6511 | $\mathrm{C}_{1}$ | 4 | 1.68194 | 6.4343 | 8.28 | AMK6511
Min35 | AMK7610 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 0.95349 | 6.7170 | 8.25 | AMK7610
Min36 | 5/5 3+3 plus 6/6 4+4 | $\mathrm{C}_{2\mathrm{v}}$ | 4 | 0.76546 | 6.8452 | 8.05 | G
Min37 | double 56/65 3+4 | $\mathrm{C}_{2\mathrm{h}}$ | 4 | 0.7584* | 6.9087 | 8.10 | G/AMK_or
Min38 | 5/6 3+4 plus 6/6 4+4 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 0.94070 | 6.9973 | 8.04 | G
Min39 | double 66/66 4+4 | $\mathrm{D}_{2\mathrm{h}}$ | 4 | 1.64030 | 7.0261 | 8.05 | L
Min40 | AMK8161 | $\mathrm{C}_{\mathrm{s}}$ | 4 | 2.51216 | 7.5823 | 8.28 | AMK8161
Min41 | double 66/66 4+4 | $\mathrm{D}_{2\mathrm{h}}$ | 6 | 2.01174 | 7.9499 | 8.16 | G
| plus 66/66 2+2 |  |  |  |  |  |
Figure 2: a) Intermolecular distances as a function of the bonding energy
computed at TPSS-Def2-TZVPP-D3BJ/B3LYP-6-31G(d,p)-D3BJ level of theory. Red
circles indicate open-shell ground states and blue full circles closed-shell
ground states. Red lines and green regions correspond to the experimental
molecular distances reported by Yamanaka et al.[9] and Buga et al.[41]
respectively. b) Asphericity as a function of the bonding energy. Blue cruxes
and red x represent the two different $\mathrm{C}_{60}$ molecules in each
dimer.

Relevantly, with more extreme pressures, higher energy minima can be reached
with reduced $\mathrm{C}_{60}$ distances, fitting the trend shown in figure 2
a) where the distances between molecules show a monotonous decrease with
increasing bonding energies. Furthermore, several of our dimers have shorter
distances than reported coalescence products, see for instance the report by
Strout et al. [4]. This indicates, that very short distances, below 8.3 Å, may
alternatively be explained by $\mathrm{C}_{60}$ molecules bonded by the
patterns reported here. Also, it is important to notice that although all the
bonding schemes yield lower HOMO-LUMO gaps compared to $\mathrm{C}_{60}$, no
clear trend between this quantity and energy, number of bonds or
intermolecular distance was found.

Interestingly, some of the found dimers are minima only with spin-polarized,
triplet or quintuplet, wave-functions. If the minimization is performed
without spin-polarization, the dimers either break into $\mathrm{C}_{60}$
molecules or transform into the more stable 66/66 2+2 cycloaddition dimer.
Dimers with spin-polarized ground states, in table LABEL:at1, are marked with
a star (*) in the HOMO-LUMO value. In addition, the value reported corresponds
to the average gap for $\alpha$ and $\beta$ electrons, all values are
available on the SI, table S2. Minima 3, 8, 10, 27 and 37 have triplet ground
states while minima 16, 17, 21, 22, 24, 25 and 32 have quintuplet ground
states. This sheds light on the potential importance of spin-polarization in
high-pressure polymerized $\mathrm{C}_{60}$ since some of these bonds may be
present in high-pressure formed structures and thus, models built with them
probably require spin-polarized Hamiltonians to be properly accounted for. In
fact, Bernasconi et al. [42] took this into consideration while computing
three-dimensional $\mathrm{C}_{60}$ polymers noticing that some of the modeled
structures had more stable spin-polarized ground states.

After the dimer formation, all the $\mathrm{C}_{60}$ molecules present some
degree of deformation with respect to their original structure. To quantify
the degree of molecular deformation we computed the asphericity of each
molecule defined as the largest difference between the distance to the
molecular center of mass for each atom and the same distance obtained for the
pristine $\mathrm{C}_{60}$, 3.51 Å. Asphericity values are summarized in table
4, relevantly, the dimers with higher symmetries than
$\mathrm{C}_{\mathrm{s}}$ have the same degree of deformation in both
molecules, i.e. the same asphericity. In this case, asphericity values range
from 0.25 Å for minima 13 and 15 to 0.75 Å for minima 37 and 39. Figure 2 b)
shows the asphericity values as a function of the bonding energy. The
asphericity qualitatively increases with increasingly endothermic bonding
energies, indicating how higher deformation is clearly related to higher
energetics and to the more extreme conditions needed for the synthesis of
structures displaying these bonding schemes. In fact, the largest asphericity
is found for one of the most energetic dimers, minimum 40, with a value of
1.30 Å.

Minima | asphericity | asphericity
---|---|---
| of molecule 1 (Å) | of molecule 2 (Å)
Min1 | 0.3018 | 0.3018
Min2 | 0.3034 | 0.3431
Min3 | 0.2977 | 0.2977
Min4 | 0.3439 | 0.3439
Min5 | 0.3470 | 0.3470
Min6 | 0.3889 | 0.3889
Min7 | 0.3831 | 0.4056
Min8 | 0.3376 | 0.4389
Min9 | 0.3980 | 0.3980
Min10 | 0.3406 | 0.3617
Min11 | 0.3365 | 0.5613
Min12 | 0.4364 | 0.4364
Min13 | 0.2504 | 0.2504
Min14 | 0.4421 | 0.4421
Min15 | 0.2513 | 0.2513
Min16 | 0.4239 | 0.4353
Min17 | 0.4334 | 0.4334
Min18 | 0.3909 | 0.3971
Min19 | 0.4140 | 0.4147
Min20 | 0.4191 | 0.4241
Min21 | 0.3786 | 0.5421
Min22 | 0.5351 | 0.3725
Min23 | 0.3899 | 0.4091
Min24 | 0.3826 | 0.3826
Min25 | 0.3717 | 0.4273
Min26 | 0.3726 | 0.3726
Min27 | 0.8917 | 0.4637
Min28 | 0.2869 | 0.2869
Min29 | 0.4599 | 0.4599
Min30 | 0.6149 | 0.4343
Min31 | 0.5692 | 0.5692
Min32 | 0.5916 | 0.4099
Min33 | 1.0139 | 0.5077
Min34 | 0.5109 | 1.1206
Min35 | 0.5816 | 1.0315
Min36 | 0.7341 | 0.7341
Min37 | 0.7595 | 0.7595
Min38 | 0.7623 | 0.7786
Min39 | 0.7520 | 0.7520
Min40 | 0.6009 | 1.3002
Min41 | 0.3764 | 0.3764
Table 4: Asphericities of each molecule in the
$\mathrm{C}_{60}$+$\mathrm{C}_{60}$ dimers.

AMK was crucial in finding exotic bonding schemes, like Minima 33 and 35 where
one of the molecules is pinched allowing the bonding between one pentagon (for
minimum 33 or hexagon for minimum 35) of one molecule to one hexagon and one
pentagon of the other molecule, see figures 3 a) and b). This pinching implies
a high asphericity value in the pinched molecule, $\sim$1.0 Å, while the other
molecule has nearly half that value in both dimers, see table 4. In both
cases, this translates to remarkably short distances, $\sim$8.25 Å, and is
illustrative for dimers with short bonding distances, $\leq 8.4$ Å.
Furthermore, AMK found another peculiar bond, the double 5/5 2+3 cycloaddition
(minimum 31), that was already present in a novel clathrate structure with
very similar distances to the ones presented here [11].

