:
1 2023 Symmetry and Geometry in Neural Representations

# Are “Hierarchical” Visual Representations Hierarchical?

Ethan Shen Ali Farhadi Aditya Kusupati
University of Washington
{ethans03, ali<EMAIL_ADDRESS>
###### Abstract

Learned visual representations often capture large amounts of semantic
information for accurate downstream applications. Human understanding of the
world is fundamentally grounded in hierarchy. To mimic this and further
improve representation capabilities, the community has explored “hierarchical”
visual representations that aim at modeling the underlying hierarchy of the
visual world. In this work, we set out to investigate if hierarchical visual
representations truly capture the human perceived hierarchy better than
standard learned representations. To this end, we create HierNet, a suite of
12 datasets spanning 3 kinds of hierarchy from the BREEDs subset of ImageNet.
After extensive evaluation of Hyperbolic and Matryoshka Representations across
training setups, we conclude that they do not capture hierarchy any better
than the standard representations but can assist in other aspects like search
efficiency and interpretability. Our benchmark and the datasets are open-
sourced at https://github.com/ethanlshen/HierNet.

##  1 Introduction and Motivation

As humans, our understanding of the world is fundamentally grounded in
hierarchy. For example, species are classified in a hierarchical manner,
starting with general orders that devolve into groups of superfamilies that
finally separate into specific species. Mirroring this tendency, many vision
datasets also reflect hierarchies. Some hierarchies are explicit — ImageNet is
based on synsets from WordNet (Miller, 1995) – while others are implicit. For
example, RedCaps can easily be thought of as a taxonomy of 12M image-text
pairs organized under 350 subreddits and finally joined at a single root node
(Desai et al., 2021).

As a result, there has been increasing interest in developing models that can
not only understand image classes but also the hierarchies that they form. One
strategy is to train embeddings so their cosine similarity reflects the known
semantic similarity between images (Barz and Denzler, 2019). Kusupati et al.
(2022) goes a step further and demonstrates using adaptive Matryoshka
Representations (MRs) that nest information hierarchically as the number of
embedding dimensions increase.

The task of learning hierarchies has also led to an interest in hyperbolic
spaces as an alternative to Euclidean spaces for the easier embedding of
complex hierarchical relationships (Nickel and Kiela, 2017). However, despite
the theoretical benefits of hierarchical embeddings, improvements in
downstream tasks such as classification and retrieval have been marginal
compared to normally learned embeddings.

In this paper, we aim to investigate whether hierarchical embeddings are
actually better at capturing the hierarchical structure of visual data. We
generate HierNet, a collection of 12 datasets spanning 3 diverse settings with
known hierarchies from the BREEDs (Santurkar et al., 2020) subset of ImageNet
(Russakovsky et al., 2015). For each dataset, we attempt to recreate their
known hierarchy by clustering image embeddings while observing for any
quantitative or qualitative benefit that hierarchical embeddings offer. In
particular, we analyze the hierarchical embeddings of two models: MERU (Desai
et al., 2023), which is a hyperbolic version of CLIP (Radford et al., 2021),
and MR-ResNet50, a version of ResNet50 trained with Matryoshka Representations
(Kusupati et al., 2022). We chose these two models because of the availability
of their non-hierarchical counterparts – trained without any constraints by
using contrastive or cross-entropy loss appropriately.

##  2 HierNet and Methodology

#### Dataset Creation:

HierNet is composed of 12 hierarchical datasets created with BREEDs, a subset
of ImageNet that conforms to both a visual and semantic hierarchy (Santurkar
et al., 2020). BREEDs contains nine levels, with nodes increasing in
specificity with depth. For example, “dog” is placed at the same level as
“cat” but higher than “bloodhound”. Each BREEDs dataset is organized under a
single root node with $n_{sup}$ superclasses and $n_{sub}$ subclasses per
superclass. Subclasses and their images are taken directly from ImageNet.

  * •

3 control datasets are sourced from level 2 root nodes & level 5 superclasses
(r2_l5). Each dataset has $n_{sup}=17$ and $n_{sub}=2$, totalling 34
subclasses (1700 images each).

  * •

We also generated 4 fine-grained datasets (level 3 root, level 5 superclass –
r3_l5). Each has $n_{sup}=10$ and $n_{sub}=2$ (1000 images each).

  * •

Finally, we created 5 high-variance datasets (level 0 root, level 5 superclass
– r0_l5), with identical sizes to the controls.

We create three types of datasets to evaluate the potential benefits of
hierarchical embeddings across a variety of use cases and granularities.
Appendix B shows the superclass-subclass composition of each dataset.

#### Cluster Quality:

For each model, we cluster image embeddings according to the number of
superclasses and subclasses present in each dataset. Similar to Nguyen and
Kornblith (2023), we generate clusters using agglomerative clustering with
Ward linkage, shown by Monath et al. (2021) to be optimal for image
embeddings. For MERU, we cluster embeddings using hyperbolic distance metric
(Section 3) and for CLIP, MR-ResNet50, and ResNet50, we use Euclidean distance
metric.

We evaluate cluster quality by comparing the discovered clusters against the
ground truth labels for superclasses and subclasses. Specifically, we track
adjusted mutual information (AMI) and purity. Adjusted mutual information
measures the mutual information between two distinct clusterings, normalized
and adjusted for chance. It quantifies to what extent two clusterings overlap.
Meanwhile, purity measures the average homogeneity of clusters. Both metrics
fall between $[0,1]$, where $1$ denotes high cluster quality and $0$ low
cluster quality. Equations for the two metrics are included in Appendix A.

#### Optimal Transport Distance:

While a good model will discover a hierarchy that reflects the true
relationship between image classes, it is also important that a hierarchy
visually aligns with the ground truth. As a result, we leverage hierarchical
optimal transport (HHOT) to measure the visual difference between discovered
clusterings and the ground truth hierarchy. HHOT was originally introduced by
Yeaton et al. (2022) as a method to measure the visual difference between
histopathology datasets that are partitioned into distinct image slides. The
technique can be easily extended to visually compare clusters, which have a
similar connotation to slides.

##  3 Hyperbolic Representations for CLIP

Figure 1: MERU and CLIP have similar accuracies across all the datasets. A
r0_l5 denotes high-variance datasets, r2_l5 denotes control datasets, and
r3_l5 denotes fine-grained datasets.

Figure 2: MERU & CLIP have a negligible difference in cluster quality across
all 12 datasets.

Figure 3: MERU and CLIP have close HHOT distances for all datasets.

To explore the utility of hyperbolic embeddings, we evaluate MERU against CLIP
using the models from Desai et al. (2023). For both superclass and subclass
accuracy, MERU and CLIP perform equally on nearly all datasets (Figure 1).
Similarly, MERU fails to demonstrate any improvement at forming superclass and
subclass clusters over CLIP, suggesting that MERU offers little actual benefit
towards reconstructing hierarchical relationships. Figure 3 supports this
conclusion. HHOT distances between MERU and CLIP are almost indistinguishable.
Our experiments show that hyperbolic embeddings, although theoretically sound,
do not necessarily improve the retention of hierarchical information.

##  4 Matryoshka Representations for ResNet50

We also compare the performance of MR-ResNet50 embeddings against those of
fixed capacity representations (FF) and PCA-reduced 2048-d FF representations.
While FF embeddings provide a non-hierarchical baseline for each dimension and
are retrained from scratch, PCA-reduced embeddings are a better comparison
when considering compute requirements. Indeed, FF embeddings require
separately trained models for each embedding dimension. On the other hand,
PCA-reduced embeddings closely mimic the behavior of MR as just an extra step
on a single model. We analyze the differences from 8 dimensional to 2048
dimensional representations.

Figure 4: On control datasets, average AMI and purity demonstrate that after
64-d, MR is more capable at clustering subclasses than FF or PCA
representations.

On control datasets, MRs are more accurate than standard FF embeddings at all
dimensions for both superclass and subclass classification, reflecting results
from Kusupati et al. (2022) (Appendix C.1). In addition, MR has the highest
purity and AMI for subclass clustering past 64 dimensions. However, there is
no benefit for superclass clustering, where MR, FF, and PCA all perform
equally (Figure 4). This suggests that in the control datasets, while MR is
better at separating the classes the models were trained on, it is no better
at identifying implicit hierarchical relationships between classes. Similarly,
in Figure 5, while the HHOT distance for MR’s subclass clustering is the
lowest of all dimensions, the gap between it and the other embeddings
disappears for superclass clusters.

Figure 5: Average HHOT distance between predicted superclass and subclass
clusters and the true hierarchy shows that MR clusters align the best with the
ground truth.

We also conducted the same experiments for the 4 fine-grained datasets and 5
high-variance datasets of HierNet. Embeddings on high-variance datasets
exhibit a similar trend to that of the control datasets (Appendix C.3).
However, on fine-grained datasets, MR outperforms FF and PCA on AMI, purity,
and HHOT distance for both superclass and subclass clusterings. While MR
embeddings do not show any general hierarchical advantage, MR embeddings seem
to possess some benefit when classes become difficult to separate. Finally, we
also note that lower-dimensional FF representations can also capture some
hierarchy while PCA-style projections from high-dimensional representations
might not preserve the underlying notion of hierarchy.

##  5 Conclusions

In this empirical investigation, we present a novel suite of hierarchical
datasets and show that “hierarchical” embeddings are not inherently better at
capturing underlying hierarchies in visual data. We also demonstrate that
standard Euclidean embeddings are able to competitively capture hierarchy
without explicit training to do so. However, hierarchical embeddings still
stand to have an impact on other axes like interpretability through entailment
in MERU and a significant reduction in search costs through Matryoshka
Representations. In the future, we see the potential for our work to be used
to benchmark and assist the development of new hierarchical representations
that possess significant performance boosts over standard Euclidean/spherical
representations.

##  6 Acknowledgements

This work is in part supported by NSF IIS 1652052, IIS 1703166, DARPA
N66001-19-2-4031, DARPA W911NF-15-1-0543 and gifts from Allen Institute for
Artificial Intelligence, Google and Apple. In addition, thanks to Aniket Rege
for his help on this project.

