* [260] Mohammed Majam, Mothepane Phatsoane, Keith Hanna, Charles Faul, Lovkesh Arora, Sarvesh Makthal, Akhil Kumar, Kashyap Jois, Samanta Tresha Lalla-Edward, Utility of a machine-guided tool for assessing risk behavior associated with contracting HIV in three sites in South Africa: protocol for an in-field evaluation , JMIR Research Protocols 10 (12), e30304, 2021
  * [261] Rachel T Esra, Jacques Carstens, Janne Estill, Ricky Stoch, Sue Le Roux, Tonderai Mabuto, Michael Eisenstein, Olivia Keiser, Mhari Maskew, Matthew P Fox, Lucien De Voux, Kieran Sharpey-Schafer, Historical visit attendance as predictor of treatment interruption in South African HIV patients: Extension of a validated machine learning model , PLOS Global Public Health 3 (7), e0002105, 2023
  * [262] Rachel Esra, Jacques Carstens, Sue Le Roux, Tonderai Mabuto, Michael Eisenstein, Olivia Keiser, Erol Orel, Aziza Merzouki, Lucien De Voux, Mhari Maskew, Kieran Sharpey-Schafer, Validation and improvement of a machine learning model to predict interruptions in antiretroviral treatment in South Africa , JAIDS Journal of Acquired Immune Deficiency Syndromes, 10.1097, 2022.
  * [263] Riyad Ismail, Onisimo Mutanga, Lalit Kumar, Modeling the Potential Distribution of Pine Forests Susceptible to Sirex Noctilio Infestations in Mpumalanga, South Africa, Transactions in GIS 14 (5), 709-726, 2010
  * [264] Rudzani Mulaudzi, Ritesh Ajoodha, Application of deep learning to forecast the South African unemployment rate: a multivariate approach , 2020 IEEE Asia-Pacific Conference on Computer Science and Data Engineering (CSDE), 1-6, 2020
  * [265] Kassahun Birhanu Tadesse, Megersa Olumana Dinka, Water Quality Class Modeling Using Machine Learning Algorithms at Roodeplaat Dam, South Africa, Preprints, 2023.
  * [266] Christopher Odoom, Eric Teye Mensah, Sarah Mensah Fobi, Daniel Maposa, Predictive Analysis of Misuse of Alcohol and Drugs using Machine Learning Algorithms: The Case of using an Imbalanced Dataset from South Africa, Alexander Boateng, Appl. Math 17 (2), 261-271, 2023
  * [267] Tshephisho Joseph Sefara, Madimetja Jonas Manamela, Promise Tshepiso Malatji, Text-based language identification for some of the under-resourced languages of South Africa,2016 International Conference on Advances in Computing and Communication Engineering (ICACCE), 303-307, 2016
  * [268] Wim Naudé, Amy Bray, Celina Lee, Crowdsourcing Artificial Intelligence in Africa: Findings from a Machine Learning Contest , IZA Discussion Paper, 2021
  * [269] Willem Fourie, Teresa Kroesen, Jurgens de Lange, Using machine learning to improve the science-policy-society interface on the SDGs in South Africa, South African SDG Hub
  * [270] Dossou, B. F. P., Tonja, A. L., Emezue, C. C., Olatunji, T., Etori, N. A., Osei, S., Adewumi, T. P., Singh, S. (2023). ”Adapting Pretrained ASR Models to Low-resource Clinical Speech using Epistemic Uncertainty-based Data Selection.” Preprint.
  * [271] Dossou, B. F. P., Houndayi, I. B., Zantou, P., Hacheme, G. (2023). ”FonMTL: Towards Multitask Learning for the Fon Language.” In EMNLP 2023.
  * [272] Dossou, B. F. P., Gbenou, Y. K. S., Nono, M. G. (2023). ”Pretrained Vision Models for Predicting High-Risk Breast Cancer Stage.” In ICLR 2023.
  * [273] Olatunji, T., Afonja, T., Dossou, B. F. P., Tonja, A. L., Emezue, C. C., Rufai, A. M., Singh, S. (2023). ”AfriNames: Most ASR models ’butcher’ African Names.” In Interspeech 2023.
  * [274] Liu, D., Jain, M., Dossou, B. F. P., Shen, Q., Lahlou, S., Goyal, A., Malkin, N., Emezue, C. C., Zhang, D., Hassen, N., Ji, X., Kawaguchi, K., Bengio, Y. (2023). ”GFlowOut: Dropout with Generative Flow Networks.” In ICML 2023.
  * [275] Ogundepo, O., Gwadabe, T., Rivera, C., Clark, J., Ruder, S., Adelani, D. I., Dossou, B. F. P., et al. (2023). ”AfriQA: Cross-lingual Open-Retrieval Question Answering for African Languages.” In EMNLP 2023.
  * [276] Olatunji, T., Afonja, T., Yadavalli, A., Emezue, C. C., Singh, S., Dossou, B. F. P., et al. (2023). ”AfriSpeech-200: Pan-African Accented Speech Dataset for Clinical and General Domain ASR.” In EMNLP 2023.
  * [277] Leong, C., Shandilya, H., Dossou, B. F. P., Tonja, A. L., Mathew, J., Omotayo, A.-H., Yousuf, O., Akinjobi, Z., Emezue, C. C., Muhammad, S., Kolawole, S., Adewumi, T. P. (2023). ”Adapting to the Low-Resource Double-Bind: Investigating Low-Compute Methods on Low-Resource African Languages.” In ICLR 2023.
  * [278] Adelani, D. I., Masiak, M., Azime, I. A., Alabi, J. O., Tonja, A. L., Mwase, C., Ogundepo, O., Dossou, B. F. P., et al. (2023). ”MasakhaNEWS: News Topic Classification for African languages.” In AACL 2023.
  * [279] Dione, C. M. B., Adelani, D. I., Nabende, P., Dossou, B. F. P., et al. (2023). ”MasakhaPOS: Part-of-Speech Tagging for Typologically Diverse African Languages.” In ACL 2023.
  * [280] Nigatu, H. H., Tonja, A. L., Kalita, J. (2023). ”The Less the Merrier? Investigating Language Representation in Multilingual Models.” In EMNLP 2023.
  * [281] Marivate, V., Mots’Oehli, M., Wagnerinst, V., Lastrucci, R., Dzingirai, I. (2023). ”PuoBERTa: Training and evaluation of a curated language model for Setswana.” In SACAIR 2023.
  * [282] Mbuvha, R., Adelani, D. I., Mutavhatsindi, T., Rakhuhu, T., Mauda, A., Maumela, T. J., Masindi, A., Rananga, S., Marivate, V., Marwala, T. ”MphayaNER: Named Entity Recognition for Tshivenda.”
  * [283] Myoya, R., Banda, F., Marivate, V., Modupe, A. (2023). ”Fine tuning multilingual pretrained african language models ” In ICLR 2023.
  * [284] Ngomane, D., Marivate, V., Abbott, J., Mabuya, R. (2023). ”Unsupervised Cross-lingual Word Embedding Representation for English-isiZulu.” In RAIL 2023.
  * [285] Siminyu, K., Abbott, J., Tubosun, K., Mthembu, A. T., Ramkilowan, A., Oladimeji, B., Cell Press Publication. ”Consultative engagement of stakeholders toward a roadmap for African language technologies.”
  * [286] Ojo, J., Ogueji, K., Stenetorp, P., Adelani, D. I.. How good are Large Language Models on African Languages?
  * [287] Obvious Nchimunya Chilyabanyama, Roma Chilengi, Michelo Simuyandi, Caroline C Chisenga, Masuzyo Chirwa, Kalongo Hamusonde, Rakesh Kumar Saroj, Najeeha Talat Iqbal, Innocent Ngaruye, Samuel Bosomprah, Performance of machine learning classifiers in classifying stunting among under-five children in Zambia , Children 9 (7), 1082, 2022
  * [288] Dhruv Grover, Sebastian Bauhoff, Jed Friedman, Using supervised learning to select audit targets in performance-based financing in health: An example from Zambia , PloS one 14 (1), e0211262, 2019
  * [289] Sipiwe Chihana, Jameson Mbale, Nchimunya Chaamwe, Leveraging Machine Learning for Ambient Air Pollutant Prediction: The Zambian Mining Environment Context , Proceedings of International Conference for ICT (ICICT)-Zambia 4 (1), 1-5, 2022
  * [290] Varun Kshirsagar, Jerzy Wieczorek, Sharada Ramanathan, Rachel Wells, Household poverty classification in data-scarce environments: A machine learning approach, arXiv preprint arXiv:1711.06813, 2017
  * [291] Kwabena Adu-Ababio, Aliisa Koivisto, Eliya Lungu, Evaristo Mwale, Jonathan Msoni, Kangwa Musole, Estimating tax gaps in Zambia, Preprint 2023.
  * [292] Sahar Kazemzadeh, Jin Yu, Shahar Jamshy, Rory Pilgrim, Zaid Nabulsi, Christina Chen, Neeral Beladia, Charles Lau, Scott Mayer McKinney, Thad Hughes, Atilla P Kiraly, Sreenivasa Raju Kalidindi, Monde Muyoyeta, Jameson Malemela, Ting Shih, Greg S Corrado, Lily Peng, Katherine Chou, Po-Hsuan Cameron Chen, Yun Liu, Krish Eswaran, Daniel Tse, Shravya Shetty, Shruthi Prabhakara, Deep learning detection of active pulmonary tuberculosis at chest radiography matched the clinical performance of radiologists, Radiology 306 (1), 124-137, 2023
  * [293] Valentina Bellemo, Zhan Wei Lim, Gilbert Lim, Quang Nguyen, Michelle YT Yip, Yuchen Xie, Xin Qi Lee, Haslina Hamzah, Jinyi Ho, Gavin Siew Wei Tan, Wynne Hsu, Mong Li Lee, Sobha Sivaprasad, Geeta Menon, Tien Yin Wong, Daniel SW Ting, Application of Deep Learning System to Screen for Diabetic Retinopathy in an Underprivileged African Population with Diabetes, Investigative Ophthalmology & Visual Science 60 (9), 1439-1439, 2019.
  * [294] John Batani, Manoj Sewak Maharaj, Towards data-driven pediatrics in Zimbabwe, 2022 International Conference on Artificial Intelligence, Big Data, Computing and Data Communication Systems (icABCD), 1-7, 2022
  * [295] Vehicle Damage model classification for Zimbabwe Insurance Sector using MobileNetV2 and DenseNet121, Pavlov Takudzwa Mpinyuri, Edmore Tarambiwa, 2022 1st Zimbabwe Conference of Information and Communication Technologies (ZCICT), 1-5, 2022
  * [296] Panashe Chiurunge, Agripah Kandiero, Framework for Prediction of Depression Among Adolescents Using Machine Learning: A Case of Zimbabwe, Integrating Digital Health Strategies for Effective Administration, 310-344, 2023.
  * [297] Daxelle Sakubu, Kelly Joelle Gatore Sinigirira, David Niyukuri, Predicting malaria dynamics in Burundi using deep Learning Models , Preprint 2023.
  * [298] CIAT, Banana pest risk assessment along banana trade axes running from low to high altitude sites, in the Eastern DR Congo and in Burundi, The Alliance of Bioversity International and the International Center for Tropical Agriculture (CIAT), 2020.
  * [299] Shafika Isaacs, Florence Ngombo , ICT in Education in the Comoros, Population, 2005
  * [300] Harriet Joseph Ottenheimer: Spelling Shinzwani: Dictionary construction and orthographic choice in the Comoro Islands, Written Language & Literacy 4 (1), 15-29, 2001
  * [301] Abdi, Hamda and Mohamed, Assabo and Omar, Abdou Idris, Sky Temperature Forecasting in Djibouti: An Integrated Approach Using Measured Climate Data and Artificial Neural Networks. Preprint SSRN, 2023.
  * [302] Denis Pastory Rubanga, Sergio Azael May-Cuevas, Yessy Arvelyna, and Sawahiko Shimada. (2023). Fracture-fault detection using deep learning with stepwise elimination from satellite images in Djibouti. Geomate Journal, 25(108), 241 - 248. Retrieved from https://geomatejournal.com/geomate/article/view/3967
  * [303] Shawn B. Hood, Matthew J. Cracknell, Michael F. Gazley, Anya M. Reading, Improved supervised classification of bedrock in areas of transported overburden: Applying domain expertise at Kerkasha, Eritrea, Applied Computing and Geosciences, 2019, 100001, ISSN 2590-1974,