Buga et al. [41] presented a series of six experimental crystalline phases
obtained by pressurizing $\mathrm{C}_{60}$ between 11.5 and 13 GPa. In their
report, structures are body-centered orthorhombic with the Immm space group
($\Gamma^{v}_{o}D^{25}_{2h}$ in Schönflies notation) and presented their
lattice parameters although no actual structure was proposed. Since their
structures are body-centered orthorhombic, each of them has three distinct
$\mathrm{C}_{60}$ distances, table 5. In figure 2 a), we show a green area
where these intermolecular distances lie. The longest distance found in this
study is $\sim$9.2 Å, in minimum 3. Note that as distances by different
computational methods may differ, if slightly, we have considered a $\pm 0.2$
Å of length ”flexibility” to our intermolecular distance values. In addition,
molecules may also have additional deformations while in the crystalline
phase, therefore hindering the direct correspondence between dimer distances
and crystalline distances. This is observed, for instance, in the 66/66 2+2
cycloaddition dimer that shows different experimental distances lying between
9.02 and 9.18 Å [5, 6]. Hence, experimental intermolecular distances larger
than 9.3 Å in table 5, may be due to either a single bond or no bond between
neighboring molecules. Interestingly, several experimental distances between
8.65 and 8.80 Å in table 5 are common distances in our dimers. Considering
this, it seems likely that some of our bonding schemes are responsible for the
distances found in these experimental phases.

Figure 3: Two distinct views of selected exotic bonding dimers found by AMK. a) Dimer AMK7077, min 33. b) Dimer AMK7610, min35. Sample Nº | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---
distance 1 (Å) | 9.07 | 8.73 | 8.69 | 8.68 | 8.67 | 8.65
distance 2 (Å) | 9.47 | 9.16 | 8.88 | 8.84 | 8.81 | 8.78
distance 3 (Å) | 9.19 | 9.05 | 8.88 | 8.85 | 8.83 | 8.79
Table 5: Intermolecular distances from the structures reported in [41].

In addition, there are other structures in the literature still to be
resolved, like the one reported by Yamanaka et al. [9]. Although a complete
crystalline structure was proposed, the structure changed considerably when
optimized with DFT without constraints [43, 42]. Namely, Yamanaka et al.
derived experimental lattice parameters of 7.86, 8.59 and 12.73 Å and a body-
centered orthorhombic structure, with corresponding distances of 7.86, 8.59
and 8.62 Å, indicated in figure 2 a) by the red lines. The authors assigned
these distances to double 66/66 4+4, no bond and 6/6 3+3 cycloaddition,
respectively. Indeed, the 6/6 3+3 cycloaddition dimer, Min 17, and 66/66 4+4
cycloaddition dimer, min 39, have bonding distances in the appropriate range
to explain this experimental data. Nevertheless, these bonding schemes were
already used by Zipoli and Bernasconi [42] to construct crystalline structures
and fail to reproduce the experimental lattice cell. Thus, other bonding
schemes must be considered. We present here several potential alternatives
within the observed range of distances.

Conversely, although the comparison of the distance is a simple, yet effective
tool, we would like to emphasize that bonding patterns, which are stable in
dimers may not be stable or not even geometrically possible in specific
crystalline structures. For instance, minimum 4, although geometrically
capable to bond to all the neighboring molecules in a crystalline fcc
structure, turns out to be unstable in such configuration [7, 8]. On the other
hand, the same minimum is present in other reported structures, such as in a
3D-rhombohedral phase [10] and in the recent graphullerene [13], proving that
the molecular environment as a whole has to be considered to evaluate the
existence of these patterns.

## Conclusion

A comprehensive exploration and in-depth characterization of a large number of
$\mathrm{C}_{60}$ dimers built from high energy molecular dynamics
trajectories through the AMK methodology was performed. Several new
$\mathrm{C}_{60}$ bonding schemes potentially relevant to understand polymeric
phases were found and were energetically, structurally and electronically
classified, increasing the number of known $\mathrm{C}_{60}$ dimers from 12 to
41. Some of the new dimers have bonding schemes matching experimental
intermolecular distances observed on high-pressure high-temperature
polymerized $\mathrm{C}_{60}$ yet to be understood, for instance in the works
of Buga et al. or Yamanaka et al. [41, 9]. Relevantly, 12 of these dimers,
i.e. 29 %, have spin-polarized ground states, which emphasizes that spin-
polarization should be explicitly taken into consideration to properly model
crystalline structures, especially with metallic or electronic exotic
properties. This work paves the way for the exploration of novel extended 1D,
2D and 3D structures based on the bonding schemes presented here obtainable by
HPHT treatment or more recent alternative routes [12, 13].

## Author Contributions

Conceptualization: M. Melle-Franco and J. Laranjeira; methodology: M. Melle-
Franco, J. Laranjeira and E. Martínez-Núñez; software: M. Melle-Franco and E.
Martínez-Núñez; validation: L. Marques, K. Strutyǹski and E. Martínez-Núñez;
formal analysis: J. Laranjeira; investigation: J. Laranjeira; resources: M.
Melle-Franco, E. Martínez-Núñez and J. Laranjeira; data curation: J.
Laranjeira; writing-original draft preparation: J. Laranjeira; writing-review
and editing: J. Laranjeira, M. Melle-Franco; visualization: J. Laranjeira;
supervision: M. Melle-Franco; funding acquisition: M. Melle-Franco, E.
Martínez-Núñez and J. Laranjeira. All authors have read and agreed to the
published version of the manuscript.

###### Acknowledgements.

This work was developed within the scope of the project CICECO-Aveiro
Institute of Materials, UIDB/50011/2020, UIDP/50011/2020 & LA/P/0006/2020,
financed by national funds through the FCT/MCTES (PIDDAC) and IF/00894/2015
finances by FCT. J. Laranjeira acknowledges a PhD grant from FCT
(SFRH/BD/139327/2018). This work was partially supported by the Consellería de
Cultura, Educación e Ordenación Universitaria (Grupo de referencia competitiva
ED431C 2021/40), and the Ministerio de Ciencia e Innovación through Grant
#PID2019-107307RB-I00. The work has been performed under the project HPC-
EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research
Innovation Action under the H2020 Programme; in particular, the author
gratefully acknowledges the support of the computer resources and technical
support provided by BSC.

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# Predicting Out-of-Distribution Error with Confidence Optimal Transport

Yuzhe Lu1, Zhenlin Wang1, Runtian Zhai1
Soheil Kolouri2, Joseph Campbell1, Katia Sycara1
1Carnegie Mellon University 2Vanderbilt University
{yuzhelu, zhenlinw, rzhai, jcampbell<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Out-of-distribution (OOD) data poses serious challenges in deployed machine
learning models as even subtle changes could incur significant performance
drops. Being able to estimate a model’s performance on test data is important
in practice as it indicates when to trust to model’s decisions. We present a
simple yet effective method to predict a model’s performance on an unknown
distribution without any addition annotation. Our approach is rooted in the
Optimal Transport theory, viewing test samples’ output softmax scores from
deep neural networks as empirical samples from an unknown distribution. We
show that our method, Confidence Optimal Transport (COT), provides robust
estimates of a model’s performance on a target domain. Despite its simplicity,
our method achieves state-of-the-art results on three benchmark datasets and
outperforms existing methods by a large margin.

##  1 Introduction

While most machine learning models are built with an i.i.d. assumption, they
often have to deal with out-of-distribution samples when deployed in the real
world. This poses serious problems on the trustworthiness of ML models as
modern architectures often give overly confident signals even in the face of
distribution shifts that significantly hurt the model’s performance. Thus, a
lot of works have focused on OOD detection, whose goal is to tell if a test
sample is from a different distribution than the training data. To name a few,
Hendrycks & Gimpel (2016) proposed to use a model’s confidence based on
softmax scores; Liang et al. (2017) found using temperature scaling Guo et al.
(2017) and small perturbations to inputs further improves OOD detection using
softmax scores; more recently, Liu et al. (2020) showed superior performance
by using energy-based scores instead of softmax scores. In a relevant but
slightly different formulation, Platanios et al. (2016; 2017); Jiang et al.
(2021); Garg et al. (2022) considers the task of directly predicting a model’s
generalization performance on OOD data. As annotating data from the target
domain would be both costly and inefficient, especially when the distribution
is non-stationary, it would be ideal to have a good estimator of a model’s
performance on the target domain in an annotation-free manner. However,
previous works have established that obtaining a consistent estimator is
impossible without further assumptions on the nature of the shift David et al.
(2010); Lipton et al. (2018).

