{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T11:08:29Z","timestamp":1784286509766,"version":"3.55.0"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T00:00:00Z","timestamp":1779408000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T00:00:00Z","timestamp":1779408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Oak Ridge National Laboratory\u2019s (ORNL) Laboratory Directed Research and Development (LDRD) Seed Program","award":["FWP ERKBP91"],"award-info":[{"award-number":["FWP ERKBP91"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Non-local games provide a versatile framework for probing quantum correlations and for benchmarking the power of entanglement. In finite dimensions, the standard method for playing several games in parallel requires a tensor product of the local Hilbert spaces, which scales additively in the number of qubits. In this work, we show that this additive cost can be reduced by exploiting algebraic embeddings. We introduce two forms of compressions. First, when a referee selects one game from a finite collection of games at random, the game quantum strategy can be implemented using a maximally entangled state of dimension equal to the largest individual game, thereby eliminating the need for repeated state preparations. Second, we establish conditions under which several games can be played simultaneously in parallel on fewer qubits than the tensor product baseline. These conditions are expressed in terms of commuting embeddings of the game algebras. Moreover, we provide a constructive framework for building such embeddings. Using tools from Lie theory, we show that aligning the various game algebras into a common Cartan decomposition enables such a qubit reduction. Beyond the theoretical contribution, our framework casts NLGs as algebraic primitives for distributed and resource-constrained quantum computations and suggested NLGs as a comparable device-independent dimension witness.<\/jats:p>","DOI":"10.1007\/s11128-026-05194-1","type":"journal-article","created":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T04:53:46Z","timestamp":1779425626000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Commuting embeddings for parallel strategies in non-local games"],"prefix":"10.1007","volume":"25","author":[{"given":"Sarah","family":"Chehade","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andrea","family":"Delgado","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elaine","family":"Wong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,22]]},"reference":[{"issue":"2","key":"5194_CR1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.104.022212","volume":"104","author":"A Pan","year":"2021","unstructured":"Pan, A.: Oblivious communication game, self-testing of projective and nonprojective measurements, and certification of randomness. Phys. Rev. A 104(2), 022212 (2021). https:\/\/doi.org\/10.1103\/PhysRevA.104.022212","journal-title":"Phys. Rev. A"},{"key":"5194_CR2","doi-asserted-by":"publisher","first-page":"337","DOI":"10.22331\/q-2020-09-30-337","volume":"4","author":"I \u0160upi\u0107","year":"2020","unstructured":"\u0160upi\u0107, I., Bowles, J.: Self-testing of quantum systems: a review. Quantum 4, 337 (2020). https:\/\/doi.org\/10.22331\/q-2020-09-30-337","journal-title":"Quantum"},{"key":"5194_CR3","doi-asserted-by":"publisher","first-page":"346","DOI":"10.22331\/q-2020-10-21-346","volume":"4","author":"D Cui","year":"2020","unstructured":"Cui, D., Mehta, A., Mousavi, H., Nezhadi, S.S.: A generalization of CHSH and the algebraic structure of optimal strategies. Quantum 4, 346 (2020). https:\/\/doi.org\/10.22331\/q-2020-10-21-346","journal-title":"Quantum"},{"key":"5194_CR4","doi-asserted-by":"publisher","unstructured":"Man\u010dinska, L., Nielsen, T.G., Prakash, J.: Glued magic games self-test maximally entangled states. arXiv preprint arXiv:2105.10658 (2021) https:\/\/doi.org\/10.48550\/arXiv.2105.10658","DOI":"10.48550\/arXiv.2105.10658"},{"issue":"6","key":"5194_CR5","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s00220-024-05001-x","volume":"405","author":"IG Todorov","year":"2024","unstructured":"Todorov, I.G., Turowska, L.: Quantum no-signalling correlations and non-local games. Commun. Math. Phys. 405(6), 141 (2024). https:\/\/doi.org\/10.1007\/s00220-024-05001-x","journal-title":"Commun. Math. Phys."