## References

  * Barz and Denzler (2019) Björn Barz and Joachim Denzler.  Hierarchy-based image embeddings for semantic image retrieval.  In _2019 IEEE winter conference on applications of computer vision (WACV)_ , pages 638–647. IEEE, 2019.
  * Desai et al. (2021) Karan Desai, Gaurav Kaul, Zubin Aysola, and Justin Johnson.  RedCaps: Web-curated image-text data created by the people, for the people.  In _NeurIPS Datasets and Benchmarks_ , 2021.
  * Desai et al. (2023) Karan Desai, Maximilian Nickel, Tanmay Rajpurohit, Justin Johnson, and Ramakrishna Vedantam.  Hyperbolic Image-Text Representations.  In _Proceedings of the International Conference on Machine Learning_ , 2023.
  * Kusupati et al. (2022) Aditya Kusupati, Gantavya Bhatt, Aniket Rege, Matthew Wallingford, Aditya Sinha, Vivek Ramanujan, William Howard-Snyder, Kaifeng Chen, Sham Kakade, Prateek Jain, et al.  Matryoshka representation learning.  In _Advances in Neural Information Processing Systems_ , December 2022\.
  * Miller (1995) George A Miller.  Wordnet: a lexical database for english.  _Communications of the ACM_ , 38(11):39–41, 1995\.
  * Monath et al. (2021) Nicholas Monath, Kumar Avinava Dubey, Guru Guruganesh, Manzil Zaheer, Amr Ahmed, Andrew McCallum, Gokhan Mergen, Marc Najork, Mert Terzihan, Bryon Tjanaka, et al.  Scalable hierarchical agglomerative clustering.  In _Proceedings of the 27th ACM SIGKDD Conference on knowledge discovery & data mining_, pages 1245–1255, 2021.
  * Nguyen and Kornblith (2023) Thao Nguyen and Simon Kornblith.  Probing Clustering in Neural Network Representations.  2023\.
  * Nickel and Kiela (2017) Maximillian Nickel and Douwe Kiela.  Poincaré embeddings for learning hierarchical representations.  _Advances in neural information processing systems_ , 30, 2017.
  * Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  In _International conference on machine learning_ , pages 8748–8763. PMLR, 2021.
  * Russakovsky et al. (2015) Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al.  Imagenet large scale visual recognition challenge.  _International journal of computer vision_ , 115:211–252, 2015.
  * Santurkar et al. (2020) Shibani Santurkar, Dimitris Tsipras, and Aleksander Madry.  Breeds: Benchmarks for subpopulation shift.  In _ArXiv preprint arXiv:2008.04859_ , 2020.
  * Yeaton et al. (2022) Anna Yeaton, Rahul G Krishnan, Rebecca Mieloszyk, David Alvarez-Melis, and Grace Huynh.  Hierarchical optimal transport for comparing histopathology datasets.  In _Medical Imaging with Deep Learning_ , 2022.  URL https://openreview.net/forum?id=gADV7sV4CMo.

##  Appendix A Metrics

###  A.1 Mutual Information

Mutual information is given by

$MI(A,B)=\sum_{i=1}^{|A|}\sum_{j=1}^{|B|}\frac{|A_{i}\cap
B_{j}|}{N}\log{\frac{N|A_{i}\cap B_{j}|}{|A_{i}||B_{j}|}}$

where $A$ and $B$ are the clusterings being compared, $N$ is the total number
of elements, and $A_{i}$ is the $i$th cluster in clustering $A$.

###  A.2 Purity

Purity is given by

$\text{purity}=\frac{1}{N}\sum_{i=1}^{|C|}\max_{j}{|C_{i}\cap G_{j}|}$

where $C$ is the discovered clustering, $C_{i}$ is the ith cluster in $C$, and
$G_{j}$ is the $j$th ground truth class.

##  Appendix B HierNet Datasets

Control Datasets

Table 1: ds5_r2_l5

| superclass | subclasses (class number)
---|---|---
0 | audio system, sound system | [loudspeaker (632), iPod (605)]
1 | bowl | [mixing bowl (659), soup bowl (809)]
2 | camera, photographic camera | [Polaroid camera (732), reflex camera (759)]
3 | digital computer | [hand-held computer (590), notebook (681)]
4 | firearm, piece, small-arm | [rifle (764), revolver (763)]
5 | glass, drinking glass | [beer glass (441), goblet (572)]
6 | gymnastic apparatus, exerciser | [horizontal bar (602), balance beam (416)]
7 | jug | [whiskey jug (901), water jug (899)]
8 | lock | [combination lock (507), padlock (695)]
9 | pen | [quill (749), ballpoint (418)]
10 | percussion instrument, percussive instrument | [steel drum (822), chime (494)]
11 | pot | [coffeepot (505), teapot (849)]
12 | stringed instrument | [acoustic guitar (402), electric guitar (546)]
13 | telephone, phone, telephone set | [dial telephone (528), cellular telephone (487)]
14 | timepiece, timekeeper, horologe | [sundial (835), hourglass (604)]
15 | weight, free weight, exercising weight | [barbell (422), dumbbell (543)]
16 | wind instrument, wind | [French horn (566), bassoon (432)]

Table 2: ds6_r2_l5

| superclass | subclasses
---|---|---
0 | bag | [plastic bag (728), purse (748)]
1 | body armor, body armour, suit of armor, suit o… | [breastplate (461), bulletproof vest (465)]
2 | cap | [shower cap (793), mortarboard (667)]
3 | coat | [trench coat (869), kimono (614)]
4 | cream, ointment, emollient | [lotion (631), sunscreen (838)]
5 | face mask | [mask (643), gasmask (570)]
6 | hat, chapeau, lid | [bonnet (452), bearskin (439)]
7 | helmet | [pickelhaube (715), crash helmet (518)]
8 | makeup, make-up, war paint | [lipstick (629), face powder (551)]
9 | necktie, tie | [Windsor tie (906), bolo tie (451)]
10 | scarf | [stole (824), feather boa (552)]
11 | sheath | [scabbard (777), holster (597)]
12 | shoe | [Loafer (630), running shoe (770)]
13 | skirt | [hoopskirt (601), overskirt (689)]
14 | sweater, jumper | [sweatshirt (841), cardigan (474)]
15 | swimsuit, swimwear, bathing suit, swimming cos… | [maillot (639), swimming trunks (842)]
16 | undergarment, unmentionable | [diaper (529), brassiere (459)]

Table 3: ds7_r2_l5

| superclass | subclasses
---|---|---
0 | salamander | [eft (27), axolotl (29)]
1 | turtle | [box turtle (37), leatherback turtle (34)]
2 | lizard | [whiptail (41), alligator lizard (44)]
3 | snake, serpent, ophidian | [night snake (60), garter snake (57)]
4 | spider | [tarantula (76), black and gold garden spider …
5 | grouse | [ptarmigan (81), prairie chicken (83)]
6 | parrot | [macaw (88), lorikeet (90)]
7 | crab | [Dungeness crab (118), fiddler crab (120)]
8 | dog, domestic dog, Canis familiaris | [bloodhound (163), Pekinese (154)]
9 | wolf | [coyote (272), red wolf (271)]
10 | fox | [grey fox (280), Arctic fox (279)]
11 | domestic cat, house cat, Felis domesticus, Fel… | [tiger cat (282), Egyptian cat (285)]
12 | bear | [sloth bear (297), American black bear (295)]
13 | beetle | [dung beetle (305), rhinoceros beetle (306)]
14 | butterfly | [sulphur butterfly (325), admiral (321)]
15 | ape | [gibbon (368), orangutan (365)]
16 | monkey | [marmoset (377), titi (380)]

Fine-Grained Datasets

Table 4: ds8_r3_l5

| superclass | subclasses
---|---|---
0 | whale | [killer whale (148), grey whale (147)]
1 | dog, domestic dog, Canis familiaris | [cairn (192), Newfoundland (256)]
2 | wolf | [timber wolf (269), red wolf (271)]
3 | wild dog | [dingo (273), dhole (274)]
4 | domestic cat, house cat, Felis domesticus, Fel… | [Egyptian cat (285), Persian cat (283)]
5 | bear | [brown bear (294), ice bear (296)]
6 | rabbit, coney, cony | [Angora (332), wood rabbit (330)]
7 | ape | [chimpanzee (367), gorilla (366)]
8 | monkey | [squirrel monkey (382), guenon (370)]
9 | elephant | [African elephant (386), Indian elephant (385)]

Table 5: ds9_r3_l5

| superclass | subclasses
---|---|---
0 | bag | [plastic bag (728), purse (748)]
1 | body armor, body armour, suit of armor, suit o… | [breastplate (461), bulletproof vest (465)]
2 | cap | [shower cap (793), mortarboard (667)]
3 | face mask | [gasmask (570), mask (643)]
4 | hat, chapeau, lid | [bearskin (439), cowboy hat (515)]
5 | helmet | [football helmet (560), crash helmet (518)]
6 | necktie, tie | [bow tie (457), Windsor tie (906)]
7 | scarf | [feather boa (552), stole (824)]
8 | sheath | [holster (597), scabbard (777)]
9 | shoe | [running shoe (770), Loafer (630)]

Table 6: ds10_r3_l5

| superclass | subclasses
---|---|---
0 | bowl | [soup bowl (809), mixing bowl (659)]
1 | glass, drinking glass | [goblet (572), beer glass (441)]
2 | jug | [water jug (899), whiskey jug (901)]
3 | keyboard instrument | [accordion (401), upright (881)]
4 | lock | [combination lock (507), padlock (695)]
5 | opener | [can opener (473), corkscrew (512)]
6 | pan, cooking pan | [frying pan (567), wok (909)]
7 | pot | [coffeepot (505), caldron (469)]
8 | stringed instrument | [harp (594), electric guitar (546)]
9 | timepiece, timekeeper, horologe | [wall clock (892), analog clock (409)]

Table 7: ds11_r3_l5

| superclass | subclasses
---|---|---
0 | bridge, span | [viaduct (888), suspension bridge (839)]
1 | column, pillar | [totem pole (863), obelisk (682)]
2 | dwelling, home, domicile, abode, habitation, d… | [monastery (663), castle (483)]
3 | fence, fencing | [chainlink fence (489), worm fence (912)]
4 | memorial, monument | [brass (458), triumphal arch (873)]
5 | mercantile establishment, retail store, sales … | [barbershop (424), shoe shop (788)]
6 | outbuilding | [barn (425), apiary (410)]
7 | place of worship, house of prayer, house of Go… | [stupa (832), church (497)]
8 | roof | [tile roof (858), vault (884)]
9 | signboard, sign | [street sign (919), traffic light (920)]

High-Variance Datasets

Table 8: ds0_r0_l5

| superclass | subclasses
---|---|---
0 | snake, serpent, ophidian | [vine snake (59), Indian cobra (63)]
1 | lobster | [spiny lobster (123), American lobster (122)]
2 | sandpiper | [red-backed sandpiper (140), redshank (141)]
3 | ape | [chimpanzee (367), gibbon (368)]
4 | monkey | [baboon (372), guenon (370)]
5 | lemur | [indri (384), Madagascar cat (383)]
6 | elephant | [Indian elephant (385), African elephant (386)]
7 | curtain, drape, drapery, mantle, pall | [shower curtain (794), theater curtain (854)]
8 | firearm, piece, small-arm | [assault rifle (413), rifle (764)]
9 | handcart, pushcart, cart, go-cart | [barrow (428), shopping cart (791)]
10 | makeup, make-up, war paint | [lipstick (629), face powder (551)]
11 | sofa, couch, lounge | [studio couch (831), park bench (703)]
12 | towel | [bath towel (434), paper towel (700)]
13 | truck, motortruck | [moving van (675), fire engine (555)]
14 | weight, free weight, exercising weight | [barbell (422), dumbbell (543)]
15 | wind instrument, wind | [panpipe (699), French horn (566)]
16 | sauce | [chocolate sauce (960), carbonara (959)]