In this work, we aim to estimate the model performance under covariate shift,
such as different appearances of the same object, differences in image quality
caused by camera degradation, etc., while assuming the label distribution of
the target domain does not change. With the covariate shift assumption, the
task of predicting model performance with only labeled source data and
unlabeled target data becomes achievable. To deal with this task, numerous
methods have been proposed. Among them, many require some labeled samples from
the target domain. For example, Guillory et al. (2021) and Deng & Zheng (2021)
train regression models based on model-based distribution distances between
the source and target domain to predict test performance. However, these
methods are suboptimal as they assume prior knowledge about the nature of the
shift and obtaining labels from the target domain is very expensive in
general. Meanwhile, more recent methods have focused on predicting target
domain performance without requiring additional labels. Baek et al.  and Jiang
et al. (2021) leverage the agreement rate of independently trained neural
networks to estimate the accuracy. Garg et al. (2022) adopts a thresholding
strategy based on scores of the softmax outputs. These methods have immediate
practical value as people can have a sense of their model’s performance on the
target domain without having to label any data beforehand. Following these
works, we assume access to labeled source data and only unlabeled target
domain data.

Given this setting, we propose Confidence Optimal Transport (COT) to provide
robust estimation of a model’s error on the target domain via the Earth
Mover’s Distance (EMD). While other methods also explored using distribution
distance such as Maximum Mean Discrepancy and Fréchet distance for the task
Guillory et al. (2021), they mainly use first and second order summary
statistics of source and target data features from a model’s penultimate
layer. Moreover, the output distances are not calibrated, meaning labeled data
from the target domain is required to learn an additional regression model,
and often underperform on shifts the regression model was not trained on Garg
et al. (2022). In contrast, our method considers the cost to transform samples
from the source to the target domain, leveraging information from every sample
instead of only the mean and covariance. By representing each sample through
its softmax vector, we empirically show that the resulting EMD between the
source and target data after de-biasing serves as a highly robust estimator of
a model’s performance on the target domain and outperforms previous methods on
three benchmark OOD datasets.

##  2 Method

###  2.1 Optimal Transport and Wasserstein Metric

In recent years, optimal transport theory has found numerous applications in
the field of machine learning. Optimal transport aims to move one distribution
of mass to another as efficiently as possible under a given cost function.
Wasserstein distances are rooted in the optimal transportation problem,
providing a robust mathematical framework for comparing probability
distributions that respect the underlying space geometry. Specifically, the
p-Wasserstein distance between two probability measures $\mu$ and $\nu$ is
defined as:

$W_{p}(\mu,\nu)=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int\|x-y\|^{p}d\pi(x,y)\right)^{1/p}$
(1)

where $\Pi(\mu,\nu)$ is the set of all possible couplings. However, in most
applications, we have to work with empirical measures. Let
$X_{1},...,X_{n}\sim P$, and $Y_{1},...,Y_{m}\sim Q$ be i.i.d. samples, and
define the empirical measures $P_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$
and $Q_{n}=\frac{1}{m}\sum_{i=1}^{m}\delta_{Y_{i}}$. Then the empirical
estimator of the p-Wasserstein distance is:

$W_{p}(P_{n},Q_{m})=\left(\inf_{\pi\in\Pi(P_{n},Q_{m})}\sum_{i,j=1}^{m,n}\|X_{i}-Y_{j}\|^{p}\pi_{ij}\right)^{1/p}$
(2)

which can be solved as a linear assignment problem. In particular, when $p=1$,
the metric (1-Wasserstein distance) is commonly known as the Earth Mover’s
Distance (EMD), and the resulting cost metric between samples is Manhattan
distance. In the following section, we will demonstrate how we leverage EMD to
tackle the problem of estimating a model’s performance on test data.

###  2.2 Confidence Optimal Transport

In this section, we introduce our method, Confidence Optimal Transport (COT),
that leverages the optimal transport framework to predict the performance of a
trained classifier on the target domain consisting of only unlabeled data
$X^{t}=\\{x^{t}_{1:m}\\}$. In its essence, COT uses the empirical estimator of
the Earth Mover’s Distance between labels from the source domain and softmax
outputs of samples from the target domain to provide highly accurate
estimates.

For COT to provide unbiased estimates, the classifier needs to be first
calibrated on a validation set from the source domain. Let $C$ be our
classifier of interest. Given validation set $\\{X^{s},Y^{s}\\}$ from the
source, we say $C$ is perfectly-calibrated if
$C(X^{s})=(\hat{Y^{s}},\hat{P^{s}})$ satisfies the following equation:

$\mathbb{P}(\hat{Y^{s}}=Y^{s}|\hat{P}=p)=p,\quad\forall p\in[0,1]$ (3)

where $\hat{Y^{s}}$ denotes class predictions, $\hat{P^{s}}$ denotes
associated confidences (probability of correctness), and probability $p$ is
over the joint distribution. In practice, Eq. 3 implies that when a model is
perfectly-calibrated, the averaged confidence should be equal to its accuracy.
We used Temperature Scaling proposed in Guo et al. (2017) to perform model
calibration.

Now suppose classifier $C$ is perfectly-calibrated on the source distribution,
and let $S^{t}=\\{softmax(C(x_{i:m}^{t}))\\}$ be the softmax outputs of our
classifier $C$ on the target dataset $X^{t}=\\{x^{t}_{1:m}\\}$. The estimated
error on the target domain $X^{t}$ via COT is defined as:

$\widehat{TestError}=\frac{1}{2}EMD(S^{t},Y^{s})=\frac{1}{2}\min_{\pi\in\Pi(S^{t},Y^{s})}\sum_{i,j=1}^{m,n}\|s_{i}-y_{j}\|\pi_{ij}$
(4)

where $Y^{s}=\\{y^{s}_{1:n}\\}$ are simply labels of the validation data from
the source domain in one-hot format. Thus, while COT computes a distance
metric between the source and target domain, it doesn’t require saving any
samples from the source domain. In fact, instead of using $Y^{s}$, we can even
avoid saving $Y^{s}$ by generating $\tilde{Y}^{s}$ on the fly given knowledge
of the label distribution.

Table 1: We compute coefficients of determination ($R^{2}$) and rank
correlations ($\rho$) to measure the linear correlation between a method’s
output quantity and the actual test error (higher the better). COT achieves
superior performance than all existing methods across different models and
datasets.