  * [304] Michael B. Richman, Lance M. Leslie, Zewdu T. Segele, Classifying Drought in Ethiopia Using Machine Learning, Procedia Computer Science, Volume 95, 2016, Pages 229-236, ISSN 1877-0509,
  * [305] Eshetu, K.D., Alamirew, T. and Woldesenbet, T.A. Interpretable machine learning for predicting evaporation from Awash reservoirs, Ethiopia. Earth Sci Inform (2023). https://doi.org/10.1007/s12145-023-01063-y
  * [306] Martin Kuradusenge, Santhi Kumaran, Marco Zennaro, ”Rainfall-induced landslide prediction using machine learning models: The case of Ngororero District, Rwanda,” International journal of environmental research and public health 17 (11), 4147, 2020
  * [307] Roger Muremyi, Dominique Haughton, Ignace Kabano, François Niragire: ”Prediction of out-of-pocket health expenditures in Rwanda using machine learning techniques,” Pan African Medical Journal 37 (1), 2020
  * [308] Yves Uwiragiye, Mbezele Junior Yannick Ngaba, Mengzhen Zhao, Ahmed S Elrys, Gerard BM Heuvelink, Jianbin Zhou, ”Modelling and mapping soil nutrient depletion in humid highlands of East Africa using ensemble machine learning: A case study from Rwanda,” Catena 217, 106499, 2022
  * [309] Francois Mbonyinshuti, Joseph Nkurunziza, Japhet Niyobuhungiro, Egide Kayitare, ”The prediction of essential medicines demand: a machine learning approach using consumption data in Rwanda,” Processes 10 (1), 26, 2021
  * [310] Similien Ndagijimana, Ignace Habimana Kabano, Emmanuel Masabo, Jean Marie Ntaganda, ”Prediction of Stunting Among Under-5 Children in Rwanda Using Machine Learning Techniques,” Journal of Preventive Medicine and Public Health 56 (1), 41, 2023
  * [311] Emmanuel Masabo, Joseph Nzabanita, Innocent Ngaruye, Charles Ruranga, Jean Paul Nizeyimana, Jeanne Uwonkunda, Denis Ndanguza, ”Early detection of students at risk of poor performance in Rwanda higher education using machine learning techniques,” International Journal of Information Technology, 1-10, 2023
  * [312] Theogene Rizinde, Innocent Ngaruye, Nathan D Cahill, ”Comparing Machine Learning Classifiers for Predicting Hospital Readmission of Heart Failure Patients in Rwanda,” Journal of Personalized Medicine 13 (9), 1393, 2023
  * [313] Jean Baptiste Nsengiyumva, Roberto Valentino, ”Predicting landslide susceptibility and risks using GIS-based machine learning simulations, case of upper Nyabarongo catchment,” Geomatics, Natural Hazards and Risk 11 (1), 1250-1277, 2020
  * [314] Paul Brimble, Patrick McSharry, Felix Bachofer, Jonathan Bower, Andreas Braun, ”Using machine learning and remote sensing to value property in Rwanda,” International Growth Centre, 2020
  * [315] Arthur Kwizera, Niranjan Kissoon, Ndidiamaka Musa, Olivier Urayeneza, Pierre Mujyarugamba, Andrew J Patterson, Lori Harmon, Joseph C Farmer, Martin W Dünser, Jens Meier, ”A machine Learning-Based triage tool for children with acute infection in a low resource setting,” Pediatric Critical Care Medicine 20 (12), e524-e530, 2019
  * [316] Janvier Niyitegeka, Didacienne Mukanyiligira, Said Rutabayiro Ngoga, Emmanuel Masabo, Louis Sibomana, Raymond Ndacyayisaba, ”Employing machine learning and internet of things for malaria outbreak prediction in Rwanda,” International Workshop Soft Computing Applications, 338-364, 2020
  * [317] Martin Kuradusenge, Eric Hitimana, Damien Hanyurwimfura, Placide Rukundo, Kambombo Mtonga, Angelique Mukasine, Claudette Uwitonze, Jackson Ngabonziza, Angelique Uwamahoro, ”Crop yield prediction using machine learning models: case of Irish potato and maize,” Agriculture 13 (1), 225, 2023
  * [318] Gatesi Jean de Dieu, Shuai Bin, Wencheng Huang, Ntakiyemungu Mathieu, ”Comparison of supervised machine learning algorithms for road traffic crash prediction models in Rwanda,” Proceedings of the Institution of Civil Engineers-Transport, 1-37, 2023
  * [319] Joseph Habiyaremye, Marco Zennaro, Chomora Mikeka, Emmanuel Masabo, Kayalvizhi Jayavel, Santhi Kumaran, ”A data-driven predictive machine learning model for efficiently storing temperature-sensitive medical products, such as vaccines: Case study: Pharmacies in Rwanda,” Journal of Healthcare Engineering 2021, 1-12, 2021
  * [320] Leopord Uwamahoro, Papias Niyigena, ”Deep Learning in Greenhouse Gases Emissions from Agriculture Activities in Rwanda Using Long Short Term Memory Recurrent Neural Network,” Int. Res. J. Eng. Technol 6 (10), 928-931, 2019
  * [321] Ines Emera, Markon Sandor, ”Creation of farmers’ awareness on fall armyworms pest detection at early stage in Rwanda using deep learning,” 2019 8th International Congress on Advanced Applied Informatics (IIAI-AAI), 538-541, 2019
  * [322] Regina Kilwenge, Julius Adewopo, Zhanli Sun, Marc Schut, ”UAV-based mapping of banana land area for village-level decision-support in Rwanda,” Remote Sensing 13 (24), 4985, 2021
  * [323] Emily Murphy, Swathi Samuel, Joseph Cho, William Adorno, Marcel Durieux, Donald Brown, Christian Ndaribitse, ”Checkbox detection on Rwandan perioperative flowsheets using convolutional neural network,” 2021 Systems and Information Engineering Design Symposium (SIEDS), 1-6, 2021
  * [324] Fidelis L Bologo, Barry Rawn, Tunmise Raji, ”Ground Truths to Support Remote-Sensing Inference of Irrigation Benefits and Effects in Rwanda,” 2023 IEEE AFRICON, 1-6, 2023
  * [325] Francis Kipkogei, Ignace H Kabano, Belle Fille Murorunkwere, Nzabanita Joseph, ”Business success prediction in Rwanda: a comparison of tree-based models and logistic regression classifiers,” SN Business & Economics 1, 1-19, 2021
  * [326] Omar H Kombo, Santhi Kumaran, Emmanuel Ndashimye, Alastair Bovim, ”An Ensemble Mode Decomposition Combined with SVR-RF Model for Prediction of Groundwater Level: The Case of Eastern Rwandan Aquifers,” Computer Science On-line Conference, 312-328, 2022
  * [327] Meron Alemnew Kifle, ”IoT and ML-Based precision agriculture: A case of Rwanda coffee,” College of science and Technology, 2022
  * [328] Esaie Dufitimana, Theophile Niyonzima, ”Leveraging the Potential of Convolutional Neural Network and Satellite Images to Map Informal Settlements in Urban Settings of the City of Kigali, Rwanda,” Rwanda Journal of Engineering, Science, Technology and Environment 5 (1), 2023
  * [329] DJB Swanborn, Nico Fassbender, VAI Huvenne, SJ Pittman, PV Stefanoudis, LC Woodall, ”Geomorphological drivers of deeper reef habitats around Seychelles,” Coral Reefs 41 (4), 1001-1016, 2022
  * [330] Benjamin Lee, Lucy Martin, Dimosthenis Traganos, Sylvanna Antat, Stacy K Baez, Annabelle Cupidon, Annike Faure, Jérôme Harlay, Matthew Morgan, Jeanne A Mortimer, Peter Reinartz, Gwilym Rowlands, ”Mapping the National Seagrass Extent in Seychelles Using PlanetScope NICFI Data,” Remote Sensing 15 (18), 4500, 2023
  * [331] Daniel R Schrider, Julien Ayroles, Daniel R Matute, Andrew D Kern, ”Supervised machine learning reveals introgressed loci in the genomes of Drosophila simulans and D. sechellia,” PLoS genetics 14 (4), e1007341, 2018
  * [332] Luo Xiao, Sally W Thurston, David Ruppert, Tanzy MT Love, Philip W Davidson, ”Bayesian models for multiple outcomes in domains with application to the Seychelles Child Development Study,” Journal of the American Statistical Association 109 (505), 1-10, 2014
  * [333] Abdullahi Ahmed Abdirahman, Abdirahman Osman Hashi, Ubaid Mohamed Dahir, Mohamed Abdirahman Elmi,Octavio Ernest Romo Rodriguez, Comparative Analysis of Machine Learning and Deep Learning Models for Sentiment Analysis in Somali Language, SSRG International Journal of Electrical and Electronics Engineering Volume 10 Issue 7, 41-52, July 2023
  * [334] Tohid Atashbar, How Nations Become Fragile: An AI-Augmented Bird’s-Eye View (with a Case Study of South Sudan), WPIEA2023167, August 11, 2023.
  * [335] Felix Sukums, Deogratias Mzurikwao, Deodatus Sabas, Rebecca Chaula, Juliana Mbuke, Twaha Kabika, John Kaswija, Bernard Ngowi, Josef Noll, Andrea S. Winkler, Sarah Wamala Andersson, The use of artificial intelligence-based innovations in the health sector in Tanzania: A scoping review, Health Policy and Technology, Volume 12, Issue 1, 2023, 100728, ISSN 2211-8837, https://doi.org/10.1016/j.hlpt.2023.100728.
  * [336] Nick Jones, Vivien Deparday, Nuala Margaret Coman, Mapping Dar es Salaam’s trees through artificial intelligence , MAY 18, 2022
  * [337] Andrew Katumba, Joyce Nakatumba-Nabende, Jenifer Winfred Namuyanja, Jeremy Tusubira Francis, Claire Babirye, Jonathan Mukiibi, Chodrine Mutebi, Hewiit Tusiime, Building NLP Text and Speech Datasets for Low Resourced Languages in East Africa, 2022.
  * [338] Emily Aiken, Suzanne Bellue, Dean Karlan, Chris Udry and Joshua E. Blumenstock, Machine learning and phone data can improve targeting of humanitarian aid. Nature 603, 864 - 870 (2022). https://doi.org/10.1038/s41586-022-04484-9
  * [339] Kohnert, Dirk. ”Machine ethics and African identities: Perspectives of artificial intelligence in Africa.” Available at SSRN 4163096 (2022).
  * [340] Fousseni, Arafat, Mawugno Koffi Kodjo, and Assiongbon Adanlete Adjanoh. ”Static and dynamic evaluation of wind potential in the Kara region of Togo using artificial neural networks.” TH Wildau Engineering and Natural Sciences Proceedings 1 (2021).
  * [341] Agbodjan, Marc Marin, Joseph Mung’atu, and Anthony Wanjoya. ”Prediction of Maize Price using Hidden Markov Chain Model: An Application on Grand market of Lome (TOGO).
  * [342] Aiken, Emily, Suzanne Bellue, Joshua Blumenstock, Dean Karlan, and Christopher R. Udry. ”Estimating Impact with Surveys versus Digital Traces: Evidence from Randomized Cash Transfers in Togo.” (2023).
  * [343] Yomo, Mawulolo, Euloge Nicaise Yalo, Masamaéya Dadja-Toyou Gnazou, Stephen Silliman, Isaac Larbi, and Khaldoon A. Mourad. ”Forecasting land use and land cover dynamics using combined remote sensing, machine learning algorithm and local perception in the Agoènyivé Plateau, Togo.” Remote Sensing Applications: Society and Environment 30 (2023): 100928.
  * [344] Dotche, Koffi Agbeblewu, Adekunlé Akim Salami, Koffi Mawugno Kodjo, Yawa Pamela CD Blu, and Yao Essemu Julien Diabo. ”Evaluating Solar Energy Harvesting using Artificial Neural Networks: A Case study in Togo.” In 2019 II International Conference on High Technology for Sustainable Development (HiTech), pp. 1-5. IEEE, 2019.