Table 9: ds1_r0_l5

| superclass | subclasses
---|---|---
0 | turtle | [box turtle (37), loggerhead (33)]
1 | crab | [fiddler crab (120), Dungeness crab (118)]
2 | stork | [black stork (128), white stork (127)]
3 | butterfly | [sulphur butterfly (325), admiral (321)]
4 | bicycle, bike, wheel, cycle | [mountain bike (671), bicycle-built-for-two (4…
5 | bottle | [water bottle (898), wine bottle (907)]
6 | cream, ointment, emollient | [lotion (631), sunscreen (838)]
7 | firearm, piece, small-arm | [assault rifle (413), rifle (764)]
8 | jug | [water jug (899), whiskey jug (901)]
9 | percussion instrument, percussive instrument | [drum (541), steel drum (822)]
10 | ship | [aircraft carrier (403), wreck (913)]
11 | signboard, sign | [street sign (919), scoreboard (781)]
12 | stringed instrument | [harp (594), cello (486)]
13 | swimsuit, swimwear, bathing suit, swimming cos… | [bikini (445), maillot (639)]
14 | telephone, phone, telephone set | [cellular telephone (487), pay-phone (707)]
15 | timepiece, timekeeper, horologe | [digital watch (531), stopwatch (826)]
16 | squash | [spaghetti squash (940), zucchini (939)]

Table 10: ds2_r0_l5

| superclass | subclasses
---|---|---
0 | phasianid | [peacock (84), quail (85)]
1 | parrot | [African grey (87), lorikeet (90)]
2 | duck | [red-breasted merganser (98), drake (97)]
3 | butterfly | [sulphur butterfly (325), admiral (321)]
4 | rabbit, coney, cony | [Angora (332), wood rabbit (330)]
5 | antelope | [hartebeest (351), impala (352)]
6 | ape | [chimpanzee (367), gibbon (368)]
7 | monkey | [baboon (372), guenon (370)]
8 | baby bed, baby’s bed | [crib (520), cradle (516)]
9 | car, railcar, railway car, railroad car | [passenger car (705), freight car (565)]
10 | curtain, drape, drapery, mantle, pall | [shower curtain (794), theater curtain (854)]
11 | gymnastic apparatus, exerciser | [balance beam (416), parallel bars (702)]
12 | memorial, monument | [brass (458), megalith (649)]
13 | pot | [caldron (469), coffeepot (505)]
14 | sofa, couch, lounge | [studio couch (831), park bench (703)]
15 | truck, motortruck | [moving van (675), fire engine (555)]
16 | squash | [spaghetti squash (940), zucchini (939)]

Table 11: ds3_r0_l5

| superclass | subclasses
---|---|---
0 | salamander | [axolotl (29), common newt (26)]
1 | snake, serpent, ophidian | [vine snake (59), Indian cobra (63)]
2 | wild dog | [dingo (273), dhole (274)]
3 | domestic cat, house cat, Felis domesticus, Fel… | [tiger cat (282), Siamese cat (284)]
4 | bear | [sloth bear (297), American black bear (295)]
5 | baby bed, baby’s bed | [crib (520), cradle (516)]
6 | camera, photographic camera | [reflex camera (759), Polaroid camera (732)]
7 | firearm, piece, small-arm | [assault rifle (413), rifle (764)]
8 | handcart, pushcart, cart, go-cart | [barrow (428), shopping cart (791)]
9 | jug | [water jug (899), whiskey jug (901)]
10 | opener | [corkscrew (512), can opener (473)]
11 | signboard, sign | [street sign (919), scoreboard (781)]
12 | sofa, couch, lounge | [studio couch (831), park bench (703)]
13 | undergarment, unmentionable | [diaper (529), brassiere (459)]
14 | weight, free weight, exercising weight | [barbell (422), dumbbell (543)]
15 | wind instrument, wind | [panpipe (699), French horn (566)]
16 | frozen dessert | [ice cream (928), ice lolly (929)]

Table 12: ds4_r0_l5

| superclass | subclasses
---|---|---
0 | phasianid | [peacock (84), quail (85)]
1 | duck | [red-breasted merganser (98), drake (97)]
2 | sandpiper | [red-backed sandpiper (140), redshank (141)]
3 | dog, domestic dog, Canis familiaris | [Australian terrier (193), Norwegian elkhound …
4 | wild dog | [dingo (273), dhole (274)]
5 | monkey | [baboon (372), guenon (370)]
6 | elephant | [Indian elephant (385), African elephant (386)]
7 | curtain, drape, drapery, mantle, pall | [shower curtain (794), theater curtain (854)]
8 | dwelling, home, domicile, abode, habitation, d… | [castle (483), cliff dwelling (500)]
9 | glass, drinking glass | [beer glass (441), goblet (572)]
10 | gymnastic apparatus, exerciser | [balance beam (416), parallel bars (702)]
11 | hat, chapeau, lid | [bonnet (452), cowboy hat (515)]
12 | outbuilding | [boathouse (449), apiary (410)]
13 | pen | [fountain pen (563), quill (749)]
14 | place of worship, house of prayer, house of Go… | [mosque (668), church (497)]
15 | towel | [bath towel (434), paper towel (700)]
16 | weight, free weight, exercising weight | [barbell (422), dumbbell (543)]

##  Appendix C Matryoshka Representations

###  C.1 Accuracies

Figure 6: Across control datasets, MR is slightly more accurate than FF when
averaged across all three control datasets. Both embeddings are more accurate
than PCA until 512 dimensions.

Figure 7: In fine grained datasets, MR is slightly more accurate than FF.

Figure 8: In high-variance datasets, while MR is still the most accurate, the
gap between it and other embeddings is smaller than before. We posit that this
is due to high-variance datasets having very diverse and thus easily separable
classes. This reduces the benefits of MR’s information packing.

###  C.2 Fine-Grained Dataset Clustering

Figure 9: In fine grained datasets, MR has consistently higher superclass and
subclass AMI/purity than FF after 128 dimensions. However, MR’s benefit over
PCA is nonexistent between 128-d and 512-d.

Figure 10: MR struggles to outperform PCA embeddings for superclass HHOT
distance. Still, as the number of dimensions increases MR easily beats out FF
in superclass and subclass distances.

###  C.3 High-Variance Dataset Clustering

Figure 11: When clustering on high-variance datasets, we find that while MR
outperforms FF embeddings on subclass clustering, it is not a better option
than PCA past 128 dimensions. Even more, FF and PCA embeddings outperform MR
in both superclass AMI and superclass purity at high dimensions.

Figure 12: MR’s subclass clusters have the best HHOT distance of all
embeddings. However, the HHOT distance of MR’s superclass clustering is
comparable to FF embeddings and definitively worse than PCA’s.


# Correlations and Symmetry of Interactions Influence Collective Dynamics of
Molecular Motors

Daniel Celis-Garza, Hamid Teimouri and Anatoly B. Kolomeisky  Department of
Chemistry and Center for Theoretical Biological Physics, Rice University,
Houston, Texas, 77005, USA<EMAIL_ADDRESS>
###### Abstract

Enzymatic molecules that actively support many cellular processes, including
transport, cell division and cell motility, are known as motor proteins or
molecular motors. Experimental studies indicate that they interact with each
other and they frequently work together in large groups. To understand the
mechanisms of collective behavior of motor proteins we study the effect of
interactions in the transport of molecular motors along linear filaments. It
is done by analyzing a recently introduced class of totally asymmetric
exclusion processes that takes into account the intermolecular interactions
via thermodynamically consistent approach. We develop a new theoretical method
that allows us to compute analytically all dynamic properties of the system.
Our analysis shows that correlations play important role in dynamics of
interacting molecular motors. Surprisingly, we find that the correlations for
repulsive interactions are weaker and more short-range than the correlations
for the attractive interactions. In addition, it is shown that symmetry of
interactions affect dynamic properties of molecular motors. The implications
of these findings for motor proteins transport are discussed. Our theoretical
predictions are tested by extensive Monte Carlo computer simulations.

##  1 Introduction

Motor proteins or molecular motors are enzymatic molecules that actively
participate in all major biological processes such as cellular transport, cell
division, transfer of genetic information, synthesis of proteins, cell
motility and signaling [2, 1, 3, 4, 5, 6]. They transform chemical energy from
specific reactions that they catalyze (usually, hydrolysis or
biopolymerization) into mechanical work to support their functions. For
example, the directed motion along linear cytoskeleton filaments by kinesin,
myosin and dynein motor proteins is fueled by the hydrolysis of adenosine
triphosphate (ATP) [6]. Biological molecular motors have been intensively
studied in recent years, and currently the single-molecule dynamics of motor
proteins is well described [4, 5, 7]. Although the properties of individual
molecules are very useful, in biological systems motor proteins typically
function in large teams. This underlines the importance of understanding the
collective behavior of molecular motors [8, 5, 9].

Experimental studies of kinesin motor proteins moving along microtubules
indicate that these molecular motors interact with each [10, 11, 12]. It was
argued that these interactions most probably are short-range and relatively
weak attractive ($1.6\pm 0.5k_{B}T$) [10]. It is reasonable to assume that
many other motor proteins have similar properties. At microscopic level,
molecular motors are involved in a variety of chemical transitions such as
binding to the filament, chemical transformations during the hydrolysis,
dissociation from the track [6]. Intermolecular interactions influence all
these processes, suggesting an important role for interactions in the
collective behavior of molecular motors. However, the underlying mechanisms
are still not well clarified [5, 8]. Existing theoretical studies of
cooperative dynamics in interacting molecular motors are mostly
phenomenological without quantitative description of relevant chemical
processes [13, 14, 15, 16].

Recently, we introduced a new theoretical approach for analyzing collective
properties of interacting molecular motors [17]. It is based on using a class
of non-equilibrium models called totally asymmetric simple exclusion processes
(TASEP), which are very powerful for studying multi-particle dynamic phenomena
[18, 19, 20]. There are many processes in Chemistry, Physics and Biology that
have been successfully analyzed by utilizing the asymmetric exclusion
processes [19, 20, 21, 22, 23, 24, 25]. TASEPs were also employed before for
investigating dynamic properties of motor proteins [19, 14, 9, 22, 26, 16],
including interacting molecular motors [14, 16, 27, 28]. But the main
advantage of our method is a procedure that describes all chemical transitions
at the single-molecule level using fundamental thermodynamic concepts [17].
This allows us to properly couple microscopic properties of interacting
molecular motors with their collective dynamic behavior.

Analyzing this theoretical approach, it was found that there is an optimal
interaction strength (weakly repulsive) that can maximize the current through
the system [17]. In addition, the calculations suggested that correlations
play important role in dynamics of interacting molecular motors. However, the
progress in understanding the cooperativity in motor proteins in this model
was limited by the following issues. Two mean-field analytical treatments were
proposed. But the first one, a simple mean-field approach (SMF), failed
completely, as compared with extensive Monte Carlo computer simulations,
producing unphysical trends in dynamic properties for strong interactions. The
second approach, a cluster mean-field (CMF), worked better, but it involved
very heavy numerical calculations. At the end, CMF was able to reproduce only
qualitatively dynamics of interacting molecular motors and not even for all
ranges of parameters. Furthermore, only symmetric splitting of interactions on
hopping rates was considered. In addition, it was not possible to extend CMF
to take into account more realistic features of motor protein’s transport such
as backward steps, bindings and unbindings from the filament, and more general
symmetries of the interactions. To understand the mechanisms of cooperativity,
it is important to have an analytical method that can successfully capture
main features of interacting molecular motors, and which can be also extended
to more complex situations.