Dataset Network AC Entropy GDE ATC ProjNorm COT $R^{2}$ $\rho$ $R^{2}$ $\rho$
$R^{2}$ $\rho$ $R^{2}$ $\rho$ $R^{2}$ $\rho$ $R^{2}$ $\rho$ CIFAR10 ResNet18
0.825 0.980 0.862 0.982 0.842 0.981 0.875 0.987 0.947 0.988 0.996 0.998
ResNet50 0.950 0.995 0.949 0.995 0.959 0.995 0.885 0.989 0.936 0.989 0.993
0.996 VGG11 0.710 0.938 0.762 0.958 0.723 0.948 0.548 0.851 0.756 0.949 0.994
0.993 Average 0.828 0.971 0.858 0.978 0.841 0.975 0.769 0.942 0.880 0.975
0.994 0.996 CIFAR100 ResNet18 0.943 0.987 0.932 0.984 0.950 0.988 0.927 0.985
0.969 0.974 0.995 0.997 ResNet50 0.957 0.987 0.948 0.984 0.962 0.989 0.955
0.991 0.982 0.991 0.992 0.996 VGG11 0.794 0.959 0.821 0.973 0.870 0.978 0.736
0.975 0.653 0.849 0.996 0.997 Average 0.898 0.978 0.900 0.980 0.927 0.985
0.873 0.984 0.868 0.938 0.994 0.997 ResNet18 0.755 0.923 0.664 0.892 0.777
0.913 0.751 0.935 0.973 0.990 0.995 0.998 Tiny ResNet50 0.844 0.975 0.776
0.954 0.892 0.981 0.800 0.962 0.983 0.990 0.994 0.997 ImageNet VGG11 0.563
0.736 0.493 0.684 0.587 0.748 0.746 0.837 0.671 0.825 0.987 0.992 Average
0.727 0.878 0.644 0.843 0.752 0.881 0.766 0.911 0.876 0.935 0.992 0.996

Figure 1: We plot the output metric value against the true test error on
CIFAR-100C using ResNet50. Compared to Entropy and ProjNorm, COT shows
stronger linear correlation.

Next, we will provide some intuitions on Eq. 4 as well as why we require the
classifier to be calibrated. Image a model with $80\%$ accuracy on the source
validation set. When it is calibrated, the model will output softmax scores
such as [0.8, 0.1, 0.1], [0.1, 0.8, 0.1] on samples from the source test set.
When calculating the coupling $\pi$ in 2 to move one distribution to another,
softmax output [0.8, 0.1, 0.1] will be matched to label [1, 0, 0] and [0.1,
0.8, 0.1] to [0, 1, 0]. Thus, when using Manhattan distance (which is ground
cost used by EMD), the cost for a single pair will be $(1-0.8)+(0.2-0)=0.4$,
which is double the error. Thus, we use EMD (1-Wasserstein distance) and
divide it by 2 in Eq. 4. At this point, it should be clear that model
calibration is a necessity: COT will underestimate the error when the model is
overconfident and overestimate when it is underconfident. As models usually
become less confident on out-of-distribution samples Hendrycks & Gimpel
(2016), we expect COT to output larger error estimates on OOD data. Meanwhile,
it is natural to expect COT to outperform simple baselines such as Averaged
Confidence, which only utilizes the average confidence instead of the whole
softmax output like COT does.

In terms of computational complexity, since COT requires calculating the EMD,
its computation time is $\mathcal{O}(n^{3})$. While such complexity seems
rather unscalable at the first glance, we will show in Sec. 3 that COT only
requires 2000 samples to provide state-of-the-art performance. Thus, when
having a large amount of data, we could break it into sufficiently-large
subsets and simply apply COT multiple times to obtain a good estimate.

##  3 Experiments

We evaluated COT on three OOD benchmark datasets and compared it against
several existing methods. COT shows state-of-the-art performance on all
evaluation metrics

#### Dataset

For image classification tasks, we utilized three synthetic-shift datasets,
CIFAR10, CIFAR100, Tiny ImageNet as in-distribution data and their corrupted
versions, CIFAR10-C, CIFAR100-C, and Tiny ImageNet-C Hendrycks & Dietterich
(2018), as out-of-distribution datasets. Each standard dataset contains
several types of common corruptions in their corrupted counterparts such as
change of brightness, existence of fog, occurrence of defocus blur, etc.,
where each corruption type has 5 levels of severity.

Table 2: We use Mean Absolute Error (MAE) to measure the difference between predicted test error and the true test error across different architectures and benchmark datasets. COT, together with AC, GDE, and ATC could provide direct estimates, in contrast to Entropy and ProjNorm, which only demonstrate linear correlation.  Dataset | Network | AC | GDE | ATC | COT
---|---|---|---|---|---
CIFAR10 | ResNet18 | 10.6 | 13.1 | 3.68 | 1.9
| ResNet50 | 6.5 | 9.65 | 3.17 | 1.50
| VGG 11 | 10.4 | 22.5 | 6.94 | 1.94
| Average | 9.2 | 15.1 | 4.6 | 1.8
CIFAR100 | ResNet18 | 11.3 | 14.9 | 3.6 | 2.9
| ResNet50 | 11.2 | 19.2 | 3.5 | 3.3
| VGG 11 | 11.7 | 31.2 | 6.2 | 2.2
| Average | 11.4 | 21.8 | 4.5 | 2.8
TinyImageNet | ResNet18 | 12.45 | 27.5 | 4.48 | 3.43
| ResNet50 | 13.7 | 29.8 | 3.49 | 3.24
| VGG 11 | 12.68 | 43.53 | 7.41 | 2.02
| Average | 12.95 | 33.60 | 5.12 | 2.90
Figure 2: We compare COT’s prediction accuracy compared to several other
methods. The scattered points are estimates against true error. Since output
metric value should be a direct estimate of the test error, we added a dashed
black line denoting $y=x$. We can see that COT provides consistent estimates,
especially when the shifts are large.

#### Architectures

We ran experiments using three different architectures ResNet18, ResNet50 He
et al. (2016), and VGG11Simonyan & Zisserman (2014), for all three datasets.
We reserved $10,000$ images as validation set from each dataset. For each
architecture, we load the weights pretrained on ImageNet1k Russakovsky et al.
(2015), and fine-tune on the target dataset. Since the datasets used in our
experiments have low resolution ($32\times 32$ for CIFAR, and $64\times 64$
for Tiny ImageNet), we upsampled the images to $224\times 224$. After fine-
tuning, we calibrated the model on the validation set using Temperature
Scaling Guo et al. (2017).

#### Methods

We consider five existing methods to compare against COT: Averaged Confidence
(AC) Hendrycks & Gimpel (2016), Entropy Wang et al. (2020), Generalized
Disagreement Equality (GDE) Jiang et al. (2021), Averaged Threshold Confidence
(ATC) Garg et al. (2022), and Projection Norm (ProjNorm) Yu et al. (2022). AC
is a simple baseline that uses the a model’s average confidence to predict
performance. Entropy shows that there is strong linear correlation between the
entropy of a model’s softmax outputs and its performance on OOD data. GDE
shows that we can evaluate a model’s generalization performance through its
disagreement rate with another model independently trained with a different
seed. ATC adopts a thresholding strategy by first learning a threshold of the
score function of a model’s softmax outputs on the validation set and then
using the percentage of samples with scores below the threshold as the
estimated error rate. Finally, ProjNorm shows that a model’s performance on
OOD data is linearly correlated to its difference in parameter space from a
reference model that’s trained with pseudo labels on the target OOD data. It’s
worth noting that Entropy and ProjNorm only demonstrates linear correlation
between the output quantity and the test error, while AC, GDE, ATC, and our
method COT not only show linear correlation but also give a direct estimate
the test error.

#### Evaluation Metrics

We utilized three evaluation metrics following prior works Yu et al. (2022);
Garg et al. (2022). The first two, coefficient of determination ($R^{2}$) and
Spearman’s rank correlation coefficient ($\rho$), measure the linear
correlation between a method’s output quantity and the actual test error. The
third one, Mean Absolute Error (MAE), measures how good a method’s estimate of
the test error is compared to the true error. We adopted $R^{2}$ and $\rho$ as
evaluation metrics as methods such as Entropy and ProjNorm do not give a
direct estimate of the error; rather, they output a quantity that’s linearly
correlated with the error. These methods are less ideal as they require
labeled OOD data to find the line, but still relevant.

#### Results

We summarize the linear correlation results in Table 1. We can see that COT
consistently outperforms all other methods in terms of $R^{2}$ and $\rho$.
Across different datasets and architectures, COT is able to maintain
$R^{2}>0.987$ and $\rho>0.992$. From Fig. 1, we observe that COT gives better
estimates than ProjNorm when the error is small, and outperforms Entropy when
the error is large. In general, we find that when the shifts are large, COT
provides more consistent estimates than AC, Entropy, GDE, and ATC, which often
underestimate the error.