  * [345] Zhuangfang NaNa Yi, Naroa Zurutuza, Do-Hyung Kim, Ruben Lopez Mendoza, Martha Morrissey, Chuck Daniels,Nick Ingalls, Jeevan Farias, Karito Tenorio, Pilar Serrano,Sajjad Anwar, Building on our success mapping 23,100 unmapped schools in eight countries , Development Seed, 2021
  * [346] Sarr, Alioune Badara, and Benjamin Sultan. ”Predicting crop yields in Senegal using machine learning methods.” International Journal of Climatology 43, no. 4 (2023): 1817-1838.
  * [347] Nyasulu, Chimango, Awa Diattara, Assitan Traore, Abdoulaye Deme, and Cheikh Ba. ”Towards Resilient Agriculture to Hostile Climate Change in the Sahel Region: A Case Study of Machine Learning-Based Weather Prediction in Senegal.” Agriculture 12, no. 9 (2022): 1473.
  * [348] MMbengue, F., Gayane Faye, Kharouna Talla, M. Adama Sarr, André Ferrari, and Modou Mbaye. ”Evaluation Of Machine Learning Classification Methods For Rice Detection Using Earth Observation Data: Case Of Senegal.” European Scientific Journal, ESJ 18, no. 17 (2022): 214.
  * [349] Bayet, Théophile, Timothée Brochier, Christophe Cambier, Alassane Bah, Christophe Denis, and N. Thiam JD Zucker. ”A Machine Learning approach to improve the monitoring of Sustainable Development Goals: a case study in Senegalese artisanal fisheries.” CNIA (2021): 30.
  * [350] Dia, Yoro, Lamine Faty, Aba Diop, Ousmane Sall, and Tony Tona Landu. ”A Hybrid Model for Predicting Road Accident Severity in Senegal.” In International Conference on Artificial Intelligence Science and Applications, pp. 1-12. Cham: Springer Nature Switzerland, 2022.
  * [351] State, Laura, Hadrien Salat, Stefania Rubrichi, and Zbigniew Smoreda. ”Explainability in Practice: Estimating Electrification Rates from Mobile Phone Data in Senegal.” In World Conference on Explainable Artificial Intelligence, pp. 110-125. Cham: Springer Nature Switzerland, 2023.
  * [352] Sarkodie, Samuel Asumadu, Emmanuel Ackom, Festus Victor Bekun, and Phebe Asantewaa Owusu. ”Energy - climate - economy - population nexus: an empirical analysis in Kenya, Senegal, and Eswatini.” Sustainability 12, no. 15 (2020): 6202.
  * [353] Kebe, M. A. M., Maodo Sene, and Nafissatou Diagne. ”An experience of detection and classification of Quality-Of-Service problems in MV/LV distribution substations using artificial intelligence: Senegal case study.” (2023): 1719-1723.
  * [354] Seck, Djamal Abdoul Nasser, and Fatima Bintou Rassoul Diakité. ”Supervised Machine Learning Models for the Prediction of renal failure in Senegal.” In 2023 International Conference on Control, Artificial Intelligence, Robotics & Optimization (ICCAIRO), pp. 94-98. IEEE, 2023.
  * [355] Lee, Kamwoo, Jeanine Braithwaite, and Michel Atchikpa. ”Understanding colonial legacy and environmental issues in Senegal through language use.” In Social, Cultural, and Behavioral Modeling: 13th International Conference, SBP-BRiMS 2020, Washington, DC, USA, October 18 - 21, 2020, Proceedings 13, pp. 23-34. Springer International Publishing, 2020.
  * [356] Dione, Cheikh M. Bamba, Jonas Kuhn, and Sina Zarrieß. ”Design and Development of Part-of-Speech-Tagging Resources for Wolof (Niger-Congo, spoken in Senegal).” In LREC. 2010.
  * [357] Alla, Lo, Dione Cheikh Bamba, Nguer Elhadji Mamadou, Ba Sileye O. Ba, and Lo Moussa. ”Using lstm to translate french to senegalese local languages: Wolof as a case study.” arXiv preprint arXiv:2004.13840 (2020).
  * [358] Dia, Yoro, Lamine Faty, Ousmane Sall, Tony Tona Landu, and Edouard Ngor Sarr. ”Predicting the Severity of the Road Accidents in Senegal: An Empirical Study.” In 2022 the 5th International Conference on Data Storage and Data Engineering, pp. 91-96. 2022.
  * [359] Dione, Doudou, Idy Diop, Ibrahima Gueye, Bassirou Ngom, and Sidi Mohamed Farssi. ”Proposal for an IoT-based e-health model in developing countries: case of Senegal.” In 2021 International Conference on Electrical, Computer and Energy Technologies (ICECET), pp. 1-7. IEEE, 2021.
  * [360] Özdogan, Mutlu, and Ajit Govind. ”Three decades of forest cover change in Senegal from remote sensing.” (2022).
  * [361] Traoré, Fousseini, Suwadu Sakho Jimbira, and Leysa Maty Sall. ”Nonlinear price transmission in the rice market in Senegal: a model-based recursive partitioning approach.” Applied Economics 54, no. 20 (2022): 2343-2355.
  * [362] Drame, M., Seck, D.A.N. and Ndiaye, B.S., 2023, August. Analysis and Forecast of Energy Demand in Senegal with a SARIMA Model and an LSTM Neural Network. In The International Conference on Deep Learning, Big Data and Blockchain (pp. 129-140). Cham: Springer Nature Switzerland.
  * [363] Sarron, Julien, Cheikh Amet Bassirou Sane, Philippe Borianne, Eric Malézieux, Thibault Nordey, Frédéric Normand, Paterne Diatta, Youga Niang, and Emile Faye. ”Is machine learning efficient for mango yield estimation when used under heterogeneous field conditions?.” In XXX International Horticultural Congress IHC2018: VII Conference on Landscape and Urban Horticulture, IV Conference on 1279, pp. 201-208. 2018.
  * [364] Diop, Papa Moussa, Jin Takamoto, Yuji Nakamura, and Morikazu Nakamura. ”A machine learning approach to classification of Okra.” In 2020 35th International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC), pp. 254-257. IEEE, 2020.
  * [365] Moustapha Mbaye, Mouhamadou, Fadi MH Abu Salem, and Nadia Tahiri. ”A new machine learning workflow to create an optimal waiting list in hospitals.” In Proceedings of the 2023 7th International Conference on Medical and Health Informatics, pp. 159-163. 2023
  * [366] Oyebode, Oladapo, and Rita Orji. ”Detecting factors responsible for diabetes prevalence in Nigeria using social media and machine learning.” In 2019 15th International Conference on Network and Service Management (CNSM), pp. 1-4. IEEE, 2019.
  * [367] Obite, Chukwudi Paul, Angela Chukwu, Desmond Chekwube Bartholomew, Ugochinyere Ihuoma Nwosu, and Gladys Ezenwanyi Esiaba. ”Classical and machine learning modeling of crude oil production in Nigeria: Identification of an eminent model for application.” Energy Reports 7 (2021): 3497-3505.
  * [368] Ighile, Eseosa Halima, Hiroaki Shirakawa, and Hiroki Tanikawa. ”Application of GIS and machine learning to predict flood areas in Nigeria.” Sustainability 14, no. 9 (2022): 5039.
  * [369] Villacis, Alexis H., Syed Badruddoza, Ashok K. Mishra, and Joaquin Mayorga. ”The role of recall periods when predicting food insecurity: A machine learning application in Nigeria.” Global Food Security 36 (2023): 100671.
  * [370] McKenzie, David J., and Dario Sansone. ”Man vs. machine in predicting successful entrepreneurs: evidence from a business plan competition in Nigeria.” (2017).
  * [371] Goni, Ibrahim, and Murtala Mohammad. ”Machine learning approach to mobile forensics framework for cyber crime detection in Nigeria.” Journal of Computer Science Research 2, no. 4 (2020): 1-6.
  * [372] Folorunso, Sakinat O., Sulaimon A. Afolabi, and Adeoye B. Owodeyi. ”Dissecting the genre of Nigerian music with machine learning models.” Journal of King Saud University-Computer and Information Sciences 34, no. 8 (2022): 6266-6279.
  * [373] Lawal, Zaharaddeen Karami, Hayati Yassin, and Rufai Yusuf Zakari. ”Flood prediction using machine learning models: a case study of Kebbi state Nigeria.” In 2021 IEEE Asia-Pacific Conference on Computer Science and Data Engineering (CSDE), pp. 1-6. IEEE, 2021.
  * [374] Nwankwo, Mmesoma Peace, Ndukaku Macdonald Onyeizu, Emmanuel Chibuogu Asogwa, Chukwuogo Okwuchukwu Ejike, and Okechukwu J. Obulezi. ”Prediction of House Prices in Lagos-Nigeria Using Machine Learning Models.” European Journal of Theoretical and Applied Sciences 1, no. 5 (2023): 313-326.
  * [375] Panjala, Pranay, Murali Krishna Gumma, Hakeem Ayinde Ajeigbe, Murtala Muhammad Badamasi, Kumara Charyulu Deevi, and Ramadjita Tabo. ”Identifying Suitable Watersheds across Nigeria Using Biophysical Parameters and Machine Learning Algorithms for Agri - Planning.” ISPRS International Journal of Geo-Information 11, no. 8 (2022): 416.
  * [376] McKenzie, David, and Dario Sansone. ”Predicting entrepreneurial success is hard: Evidence from a business plan competition in Nigeria.” Journal of Development Economics 141 (2019): 102369.
  * [377] Gladys, Ajibolu Oyinkansola, and Akinola Solomon Olalekan. ”A machine learning model for predicting colour trends in the textile fashion industry in south-west Nigeria.” International Journal on Integrated Education 4, no. 2 (2021): 174-188.
  * [378] Odeniyi, Olufemi A., Mabel E. Adeosun, and Tayo P. Ogundunmade. ”Prediction of terrorist activities in Nigeria using machine learning models.” Innovations 71 (2022): 87-96.
  * [379] Ogundunmade, T. P., and A. A. Adepoju. ”Modelling Liquefied Petroleum Gas Prices in Nigeria Using Time Series Machine Learning Models.” Mod Econ Manag 1, no. 5 (2022): 3.
  * [380] Ekubo, Ebiemi Allen, and Bukohwo Michael Esiefarienrhe. ”Using machine learning to predict low academic performance at a Nigerian university.” The African Journal of Information and Communication 2022, no. 30 (2022): 1-33.
  * [381] Ekubo, Ebiemi Allen, and Bukohwo Michael Esiefarienrhe. ”Using machine learning to predict low academic performance at a Nigerian university.” The African Journal of Information and Communication 2022, no. 30 (2022): 1-33.
  * [382] Muhammad, Bilyaminu, and Asaf Varol. ”A Symptom-Based Machine Learning Model for Malaria Diagnosis in Nigeria.” In 2021 9th International Symposium on Digital Forensics and Security (ISDFS), pp. 1-6. IEEE, 2021.
  * [383] Salele, Badiya, Yakubu Aminu Dodo, Dalhatu Aliyu Sani, Mohammed Awad Abuhussain, Barno Sayfutdinovna Abdullaeva, and Adam Brysiewicz. ”Run-off modelling of pervious and impervious areas using couple SWAT and a novel machine learning model in cross-rivers state Nigeria.” Water Science & Technology 88, no. 7 (2023): 1893-1909.
  * [384] Adeeyo, Y. A., and A. A. Osinaike. ”Modeling the Oil Viscosity of Nigerian Crudes Using Machine Learning.” In SPE Nigeria Annual International Conference and Exhibition, p. D021S002R004. SPE, 2023.
  * [385] Jean, Neal, Marshall Burke, Michael Xie, W. Matthew Davis, David B. Lobell, and Stefano Ermon. ”Combining satellite imagery and machine learning to predict poverty.” Science 353, no. 6301 (2016): 790-794.