In this paper, a new theoretical framework for analyzing complex dynamics of
interacting molecular motors via TASEP is presented. We develop a modified
cluster mean-field (MCMF) approach that accounts for some correlations in the
system. This provides a direct way of analytically calculating all dynamic
properties in the system, and the results agree quite well with computer
simulations. The method allows us to explicitly analyze the role of
interactions in dynamics of interacting molecular motors. More specifically,
it is found that correlations are weaker and more short-range for repulsive
interactions while for attractions they are stronger and more long-range. We
also investigate the role of the symmetry of interactions and show that it
might dramatically modify the dynamic behavior. But most importantly, the
developed framework allows us to understand the microscopic origin of dynamic
phenomena in motor proteins and it can be easily generalized to account for
more complex processes associated with molecular motors.

##  2 Theoretical Description

###  2.1 Model

In our model, the transport of molecular motors along linear filaments is
viewed as a motion of multiple particles on a lattice with $L$ ($L\gg 1$)
sites, as shown in Fig. 1. The state of occupancy for each lattice site $i$
$(1\leq i\leq L)$ is characterized by an occupation number $\tau_{i}$. If the
site $i$ is occupied then $\tau_{i}=1$, if it is empty then $\tau_{i}=0$. Each
lattice site can accommodate only one particle.

In addition to exclusions, molecular motors can interact with each other via a
short-range potential. Here we assume that any two neighboring particles
interact with each other with an energy $E$. The case of positive $E$ defines
attractive interactions, while $E<0$ corresponds to repulsions. In other
words, any bond connecting two neighboring particles on the lattice is
associated with the energy $E$. In our system transition rates depend if these
bonds are broken or created. Any forward motion of the individual molecular
motor that does not change the number of bonds is taking place with the rate
1. It can be done by a single molecule that do not have any neighbors (see
Fig. 1), or it can involve breaking one bond and creating another one (Fig.
1). In both cases, there is no energy change in the system. However, the
forward transition associated with creating a new bond has a rate $q\neq 1$.
In this case, the molecular motor joins an existing cluster of particles: see
Fig. 1. Similarly, the transition that is coupled with breaking the bond has a
rate $r\neq 1$. Here the particle dissociates from the cluster but
simultaneously it does not bind to another cluster ( Fig. 1). The transition
rates $q$ and $r$ are associated with changes in energy.

It has been argued that creating and breaking such bonds (or pairs of
particles) can be viewed as opposing chemical transitions, which justifies the
application of the detailed balance arguments [17]. This leads to the
following relation between the transitions rates,

$\frac{q}{r}=\exp{\left(\frac{E}{k_{B}T}\right)}.$ (1)

To evaluate dynamic properties of molecular motors we need to know the
explicit values for rates $q$ and $r$. The interaction energy can be split in
the following way,

$q=\exp{\left(\frac{\theta E}{k_{B}T}\right)},\quad
r=\exp{\left(\frac{(\theta-1)E}{k_{B}T}\right)},$ (2)

where a dimensionless parameter $\theta$ ($0\leq\theta\leq 1$) specifies how
the energy affects these transition rates. Previously, only a symmetric
splitting of interactions ($\theta=1/2$) has been considered [17].

It is easy to understand the physical meaning of Eq. (2) [17]. When the
interactions between molecular motors are attractive ($E>0$), the rate of
creating the bond is larger ($q>1$), while the rate of breaking the bond is
smaller ($r<1$). For repulsive interactions ($E<0$) the trend is opposite — it
is faster to break the particle cluster ($r>1$) than to increase the cluster
size ($q<1$). In the case of no interactions ($E=0$) these transitions rates
are the same ($q=r=1$) and the model reduces to standard TASEP with only hard-
core exclusions.

Figure 1: Schematic view of the TASEP model for interacting molecular motors.
Binding corresponds to particle joining the cluster, while unbinding describes
the breaking from the cluster.

In our model, particles enter the system from the left side of the lattice and
they leave the system from the right side of the lattice: see Fig. 1.
Interactions are also modifying the entrance and exit rates in comparison with
the original TASEP model. When entering the system does not lead to creating a
pair of particles the rate for this process is $\alpha$ (Fig. 1). However,
entering with creating the bond has a rate $q\alpha$. Similarly, the exit rate
for the case when no bond breaking is involved is equal to $\beta$, while the
dissociating from the cluster is taking place with the rate $r\beta$ (Fig. 1).

###  2.2 Modified Cluster Mean-Field Theory

Previous theoretical studies indicated that correlations are important for the
system with multiple particles [17]. Neglecting correlations leads to
unphysical behavior for strong interactions between molecular motors [17].
This indicates that any successful theoretical treatment must take
correlations into account. This is the main idea of our approach that we call
a modified cluster mean-field.

To account for correlations we analyze bulk clusters of two neighboring sites
on the lattice. Each cluster can be found in one of four possible states. We
label a configuration with 2 empty sites as $(0,0)$, with two occupied sites
as $(1,1)$, and two half-occupied clusters are labeled as $(1,0)$ and $(0,1)$.
Next, we introduce functions $P_{11}$, $P_{10}$, $P_{01}$ and $P_{00}$ as
probabilities of finding the configurations $(1,1)$, $(1,0)$, $(0,1)$ and
$(0,0)$, respectively. The conservation of probability for these functions
requires that

$P_{11}+P_{10}+P_{01}+P_{00}=1.$ (3)

In addition, we have

$P_{10}+P_{11}=\rho,\quad P_{01}+P_{11}=\rho,$ (4)

where $\rho$ is the bulk density (or the probability to find the particle at
given site). Here it was assumed also that the bulk density is uniform and
independent of the position on the lattice if the two-site cluster is far away
from the boundaries. Combining Eqs. (3) and (4) leads to $P_{10}=P_{01}$ and

$P_{10}+P_{00}+\rho=1.$ (5)

Figure 2: Four-sites bulk lattice segments that are utilized for calculating
the particle currents in the system.

Let us consider a particle flux in the bulk of the system at large times when
stationary conditions are achieved. We can concentrate on the segment of 4
consecutive sites as shown in Fig. 2. To measure the current, only transitions
between the second and the third sites of the segment are counted. Then there
are four possible configurations that support the particle current, as
illustrated in Fig. 2. Correspondingly, the total flux have 4 contributions
from each configurations, $J_{bulk}=J_{1}+J_{2}+J_{3}+J_{4}$. The first
contribution from configuration 1 (Fig. 2) can be written as

$J_{1}=\gamma P_{10}\left(\frac{P_{00}}{\rho+P_{00}}\right),$ (6)

where $\gamma=\frac{1}{1+\exp{\left(\frac{E}{k_{B}T}\right)}}$. This
expression is an approximation and it can be understood in the following way.
The second factor ($P_{10}$) gives the probability that the cluster consisting
of the second and third sites (see Fig. 2) is in the state $(1,0)$. The first
factor ($\gamma$) gives a probability that the first site is empty, i.e., it
is just a Boltzmann’s factor. The third factor ($\frac{P_{00}}{\rho+P_{00}}$)
is a probability to have the last site empty. If we have the configuration
$(1,0)$ in the middle cluster then the last site can be found in one of two
states: it can be occupied with the probability $\rho$ or it can be empty with
the probability $P_{00}$. This is because in this case the cluster consisting
of the sites 3 and 4 can only be found in configurations $(0,0)$ or $(0,1)$.
Then the particle current from the second configuration (Fig. 2) is equal to

$J_{2}=(1-\gamma)rP_{10}\left(\frac{P_{00}}{\rho+P_{00}}\right).$ (7)

Here
$(1-\gamma)=\frac{\exp{\left(\frac{E}{k_{B}T}\right)}}{1+\exp{\left(\frac{E}{k_{B}T}\right)}}$
is a probability to have the first site occupied, and $r$ is the transition
rate for this configuration. Similar arguments can be presented for
contributions from configurations 3 and 4, yielding

$J_{3}=\gamma qP_{10}\left(\frac{\rho}{\rho+P_{00}}\right),$ (8)

and

$J_{4}=(1-\gamma)P_{10}\left(\frac{\rho}{\rho+P_{00}}\right).$ (9)

Combining together Eqs. (6), (7), (8) and (9) we obtain the expression for the
total bulk current,

$\displaystyle J_{bulk}$ $\displaystyle=$
$\displaystyle\gamma\left(\frac{P_{10}P_{00}}{\rho+P_{00}}\right)+(1-\gamma)r\left(\frac{P_{10}P_{00}}{\rho+P_{00}}\right)$
(10) $\displaystyle+q\gamma\left(\frac{\rho
P_{10}}{\rho+P_{00}}\right)+(1-\gamma)\left(\frac{\rho
P_{10}}{\rho+P_{00}}\right).$

This equation can be also written in the following form,

$J_{bulk}=A\left(\frac{P_{10}P_{00}}{1-P_{10}}\right)+B\left(\frac{\rho
P_{10}}{1-P_{10}}\right),$ (11)

where auxiliary functions $A$ and $B$ are defined as

$\displaystyle
A=\frac{1+r\exp{\left(\frac{E}{k_{B}T}\right)}}{1+\exp{\left(\frac{E}{k_{B}T}\right)}},\quad
B=\frac{q+\exp{\left(\frac{E}{k_{B}T}\right)}}{1+\exp{\left(\frac{E}{k_{B}T}\right)}}.$
(12)

To calculate explicitly dynamic properties in the system we have to express
everything in terms of the bulk density $\rho$ and the interaction energy $E$.
Eq. (5) gives the connection between $P_{10}$, $P_{00}$ and $\rho$, and one
more additional relation is needed in order to have all equations only in
terms of $\rho$ and $E$. We can approximate the function $P_{10}$ as

$P_{10}\simeq\frac{\rho(1-\rho)}{1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}}.$
(13)

The physical meaning of this approximation can be explained if we note that
$P_{10}$ is the probability to have the two-site cluster in the configuration
$(1,0)$. This probability is equal to the product of two terms: one is the
probability to have the first site occupied ($\rho$) and the second term is
the probability that the second site is empty given that the first one is not
($(1-\rho)/[1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}]$). It can be
argued that the situation when two sites in the cluster are occupied is
affected by the interactions between them, and we approximate it via the usual
Boltzmann’s factor. One can see that this equation leads to a very reasonable
behavior at the limiting cases. When there is no interactions ($E=0$) it
predicts that $P_{10}=\rho(1-\rho)$, as expected. For very strong repulsions
($E\rightarrow-\infty$) it gives $P_{10}=\rho$, which is a correct result
since in this limit our problem is identical to the motion of non-interacting
dimers on the lattice [17, 30]. For very large attractions
($E\rightarrow\infty$) the prediction is that $P_{10}\rightarrow 0$. This is
again seems to be a reasonable result because in this case one is expecting to
have the whole system fully occupied without any vacancies.