As for the estimation error measured by mean absolute error (MAE), we compare
COT with AC, GDE, ATC, which provide direct error estimates. We summarize the
results in Table 2 and Fig. 2. Again, we see that COT demonstrates notable
improvements over previous methods, especially when encountering large shifts.
In addition, we would like to point out two observations. Firstly, while COT
performs consistent estimates over different architectures, other methods
often have difficulty estimating VGG11’s error despite similar source
performance. This makes COT more broadly applicable to different
architectures. Meanwhile, it would be interesting to further understand the
root of such discrepancy. Secondly, in our experiment setting, GDE seems to
systematically underestimate the test error. The reason is likely that both
models started from the same pretrained weights, so the only stochasticity we
could impose was the training data order. One would expect the two models
fine-tuned from the same pretrained weights might be more similar to each
other than two with different initialization trained from scratch are. We plan
to perform experiments with models trained from scratch in the future, but we
do want to point out that this phenomenon undermines the practicality of GDE
as transfer learning is frequently used to improve models’ performance.

Figure 3: We analyzed the sample complexity of COT and compared with ATC. We
found that COT is more sensitive to the number of test samples, but
outperforms ATC with 1000 or more samples.

#### Sample Complexity

In this section, we answer the question: how many test samples are required
for our method to provide accurate error estimates? We vary the number of test
samples from 100 to 10000 and conduct the experiments using the ResNet50
architecture. The main results are summarized in Fig 3. We can see that COT
can still outperform ATC with as few as 1000 samples. However, COT
deteriorates when there are only 1 or 2 samples for each class (CIFAR-100 and
Tiny-ImageNet). This is expected as an good empirical estimation of
distribution distances would require a sufficient sample size. Empirically,
COT works the best when there are at least 10 samples for each class on
average. By contrast, ATC is insensitive to the number of test samples, which
is unsurprising, since ATC uses a score function to reduce the softmax outputs
to a scalar and adopts a thresholding strategy to estimate the error. To
summarize, COT provides superior error estimates when there are at least 2000
test samples. While computing the EMD generally requires a computation time of
$\mathcal{O}(n^{3})$, current open-source solvers Flamary et al. (2021) can
find the solution of this scale ($n=2000$) within seconds. When we have a
large number of samples from the target domain, we can essentially break them
down into small batches and take the average of batch estimates as the final
estimate.

##  4 Conclusion and Future Work

In this work, we proposed COT, a novel method providing robust estimations of
a model’s performance on out-of-distribution target domain without any
additional annotation. COT consistently outperforms existing methods on three
synthetic shifts datasets across various architectures. For next steps, we
plan to evaluate our method on natural shifts due to difference in the image
collection process. In terms of computational complexity, while solving for
the EMD in COT requires cubic time, we showed that COT only requires a batch
size of 2000 to provide superior error predictions. Thus, simply estimating
with COT on subsets instead of the whole dataset allows us to scale it to a
large number of test samples. In future work, we plan to improve COT’s sample
complexity to make COT more efficient. Finally, we admit that the current
formulation COT is mostly based on heuristics and empirical observations. We
are seeking a more formal understanding of why COT works well.

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# Observing the “quantum Cheshire cat” effect with noninvasive weak
measurement

Yosep Kim  Department of Physics, Pohang University of Science and Technology
(POSTECH), Pohang 37673, Korea  Dong-Gil Im  Department of Physics, Pohang
University of Science and Technology (POSTECH), Pohang 37673, Korea  Yong-Su
Kim  Center for Quantum Information, Korea Institute of Science and Technology
(KIST), Seoul 02792, Korea  Sang-Wook Han  Center for Quantum Information,
Korea Institute of Science and Technology (KIST), Seoul 02792, Korea  Sung
Moon  Center for Quantum Information, Korea Institute of Science and
Technology (KIST), Seoul 02792, Korea  Yoon-Ho Kim<EMAIL_ADDRESS>Department of Physics, Pohang University of Science and Technology (POSTECH),
Pohang 37673, Korea  Young-Wook Cho<EMAIL_ADDRESS>Center for
Quantum Information, Korea Institute of Science and Technology (KIST), Seoul
02792, Korea

###### Abstract

One of the common conceptions of nature, typically derived from the
experiences with classical systems, is that attributes of the matter coexist
with the substance. In the quantum regime, however, the quantum particle
itself and its physical property may be in spatial separation, known as the
quantum Cheshire cat effect. While there have been several reports to date on
the observation of the quantum Cheshire cat effect, all such experiments are
based on first-order interferometry and destructive projection measurement,
thus allowing simple interpretation due to measurement-induced disturbance and
also subject to trivial interpretation based on classical waves. In this work,
we report a genuine experimental observation of the quantum Cheshire cat
effect with noninvasive weak quantum measurement as originally proposed. The
use of the weak-measurement probe has allowed us to identify the location of
the single-photon and that of the disembodied polarization state in a quantum
interferometer. We furthermore elucidate the paradox of the quantum Cheshire
cat effect as quantum interference of the transition amplitudes for the photon
and the polarization state which are directly obtained from the measurement
outcomes or the weak values.

Everyday experiences, typically derived from observing classical systems,
shape up our common conceptions of nature. Quantum effects, on the other hand,
often reveal peculiar counter-intuitive phenomena. Among such counter-
intuitive phenomena, the quantum Cheshire cat effect, in which the quantum
particle itself and its physical property are in spatial separation, most
vividly illustrates the stark difference between the quantum system and the
classical system Aharonov13 . The disembodiment of the physical property
(i.e., the state) from the particle itself is not only conceptually
interesting, but may also provide a novel way to avoid local decoherence on a
certain physical state Richter18 .

According to the original proposal for the quantum Cheshire cat effect
Aharonov13 , the observation of the disembodiment effect requires probing the
particle itself and the disembodied physical state with noninvasive weak
quantum measurement Banaszek01 ; Baek08 . Otherwise, i.e., if observed with
projective measurement, measurement-induced disturbance plays a prominent
role, thus making the quantum Cheshire cat effect argument inconsequential
Cho19 . In the literature, the quantum Cheshire cat effect was reported via a
neutron experiment Denkmayr14  and a single-photon experiment Ashby16 , both
based on essentially identical Mach-Zehnder interferometry setup shown in Fig.
1. An absorber (ABS) or a half-wave plate (HWP) inserted in one of the
interferometric paths $a$ or $b$ acts as the probe for the particle itself or
the disembodied physical state (i.e., spin or polarization). An argument in
support of the quantum Cheshire cat effect may be made by observing the change
of detection rate induced by the probe. For instance, if the photon travels
along path $a$ (and its polarization state in path $b$), an ABS inserted in
path $b$ would not affect the detection probability at all. While these early
experiments do provide some insights on the quantum Cheshire cat effect
phenomenologically, they are based on simple first-order interferometry and
destructive projection measurement Denkmayr14 ; Ashby16 . The original
requirement of the noninvasive weak quantum measurement, necessary for
unambiguous observation of the quantum Cheshire cat effect, was not satisfied
Duprey18 ; Quach18 ; Sokolovski16 ; Correa15 ; Michielsen15 . Therefore, the
quantum Cheshire cat experiments reported to date are all prone to
interpretation based on measurement-induced disturbance and also subject to
trivial interpretation based on classical waves. In fact, it is simple to
reproduce the Mach-Zehnder type quantum Cheshire cat experiment by using
classical light waves, thus requiring no quantum interpretation at all
Stuckey16 ; Atherton15 .