  * [386] Ibrahim, Esther Shupel, Claas Nendel, Bahareh Kamali, Efron Nduke Gajere, and Patrick Hostert. ”Predicting potato diseases in smallholder agricultural areas of Nigeria using machine learning and remote sensing-based climate data.” PhytoFrontiers ja (2023).
  * [387] Oyebode, Oladapo, and Rita Orji. ”Social media and sentiment analysis: the Nigeria presidential election 2019.” In 2019 IEEE 10th Annual Information Technology, Electronics and Mobile Communication Conference (IEMCON), pp. 0140-0146. IEEE, 2019.
  * [388] Eneanya, Obiora A., Jorge Cano, Ilaria Dorigatti, Ifeoma Anagbogu, Chukwu Okoronkwo, Tini Garske, and Christl A. Donnelly. ”Environmental suitability for lymphatic filariasis in Nigeria.” Parasites & vectors 11, no. 1 (2018): 1-13.
  * [389] Obulezi, Okechukwu J., Eberechukwu Q. Chinedu, Dorathy O. Oramulu, Harrison O. Etaga, Ndukaku M. Onyeizu, and Chukwuogo Okwuchukwu Ejike. ”Machine Learning models for predicting transportation costs inflated by fuel subsidy removal policy in Nigeria.” International Research Journal of Modernization in Engineering Technology and Science 5: 1053-1070.
  * [390] Mbaoma, Oliver C., Akinyemi O. Ogunkeyede, Adedoyin A. Adebayo, and Prekeyi Tawari-Fufeyin. ”Geospatial and Machine Learning Driven Air Pollution Modelling Using Vehicular Load and Climatic Variables in Agbarho, Delta State, Nigeria.” European Journal of Environment and Earth Sciences 3, no. 3 (2022): 34-41.
  * [391] Achara, Miriam, Emeka J. Okereke, Nwulu Stephen Onyemere, and Ufuoma Earnest Ofierohor. ”Financial Institution Readiness and Adoption of Machine Learning Algorithm and Performance of Select Banks in Rivers State, Nigeria.” Asian Journal of Economics, Finance and Management (2023): 180-192.
  * [392] Oyewola, David O., Asabe Ibrahim, Joshua A. Kwanamu, and Emmanuel Gbenga Dada. ”A new auditory algorithm in stock market prediction on oil and gas sector in Nigerian stock exchange.” Soft computing letters 3 (2021): 100013.
  * [393] Odeniyi, Olufemi A., Mabel E. Adeosun, and Tayo P. Ogundunmade. ”Prediction of terrorist activities in Nigeria using machine learning models.” Innovations 71 (2022): 87-96.
  * [394] Ogundunmade, T. P., and A. A. Adepoju. ”Modelling Liquefied Petroleum Gas Prices in Nigeria Using Time Series Machine Learning Models.” Mod Econ Manag 1, no. 5 (2022): 3.
  * [395] Ekubo, Ebiemi Allen, and Bukohwo Michael Esiefarienrhe. ”Using machine learning to predict low academic performance at a Nigerian university.” The African Journal of Information and Communication 2022, no. 30 (2022): 1-33.
  * [396] Muhammad, Bilyaminu, and Asaf Varol. ”A Symptom-Based Machine Learning Model for Malaria Diagnosis in Nigeria.” In 2021 9th International Symposium on Digital Forensics and Security (ISDFS), pp. 1-6. IEEE, 2021.
  * [397] Hamidou Mamane Moumouni, Noma Talibi Soumaïla, Boureima Seibou, Adekunlé Akim Salami, Attoumane Kosso Moustapha, and Madougou Saïdou. ”Electrical Charge of Niamey City Modelisation by Neural Network.” Science 7, no. 1 (2019): 13-19.
  * [398] Schulz, Dario, He Yin, Bernhard Tischbein, Sarah Verleysdonk, Rabani Adamou, and Navneet Kumar. ”Land use mapping using Sentinel-1 and Sentinel-2 time series in a heterogeneous landscape in Niger, Sahel.” ISPRS Journal of Photogrammetry and Remote Sensing 178 (2021): 97-111.
  * [399] Painter, Thomas M. ”Adult Literacy and Cooperative Training Programs inthe Niamey Department Development Project (Niger Republic): Evaluation of Phase I Activities and Recommendations for Phase II.” (1980).
  * [400] Eli, Taleb and Sidi Hamou, Lalla Aisha. ”Investigating the Factors That Influence Students’ Choice of English Studies As a Major: the Case of University of Nouakchott Al Aasriya, Mauritania.” International Journal of Technology, Innovation and Management (IJTIM), 2(1), 2022.
  * [401] Aragón, Diego Gómez. ”Identifying Desert Locust Breeding Areas by Means of Earth Observation in Mauritania.” PhD diss., Universidad de Valladolid, 2019.
  * [402] El Arby Chrif, Mohamed El Moustapha, Saleck, Moustapha Mohamed, N’Diaye, Aichetou Cheikh Mohamedou, and Mahmoud, El Benany Mohamed. ”Business Intelligence Models for e-Government in Mauritania: A Survey.” In The International Conference on Artificial Intelligence and Smart Environment, pp. 307-312. Cham: Springer International Publishing, 2022.
  * [403] Abeidi, Hassine Ali, Kerim, Bedine, and Nanne, Mohamedade Farouk. ”A Study for Remote Monitoring of Water Points in Mauritania Based on IoT (LoRa) Technology.” Indonesian Journal of Electrical Engineering and Informatics (IJEEI), 10(2), 2022, pp. 398-409.
  * [404] Gómez-Escalonilla, Victor, Pedro Martinez-Santos, and Miguel Martin-Loeches. ”Preprocessing approaches in machine-learning-based groundwater potential mapping: an application to the Koulikoro and Bamako regions, Mali.” Hydrology and Earth System Sciences, 26(2), 2022, pp. 221-243.
  * [405] Gómez-Escalonilla, Victor, Oumou Diancoumba, D. Y. Traoré, E. Montero, M. Martin-Loeches, and P. Martinez-Santos. ”Multiclass spatial predictions of borehole yield in southern Mali by means of machine learning classifiers.” Journal of Hydrology: Regional Studies, 44, 2022, 101245.
  * [406] Boudinaud, Laure, and Sacha Alex Orenstein. ”Assessing Cropland Abandonment from Violent Conflict in Central Mali with SENTINEL-2 and Google Earth Engine.” The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 46, 2021, pp. 9-15.
  * [407] Traore, Boukaye Boubacar, Bernard Kamsu-Foguem, and Fana Tangara. ”Deep convolution neural network for image recognition.” Ecological informatics, 48, 2018, pp. 257-268.
  * [408] Rodriguez-Gonzalez, Alejandro, Massimiliano Zanin, and Ernestina Menasalvas-Ruiz. ”Public health and epidemiology informatics: can artificial intelligence help future global challenges? An overview of antimicrobial resistance and impact of climate change in disease epidemiology.” Yearbook of Medical Informatics, 28(01), 2019, pp. 224-231.
  * [409] Rajanarayanan, S., Lea Sorilla Nisperos, and J. R. Basal. ”Recognition of pathogens using image classification based on improved recurrent neural network with LSTM.” ICTACT Journal on Soft Computing, 9(2), 2019.
  * [410] Bizimana, Jean-Claude, David A. Bessler, and Francis Keita. ”Three decades of market liberalization policy in Mali: An analysis of the grain markets.” No. 333-2016-14672, 2016.
  * [411] Briggs, Morgan. ”SatDash: An Interactive Dashboard for Assessing Land Damage in Nigeria and Mali.” In ACM SIGCAS Conference on Computing and Sustainable Societies, 2021, pp. 100-114.
  * [412] Hirblinger, A.T., Wählisch, M., Keator, K., McNaboe, C., Duursma, A., Karlsrud, J., Sticher, V., Verjee, A., Kyselova, T., Kwaja, C.M. and Perera, S. ”Making peace with uncertainty: Reflections on the role of digital technology in peace processes beyond the data hype.” 2023\.
  * [413] Diallo, Abdoul Karim, and Kheireddine Abainia. ”Offensive Language Detection in Code-Mixed Bambara-French Corpus: Evaluating machine learning and deep learning classifiers.” In 2023 International Conference on Decision Aid Sciences and Applications (DASA), IEEE, 2023, pp. 121-125.
  * [414] Olkers, Tim, Shuang Liu, and Oliver Musshoff. ”A typology of Malian farmers and their credit repayment performance-An unsupervised machine learning approach.” 2023\.
  * [415] Traoré, Mohamed, and Hamadoun Bokar. ”Principal Component Analysis of Hydrochemical Data From Groundwater Resources in Bamako.”
  * [416] de Sousa, Celio, Lola Fatoyinbo, Christopher Neigh, Farrel Boucka, Vanessa Angoue, and Trond Larsen. ”Cloud-computing and machine learning in support of country-level land cover and ecosystem extent mapping in Liberia and Gabon.” PLoS One, 15(1), 2020, e0227438.
  * [417] Blair, Robert A., Christopher Blattman, and Alexandra Hartman. ”Predicting local violence: Evidence from a panel survey in Liberia.” Journal of Peace Research, 54(2), 2017, pp. 298-312.
  * [418] Burch Jones, Evelyn J. ”Remote Sensing, Machine Learning, and Change Detection Applications for Land Cover Studies: A Geospatial Analysis of Two Liberian Proposed Protected Areas.” 2021\.
  * [419] Yazdani, Mehrdad, Mai H. Nguyen, Jessica Block, Daniel Crawl, Naroa Zurutuza, Dohyung Kim, Gordon Hanson, and Ilkay Altintas. ”Scalable detection of rural schools in Africa using convolutional neural networks and satellite imagery.” In 2018 IEEE/ACM International Conference on Utility and Cloud Computing Companion (UCC Companion), IEEE, 2018, pp. 142-147.
  * [420] Miller, E., Luo, D., Deparday, V., Giovando, C., Gevaert, C.M., Emanuele, R., Kassel, S., Needham, E., Lipstein, G., Bull, P. and Slavitt, I. ”An open machine learning challenge to map urban development and resilience in diverse African cities from aerial imagery.” In AGU Fall Meeting 2019, December 2019.
  * [421] Carreiras, Joao MB, Maria J. Vasconcelos, and Richard M. Lucas. ”Understanding the relationship between aboveground biomass and ALOS PALSAR data in the forests of Guinea-Bissau (West Africa).” Remote Sensing of Environment, 121, 2012, pp. 426-442.
  * [422] Pereira, Sofia C., Catarina Lopes, and Joao Pedro Pedroso. ”Mapping Cashew Orchards in Cantanhez National Park (Guinea-Bissau).” Remote Sensing Applications: Society and Environment, 26, 2022, 100746.
  * [423] Martiarena, Merlin Leunda, and Marina Padrao Temudo. ”Endogenous learning and innovation in African smallholder agriculture: lessons from Guinea-Bissau.” The Journal of Agricultural Education and Extension, 2023, pp. 1-19.
  * [424] Badji, Aliou, Youssou Dieng, Ibrahima Diop, Papa Alioune Cisse, and Boubacar Diouf. ”Automatic Speaker Recognition (ASR) Application in the monitoring of PLHIV in the cross-border area between the Gambia, Guinea-Bissau and Senegal.” In Proceedings of the 10th International Conference on Information Systems and Technologies, 2020, pp. 1-9.
  * [425] Yehadji, Dègninou, Geraldine Gray, Carlos Arias Vicente, Petros Isaakidis, Abdourahimi Diallo, Saa André Kamano, and Thierno Saidou Diallo. ”Development of machine learning algorithms to predict viral load suppression among HIV patients in Conakry (Guinea).” (2023).