Finally, taking into account all approximations we obtain the general
expression for the bulk current only in terms of the particle density $\rho$
and the interaction $E$,

$\displaystyle J_{bulk}$ $\displaystyle=$
$\displaystyle\frac{A\rho(1-\rho)^{2}\left[1-2\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}{\left[(1-\rho)^{2}+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]\left[1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}+$
(14)
$\displaystyle\frac{B\rho^{2}(1-\rho)}{\left[(1-\rho)^{2}+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}$

For the case of zero interactions this equation suggests that
$J_{bulk}=\rho(1-\rho)$, the known result form the original TASEP model [18,
19]. For strong repulsions ($E\rightarrow-\infty$) we predict that

$J_{bulk}=\frac{\rho(1-2\rho)}{1-\rho}.$ (15)

This is identical to the expression that was derived earlier for TASEP of
dimers [30]. For large attractions ($E\rightarrow\infty$) it predicts that the
bulk current vanishes, $J_{bulk}=0$, and this is expected because particles
will not be able to move since they will be stuck together in one large
cluster.

At boundaries the dynamics in the system is governed by exit and entrance
rates. Using the same approximations as explained above for the bulk fluxes it
can be shown that the expressions for entrance current is given by

$J_{entr}=\frac{\alpha(1-\rho)\left[1-2\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]+\alpha
q\rho(1-\rho)}{1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}}.$ (16)

For $E=0$ this equation reduces to $J_{entr}=\alpha(1-\rho)$, which is
expected for this situation. For $E\rightarrow-\infty$it predicts that
$J_{entr}=\alpha(1-2\rho)$, in agreement with known results on TASEP of
extended objects [30]. For strong attractions ($E\rightarrow\infty$) the
current disappears, $J_{entr}\rightarrow 0$. Similarly, for the exit current
we obtain

$J_{exit}=\frac{\beta\rho\left[1-\rho+r\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}{1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}}.$
(17)

Again, for $E=0$ and for $E\rightarrow-\infty$ it produces the expected
result, $J_{exit}=\beta\rho$, while for strong attractions it leads to
$J_{exit}\rightarrow 0$.

###  2.3 Phase Diagrams

Similarly to the original TASEP, it can be argued that in the system of
interacting molecular motors there are three dynamic phases at stationary
conditions. When the rate limiting step is the entrance into the system we
have a low-density (LD) phase. For the case of exiting being small a high-
density (HD) phase will be realized. Finally, when bulk processes are the most
important, the system is in a maximal-current (MC) phase.

Our analytical theory can calculate explicitly the phase boundaries. The MC
phase is characterized by a condition that $\frac{\partial
J_{bulk}}{\partial\rho}=0$, which leads to the following expression:

$\displaystyle\left(2\rho^{2}-4\rho+1\right)(\rho-1)^{4}-\rho^{4}\exp\left((\theta+3)\frac{E}{k_{B}T}\right)-\rho^{2}(2\rho-1)(\rho-1)^{2}\exp\left((\theta+2)\frac{E}{k_{B}T}\right)$
$\displaystyle+(\rho-1)^{6}\exp\left(\theta\frac{E}{k_{B}T}\right)-(\rho-2)\rho(\rho-1)^{4}\exp\left((\theta+1)\frac{E}{k_{B}T}\right)+\left(\rho^{4}-2\rho^{5}\right)\exp\left(4\frac{E}{k_{B}T}\right)$
$\displaystyle-\rho\left(4\rho^{3}-16\rho^{2}+15\rho-4\right)(\rho-1)^{2}\exp\left(\frac{E}{k_{B}T}\right)-\rho^{3}\left(\rho^{3}-10\rho^{2}+13\rho-4\right)\exp\left(3\frac{E}{k_{B}T}\right)$
$\displaystyle+\rho^{2}\left(3\rho^{4}-22\rho^{3}+39\rho^{2}-26\rho+6\right)\exp\left(2\frac{E}{k_{B}T}\right)=0.$
(18)

For $E=0$ this complicated expression reduces to the following formula,

$-4\rho^{5}+10\rho^{4}-16\rho^{3}+14\rho^{2}-8\rho+2=0,$ (19)

which has only one real root, $\rho=\frac{1}{2}$. For very large repulsions
($E\rightarrow-\infty$) one can obtain from Eq.(2.3),

$\left(2\rho^{2}-4\rho+1\right)(\rho-1)^{4}=0.$ (20)

This equation has three roots but only one of them is physically reasonable,
$\rho=1-1/\sqrt{2}$, which leads to a nonzero flux in the system. (The root
$\rho=1$ does not support the nonzero flux through the system and it can be
neglected.) Substituting this density into Eq. (15) leads to a prediction that
the particle flux in this case is equal to $J=3-2\sqrt{2}\approx 0.17$. For
large attractions ($E\rightarrow\infty$) Eq.(2.3) predicts that $\rho=1/2$,
but the current here is approaching zero, as was discussed above. For general
conditions, this equation can be always solved numerically and after choosing
the physically relevant root for the density in the MC phase, $\rho_{MC}$, the
particle fluxes can be calculated using Eq. (14). The phase diagrams
calculated via this method are presented in Fig. 3 for various sets of
parameters.

Figure 3: Stationary phase diagrams of TASEP with interacting particles for
various interaction strengths and interaction splittings: (a) $E=-1.7$kBT,
$\theta=0.25$; (b) $E=-1.2$kBT, $\theta=0.5$; (c) $E=-0.7$kBT, $\theta=0.75$;
(d) $E=3$kBT, $\theta=0.5$. Lines are theoretical predictions, symbols are
from Monte Carlo computer simulations.

To determine the density of molecular motors in the LD phase and the boundary
lines separating the low-density and the maximal-current phases, we use the
continuity of the stationary currents at the transition line,
$J_{bulk}=J_{entr}$. Combining Eqs. (14) and (16) yields the following
expression for the entrance rate $\alpha$,

$\alpha=\frac{A\rho(1-\rho)\left[1-2\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]+B\rho^{2}\left[1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}{\left[1-2\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}+q\rho\right]\left[(1-\rho)^{2}+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}.$
(21)

Solving this equation for $\rho$ provides an estimate for the particle density
in the bulk of the system in the LD phase. Increasing the entrance rate
$\alpha$ leads to large bulk densities, and the phase boundary between LD and
MC phase is achieved when $\rho=\rho_{MC}$. For example, for zero
interactions, $E=0$, from Eq. (21) it follows that $\rho_{LD}=\alpha$, and the
phase boundary corresponds to $\alpha=0.5$. These estimates fully agree with
results from the original TASEP with non-interacting particles [18, 19]. For
the case of very strong repulsions, $E\rightarrow-\infty$, we derive from Eq.
(21) that $\rho_{LD}=\alpha/(1+\alpha)$, and the phase boundary between LD and
MC phase corresponds to $\alpha=\sqrt{2}-1\approx 0.41$. This again agrees
with known results on TASEP of extended objects [30]. In the opposite limit of
very large attractions, $E\rightarrow\infty$, the low-density phase cannot be
realized for any nonzero values of $\alpha$.

Similar calculations can be performed for obtaining properties of the HD phase
and the boundaries between high-density and maximal-current phase. The exit
rate $\beta$ is coupled with the density $\rho$ via

$\beta=\frac{A(1-\rho)^{2}\left[1-2\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]+B\rho(1-\rho)\left[1-\rho+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}{\left[(1-\rho)^{2}+\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]\left[1-\rho+r\rho\exp{\left(\frac{E}{k_{B}T}\right)}\right]}.$
(22)

This expressions allows us to calculate the bulk particle density in the HD
phase. One can see that increasing the exit rate $\beta$ lowers the bulk
density until the phase boundary with the MC phase is reached at
$\rho=\rho_{MC}$. This can be illustrated by again considering the limiting
cases. When motor proteins do not interact with each other ($E=0$) Eq.(22)
yields $\rho_{HD}=1-\beta$ and the phase boundary between HD and MC phase can
be found at $\beta=0.5$. This is fully consistent with known results for TASEP
of non-interacting particles [18, 19]. For strong repulsions we predict that
$\rho_{HD}=(1-\beta)/(2-\beta)$, and the phase boundary between HD and MC
phases is observed at $\beta=2-\sqrt{2}\approx 0.59$. Note that these results
slightly differ from calculations for TASEP of dimers because of the different
exiting rules [30]. For strong attractions the flux through the system is
vanishing and the high-density is always observed.

The phase boundary between LD and HD phases can be estimated from the
condition that at this line the particle currents from both phases become
equal, $J_{LD}=J_{HD}$. It can be shown from Eqs. (16) and (17) that

$\frac{\beta}{\alpha}=\left[\frac{q\rho_{LD}(1-\rho_{LD})+(1-\rho_{LD})(1-2\rho_{LD}+\rho_{LD}\exp{\left(\frac{E}{k_{B}T}\right)})}{\rho_{HD}(1-\rho_{HD}+r\rho_{HD}\exp{\left(\frac{E}{k_{B}T}\right)})}\right]\left[\frac{1-\rho_{HD}+\rho_{HD}\exp{\left(\frac{E}{k_{B}T}\right)}}{1-\rho_{LD}+\rho_{LD}\exp{\left(\frac{E}{k_{B}T}\right)}}\right].$
(23)

In this expression, densities $\rho_{HD}$ and $\rho_{LD}$ are obtained by
solving Eqs. (21) and (22), respectively, for specific values of $E$ and
$\theta$. For $E=0$ we find that the LD-HD phase boundry is given by
$\beta=\alpha$, and the triple point (where LD, HD and MC phases meet
together) is found at $\beta_{c}=\alpha_{c}=0.5$, as expected for the standard
TASEP [18, 19].

Figure 4: Maximal particle currents as a function of the interaction energy
for different energy splittings: (a) $\theta=0$, (b) $\theta=0.25$, (c)
$\theta=0.75$ and (d) $\theta=1$. In simulations $\alpha=\beta=1$ was
utilized.

##  3 Monte Carlo Simulations and Discussions

Because of the approximate nature of our method, it is important to test these
theoretical predictions. It was done in this work by running extensive
computer Monte Carlo simulations. We utilized the Monte Carlo algorithm known
as a random sequential update. In our simulations we used a lattice of size
$L=1000$ to minimize any finite size and boundary effects. The particle
current and density profiles of molecular motors were averaged over $10^{8}$
Monte Carlo steps. To ensure that the system is at the stationary-state
conditions, the first $20\%$ of events were discarded. We have used a
precision of $0.01$ when comparing density profiles to construct accurate
phase diagrams. The error in calculating the phase boundaries by our method
was estimated to be less than $1\%$.

In Fig. 3 we compare theoretically calculated phase diagrams with results
obtained in Monte Carlo computer simulations. One can see that for relatively
weak interactions theory agrees quite well with computer simulations (Fig.
3c), while for stronger interactions (attractive or repulsive) the agreement
is mostly qualitative (although still better for repulsions): see Figs. 3a, 3b
and 3d. Comparing phase behavior at different interactions, one can notice
that the LD phase is dominating at repulsions, but the HD phase is more
prevalent for attractions. These observations are consistent with expectations
that repulsions lead to smaller particle clusters and lower density while
attractions stimulate the formation of large clusters, which corresponds to
higher density.