},{"key":"5194_CR6","doi-asserted-by":"publisher","unstructured":"Mousavi, H., Nezhadi, S.S., Yuen, H.: Nonlocal games, compression theorems, and the arithmetical hierarchy. In: Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp. 1\u201311 (2022). https:\/\/doi.org\/10.1145\/3519935.3519949","DOI":"10.1145\/3519935.3519949"},{"key":"5194_CR7","doi-asserted-by":"publisher","unstructured":"Miller, C.A., Shi, Y.: Optimal robust quantum self-testing by binary nonlocal xor games. arXiv preprint arXiv:1207.1819 (2012) https:\/\/doi.org\/10.48550\/arXiv.1207.1819","DOI":"10.48550\/arXiv.1207.1819"},{"issue":"11","key":"5194_CR8","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1145\/3485628","volume":"64","author":"Z Ji","year":"2021","unstructured":"Ji, Z., Natarajan, A., Vidick, T., Wright, J., Yuen, H.J.: MIP* = RE. Commun. ACM 64(11), 131\u2013138 (2021). https:\/\/doi.org\/10.1145\/3485628","journal-title":"Commun. ACM"},{"issue":"6","key":"5194_CR9","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s11128-025-04799-2","volume":"24","author":"H Chen","year":"2025","unstructured":"Chen, H., Jia, W.: Dynamics of quantum bertrand duopoly games with asymmetric information: H. Chen, W. Jia. Quantum Inf. Process. 24(6), 177 (2025)","journal-title":"Quantum Inf. Process."},{"issue":"15","key":"5194_CR10","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","volume":"23","author":"JF Clauser","year":"1969","unstructured":"Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23(15), 880 (1969). https:\/\/doi.org\/10.1103\/PhysRevLett.23.880","journal-title":"Phys. Rev. Lett."},{"key":"5194_CR11","doi-asserted-by":"publisher","unstructured":"Arkhipov, A.: Extending and characterizing quantum magic games. arXiv preprint arXiv:1209.3819 (2012) https:\/\/doi.org\/10.48550\/arXiv.1209.3819","DOI":"10.48550\/arXiv.1209.3819"},{"issue":"4","key":"5194_CR12","doi-asserted-by":"publisher","DOI":"10.1088\/2058-9565\/adf1c0","volume":"10","author":"J Furches","year":"2025","unstructured":"Furches, J., Chehade, S., Hamilton, K., Wiebe, N., Marrero, C.O.: Application-level benchmarking of quantum computers using nonlocal game strategies. Quantum Sci. Technol. 10(4), 045002 (2025). https:\/\/doi.org\/10.1088\/2058-9565\/adf1c0","journal-title":"Quantum Sci. Technol."},{"key":"5194_CR13","doi-asserted-by":"publisher","unstructured":"Chehade, S., Delgado, A., Wong, E.: A game-theoretic quantum algorithm for solving magic squares. arXiv preprint arXiv:2505.13366 (2025) https:\/\/doi.org\/10.48550\/arXiv.2505.13366","DOI":"10.48550\/arXiv.2505.13366"},{"key":"5194_CR14","doi-asserted-by":"publisher","first-page":"3373","DOI":"10.1103\/PhysRevLett.65.3373","volume":"65","author":"ND Mermin","year":"1990","unstructured":"Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373\u20133376 (1990). https:\/\/doi.org\/10.1103\/PhysRevLett.65.3373","journal-title":"Phys. Rev. Lett."},{"issue":"1\u20133","key":"5194_CR15","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/S0301-0104(01)00318-4","volume":"267","author":"N Khaneja","year":"2001","unstructured":"Khaneja, N., Glaser, S.J.: Cartan decomposition of su ($$2^n$$) and control of spin systems. Chem. Phys. 267(1\u20133), 11\u201323 (2001). https:\/\/doi.org\/10.1016\/S0301-0104(01)00318-4","journal-title":"Chem. Phys."},{"issue":"3","key":"5194_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.3.030341","volume":"3","author":"M Larocca","year":"2022","unstructured":"Larocca, M., Sauvage, F., Sbahi, F.M., Verdon, G., Coles, P.J., Cerezo, M.: Group-invariant quantum machine learning. PRX quantum 3(3), 030341 (2022). https:\/\/doi.org\/10.1103\/PRXQuantum.3.030341","journal-title":"Group-invariant quantum machine learning. PRX quantum"},{"key":"5194_CR17","doi-asserted-by":"publisher","unstructured":"Hall, B.C.: Lie groups, lie algebras, and representations. In: Quantum theory for mathematicians, pp. 333\u2013366. Springer, Berlin (2013). https:\/\/doi.org\/10.1007\/978-3-319-13467-3","DOI":"10.1007\/978-3-319-13467-3"},{"issue":"15","key":"5194_CR18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.110.150501","volume":"110","author":"N Brunner","year":"2013","unstructured":"Brunner, N., Navascu\u00e9s, M., V\u00e9rtesi, T.: Dimension witnesses and quantum state discrimination. Phys. Rev. Lett. 110(15), 150501 (2013). https:\/\/doi.org\/10.1103\/PhysRevLett.110.150501","journal-title":"Phys. Rev. Lett."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05194-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05194-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05194-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T10:49:56Z","timestamp":1784285396000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05194-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,22]]},"references-count":18,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026,6]]}},"alternative-id":["5194"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05194-1","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,22]]},"assertion":[{"value":"3 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 May 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest. The authors have no financial or non-financial interests that are directly or indirectly related to the work submitted.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"185"}}