  * [426] Colubri, A., Silver, T., Fradet, T., Retzepi, K., Fry, B. and Sabeti, P., 2016. Transforming clinical data into actionable prognosis models: machine-learning framework and field-deployable app to predict outcome of Ebola patients. PLoS neglected tropical diseases, 10(3), p.e0004549.
  * [427] Fatima Ezzahra SALAMATE, Mohamed EL AZHARI, Jamal Zahi: Machine learning in epidemiology: Characterization of risk factors related to the occurrence of pulmonary and extra pulmonary tuberculosis in the province of Settat, International Journal of Computer Engineering and Data Science (IJCEDS) 2 (3), 2022
  * [428] Nyamekye, Clement, Samuel Kwofie, Benjamin Ghansah, Emmanuel Agyapong, and Linda Appiah Boamah. ”Assessing urban growth in Ghana using machine learning and intensity analysis: A case study of the New Juaben Municipality.” Land Use Policy, 99, 2020, 105057.
  * [429] Abdul Muhsin Zambang, M., Jiang, H., and Wahab, L. ”Modeling vehicle ownership with machine learning techniques in the Greater Tamale Area, Ghana.” PloS one, 16(2), 2021, p.e0246044.
  * [430] Andoh, John, Louis Asiedu, Anani Lotsi, and Charlotte Chapman-Wardy. ”Statistical analysis of public sentiment on the Ghanaian government: a machine learning approach.” Advances in Human-Computer Interaction, 2021, pp. 1-7.
  * [431] Twumasi, Clement, and Juliet Twumasi. ”Machine learning algorithms for forecasting and backcasting blood demand data with missing values and outliers: A study of Tema General Hospital of Ghana.” International Journal of Forecasting, 38(3), 2022, pp. 1258-1277.
  * [432] Owusu-kumih, Eric, and Ebenezer Asare Aboagye. ”Historical Analysis and Time Series Forecasting of Internet Data Usage and Revenues in Ghana Using Machine Learning Based Facebook Prophet Model.” Available at SSRN 4290081, 2022.
  * [433] Wahab, Lukuman, and Haobin Jiang. ”Severity prediction of motorcycle crashes with machine learning methods.” International journal of crashworthiness, 25(5), 2020, pp. 485-492.
  * [434] Guenther, Martin. ”Local effects of artisanal mining: Empirical evidence from Ghana.” In Presentation at the International Conference’Environmental Economics: A Focus on Natural Resources’, Orleans, 2018.
  * [435] Adu, Kwabena, Patrick Kwabena Mensah, Mighty Abra Ayidzoe, Obed Appiah, Ebenezer Quayson, Christopher Bombie Ninfaakang, and Michael Opoku. ”GC3558: An open-source annotated dataset of Ghana currency images for classification modeling.” Data in Brief, 45, 2022, 108616.
  * [436] Agbodza, Paul A. ”Ito calculus-machine learning projection of forward US dollar-Ghana cedi rates.” In 2019 International Conference on Computing, Computational Modelling and Applications (ICCMA), IEEE, 2019, pp. 102-1022.
  * [437] Barenblitt, Abigail, Amanda Payton, David Lagomasino, Temilola Fatoyinbo, Kofi Asare, Kenneth Aidoo, Hugo Pigott et al. ”Spatial distribution of artisanal goldmining in Ghana: Using machine learning and Google Earth Engine to quantify conversion of vegetation to gold mines.” Authorea Preprints, 2022.
  * [438] Agbenyega, Diana Ayorkor, John Andoh, Samuel Iddi, and Louis Asiedu. ”Modelling Customs Revenue in Ghana Using Novel Time Series Methods.” Applied Computational Intelligence and Soft Computing, 2022.
  * [439] Agbodza, Paul A. : ”Machine Learning of Jump Dynamics in US Dollar-Ghana Cedi Exchange Returns.” In 2019 International Conference on Computer, Data Science and Applications (ICDSA), IEEE, 2019, pp. 1-6.
  * [440] Signé, Landry, and Chelsea Johnson. Africa’s Mining Potential: Trends, Opportunities, Challenges and Strategies. (2021).
  * [441] Gallwey, Jane, Carlo Robiati, John Coggan, Declan Vogt, and Matthew Eyre. ”A Sentinel-2 based multispectral convolutional neural network for detecting artisanal small-scale mining in Ghana: Applying deep learning to shallow mining.” Remote Sensing of Environment 248 (2020): 111970.
  * [442] Kansake, Bruno Ayaga, Felix Adaania Kaba, Nelson Kofi Dumakor-Dupey, and Clement Kweku Arthur. ”The future of mining in Ghana: Are stakeholders prepared for the adoption of autonomous mining systems?.” Resources Policy 63 (2019): 101411.
  * [443] Forson, Eric Dominic, and Prince Ofori Amponsah. ”Mineral prospectivity mapping over the Gomoa Area of Ghana’s southern Kibi-Winneba belt using support vector machine and naive bayes.” Journal of African Earth Sciences 206 (2023): 105024.
  * [444] Arhinful, Kofi Owusu, and George Agyei. ”Fuel Consumption Prediction in Shovel-Truck System of Surface Mine Using Artificial Neural Network.” Journal of Science and Technology Research 5, no. 2 (2023).
  * [445] Issaka, Yakubu, and Bernard Kumi-Boateng. ”Artificial intelligence techniques for predicting tidal effects based on geographic locations in Ghana.” Geodesy and Cartography 46, no. 1 (2020): 1-7.
  * [446] Elvis Korku Avenyo, Learning to innovate in urban informal enterprises of Ghana: Role of apprenticeship and formal interaction, Scientific African, Volume 20, 2023, e01740, ISSN 2468-2276,
  * [447] John Bosco Dramani, Prince Boakye Frimpong, Kwame Ansere Ofori-Mensah, Modelling the informal sector and energy consumption in Ghana, Social Sciences & Humanities Open, Volume 6, Issue 1, 2022, 100354, ISSN 2590-2911,
  * [448] Franic, J. What do we really know about the drivers of undeclared work? An evaluation of the current state of affairs using machine learning. AI & Soc (2022). https://doi.org/10.1007/s00146-022-01490
  * [449] Mohammed, Nuredin I., Alexander Jarde, Grant Mackenzie, Umberto D’Alessandro, and David Jeffries. ”Deploying Machine Learning Models Using Progressive Web Applications: Implementation Using a Neural Network Prediction Model for Pneumonia Related Child Mortality in The Gambia.” Frontiers in Public Health, 9 (2022): 772620.
  * [450] Ingram, Will, Leen Saeb, Seyed Sadr, Rob Hygate, and Fayyaz Ali Memon. ”A preliminary investigation of smart rural water distribution systems in the Gambia.” (2018).
  * [451] Ferraris, Stefano, Rosa Meo, Stefano Pinardi, Matteo Salis, and Gabriele Sartor. ”Machine Learning as a Strategic Tool for Helping Cocoa Farmers in Côte D’Ivoire.” Sensors 23, no. 17 (2023): 7632.
  * [452] Eneanya, O.A., Koudou, B.G., Aboulaye, M., Elvis, A.A., Souleymane, Y., Kouakou, M.M., Weil, G.J. and Fischer, P.U., 2021. Progress towards onchocerciasis elimination in Côte d’Ivoire: A geospatial modelling study. PLoS neglected tropical diseases, 15(2), p.e0009091.
  * [453] Gong, Xianda, Heike Wex, Thomas Müller, Silvia Henning, Jens Voigtländer, Alfred Wiedensohler, and Frank Stratmann. ”Understanding aerosol microphysical properties from 10 years of data collected at Cabo Verde based on an unsupervised machine learning classification.” Atmospheric Chemistry and Physics, 22, no. 8 (2022): 5175-5194.
  * [454] Varela, Danilson, Maria M. Romeiras, and Luis Silva. ”Implications of climate change on the distribution and conservation of Cabo Verde endemic trees.” Global Ecology and Conservation, 34 (2022): e02025.
  * [455] Guzman-Vilca, Wilmer Cristobal, Manuel Castillo-Cara, and Rodrigo M. Carrillo-Larco. ”Development, validation, and application of a machine learning model to estimate salt consumption in 54 countries.” Elife, 11,(2022): e72930.
  * [456] Tiengo, Rafaela, José MR Pacheco, Jéssica Garcia Uchôa, and Artur José Freire Gil. ”Using Sentinel-1 GRD SAR data for volcanic eruptions monitoring: the case-study of Fogo Volcano (Cabo Verde) in 2014/2015.” Intercontinental Geoinformation Days, (2021): 22-25.
  * [457] Harvey, David, Wessel Valkenburg, and Amara Amara. ”Predicting malaria epidemics in Burkina Faso with machine learning”. PLoS One, 16, no. 6 (2021): e0253302.
  * [458] Schug, Franz, Akpona Okujeni, Janine Hauer, Patrick Hostert, Jonas O. Nielsen, and Sebastian van der Linden. ”Mapping patterns of urban development in Ouagadougou, Burkina Faso, using machine learning regression modeling with bi-seasonal Landsat time series.” Remote Sensing of Environment, 210 (2018): 217-228.
  * [459] Yameogo, Baowendsomme Armel, Tounwendyam Frédéric Ouedraogo, and Constantin Zongo. On the Machine Learning Models to Predict Town-Scale Energy Consumption in Burkina Faso. In International Conference on Emerging Technologies for Developing Countries, pp. 65-73. Cham: Springer Nature Switzerland, 2022.
  * [460] Bournas, Nasreddine, Abdoulaye Touré, Mahamoud Balboné, Palalé Sama Zagré, Abdoulaye Ouédraogo, Kanita Khaled, Alexander Prikhodko, and Jean Legault, Use of machine learning techniques on airborne geophysical data for mineral resources exploration in Burkina Faso, ASEG Extended Abstracts, 2019, no. 1 (2019), 1-4.
  * [461] Zongo, Constantin, Frederic T Ouedraogo, and M. A. L. O. Sadouanouan, Forecasting models for monthly energy consumption using machine learning in Burkina Faso, In 2023 IEEE AFRICON, pp. 1-3. IEEE, 2023.
  * [462] Hounkpatin, Kpade OL, Aymar Y. Bossa, Yacouba Yira, Mouïnou A. Igue, and Brice A. Sinsin, Assessment of the soil fertility status in Benin (West Africa) - Digital soil mapping using machine learning, Geoderma Regional, 28 (2022): e00444.
  * [463] Selvaraj, M.G., Vergara, A., Montenegro, F., Ruiz, H.A., Safari, N., Raymaekers, D., Ocimati, W., Ntamwira, J., Tits, L., Omondi, A.B. and Blomme, G.: Detection of banana plants and their major diseases through aerial images and machine learning methods: A case study in DR Congo and Republic of Benin, ISPRS Journal of Photogrammetry and Remote Sensing,169, pp.110-124, 2020.
  * [464] Guenoupkati, Agbassou, Adekunlé Akim Salami, Mawugno Koffi Kodjo, and Kossi Napo. Short-Term Electricity Generation Forecasting Using Machine Learning Algorithms: A Case Study of the Benin Electricity Community (CEB), TH Wildau Engineering and Natural Sciences Proceedings, 1,(2021).
  * [465] Agossou, C., Atchadé, M.N., Djibril, A.M. and Kurisheva, S.V., Mathematical modeling and machine learning for public health decision-making: the case of breast cancer in Benin, Mathematical Biosciences and Engineering, 19(2), pp.1697-1720, 2022
  * [466] Akpoti, Komlavi, Amos T. Kabo-Bah, Elliott R. Dossou-Yovo, Thomas A. Groen, and Sander J. Zwart. Mapping suitability for rice production in inland valley landscapes in Benin and Togo using environmental niche modeling, Science of the total environment, 709 (2020), 136-165.