Figure 1: Essential schematic of the quantum Cheshire cat experiments based on
the Mach-Zehnder interferometer Denkmayr14 ; Ashby16 . A polarized beam is
split into two paths with a non-polarizing beam splitter (NPBS). With half-
wave plates (HWP), the polarizations in the upper and lower paths,
respectively, are set at $|D\rangle$ and $|A\rangle$. The polarizing beam
splitter (PBS) ensures that only the $|D\rangle$ polarization reaches the
detector. To probe the locations of the photon itself and its polarization
state, an absorber (ABS) or a HWP is inserted in paths $a$ or $b$.

In this work, we report, to the best of our knowledge, the first genuine
experimental observation of the quantum Cheshire cat effect with noninvasive
weak quantum measurement as originally proposed Aharonov13 . The location of
the single-photon and that of the disembodied polarization state in a quantum
interferometer have been identified by using the noninvasive weak-measurement
probes. In our work, the weak measurement interaction is implemented by
exploiting linear optical entangling gates via two-photon quantum
interference, which cannot be explained classically Pryde05 ; Pryde04 ;
Kiesel05 . We furthermore elucidate the paradox of the quantum Cheshire cat
effect as quantum interference of the transition amplitudes for the photon and
the polarization state which are directly obtained from the measurement
outcomes or the weak values Sokolovski16 .

Figure 2: (a) Experimental schematic to observe the quantum Cheshire cat
effect. The system photon and its polarization are disembodied during the
transit through the optical paths. The system states are prepared as presented
in figure using HWP, beam displacers (BD), and a PBS. To probe the presence of
the photon and the disembodied polarization state at the lower path, the
system photon is weakly coupled to the pointer photon via the unitary
interaction $\hat{U}_{\mathcal{M}}$. The quantum circuit for the unitary
interaction is shown in (b) and its quantum optical implementation is shown in
(c). Note that the controlled-Z (CZ) gate is implemented via two-photon
quantum interference at a partial polarizing beam splitter (PPBS) having
polarization-dependent transmissions ($T_{H}=1$, $T_{V}=1/3$), and HWP for the
CZ gate set at $45^{\circ}$. The pointer state is finally measured with a
quarter-wave plate (QWP), a HWP, and a PBS.

We illustrate the experimental schematic to observe the quantum Cheshire cat
effect in Fig. 2(a). The system photon represents the Cheshire cat and its
horizontal $|H\rangle$ and vertical $|V\rangle$ polarization states correspond
to the status of her grin. The disembodiment of the grin (the polarization
state) from the cat (the single photon) occurs as the system photon propagates
between beam displacers (BD). The initial state of the photon in Fig. 2(a)
after the first BD and HWP is given as, Kim03

$|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}=\frac{1}{\sqrt{2}}(|u\rangle\otimes|D\rangle+|l\rangle\otimes|A\rangle),$
(1)

where $|D\rangle=(|H\rangle+|V\rangle)/\sqrt{2}$ and
$|A\rangle=(|H\rangle-|V\rangle)/\sqrt{2}$. To probe the presence of the
photon and its disembodied polarization state at the lower path, the pointer
photon $|\Phi_{\mathrm{i}}\rangle_{\mathrm{p}}$, initialized in $|H\rangle$,
is weakly coupled to the system photon for noninvasive measurement. Since
there are only two possible optical paths, the information gained on the lower
path $|l\rangle$ with the noninvasive measurement is sufficient to make
complementary conclusions on the upper path $|u\rangle$. The system photon is
eventually subject to projection measurement in the basis
$|\Psi_{\mathrm{f}}\rangle_{\mathrm{s}}$,

$|\Psi_{\mathrm{f}}\rangle_{\mathrm{s}}=\frac{1}{\sqrt{2}}(|u\rangle+|l\rangle)\otimes|D\rangle,$
(2)

and the state of the pointer photon is measured with a QWP, a HWP, and a PBS.

As shown in Fig. 2(a), the system photon in the lower path $|l\rangle$ is
weakly probed for the observable $\hat{\Pi}_{a}=|a\rangle\langle a|$ via the
measurement interaction of $\hat{U}_{\mathcal{M}}$ Pryde05 ; Pryde04 . The
measurement interaction imparts a rotating operation $R(g/2)$ on the pointer
state, conditioned on the system photon’s polarization state $|a\rangle$,
i.e.,
$\hat{U}_{\mathcal{M}}=(\hat{\mathbb{I}}-\hat{\Pi}_{a})\otimes\hat{\mathbb{I}}+\hat{\Pi}_{a}\otimes
R(g/2)$. Here, $R(g/2)$ is defined by $R(g/2)|H\rangle\rightarrow\cos
g|H\rangle+\sin g|V\rangle$ and $R(g/2)|V\rangle\rightarrow\sin
g|H\rangle-\cos g|V\rangle$. The quantum circuit for the unitary interaction
$\hat{U}_{\mathcal{M}}$ is shown in Fig. 2(b) and its quantum optical
implementation is shown in Fig. 2(c). The interaction strength $g=4\theta_{g}$
and the observable $\hat{\Pi}_{a}$ are set by the HWP angles $\theta_{g}$ and
$\theta_{a}$. The controlled-Z (CZ) gate is implemented via two-photon quantum
interference at a partial polarizing beam splitter (PPBS) having polarization-
dependent transmissions ($T_{H}=1$, $T_{V}=1/3$), and HWP for the CZ gate set
at $45^{\circ}$ Hong87 ; Kiesel05 .

The total unitary operation acting on the three-mode system-pointer state
$|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}\otimes|\Phi_{\mathrm{i}}\rangle_{\mathrm{p}}$
is given by Kim18

$\displaystyle\hat{U}_{\mathrm{tot}}$
$\displaystyle=\hat{\Pi}_{u}\otimes\hat{\mathbb{I}}\otimes\hat{\mathbb{I}}+\hat{\Pi}_{l}\otimes\hat{U}_{\mathcal{M}}$
(3)
$\displaystyle=(\mathbb{\hat{I}}\otimes\mathbb{\hat{I}}-\hat{\Pi}_{l}\otimes\hat{\Pi}_{a})\otimes\mathbb{\hat{I}}+\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}\otimes
R(g/2).$

For $g=\pi/2$, the projection operators on the system state,
$\mathbb{\hat{I}}\otimes\mathbb{\hat{I}}-\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}$
and $\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}$, are perfectly distinguished by the
pointer state and the outcomes of the projection measurement on the system can
be extracted by analyzing the state of the pointer. In contrast, for $|g|\ll
1$, the state of the system photon is weakly coupled to the pointer state,
realizing the noninvasive weak measurement which is essential for the genuine
observation of the quantum Cheshire cat effect.

In the limit of weak measurement, $|g|\ll 1$, the system-pointer evolution
becomes
$\hat{U}_{\mathrm{tot}}|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}|\Phi_{\mathrm{i}}\rangle_{\mathrm{p}}$
and is calculated to be,

$\hat{U}_{\mathrm{tot}}|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}|\Phi_{\mathrm{i}}\rangle_{\mathrm{p}}=|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}|H\rangle_{\mathrm{p}}+g\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}|\Psi_{\mathrm{i}}\rangle_{\mathrm{s}}|V\rangle_{\mathrm{p}}.$
(4)

Note that the state of the system photon is negligibly disturbed. Upon
projection measurement of the system state onto the final state
$|\Psi_{\mathrm{f}}\rangle_{\mathrm{s}}$, the pointer state becomes

$|\Phi_{\mathrm{f}}\rangle_{\mathrm{p}}\propto|H\rangle_{\mathrm{p}}+g\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}\rangle_{\mathrm{w}}|V\rangle_{\mathrm{p}},$
(5)

where $\langle\hat{O}\rangle_{\mathrm{w}}$ indicates the weak value, defined
as Aharonov88 ; Dressel14 ; Kim18 ; Piacentini161 ; Thekkadath16 ; Cho10