However, our method works much better for predicting particle fluxes in the MC
phase, as illustrated in Fig. 4. The theory correctly describes the fluxes for
repulsive interactions for all ranges of parameters (with the exception of the
special case corresponding to $\theta=0$). But for attractions the good
agreement is found only for small $\theta$. For larger values of the energy
splittings ($\theta>0.25$) there is only a qualitative agreement on the
overall trends: the fluxes decrease to zero with increasing the interaction
strength (with the exception of the special case corresponding to $\theta=1$).

These observations suggest that correlations are important for understanding
dynamic properties of interacting molecular motors. To quantify this effect,
we investigated a correlation function $C_{i}$ defined as

$C=\langle\tau_{i}\tau_{i+1}\rangle-\langle\tau_{i}\rangle\langle\tau_{i+1}\rangle,\quad
i=1,...,L-1$ (24)

where two-point and one-point density functions are given by

$\displaystyle\langle\tau_{i}\tau_{i+1}\rangle=\sum\limits_{\tau_{i}}\sum\limits_{\tau_{i+1}}\tau_{i}\tau_{i+1}P(\tau_{i},\tau_{i+1})=P_{11},$
(25)
$\displaystyle\langle\tau_{i}\rangle=\sum\limits_{\tau_{i}}\tau_{i}P(\tau_{i})=\rho.$
(26)

The physical meaning of the correlation function $C_{i}$ is that it gives a
measure of how the presence of the particle at site $i$ affects the occupation
of the neighboring site $i+1$. Using the definition together with the
normalization condition and Eq. (13), we obtain the following analytical
expression for $C$,

$C(E)=\frac{\rho^{2}(1-\rho)\left[\exp\left(\frac{E}{k_{B}T}\right)-1\right]}{1+\rho\left[\exp\left(\frac{E}{k_{B}T}\right)-1\right]}.$
(27)

Note that $C$ is uniform in the bulk of the system. For the case of zero
interactions ($E=0$) it predicts that $C=0$. This fully agrees with what we
know about TASEP for noninteracting particles. Here a simple mean-field
theory, that completely neglects any correlations, works quantitatively well
and it correctly describes the majority of dynamic properties of the system
[18, 19]. For strong repulsions ($E\rightarrow-\infty$) it gives
$C=-\rho^{2}$, while for strong attractions ($E\rightarrow\infty$) we have
$C=\rho(1-\rho)$.

Figure 5: Correlations as a function of the interaction energy for: (a)
$\theta=0.25$, (b) $\theta=0.5$, (c) $\theta=0.75$. In simulations
$\alpha=\beta=1$ was utilized.

The correlation functions predicted in our method and obtained from computer
simulations are presented in Fig. 5. It is interesting to analyze these data.
The physical meaning of the correlation function $C_{i}$ is that it gives a
measure of how the presence of the particle at site $i$ affects the occupation
of the neighboring site $i+1$. When there are no correlations we have $C=0$.
Negative correlation functions ($C<0$) indicate that there is a less
probability to find the particle next to the already occupied site. This is
the case for repulsive interactions. In contrary, positive values for $C$
suggest that the presence of the particle at given site enhances the
probability to find the particle at the neighboring site. It is clear that
this situation can be realized for attractive interactions. Comparing
theoretical predictions with Monte Carlo simulations (Fig. 5) again indicates
that our theory works very well for repulsive interactions, while for
attractions, although the trends are correctly picked up, there are
deviations.

The analysis of results presented in Figs. 3, 4 and 5 strongly indicates that
correlations are important for understanding the mechanisms of interacting
molecular motors. However, it also raises a question of why our theoretical
approach, that explicitly takes into account some correlations, is able to
correctly describe the stationary properties only for repulsive interactions
and for weak attractions. To answer this question we note that the dynamic
behavior strongly depends on the sign of the interactions. For $E<0$, the
presence of the particle at the site $i$ leads to a lower probability of
finding another particle at the site $i+1$. Then if there is nothing at the
site $i+1$ the occupancy of the site $i+2$ will be independent of the fact
that there is the particle at the site $i$. These arguments suggest that
correlations for repulsive interactions are short-range and relatively weak.
For $E>0$ the situation is very different. Here the presence of the particle
at the site $i$ stimulates the occupancy of the site $i+1$, and consequently
the occupation state of the site $i+2$ depends on the state of the site $i$.
This is consistent with long-range and strong correlations. By construction
(see Sec. 2.2), our theory accounts only for short-range correlations because
the evolution of two-site clusters is monitored. This is the main reason why
our approach is so successful for repulsive interactions, while providing
mostly a qualitative description for attractive interactions.

One of the main advantages of our theoretical method is the fact that it can
be easily extended to account for more realistic features of the motor
proteins transport. To illustrate this we analyze the effect of varying the
splitting coefficient $\theta$ on multi-particle dynamics of interacting
molecular motors. The results are presented in Figs. 4 and 6. One can see that
dynamics is different for small and large values of the interaction
splittings. It can be concluded from Eq. (2) that small $\theta$ describe the
situation when the formation of the particle clusters is weakly affected by
the interactions. At the same time, the breaking of the particle bonds is
strongly influenced by interactions. For $\theta\approx 1$ the trend is
opposite: the particle cluster formation depends strongly on interactions,
while the bond breaking is not.

Fig. 4 shows that the particle current (in the MC phase) is generally a non-
monotonic function of interactions. At large repulsions the current saturates,
while for large attractions the fluxes are going to zero. The maximal particle
current is achieved for relatively weak repulsive interactions
($E\simeq-(1-2)$ kBT). The monotonic decrease in the particle current is only
observed for the special case of $\theta=0$. Similar dynamics is observed for
large $\theta>0.9$ (see Fig. 6), but here the most optimal conditions are
reached now for positive interactions. The case of $\theta=1$ is again a
special one, and the monotonic increase in the current is observed for all
range of interactions.

Figure 6: Maximal particle currents as a function of the interaction energy
for large energy splittings. Lines are theoretical predictions, symbols are
from computer simulations.

In light of these findings, it is important to discuss the effect of
intermolecular interactions for real motor proteins. These interactions have
been measured experimentally for kinesins, indicating weak attractions of
order of $E=(1.6\pm 0.5)$ kBT [10]. Previous theoretical studies suggested
that kinesins function at the conditions that do support the maximal current,
but the analysis was based on the symmetric splitting of interactions for
transitions rates ($\theta=0.5$) [17]. Our new results presented in Figs. 4
and 6 indicate that this is probably a reasonable description of multi-
particle dynamics of kinesins for for most interaction splittings
($\theta<0.9$). In this case the kinesin might operate at the conditions where
small changes in interactions lead to large modification in the particle
dynamics. It has been argued that this might be important for maintaining
robust cellular transport [5, 8, 17]. However, our results (Fig. 6) also
suggest another intriguing possibility that the kinesin fluxes might be
optimized if the splitting affects more the formation of particle clusters
($\theta>0.9$). The parameter $\theta$ is a microscopic property that cannot
be obtained from our mesoscopic theoretical method. To test this idea it will
be important to measure and calculate this quantity in more advanced
experimental and theoretical investigations.

##  4 Summary and Conclusions

We developed a new theoretical approach to analyze the role of intermolecular
interactions in the collective dynamics of molecular motors that move along
linear filaments. Our method is based on utilizing totally asymmetric
exclusion processes, which have been successfully applied for studying various
processes in Chemistry, Physics and Biology. It modifies the transition rates
by interactions via fundamental thermodynamic arguments. A simple theoretical
framework, that we call the modified cluster mean-field and that takes into
account some correlations, is presented and fully discussed. It allows us to
calculate analytically or numerically exactly all dynamic properties of
interacting molecular motors. We find that interactions induce correlations in
the system of collectively moving motor proteins, and the strength of
correlations depends on the sign of the interactions. It was argued that for
repulsions the correlations are short-range and relatively weak, while for
attractions the range and amplitude of correlations are larger. This also
leads to different dynamic behavior of interacting molecular motors. For
repulsions the dynamics is weakly affected by the strength of interactions,
however for attractions the dynamics is modified much stronger. We also
investigated the effect of the symmetry of interaction by analyzing splittings
between different transitions. It was found that when the formation of
particle clusters is weakly affected by interactions the most optimal fluxes
can be realized for weakly repulsive interaction. But when breaking of bonds
between neighboring particles is strongly influenced, the maximal current can
be achieved for attractive interactions. Furthermore, the importance of these
results for kinesin motor proteins has been discussed.

The most important advantage of our method is that it can be easily extended
to investigate additional more realistic features of molecular motors
transport such as backward steps, binding and unbinding of motor proteins at
all sites, multiple parallel pathways and limited resources of motor proteins
in the surrounding solution. It will be very interesting to generalize our
approach to study these phenomena if we want to understand better mechanisms
of cellular transport processes. However, despite its simplicity and the
successful application for the interacting molecular motors, it should be
noted that our method is still approximate and many important features are not
well described. For example, for a large range of parameters the effect of
attractive interactions is given only qualitatively. Thus it will be important
to test our theory in experiments and in more advanced theoretical models.

## Acknowledgments

The work was supported by grants from the Welch Foundation (C-1559), from the
NSF (Grant CHE-1360979), and by the Center for Theoretical Biological Physics
sponsored by the NSF (Grant PHY-142765).

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# Generation of heralded optical ‘Schrödinger cat’ states by photon-addition

Yi-Ru Chen,1 Hsien-Yi Hsieh,1 Jingyu Ning,1 Hsun-Chung Wu,1 Hua Li Chen,2 Zi-
Hao Shi,1 Popo Yang,1 Ole Steuernagel,1 Chien-Ming Wu,1 and Ray-Kuang
Lee1,2,3,4<EMAIL_ADDRESS>1Institute of Photonics Technologies, National
Tsing Hua University, Hsinchu 30013, Taiwan
2Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
3Physics Division, National Center for Theoretical Sciences, Taipei 10617,
Taiwan
4Center for Quantum Technology, Hsinchu 30013, Taiwan

###### Abstract

Optical ‘Schrödinger cat’ states, the non-classical superposition of two
quasi-classical coherent states, serve as a basis for gedanken experiments
testing quantum physics on mesoscopic scales and are increasingly recognized
as a resource for quantum information processing. Here, we report the first
experimental realization of optical ‘Schrödinger cats’ by adding a photon to a
squeezed vacuum state, so far only photon-subtraction protocols have been
realized. Photon-addition gives us the advantage of using heralded signal
photons as experimental triggers, and we can generate ‘Schrödinger cats’ at
rates exceeding $8.5\times 10^{4}$ counts per second; at least one order of
magnitude higher than all previously reported realizations. Wigner
distributions with pronounced negative parts are demonstrated at down to -8.89
dB squeezing, even when the initial squeezed vacuum input state has low
purity. Benchmarking against such a degraded squeezed input state we report a
maximum fidelity of more than $80\%$ with a maximum cat amplitude of
$|\alpha|\approx 1.66$. Our experiment uses photon-addition from pairs, one of
those photons is used for monitoring, giving us enhanced control; moreover the
pair production rates are high and should allow for repeated application of
photon-addition via repeat-stages.