All quantum mixtures are proper

Leonardo Castellani

Dipartimento di Scienze e Innovazione Tecnologica
Università del Piemonte Orientale, viale T. Michel 11, 15121 Alessandria,
Italy
INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy
Arnold-Regge Center, via P. Giuria 1, 10125 Torino, Italy

It is argued that proper and improper quantum mixed states have no observable
differences, and hence should not be distinguished. This has implications for
subjective approaches to quantum mechanics, and invalidates one of the main
motivations for relational interpretations of QM.

<EMAIL_ADDRESS>
##  1 Introduction

Measurements by definition presuppose a separation between the observed system
S and the observer O. This separation can be moved towards S or towards O,
thus redefining S and O, as was realized from the early days of quantum
mechanics (the “Heisenberg cut”). Once the separation is decided, so that
subsystems S and O are precisely identified, the quantum mechanical rules
provide statistical predictions on the results of measurements performed by O
on the system S.

Take for simplicity S to be a qubit, i.e. a two-dimensional quantum system,
and A to be an observable of S with orthonormal eigenvectors $|0\rangle$ and
$|1\rangle$. If S is in the pure state
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, and O measures A without
registering the result, the pure state collapses into a mixture described by
the density matrix

$|\psi\rangle\langle\psi|\longrightarrow|\alpha|^{2}|0\rangle\langle
0|+|\beta|^{2}|1\rangle\langle 1|$ (1.1)

This statistical mixture indicates that S is in the state $|0\rangle$ with
probability $|\alpha|^{2}$ and in the state $|1\rangle$ with probability
$|\beta|^{2}$, and reflects the ignorance of the measurement result. The
collapse (1.1) is the “ontic part” of the process, being independent of the
knowledge of the measurement result. The “epistemic part” is in the sharper
description $|0\rangle\langle 0|$ or $|1\rangle\langle 1|$, depending on this
knowledge. Thus we can describe the measurement as a two step process, an
“ontic” collapse followed by an “epistemic” collapse:

$(\alpha|0\rangle+\beta|1\rangle)(\langle 0|\alpha^{*}+\langle
1|\beta^{*})\xrightarrow{``ontic"collapse}|\alpha|^{2}|0\rangle\langle
0|+|\beta|^{2}|1\rangle\langle 1|$ (1.2) $|\alpha|^{2}|0\rangle\langle
0|+|\beta|^{2}|1\rangle\langle
1|\xrightarrow[(knows~{}that~{}result~{}is~{}1)]{``epistemic"~{}collapse}|1\rangle\langle
1|$ (1.3)

In what follows, our concern will be with the ontic part, since it is the
genuinely quantum mechanical part of the measurement process. The epistemic
part is essentially the same as in classical situations, for example when
flipping a coin and looking or not looking at the result.

Here we assume that all physical systems, including observers, are described
by quantum mechanics, and that “classicality” arises as an emergent
phenomenon. By “observer” we mean a physical system (human or not) that can
interact with S, and due to this interaction becomes correlated with S. By
“measurement” we mean this interaction, occurring at a definite time, and
producing the ontic collapse.