$\langle\hat{O}\rangle_{\mathrm{w}}=\frac{\langle\Psi_{\mathrm{f}}|\hat{O}|\Psi_{\mathrm{i}}\rangle}{\langle\Psi_{\mathrm{f}}|\Psi_{\mathrm{i}}\rangle}.$
(6)

The weak value is extracted by analyzing the final pointer state in Eq. (5) as
follows,

$\displaystyle\langle\hat{\sigma}_{x}^{a}\rangle_{\mathrm{p}}$
$\displaystyle=$ $\displaystyle
2g\real\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}\rangle_{\mathrm{w}},$ (7)
$\displaystyle\langle\hat{\sigma}_{y}^{a}\rangle_{\mathrm{p}}$
$\displaystyle=$ $\displaystyle
2g\imaginary\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{a}\rangle_{\mathrm{w}},$

where $\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$ are Pauli operators and the
expectation values are defined as
$\langle\hat{\sigma}^{a}_{k}\rangle_{\mathrm{p}}={{}_{\mathrm{p}}\langle\Phi_{\mathrm{f}}|\hat{\sigma}_{k}|\Phi_{\mathrm{f}}\rangle_{\mathrm{p}}}/{}_{\mathrm{p}}\langle\Phi_{\mathrm{f}}|\Phi_{\mathrm{f}}\rangle_{\mathrm{p}}$.

In the experiment, the system and the pointer photons at 780 nm are produced
via spontaneous parametric down conversion from a type-II beta-barium borate
(BBO) crystal pumped by a 390 nm pulsed laser. The single-photons are
delivered to the experimental setup shown in Fig. 2(a) via the single-mode
optical fibers and, as described earlier, the measurement interaction
$\hat{U}_{\mathcal{M}}$ is based on two-photon quantum interference. To ensure
high degree of spectral indistinguishability, necessary for high-visibility
quantum interference, between the two single-photons, 1-nm bandwidth
interference filters are placed in front of the detector D1 and D2.

For the quantum Cheshire cat effect, the relevant observables are
$\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}$ and
$\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}$, which represent the existence of the
system photon itself and the presence of the photon’s polarization state,
respectively, in the lower path $|l\rangle$. The measurement outcomes
$\langle\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}\rangle_{\mathrm{w}}=0$ and
$\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}=1$ indicates
the observation of the quantum Cheshire cat effect: the polarization state is
found in the path in which the system photon does not exist. We obtain the
weak values of $\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}$ and
$\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}$ from the linear combinations of the
weak values of $\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}$ and
$\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}$ by making use of the relations
$\mathbb{\hat{I}}=\hat{\Pi}_{H}+\hat{\Pi}_{V}$ and
$\hat{\sigma}_{z}=\hat{\Pi}_{H}-\hat{\Pi}_{V}$ as follows:

$\displaystyle\langle\hat{\Pi}_{l}\otimes\hat{\mathbb{I}}\rangle_{\mathrm{w}}=\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}\rangle_{\mathrm{w}}+\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}\rangle_{\mathrm{w}},$
(8)
$\displaystyle\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}=\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}\rangle_{\mathrm{w}}-\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}\rangle_{\mathrm{w}}.$

According to Eq. (7), the real and imaginary parts of
$\langle\hat{\Pi}_{l}\otimes\hat{\mathbb{I}}\rangle_{\mathrm{w}}$ are obtained
from
$(\langle\hat{\sigma}_{x}^{H}\rangle+\langle\hat{\sigma}_{x}^{V}\rangle)/2g$
and
$(\langle\hat{\sigma}_{y}^{H}\rangle+\langle\hat{\sigma}_{y}^{V}\rangle)/2g$
at $|g|\ll 1$, and
$\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}$ is estimated
similarly.

The experimental confirmation for the observation of the quantum Cheshire cat
effect is shown in Fig. 3 in which the pointer measurements
$\langle\hat{\sigma}_{x}^{H}\rangle_{\mathrm{p}}+\langle\hat{\sigma}_{x}^{V}\rangle_{\mathrm{p}}$
and
$\langle\hat{\sigma}_{x}^{H}\rangle_{\mathrm{p}}-\langle\hat{\sigma}_{x}^{V}\rangle_{\mathrm{p}}$
are shown as a function of the measurement strength $g$. Each measurement
interaction for $\langle\hat{\sigma}_{x}^{H}\rangle_{\mathrm{p}}$ and
$\langle\hat{\sigma}_{x}^{V}\rangle_{\mathrm{p}}$ is implemented by setting
the HWP angle $\theta_{a}$ in Fig. 2(c) as 45∘ and 0∘, respectively. Then, the
pointer state, conditioned on the projection measurement of the system onto
the state $|\Psi_{\mathrm{f}}\rangle_{\mathrm{s}}$ at detector D1, is analyzed
from the coincident detection events of D1 and D2 with the set of a QWP, a
HWP, and a PBS at detector D2. The expectation values of
$\langle\hat{\sigma}_{x}^{H}\rangle_{\mathrm{p}}$ and
$\langle\hat{\sigma}_{x}^{V}\rangle_{\mathrm{p}}$ are obtained at each $g$,
and the sum and the difference are given as the data points in Figs. 3(a) and
(b) for the real parts of
$\langle\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}\rangle_{\mathrm{w}}$ and
$\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}$,
respectively. Note that the imaginary parts have zero value, so the results
for $\langle\hat{\sigma}_{y}^{a}\rangle_{\mathrm{p}}$ are not presented.

The weak values are extracted from the experimental data in Fig. 3 from the
slope at $g=0$ by using the polynomial fit to the data according to the
relation in Eq. (7). The experimentally obtained weak values are
$\langle\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}\rangle_{\mathrm{w}}=0.018\pm
0.206$ and
$\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}=1.085\pm
0.206$ are in good agreement with the theoretical prediction and clearly
demonstrate the quantum Cheshire cat effect.

Figure 3: Experimental observation of the quantum Cheshire cat effect. The
observation is realized by measuring weak values. Measurements of pointer
states (solid circle) are recorded as a function of $g$ for observables: (a)
$\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}$ and (b)
$\hat{\Pi}_{l}\otimes\hat{\sigma}_{x}$. Note that the imaginary parts have
zero value, so the results for
$\langle\hat{\sigma}_{y}^{a}\rangle_{\mathrm{p}}$ are not presented. One
standard deviation due to Poissonian counting statistics are considered as
error bars. The black solid lines are the exact theoretical predictions. For
given observables, weak values are extracted by taking the first order
derivative with polynomial curve fit at $g$=0. The measurement results of
$\langle\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}\rangle_{\mathrm{w}}=0.018\pm
0.206$ and
$\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}\rangle_{\mathrm{w}}=1.085\pm
0.206$ indicate the quantum Cheshire cat effect that the physical property
(polarization) can be found in the path where the physical carrier (photon)
does not exist.

While the quantum Cheshire cat effect may look paradoxical, we may interpret
the effect as quantum interference of the transition amplitudes for the photon
and the polarization state. The weak value of Eq. (6), formally, can be
interpreted as the transition amplitude
$\langle\Psi_{\mathrm{f}}|\hat{O}|\Psi_{\mathrm{i}}\rangle$ along the virtual
path defined by $\hat{O}$ from the initial state $|\Psi_{\mathrm{i}}\rangle$
to the final state $|\Psi_{\mathrm{f}}\rangle$, which is normalized by the
total transition amplitude $\langle\Psi_{\mathrm{f}}|\Psi_{\mathrm{i}}\rangle$
Sokolovski16 . Considering the spatial modes $\hat{\Pi}_{u}$ and
$\hat{\Pi}_{l}$ and the polarization modes $\hat{\Pi}_{H}$ and
$\hat{\Pi}_{V}$, there are four possible virtual transition paths as follows,
see Fig. 4(a).