Coherent laser fields described by the quantum state $|\alpha\rangle$ can be
put into a mesoscopic superposition of two such states
$|\alpha\rangle-|-\alpha\rangle$ Marek  _et al._ (2008); Navarrete-Benlloch
_et al._ (2012); Chabaud  _et al._ (2017). Here the light field points in
opposite directions $\pm\alpha$ at the same time, and this is superposed in
such a way that the relative phase between the individual components leads to
destructive interference at the center. Such superposition states are called
optical Schrödinger “cat states”, in honor of Erwin Schrödinger’s famous
gedanken experiments to illustrate paradoxa of quantum physics Schrödinger
(1935).

Such states are important as tools for testing macroscopic limits of quantum
physics and also for many quantum computation and communication protocols
Braunstein and van Loock (2005); Weedbrook  _et al._ (2012). But they are hard
to generate and so far only available at low production rate and low
amplitudes, $\alpha$, when using photon-subtraction schemes Ourjoumtsev  _et
al._ (2006).

To go beyond these constrains, here, we report the first experimental
realization of photon-addition for the generation of ‘negative cat’ mesoscopic
superposition states, of the form

$\Psi_{-}=N_{\alpha}(|\alpha\rangle-|-\alpha\rangle)\;,$ (1)

from highly squeezed input states. $N_{\alpha}$ is the normalization constant.

Traditionally, the constituent states $|\alpha\rangle$ are described as weakly
excited Glauber coherent states but in our case the input states can be so
highly squeezed, that we provide an additional modified description to
determine the quantum fidelity of the performance of our setup. We do this by
benchmarking against experimentally determined input states which are
subjected to photon addition, $\hat{\rho}_{\rm sq}^{\rm add}$, as well as
Glauber state superpositions $\Psi_{-}$ (1).

Our approach, using photon-addition, has three immediate advantages over the
traditional photon-subtraction approach: the generation rate increases by an
order of magnitude, the states are non-classical even for high initial
squeezing and low initial state purity, and the amplitude $|\alpha|$ of the
final state, very desirably, increases when using photon-addition, instead of
decreasing it, as is unavoidable when using subtraction.

The earliest theoretical proposal for photon-addition to coherent states
Agarwal and Tara (1991) was soon followed by those for addition to a squeezed
vacuum state Quesne (2001); Zhang and Fan (1992) and an experimental
demonstration of photon-addition and the quantum-to-classical transition for
coherent states of light Zavatta  _et al._ (2004). Also, quantum commutation
rules for thermal states were probed experimentally Parigi  _et al._ (2007);
Zavatta  _et al._ (2007). Here, we perform the first addition of a single
photon to a squeezed vacuum state, both constituent states have no classical
counterpart.

Using an optical parametric oscillator (OPO) in the continuous wave regime,
photon-subtraction from a squeezed vacuum state has been implemented for the
generation of ‘Schrödinger cats’ Neergaard-Nielsen  _et al._ (2006); Wakui
_et al._ (2007). Production of cat states based on controlled removal of a
single photon has to use weakly reflecting mirrors, to avoid damage to the
desired output-states. This badly limits the success rate of such subtraction
approaches.

To boost the state’s usefulness for fundamental and practical applications,
several schemes have been attempted to enlarge the size of cat states, such as
using two-photon Fock states Ourjoumtsev  _et al._ (2007), ancilla-assisted
photon-subtraction Takahashi  _et al._ (2008), and optical synthesis Huang
_et al._ (2015). Even though high purity and entanglement of optical cat
states Asavanant  _et al._ (2017); Sychev  _et al._ (2017), as well as remote
preparation of non-local superpositions Ourjoumtsev  _et al._ (2009), have
been demonstrated, a high generation rate is crucial for practical
applications of such superpositions in fundamental tests and quantum
information protocols.

Contrasted with photon-subtraction, photon-addition from an entangled photon-
pair allows us to use one of the pair as a heralding photon to monitor success
whilst also substantially increasing production rates. Therefore, only through
photon-addition can we reasonably hope to add more photons in further stages
whilst preserving, or even enhancing further, the non-classical nature of the
final state.

A pure squeezed vacuum state, $|\xi\rangle$, to which $m$ photons are added,
has the form
$|\xi,m\rangle=\frac{1}{\sqrt{N_{m}(|\xi|)}}(\hat{a}^{\dagger})^{m}|\xi\rangle$.
Here, $N_{m}(|\xi|)=m!(1-|\xi|^{2})^{-m/2}P_{m}[(1-|\xi|^{2})^{-1/2}]$ is the
normalization factor with Legendre polynomials $P_{m}$ Zhang and Fan (1992).

In the case of one added photon ($m=1$), the single-photon–added pure squeezed
vacuum state is
$|\xi,1\rangle=\frac{\hat{a}^{\dagger}|\xi\rangle}{\sqrt{N_{1}(|\xi|)}}$ and
only a negative-cat superposition state $\Psi_{-}$ (1) can result Dakna  _et
al._ (1998a, b). This is achieved using the signal photon of photon-pairs from
spontaneous parametric down conversion (SPDC), formally

$\\!\\!\\!\\!\\!\\!\\!\\!\\!|\psi\rangle\approx(1+g\,\hat{a}_{s}^{\dagger}\hat{a}_{i}^{\dagger})|\xi\rangle_{s}|0\rangle_{i}=|\xi\rangle_{s}|0\rangle_{i}+g\,\hat{a}_{s}^{\dagger}|\xi\rangle_{s}|1\rangle_{i}\;.\quad$
(2)

Here, $g$ is the gain factor of the SPDC process, and $\hat{a}_{s}^{\dagger}$
and $\hat{a}_{i}^{\dagger}$ are signal and idler photon creation operators,
respectively.

In our experiment, the squeezed vacuum state, $\varrho_{\rm sq}^{\rm OPO}$, at
the output of the OPO, see Fig. 1, is degraded. We reach squeezing levels as
low as -8.89 dB, with a purity of 0.49. Yet, after addition of a single photon
we reach a ‘relative fidelity’, $F_{\text{sq}}^{\text{add}}$, of 80.1% with a
maximum cat size $|\alpha|\approx$ 1.66 at a generation rate of 8.5 $\times
10^{4}$ counts per second, this is at least one order of magnitude higher than
all previously reported rates (which are all based on photon-subtraction).

Figure 1:  Schematic diagram of the experimental setup. (a) The squeezed vacuum beam $|\xi\rangle_{s}$ is produced by the OPO (optical parametric oscillator) and injected into the type-II SPDC (spontaneous parametric down conversion) to perform the photon-addition scheme $\hat{a}_{s}^{\dagger}|\xi\rangle_{s}$. (b) After passing through the filter cavity (FC), the idler single photon state $|1\rangle_{i}$ is used as the heralding signal, collected by the superconducting nanowire single photon detector (SNSPD). The optical ‘Schrödinger cats’, i.e., photon-added squeezed state, is characterized by homodyne tomography with the balanced homodyne detectors (BHD). The setup also includes BS (beam splitter), DM (dichroic mirrors), PBS (polarization beam splitter), SHG (second-harmonic generator), and OSC (oscilloscope). Figure 2:  Wigner distributions of optical Schrödinger cats: (a)-(d) for the theoretical photon-added squeezed states, $W[\hat{\rho}_{\text{sq}}^{\text{add}}]$ of Eq. (3); and (e)-(h) for experimental data, $W[\varrho_{\text{sq}}^{\text{add}}]$, obtained with OPO pump-powers at $P_{\text{OPO}}=5$, $20$, $40$, and $60$ mW, respectively.  $P_{\text{OPO}}$ ($P_{\text{rel}}$%) | SQ : ASQ ($\varrho_{\text{sq}}^{\text{OPO}}$) | Purity ($\varrho_{\text{sq}}^{\text{OPO}}$) | SQ : ASQ ($\varrho_{\text{sq}}^{\text{empty}}$) | Purity ($\varrho_{\text{sq}}^{\text{add}}$) | $W[\varrho_{\text{sq}}^{\text{add}}](0,0)$ | $F_{\text{cat}}$ | $F_{\text{sq}}^{\text{add}}$ | Cat size ($|\alpha|$) |  $\Gamma$(counts/s)
---|---|---|---|---|---|---|---|---|---
5 mW (5%) | -3.76 : 3.89 dB | 0.99 | -0.33 : 0.50 dB | 0.54 | -0.095 | 64.9% | 66.3% | 0.28 | 4.4 $\times 10^{4}$
20 mW (19%) | -6.27 : 7.31 dB | 0.89 | -1.42 : 1.78 dB | 0.55 | -0.104 | 65.9% | 68.7% | 0.62 | 4.5 $\times 10^{4}$
40 mW (38%) | -7.59 : 11.55 dB | 0.63 | -3.42 : 4.87 dB | 0.54 | -0.090 | 61.8% | 68.7% | 1.02 | 5.1 $\times 10^{4}$
60 mW (58%) | -8.89 : 15.13 dB | 0.49 | -4.17 : 9.85 dB | 0.30 | -0.023 | 42.4% | 80.1% | 1.66 | 8.5 $\times 10^{4}$
Table 1:  Results summary for a fixed SPDC pump-power of 10 mW at different
OPO pump-power levels $P_{\text{OPO}}$ (OPO threshold power of
$P_{\text{th}}=$ 103.9 mW, hence, power relative to threshold
$P_{\text{rel}}=P_{\text{OPO}}/P_{\text{th}}$ in %), column 1. Columns 2 and
3, squeezing (SQ) and anti-squeezing (ASQ) levels for state
$\varrho_{\text{sq}}^{\text{OPO}}$ right after the OPO and its purity. Column
4, SQ : ASQ values obtained after the empty SPDC cavity (without pump field,
i.e., for $\varrho_{\text{sq}}^{\text{empty}}$). Column 5, purity of the
output-state, $\varrho_{\text{sq}}^{\text{add}}$, after performing the photon-
addition and, column 6, its negative Wigner distribution values
$W[\varrho_{\text{sq}}^{\text{add}}](0,0)$. Fidelities w.r.t. Glauber cat
states (1) and photon-added impure squeezed states (3), columns 7 and 8. Cat
size $|\alpha|$, and generation rates $\Gamma$, columns 9 and 10. All
squeezing data are averaged over $10$ measurements with a statistical error of
0.12 dB.

As sketched in Fig. 1 (a), we initially generate squeezed vacuum states in a
bow-tie OPO cavity enclosing a periodically poled nonlinear KTiOPO4 (PPKTP)
crystal with second order nonlinear susceptibility, $\chi^{(2)}$, operated
below the lasing threshold, at wavelength 1064 nm Wu  _et al._ (2019). This
OPO-cavity has an optical path length of approximately $285$ mm and thus a
free spectral range of $1.052$ GHz, with a finesse of $19.8$ at $532$ nm and
of $33.4$ at $1064$ nm. The overall efficiency, $1-L$ (where $L$ is the loss),
is $82.2\pm 0.35\%$ and the phase noise is $34.50\pm 1.26$ mrad.

The OPO-generated squeezed vacuum states are subsequently injected into a
type-II SPDC-crystal which is inside another bow-tie cavity of the same
configuration as the OPO-cavity, i.e., also $285$ mm in length, with full-
width at half maximum (FWHM) of 31.8 MHz. It performs the photon-addition
[formally $\hat{a}_{s}^{\dagger}|\xi\rangle$, compare Fig. 1 (b)].