We also assume that the state of a system is completely described by its
density matrix, as introduced in [1] (see also [2]). In particular we consider
the state of a subsystem in a composite system to be completely described by
its reduced density matrix.

##  2 Measurement as interaction

A central tenet of QM is the unitary evolution of an isolated system S,
according to the Schrödinger equation. On the other hand, the measurement of
an observable on S implies an interaction with an external measuring apparatus
O: the system S in this case is not isolated, and its evolution during the
measurement is described by a (nonunitary) projection. The clash between
unitary evolution (Process 2 in von Neumann terminology [1]) and projection
(Process 1) is at the core of the so called “measurement problem” in quantum
mechanics (see for ex. [3, 4] and ref.s therein). We will argue that the
problem disappears if we consider only the ontic part of the collapse, as
described by (1.2) in the previous Section, where a pure state collapses into
a mixture. The epistemic part we consider irrelevant to the discussion, since
it is intrinsic to all statistical ensembles (not necessarily quantum).

More precisely, the problem seems to arise when we consider the (isolated)
composite system S+O and want to describe its evolution during the
measurement. The composite system, being isolated from external influences,
must evolve unitarily. But how can we describe unitarily a measurement that
takes place inside S+O, involving a nonunitary process on the subsystem S ?
The solution is provided by the reduced density matrix formalism. Indeed the
reduced density matrix for the subsystem S, after the interaction with the
measuring apparatus O, describes exactly the same statistical mixture as in
(1.1), with weights equal to the Born probabilities. This is a remarkable feat
of the formalism of textbook QM, and allows the interpretation of S as a
“measured system” within a purely unitary evolution.

To be concrete, consider S and O to be two qubits, i.e. two-dimensional
quantum systems, each with orthonormal basis vectors $|0\rangle$ and
$|1\rangle$. Moreover define observables A on S and B on O, both with
eigenvectors given by $|0\rangle$ and $|1\rangle$. The unitary evolution must
entangle S with O, since states of S must be correlated with states of the
apparatus O. This can be realized with a CNOT gate, i.e. a unitary operation
on the composite system acting on the basis vectors as
$|0\rangle|0\rangle\rightarrow|0\rangle|0\rangle$,
$|0\rangle|1\rangle\rightarrow|0\rangle|1\rangle$,
$|1\rangle|0\rangle\rightarrow|1\rangle|1\rangle$,
$|1\rangle|1\rangle\rightarrow|1\rangle|0\rangle$.

Consider as initial state of S+O the product state
$(\alpha|0\rangle+\beta|1\rangle)|0\rangle$. Then the CNOT gate transforms it,
by linearity, into the entangled state
$\alpha|0\rangle|0\rangle+\beta|1\rangle|1\rangle$, cf. Fig. 1. Thus, results
of measurements of A on S are correlated with those of B on O.

Fig. 1 Circuit modeling the measurement of an observable with eigenvectors
$|0\rangle$ and $|1\rangle$. Both the measured system S and the observer (or
apparatus) O are taken to be 1-qubit systems.

Is this a good model for quantum measurements, if B is identified with the
pointer position of a measuring apparatus O ? At first sight there seems to be
a problem, since the effect of a measurement on S should project it into one
of the two base states $|0\rangle,|1\rangle$, with probabilities
$|\alpha|^{2}$ and $|\beta|^{2}$ respectively. Here instead we have a final
entangled state for the composite system. How can we resolve this apparent
disagreement ?

The answer is to focus on the state of the subsystem S. This state is
completely described by the reduced density matrix $\rho^{S}$, obtained by
tracing the density matrix of S+O over system O. The partial trace yields

$\rho^{S}=Tr_{O}[(\alpha|0\rangle|0\rangle+\beta|1\rangle|1\rangle)(\alpha^{*}\langle
0|\langle 0|+\beta^{*}\langle 1|\langle 1|)]=|\alpha|^{2}|0\rangle\langle
0|+|\beta|^{2}|1\rangle\langle 1|$ (2.1)

and describes the state of subsystem S after the CNOT interaction. Note that
$\rho^{S}$ describes a mixed state of S. Before the CNOT interaction, S was in
the pure state $\alpha|0\rangle+\beta|1\rangle$. Therefore the effect of the
interaction on the subsystem S has been a nonunitary transformation, even if
S+O evolves unitarily.

The statistical mixture (2.1) coincides with (1.1). Thus the ontic part of the
measurement process can be described by a unitary evolution of S+O.

Note: the quantum circuit in Fig. 1 implements a measurement in the
computational basis also on mixed states. Indeed we can replace the pure state
$|\psi\rangle$ (input state for the first qubit) with an input density matrix
$\rho_{1}$, while keeping the state of the second qubit equal to $|0\rangle$.
Take for example $\rho_{1}=p_{a}|a\rangle\langle a|+p_{b}|b\rangle\langle b|$
with $p_{a}+p_{b}=1$, the generalization to an arbitrary ensemble being
straightforward. The pure states $|a\rangle,|b\rangle$ do not need to be
orthogonal, and are expanded on the computational basis as

$|a\rangle=\alpha|0\rangle+\beta|1\rangle,~{}~{}~{}|b\rangle=\gamma|0\rangle+\delta|1\rangle$
(2.2)

Then the density matrix $\rho$ for the 2-qubit system evolves as

$\rho=(p_{a}|a\rangle\langle a|+p_{b}|b\rangle\langle
b|)\otimes|0\rangle\langle 0|\longrightarrow\rho^{\prime}={\rm
CNOT}~{}\rho~{}{\rm CNOT}$ (2.3)

under the action of the circuit in Fig. 1. Recalling that CNOT =
$|0\rangle\langle 0|\otimes I+|1\rangle\langle 1|\otimes X$ where $X$ is the
quantum NOT gate, i.e. $X|0\rangle=|1\rangle,X|1\rangle=|0\rangle$, it is an
easy exercise to find $\rho^{\prime}$ as given by (2.3). Taking then its
partial trace yields the reduced density matrix $\rho^{\prime}_{1}$ for the
first qubit:

$\rho^{\prime}_{1}=Tr_{2}(\rho^{\prime})=(p_{a}|\alpha|^{2}+p_{b}|\gamma|^{2})|0\rangle\langle
0|+(p_{a}|\beta|^{2}+p_{b}|\delta|^{2})|1\rangle\langle 1|$ (2.4)

This is exactly the same density matrix one obtains for the first qubit after
measuring it (and ignoring the result), using the textbook rule

$\rho^{\prime}_{1}=\sum_{m}P_{m}\rho_{1}P_{m}$ (2.5)

$P_{m}$ being the projectors $|0\rangle\langle 0|$ and $|1\rangle\langle 1|$.
The transformation $\rho_{1}\rightarrow\rho^{\prime}_{1}$ is in general not
unitary (quick proof: $Tr(\rho^{\prime}_{1})^{2}$ in general differs from
$Tr(\rho_{1})^{2}$), and represents the ontic collapse of the density matrix
under a measurement. Thus the ontic part of the measurement is described by a
unitary evolution of the S+O density matrix.

##  3 Objections

An often heard objection [5, 6] relies on the distinction between proper and
improper mixtures, a terminology introduced by D’ Espagnat [5]. A statistical
mixture due to ignorance is called proper, whereas the same mixture describing
a subsystem of a composite system in an entangled pure state is called
improper. In the example of the preceding Section, the objection runs as
follows. We cannot interpret the (improper) statistical mixture (2.1) to
describe a system which is in one of the two states $|0\rangle,|1\rangle$,
with probabilities $|\alpha|^{2}$ and $|\beta|^{2}$ respectively: if this were
the case, the total system should find itself in one of the two states
$|0\rangle|0\rangle$, $|1\rangle|1\rangle$ with probabilities $|\alpha|^{2}$
and $|\beta|^{2}$ respectively, implying for its density matrix to be

$|\alpha|^{2}|0\rangle|0\rangle\langle 0|\langle
0|+|\beta|^{2}|1\rangle|1\rangle\langle 1|\langle 1|$ (3.1)

in contradiction with the density matrix of S+O after the CNOT interaction

$(\alpha|0\rangle|0\rangle+\beta|1\rangle|1\rangle)(\alpha^{*}\langle
0|\langle 0|+\beta^{*}\langle 1|\langle 1|)$ (3.2)

Another objection invokes “basis ambiguity”, meaning that the same density
matrix $\rho$ can correspond to different statistical ensembles (for an
exhaustive study see [7]). Therefore knowing $\rho$ does not imply knowing
which observable has been measured. For example in the case
$\alpha=\beta=1/\sqrt{2}$, the reduced density matrix for S in (2.1) becomes

${1\over 2}(|0\rangle\langle 0|+|1\rangle\langle 1|)={1\over
2}(|a\rangle\langle a|+|b\rangle\langle b|)$ (3.3)

for any orthonormal vectors $|a\rangle,|b\rangle$ connected to
$|0\rangle,|1\rangle$ by an orthogonal transformation. Thus the same reduced
density matrix would describe the state of S after a measurement, where the
measured observable could be one of an infinite set of mutually incompatible
(noncommuting) observables.

Note that the same basis ambiguity holds for the state of the composite S+O
system after the CNOT interaction, when $\alpha=\beta=1/\sqrt{2}$. This is
often referred to as the “preferred-basis problem”.

##  4 Counterobjections

Here we counter both objections, on logical and physical grounds. There have
been numerous rebuttals and counter-rebuttals over the years, see for ex. [8,
9, 10, 11], and most of the present Section is far from original. If the
density matrix $\rho^{S}$ gives a complete description of the state of S, it
makes no sense to distinguish states corresponding to the same $\rho^{S}$. If
only S is accessible to observations, no experiment can distinguish between a
proper or an improper mixture. The fact that S is part of a larger system S+O,
and that this larger system is in a pure entangled state, can only be revealed
by measurements on the whole S+O. Only then one can discriminate between the
two states of the composite system S+O, given in (3.1) and in (3.2),
respectively a pure entangled state and a statistical mixture. These two
different states give rise to the same reduced density matrix for S, and this
matrix defines a unique physical state, i.e. a statistical mixture, neither
proper nor improper. All measurements made on S are consistent with the
ignorance interpretation, and therefore $\rho^{S}$ in (2.1) has all the
markings of a state of a measured system.