$\hat{\Pi}_{u}\otimes\hat{\Pi}_{H},\ \hat{\Pi}_{u}\otimes\hat{\Pi}_{V},\
\hat{\Pi}_{l}\otimes\hat{\Pi}_{H},\ \hat{\Pi}_{l}\otimes\hat{\Pi}_{V}.$ (9)

The weak values, namely the normalized transition amplitudes
$\langle\hat{O}\rangle_{\textrm{w}}=\langle\Psi_{\mathrm{f}}|\hat{O}|\Psi_{\mathrm{i}}\rangle/\langle\Psi_{\mathrm{f}}|\Psi_{\mathrm{i}}\rangle$,
for the initial and final states in Eqs. (1) and (2) are given as,

$\displaystyle\langle\hat{\Pi}_{u}\otimes\hat{\Pi}_{H}\rangle_{\mathrm{w}}=0.5,~{}~{}\langle\hat{\Pi}_{u}\otimes\hat{\Pi}_{V}\rangle_{\mathrm{w}}=0.5,$
$\displaystyle\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}\rangle_{\mathrm{w}}=0.5,~{}~{}\langle\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}\rangle_{\mathrm{w}}=-0.5.$
(10)

Note that the sum of all the normalized transition amplitudes is equal to
unity because the observables sum to the identity operator.

Figure 4: Conceptual Feynman diagrams for the transition from
$|\Psi_{\mathrm{i}}\rangle$ to $|\Psi_{\mathrm{f}}\rangle$. The virtual paths
defined by $\hat{O}_{k}$ can be arbitrarily set to satisfy
$\sum_{k}\hat{O}_{k}=\hat{\mathbb{I}}$ so that
$\langle\Psi_{\mathrm{f}}|\Psi_{\mathrm{i}}\rangle=\sum_{k}\langle\Psi_{\mathrm{f}}|\hat{O}_{k}|\Psi_{\mathrm{i}}\rangle$.
(a)
$\hat{O}_{k}=\\{\hat{\Pi}_{u}\otimes\hat{\Pi}_{H},\hat{\Pi}_{u}\otimes\hat{\Pi}_{V},\hat{\Pi}_{l}\otimes\hat{\Pi}_{H},\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}\\}$,
(b)
$\hat{O}_{k}=\\{\hat{\Pi}_{u}\otimes\hat{\mathbb{I}}/2,\hat{\Pi}_{u}\otimes\hat{\sigma}_{z}/2,\hat{\Pi}_{l}\otimes\hat{\mathbb{I}}/2,\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}/2\\}$,
where $\hat{\Pi}_{u}$ and $\hat{\Pi}_{l}$ represent the spatial modes of the
state and $\hat{\Pi}_{H}$ and $\hat{\Pi}_{V}$ represent the polarization modes
of the state. The transition amplitudes along the virtual paths in (a) and (b)
are related with each other due to
$\hat{\mathbb{I}}=\hat{\Pi}_{H}+\hat{\Pi}_{V}$ and
$\hat{\sigma}_{z}=\hat{\Pi}_{H}-\hat{\Pi}_{V}$. The line color shows the
normalized transition amplitude of each virtual paths by the total transition
amplitude $\langle\Psi_{\mathrm{f}}|\Psi_{\mathrm{i}}\rangle$ for the initial
and final states in Eqs. (1) and (2).

Similarly, as shown in Fig. 4(b), another complete set of virtual transition
paths exists as the following observables,

$\displaystyle\hat{\Pi}_{u}\otimes\mathbb{\hat{I}}/2,~{}~{}\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}/2,~{}~{}\hat{\Pi}_{u}\otimes\hat{\sigma}_{z}/2,~{}~{}\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}/2,$
(11)

where the observables also sum to the identity operator. The corresponding
normalized transition amplitudes for the initial and final states in Eqs. (1)
and (2) are calculated as,

$\displaystyle\langle\hat{\Pi}_{u}\otimes\mathbb{\hat{I}}/2\rangle_{\mathrm{w}}=0.5,~{}~{}\langle\hat{\Pi}_{u}\otimes\hat{\sigma}_{z}/2\rangle_{\mathrm{w}}=0,$
$\displaystyle\langle\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}/2\rangle_{\mathrm{w}}=0,~{}~{}\langle\hat{\Pi}_{l}\otimes\hat{\sigma}_{z}/2\rangle_{\mathrm{w}}=0.5.$
(12)

The normalized transition amplitudes signify that the system photon can be
found in the only upper path $|u\rangle$ while the polarization appears in the
only lower path $|l\rangle$ during the transition. This paradoxical results
can be understood as the interference between the fundamental transition
amplitudes in Eq. (10) Sokolovski16 . The observables in Eq. (11) can be
expressed as the linear combination of the observables in Eq. (9), e.g.,
$\hat{\Pi}_{u}\otimes\mathbb{\hat{I}}=\hat{\Pi}_{u}\otimes\hat{\Pi}_{H}+\hat{\Pi}_{u}\otimes\hat{\Pi}_{V}$
and
$\hat{\Pi}_{u}\otimes\hat{\sigma}_{z}=\hat{\Pi}_{u}\otimes\hat{\Pi}_{H}-\hat{\Pi}_{u}\otimes\hat{\Pi}_{V}$.
Consequently, it is possible to interpret the transition amplitudes in Eq.
(12) as the outcomes of constructive and destructive interference between the
transition amplitudes in Eq. (10). For instance, the null transition amplitude
for the observable $\hat{\Pi}_{l}\otimes\mathbb{\hat{I}}$ is because of a
destructive interference of two transition paths
$\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}$ and $\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}$.

While the weak quantum measurement has allowed us to investigate the
counterfactual quantum Cheshire cat effect, it is important to remember that,
unlike projection measurement, irreversible quantum state collapse has not
occurred. If one measures the transition probability via strong projection
measurement, not the amplitudes of the observables in Eq. (9), the post-
measurement state would be fully collapsed into one of the eigenstates of the
measurement observable and, therefore, the sum of the transition probabilities
$|\langle\Psi_{\mathrm{f}}|\hat{\Pi}_{l}\otimes\hat{\Pi}_{H}|\Psi_{\mathrm{i}}\rangle|^{2}+|\langle\Psi_{\mathrm{f}}|\hat{\Pi}_{l}\otimes\hat{\Pi}_{V}|\Psi_{\mathrm{i}}\rangle|^{2}$
would not exhibit quantum interference. This outcome is also completely
consistent with the classical wave theory.

To conclude, we have reported, to the best of our knowledge, the first genuine
experimental observation of the quantum Cheshire cat effect. As suggested in
the original proposal Aharonov13 , we have probed the photon’s existence and
its polarization property using a noninvasive weak measuring apparatus during
the state transition. The noninvasive weak measuring apparatus was realized by
coupling the quantum system with the quantum pointer by utilizing another
single-photon. The quantum pointer reveals the quantum Cheshire cat effect in
the framework of the weak value measurement and the apparent quantum Cheshire
cat paradox was explained as quantum interference of virtual transition paths.
Our experimental apparatus can be applied to investigate other paradoxical
phenomena based on weak value such as Hardy’s paradox Aharonov02 ; Lundeen09 ;
Yokota09 , Leggett-Garg inequality Leggett85 ; Palacios10 ; Goggin11 , and
quantum pigeonhole paradox Aharonov16 ; Chen19 .

This work was supported by the National Research Foundation of Korea (Grant
Nos. 2019R1A2C3004812, 2019M3E4A107866011, and 2019M3E4A1079777) and KIST
institutional programs (Project No. 2E30620). Y.K. acknowledges support from
the Global Ph. D. Fellowship by the National Research Foundation of Korea
(Grant No. 2015H1A2A1033028).

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