The OPO-cavity’s free spectral range is $1.052$ GHz giving rise to neighboring
modes which we suppress with filter cavities, consisting of one triangle
cavity followed by two Fabry-Perot cavities, with FWHM of 16.3, 131, and 101
MHz, respectively. This filtering plays a critical role in our experiments
since it ensures that, at a time, only a single signal-photon is added into
the squeezed vacuum. The SPDC-pair’s idler photon gets detected by the
superconducting nanowire single-photon detector, generating the heralding
trigger signal for the homodyne detector, the output of which is used to
tomographically reconstruct the Schrödinger cats, see Fig. 1.

The magnitudes of squeezing (SQ) versus anti-squeezing (ASQ), at the output of
the OPO, $\varrho_{\rm sq}^{\rm OPO}$, are almost the same at a low pump-power
of 5 mW, specifically Hsieh  _et al._ (2022): SQ : ASQ = $-3.77$ : $3.89$ dB.
At the highest pump-power of $60$ mW, the squeezing ratio is SQ : ASQ =
$-8.89$ : $15.13$ dB. Simultaneously, the purity of the squeezed vacuum drops
from 0.99 to 0.49, see columns 2 and 3 of Table 1 below.

Note, we use the denotation ‘$\hat{\rho}$’ for theoretical and ‘$\varrho$’ for
experimentally reconstructed quantum states.

The SQ : ASQ levels are additionally measured after the SPDC cavity, without
applying SPDC pump power, yielding the quantum state
$\varrho_{\text{sq}}^{\text{empty}}={\cal L}[\varrho_{\rm sq}^{\rm OPO}]$,
where $\cal L$ is a Lindbladian superoperator describing the effects of the
empty SPDC cavity, see column 4 of Table 1. We do not model these effects
explicitly but determine the losses due to the presence of the empty SPDC-
cavity in the experimental reconstruction of
$\varrho_{\text{sq}}^{\text{empty}}$, which subsequently serves as our
benchmark for the quantification of the ‘relative fidelity’,
$F_{\text{sq}}^{\text{add}}$, of the photon-addition process.

At an SPDC pump-power of $10$ mW, we can ensure that the purity of our signal
photon (which is to be added) shows a second-order correlation function of
$g^{(2)}(0)\approx 0.55$, confirming that we avoid detrimental multiple-photon
events. This SPDC pump-power of $10$ mW is therefore from now on chosen and
kept fixed while we generate photon-added squeezed states at different OPO
pump-powers, namely at $5$, $20$, $40$, and $60$ mW, see Table 1. In Fig. 2
(a)-(d) we display the Wigner distributions,
$W[\hat{\rho}_{\text{sq}}^{\text{add}}]$, for the theoretical description
$\hat{\rho}_{\text{sq}}^{\text{add}}$ of photon-added impure squeezed states,
formally generated from the expression

$\displaystyle\hat{\rho}_{\text{sq}}^{\text{add}}=\hat{a}_{s}^{\dagger}\,\varrho_{\text{sq}}^{\text{empty}}\,\hat{a}_{s}\;.$
(3)

The calculation of states (3) is based on the fact that we use degraded
squeezed vacuum states, $\varrho_{\text{sq}}^{\text{empty}}$ (note that we
have dropped explicit reference to state normalization). Tomographic
reconstructions of the Wigner distributions
$W[\varrho_{\text{sq}}^{\text{add}}]$ (from BHD-data, see Fig. 1) are shown in
Fig. 2 (e)-(h).

In Fig. 2 (a) and (e), we display the Wigner distribution when using low OPO
pump-power of $P_{\text{OPO}}=$ 5 mW. The experimentally reconstructed state
$\varrho_{\text{sq}}^{\text{add}}$ has a purity of $0.54$ and a negative value
of $W[\varrho_{\text{sq}}^{\text{add}}](0,0)=-0.095$. Comparison between
experimental Wigner distributions and Glauber-cat reference states $\Psi_{-}$
of Eq. (1) give good agreement at such low pump-power levels, as the
degradation in the injected squeezed vacuum states is not severe. Maximizing
the fidelity with respect to an ideal Glauber cat state $\Psi_{-}$ allows us
to estimate the amplitude of our cat state as $|\alpha|=0.28$, see Table 1
column 9.

When we increase the OPO pump-power to 20 and 40 mW, the OPO output state,
$\varrho_{\text{sq}}^{\text{OPO}}$, starts to show degraded squeezing levels,
SQ : ASQ = -6.27 : 7.31 dB and -7.59 : 11.55 dB, see Table 1 column 2. Yet,
despite this degradation, the purity in the target cat states remains as high
as that from the nearly perfectly pure squeezed state, i.e., 0.55 and 0.54,
respectively, see Table 1 column 7. For both cases, the negativity remains
pronounced and the amplitude of our cat states enlarges to $|\alpha|=0.62$ and
$1.02$, respectively. See Table 1 columns 6 and 9, and Fig. 2 (f), (g) for the
associated Wigner distributions $W[\varrho_{\text{sq}}^{\text{add}}]$.

Finally, we reach an OPO pump-power of 60 mW. Now the OPO output-state,
$\varrho_{\text{sq}}^{\text{OPO}}$, shows further degraded squeezing levels
and a low purity of $0.49$, see Table 1 column 3. Therefore, the final optical
Schrödinger cat state, $\varrho_{\text{sq}}^{\text{add}}$, also has a low
purity of $0.30$, see Table 1 column 5. The associated ‘cat amplitude’ is
large, $|\alpha|=1.66$, see Table 1 column 9 and Fig. 2 (h):

Most surprisingly, however, despite the input state
$\varrho_{\text{sq}}^{\text{OPO}}$’s low purity (at 60 mW OPO pump-power) and
a noisy environment, using photon-addition lets the optical cat ‘survive’: it
shows a clear negative value of
$W[\varrho_{\text{sq}}^{\text{add}}](0,0)=-0.023$.

Overall, we observe a maximum negativity of $-0.104$ at an OPO pump-power of
$20$ mW when the purity of our generated optical cat stats also reaches its
largest value.

It is known that, when applying photon-subtraction, the input squeezed states
must be pure enough, practically limiting the input state’s squeezing to no
less than $-5$ dB. Additionally, repeated application of photon-subtraction
drives states towards the vacuum state Hu  _et al._ (2010).

By contrast, our photon-addition scheme can support the generation of optical
cat states with higher squeezing levels, even as the input squeezed states
become increasingly impure.

Photon-addition is so tolerant to experimental imperfections that we still
observe pronounced negative parts in the Wigner distribution all the way down
to $-8.89$ dB in squeezing, even though the input squeezed state has low
purity, see Table 1 column 6.

We compare our experimentally reconstructed state
$\varrho_{\text{sq}}^{\text{add}}$ with the ideal Glauber cat state
($|\alpha\rangle-|-\alpha\rangle$) by calculating its fidelity
$F_{\text{cat}}$. This fidelity remains higher than $60\%$ even when working
at the OPO pump-power of 40 mW, but at 60 mW pump-power it drops to $42.4\%$,
see Table 1 column 7. We will now explain that this drop in fidelity is a poor
measure of the good performance of photon-addition; the latter is better
quantified by the relative fidelity $F_{\text{sq}}^{\text{add}}$ with respect
to state (3), see Table 1 column 8:

When we introduced the negative cat state $\Psi_{-}$ of Eq. (1), based on
Glauber states, we emphasized that our squeezing levels are so high and
therefore, at the current state of technology for the generation of squeezed
states, the impurity so large, that descriptions by Glauber states as
constituents become ill-matched. To address this shortcoming in the
_theoretical analysis_ we therefore directly take into account the degradation
in the injected squeezed states by determining it experimentally as
$\varrho_{\text{sq}}^{\text{empty}}$ Hsieh  _et al._ (2022). We then extract
the fidelity due to photon-addition by using the _experimentally
reconstructed_ density matrix $\varrho_{\text{sq}}^{\text{add}}$ and comparing
it to the ideal photon-addition upon $\varrho_{\text{sq}}^{\text{empty}}$,
namely by comparison with the state $\hat{\rho}_{\text{sq}}^{\text{add}}$ of
Eq. (3). This, gives us the benchmark for the extraction of the relative
fidelity, $F_{\text{sq}}^{\text{add}}$, of only the photon-addition operation
by itself.

To our delight, this analysis confirms that photon-addition is robust. Adding
a photon to a highly squeezed impure input state has a high relative fidelity
$F_{\text{sq}}^{\text{add}}$-value of $80.1\%$ at 60 mW, see Table 1 column 8.

We tentatively attribute this good performance to the fact that the highly
squeezed impure input state $\varrho_{\text{sq}}^{\text{empty}}$ has broad
support in phase space and thus lower peak values than when it is generated at
lower OPO pump-power, see Fig. 2 (d) and (h). This should make it easier for
the added photon to ‘punch a negative hole’ into its Wigner distribution, thus
increasing the relative fidelity.

These interpretations are in agreement with our finding that the ‘cat
survives’ to extremely high levels of squeezing whilst being generated from a
rather impure squeezed input state.

We have shown that photon-addition can be applied to a highly squeezed input
state with a squeezing level, SQ, of $-8.89$ dB. This corresponds to a cat
state with an amplitude of 1.66 and a fidelity for the photon-addition process
of $80.1\%$ whilst maintaining control through the use of idler photons at a
generation rate of 8.5 $\times 10^{4}$ per second, this is at least one order
of magnitude higher than all previously reported rates.

Our experiments thus overturn the common belief that sequential application of
photon-addition through multiple OPOs would involve significant losses and is
therefore not advisable in experimental practice Lvovsky  _et al._ (2020).

Instead, our successful generation of heralded optical Schrödinger cats by
photon-addition opens up the possibility towards generation of still larger
cat states by repeatedly adding photons Arman  _et al._ (2021).

We emphasize that imperfections of our experiment are to a considerable extent
due to the fact that the squeezed states we start from have reduced levels of
purity when they are highly squeezed. In other words, the synthesis of highly
squeezed states with high purity remains a desirable and partially unfulfilled
goal of the quantum optics community. By contrast, we establish here that the
addition of a photon can already be controlled to a surprisingly good degree,
we quantified this using a well-matched ‘relative fidelity’ measure.

Controlled photon-addition promises to become a new powerful building block
for advanced quantum state synthesis. With such a high generation rate, our
photon-addition approach also promises to facilitate applications in quantum
information processing using cat codes Chamberland  _et al._ (2022); Gravina
_et al._ (2023), or preparing error-correcting codes Gottesman  _et al._
(2001); Vasconcelos  _et al._ (2010); Bourassa  _et al._ (2021); Eaton  _et
al._ (2019); Tzitrin  _et al._ (2020).

This work is partially supported by the Ministry of Science and Technology of
Taiwan (Nos. 108-2923-M-007-001-MY3 and 110-2123-M-007-002), Office of Naval
Research Global, US Army Research Office (ARO), and the collaborative research
program of the Institute for Cosmic Ray Research (ICRR), the University of
Tokyo.

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