The second objection in the preceding Section questions the interpretation of
the mixed state (3.3) as the state of a measured system, since it seems that
in this case the measured observable cannot be specified uniquely,111more
precisely, its eigenvectors cannot be specified uniquely. due to basis
ambiguity. To this we answer as follows:

i) Basis ambiguity does not compromise in any way the interpretation of (3.3)
as the state of a measured system: it may very well be the state obtained
after measuring any of a whole family of possibly noncommuting observables.
The fact that the measurements of any of those observables lead to the same
statistical mixture does not mean that the mixture does not describe correctly
the state after the measurement.

ii) We can assume that the observer knows which observable is being measured,
i.e. what particular interaction takes place between the apparatus and the
system. In our simplified two-qubit example this interaction is given by a
CNOT gate, and models a measurement of an observable in the computational
basis, i.e. of an observable with eigenvectors $|0\rangle$ and $|1\rangle$.
This removes the basis ambiguity: the basis, in which the mixed state (2.1) is
to be interpreted as a state of a measured system, is the computational one.
It is really the measuring apparatus, with its specific interaction with the
system, that dictates the “interpretation basis”. If the observer wants to
measure a different observable, for example with eigenvectors $|+\rangle$ and
$|-\rangle$, then the circuit implementing the system-apparatus interaction
must be modified as follows:

Fig. 2 Circuit modeling the measurement of an observable with eigenvectors
$|+\rangle$, $|-\rangle$. The gate $H$ is the Hadamard gate, defined by
$H|0\rangle=|+\rangle$, $H|1\rangle=|-\rangle$ .

In this case, with $\alpha=\beta=1/\sqrt{2}$, the reduced density matrix for
the first qubit, after the interaction, is
$\rho^{S}=1/2(|+\rangle\langle+|+|-\rangle\langle-|)$, equal to the one
obtained in the circuit of Fig.1, i.e. $\rho^{S}=1/2(|0\rangle\langle
0|+|1\rangle\langle 1|)$. However the interpretation basis is different, and
is given by the orthonormal couple $|+\rangle,|-\rangle$. Indeed the new
circuit in Fig. 2 correlates the states $|+\rangle$, $|-\rangle$ of the system
to the states $|+\rangle$, $|-\rangle$ of the apparatus, respectively, and
thus models the measurement of an observable with $|+\rangle$ and $|-\rangle$
eigenvectors. Thus the choice of an interpretation basis is not ambiguous, but
can be considered epistemic, since it relies on the observer’s knowledge of
which observable is being measured.

##  5 Refinements

With the same logic, we can describe a measurement on a spatially extended
2-qubit system S in the entangled state
${1\over\sqrt{2}}(|0\rangle|0\rangle+|1\rangle|1\rangle)$, where now the
observer (call her Alice) has access only to the first qubit. Standard quantum
mechanics tells us that Alice has a probability $1/2$ to obtain 0 or 1, and
that the measurement produces a corresponding collapse of S into one of the
states $|0\rangle|0\rangle$ or $|1\rangle|1\rangle$. Again we can describe the
ontic part of this measuring process by a specific interaction between Alice
and her qubit, the only accessible (to Alice) part of S. Modeling Alice as a
single qubit, the interaction is given by the same CNOT gate used in Section
2:

Fig. 3 Circuit modeling the measurement by Alice on an entangled state of
system S, where she has access only to the first qubit. She interacts with the
first qubit via a CNOT gate.

The state of Alice is initialized to $|0\rangle$ (the “ready state”). The
output is the maximally entangled 3-qubit state given in Fig. 3. To find the
state of the subsystem S after the interaction one must trace over A the total
density matrix, which yields the reduced density matrix for S:

$\rho^{S}=Tr_{A}(\rho^{SA})={1\over 2}Tr_{A}(|000\rangle+|111\rangle)(\langle
000|+\langle 111|)={1\over 2}|00\rangle\langle 00|+{1\over 2}|11\rangle\langle
11|$ (5.1)

This reduced matrix describes system S as a system that has collapsed to the
state $|00\rangle$ or $|11\rangle$ with probability $1/2$: this is the “ontic”
collapse, and occurs precisely at the time of the interaction. If Alice
“looks” at the result after the measurement, she will know whether the system
S is in the state $|00\rangle$ or in the state $|11\rangle$, thus producing
the “epistemic” collapse. This part of the collapse occurs instantaneously on
the whole extended system S, without any paradox since it is related to a
“state of knowledge” of Alice.

We can adapt this discussion to a Schrödinger’s cat situation, imagining that
the entangled initial state of S has been produced by an interaction between a
decaying atom and the cat. Then Alice “measures the cat”, in the sense that
her state becomes entangled with the cat state ($|alive\rangle$ or
$|dead\rangle$). This ontic measurement produces the final state in Fig. 3.
Tracing over Alice’s degrees of freedom yields a mixed state for the subsystem
cat + atom. This mixed state is a measured state, not a superposition state:
the cat is either alive or dead, with probabilities depending on the
particular interaction between the cat and the atom (in Fig. 3 these
probabilities are equal). Thus no superposition of $|alive\rangle$ and
$|dead\rangle$ states is left, and the “paradox” disappears.

##  6 A critique of state subjectivism

In most subjectivist or relational interpretations of QM (see for ex. [12, 13,
14, 15]) the alleged observer dependence of quantum states is claimed to be a
consequence of the usual quantum mechanical rules, when these are applied to
“third person” situations, as in Wigner’s friend thought experiment [16]. This
involves an isolated lab L, where an observer F (the “friend”) measures a
system S, and an external observer W (“Wigner”) outside of L. It is claimed
that F and W give different descriptions of the state of S after a measurement
by F, essentially because for F the system S has collapsed into a projected
state (eigenvector of the measured observable) whereas for W the whole system
in L, being isolated, must evolve unitarily. Again the contradiction is only
apparent, since W and F should compare their descriptions of the same system
S: if they do so, they will both agree that S, after the measurement, is in a
mixed state, for the same reason discussed in Section 2. Modeling S and F as
one-qubit systems, and (the ontic part of) the measurement as a CNOT gate, the
reduced density matrix for S describes its state as the same mixture for any
observer. If, in addition, F also knows the result of the measurement, he will
be able to describe the state of S as a particular state of the mixture. This
further sharpening, due to additional knowledge of F (W has no access to the
inner contents of L), is what produces the “epistemic collapse”, common to all
situations (classical or quantum) where one has incomplete information on the
state of a system, and encodes this ignorance in a statistical ensemble.

To defend their thesis, supporters of the non-objectivity of quantum states
often insist on the distinction between proper and improper mixtures, and on
the claim that only proper mixtures can describe the state of a measured
system (with ignorance of the result). One can understand why this is a very
sensitive point in any discussion regarding the “objectivity of the wave
function”. In Section 4 we have argued that the proper/improper distinction is
unphysical, i.e. not detectable by any experiment on S. Therefore we conclude
that Wigner’s friend experiment cannot be taken as cornerstone of a relational
interpretation of QM, as done for example in [15].

##  7 Conclusions

We have presented a case for the objectivity of quantum states, further
elaborating and developing some arguments discussed in [17]. For this we have
recalled the description of a measured system as a mixture, coinciding with
the one obtained by tracing over the observer’s degrees of freedom. The only
epistemic ingredient of a quantum mixture is due to the incomplete
information, encoded in the statistical ensemble of the mixture. This
ingredient is intrinsic to the very definition of a statistical ensemble, and
is not a characteristic of the quantum world. For the whole argument we used
the standard rules of QM: in particular the reduced density matrix formalism
allows to describe measurement as an unitary process, i.e an interaction
between observer and system. In our opinion many of the so-called paradoxes or
interpretative problems of QM are resolved by making use of the full power of
its formalism. In other words, QM is quite capable to cure itself from
apparent internal contradictions.

## Acknowledgements

This work is supported by the research funds of the Eastern Piedmont
University and INFN - Torino Section.

## References

  * [1] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,“ Princeton University Press, Princeton, N. J., 1955, R.T. Beyer, tr; originally published as Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
  * [2] L. Landau, “Das Dämpfungsproblem in der Wellenmechanik”, Z. Phys. 45 (1927) 430-441.
  * [3] M. Schlosshauer, “Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics,” Rev. Mod. Phys. 76, 1267-1305 (2004) [arXiv:quant-ph/0312059 [quant-ph]].
  * [4] C. Brukner, “On the quantum measurement problem.” In R. Bertlmann and A. Zeilinger, editors, Quantum [Un]Speakables II: Half a Century of Bell’s Theorem, pages 95–117. Springer, 2017; arXiv:1507.05255.
  * [5] B. d’Espagnat, “Conceptual Foundations of Quantum Mechanics,” 2nd ed., W. A. Benjamin, Menlo Park, CA, 1976.
  * [6] R. I. G. Hughes, “The Structure and Interpretation of Quantum Mechanics,” Harvard University Press, Cambridge, MA., 1989.
  * [7] L. Hughston, R. Jozsa and W. Wootters, “A Complete Classification of Quantum Ensembles Having a Given Density Matrix.“ Physics Letters A, 183, 14 - 18 (1993).
  * [8] J. Anandan and Y. Aharonov, “Meaning of the density matrix,” Found. Phys. Lett. 12, no.6, 571-578 (1999) doi:10.1023/A:1021699226154 [arXiv:quant-ph/9803018 [quant-ph]].
  * [9] B. d’Espagnat, Reply to Aharonov and Anandan’s “Meaning of the Density Matrix”, arXiv:quant-ph/9804063, 1998.
  * [10] K. A. Kirkpatrick, “Indistinguishability and improper mixtures”, arXiv:quant-ph/0109146 (2001)
  * [11] B. d’ Espagnat, “Reply to K A Kirkpatrick,” arXiv:quant-ph/0111081, 2001
  * [12] C. A. Fuchs, R. Schack, “Quantum-Bayesian coherence,” Rev. Mod. Phys. 85, 1693 (2013).
  * [13] N. D. Mermin, “Physics: QBism puts the scientist back into science,” Nature 507, 421–423 (2014).
  * [14] H. Everett, “Relative state formulation of quantum mechanics,” Rev. Mod. Phys. 29, 454–462 (1957).
  * [15] C. Rovelli, “Relational Quantum Mechanics,” International Journal of Theoretical Physics 35 (1996) 1637; arXiv:quant-ph/9609002.
  * [16] E. P. Wigner, in The Scientist Speculates (ed. Good, I. J.) 284–302 (Heinemann, 1961).
  * [17] L. Castellani, “No Relation for Wigner’s Friend,” Int. J. Theor. Phys. 60, no.6, 2084-2089 (2021).