Performance Analysis of Quantum CSS Error-Correcting Codes via MacWilliams Identities
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AbstractWe analyze the performance of quantum stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis, allowing the derivation of the exact asymptotic error rate for short codes. For example, on a depolarizing channel with physical error rate $\rho \to 0$, the logical error rate $\rho_\mathrm{L}$ is asymptotically $\rho_\mathrm{L} \approx 16 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \approx 16.3 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \approx 18.7 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \approx 149.3 \rho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $\rho_\mathrm{L} \approx 1215 \rho^4$ and $\rho_\mathrm{L} \approx 663 \rho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively. Finally, we extend our analysis to include realistic, noisy syndrome extraction circuits by modeling error propagation throughout gadgets. This enables estimation of logical error rates under faulty measurements. The performance analysis serves as a design tool for developing fault-tolerant quantum systems by guiding the selection of quantum codes based on their error correction capability. Additionally, it offers a novel perspective on quantum degeneracy, showing it represents the fraction of non-correctable error patterns shared by multiple logical operators.Featured image: Example of errors leading to a logical operator of weight $w=4$ for the $[[ 13,1,3 ]]$ surface code.Popular summaryThe exploitation of the unique features of quantum mechanics has opened new perspectives on how we can sense, process, and communicate information. From an engineering point of view, there are many challenges to solve, calling for both theoretical and experimental research studies. The aim is to progress towards the already known possible applications of quantum information technologies, as well as those currently still unforeseen, that will arise when practical implementations become available. One of the main challenges is how to deal with the noise caused by unwanted interaction of the quantum information with the environment. Quantum error correcting codes, where a redundant representation of quantum states protects from certain types of errors, are therefore of paramount importance for quantum computation, quantum memories, and quantum communication systems. In this paper we provide an analytical evaluation of the performance of stabilizer codes, like quantum low-density parity-check (QLDPC) codes and surface codes. We propose a framework for the performance investigation of stabilizer codes by means of the quantum MacWilliams identities. Moreover, we develop a logical operator analysis leading to exact expressions for the logical error rates, assuming complete decoders (decoders that always attempt to correct the error). Specifically, we analyze minimum weight (MW) decoding, which finds the lowest weight error consistent with the syndrome. The analysis is conducted for both symmetric and asymmetric models of quantum channel errors. In practical quantum systems, however, syndrome extraction is a critical yet error-prone component of quantum error correction. Measurements are inherently noisy and typically require repetition to ensure reliability. Additionally, faults during extraction can propagate, causing high weight correlated errors. Thus, we introduce a framework that models the full syndrome extraction process, incorporating gate-specific noise and measurement imperfections. This enables the estimation of logical error rates under realistic circuit-level noise assumptions. The key contributions of the paper can be summarized as follows: – we derive the weight enumerator L(z) for the undetectable errors of arbitrary stabilizer codes via MacWilliams identities; – we derive theoretical upper bounds for the error correction capability of CSS stabilizer codes; – we derive closed form expressions for the coefficients of L(z) which significantly impact the performance of surface codes for any code distance; – we derive the exact performance of stabilizer codes under MW decoding, including surface codes under MWPM decoding, over symmetric and asymmetric channels; – we introduce a novel perspective on quantum degeneracy, analyzing its influence on the error correction capability of a quantum code.► BibTeX data@article{Forlivesi2025performanceanalysis, doi = {10.22331/q-2025-12-16-1950}, url = {https://doi.org/10.22331/q-2025-12-16-1950}, title = {Performance {A}nalysis of {Q}uantum {CSS} {E}rror-{C}orrecting {C}odes via {M}ac{W}illiams {I}dentities}, author = {Forlivesi, Diego and Valentini, Lorenzo and Chiani, Marco}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1950}, month = dec, year = {2025} }► References [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases and Bound Entanglement, Phys. Rev. Lett. 82, 5385 (1999). https://doi.org/10.1103/PhysRevLett.82.5385 [3] K. R. Parthasarathy, On the maximal dimension of a completely entangled subspace for finite level quantum systems, Proc. Math. Sci. 114, 365 (2004). https://doi.org/10.1007/BF02829441 [4] B. V. R. Bhat, A completely entangled subspace of maximal dimension, Int. J. Quantum Inf. 4, 325 (2006). https://doi.org/10.1142/S0219749906001797 [5] J. Walgate and A. J. Scott, Generic local distinguishability and completely entangled subspaces, J. Phys. A: Math. Theor. 41, 375305 (2008). https://doi.org/10.1088/1751-8113/41/37/375305 [6] R. Augusiak, J. Tura, and M. Lewenstein, A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces, J. Phys. A: Math. Theor. 44, 212001 (2011). https://doi.org/10.1088/1751-8113/44/21/212001 [7] N. Johnston, Non-positive-partial-transpose subspaces can be as large as any entangled subspace, Phys. Rev. A 87, 064302 (2013). https://doi.org/10.1103/PhysRevA.87.064302 [8] R. Sengupta, Arvind, and A. I. Singh, Entanglement properties of positive operators with ranges in completely entangled subspaces, Phys. Rev. A 90, 062323 (2014). https://doi.org/10.1103/PhysRevA.90.062323 [9] M. Brannan and B. Collins, Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups, Commun. Math. Phys. 358, 1007 (2018). https://doi.org/10.1007/s00220-017-3023-6 [10] N. Alon and L. Lovász, Unextendible Product Bases, J. Combinat. Theor. Series A 95, 169 (2001). https://doi.org/10.1006/jcta.2000.3122 [11] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement, Commun. Math. Phys. 238, 379 (2003). https://doi.org/10.1007/s00220-003-0877-6 [12] A. O. Pittenger, Unextendible product bases and the construction of inseparable states, Linear Algebra Appl. 359, 235 (2003). https://doi.org/10.1016/S0024-3795(02)00423-8 [13] N. Johnston, The structure of qubit unextendible product bases, J. Phys. A: Math. Theor. 47, 424034 (2014). https://doi.org/10.1088/1751-8113/47/42/424034 [14] M. Demianowicz and R. Augusiak, From unextendible product bases to genuinely entangled subspaces, Phys. Rev. A 98, 012313 (2018). https://doi.org/10.1103/PhysRevA.98.012313 [15] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000). https://doi.org/10.1103/PhysRevA.62.062314 [16] P. Hayden, D. W. Leung, and A. Winter, Aspects of Generic Entanglement, Commun. Math. Phys. 265, 95 (2004). https://doi.org/10.1007/s00220-006-1535-6 [17] P. Hayden, Entanglement in Random Subspaces, AIP Conf. Proc. 734, 226 (2004). https://doi.org/10.1063/1.1834421 [18] M. Demianowicz and R. Augusiak, An approach to constructing genuinely entangled subspaces of maximal dimension, Quantum Inf. Process. 19, 199 (2020). https://doi.org/10.1007/s11128-020-02688-4 [19] M. Demianowicz, Universal construction of genuinely entangled subspaces of any size, Quantum 6, 854 (2022). https://doi.org/10.22331/q-2022-11-10-854 [20] J. M. Leinaas, J. Myrheim, and P. O. Sollid, Low-rank extremal positive-partial-transpose states and unextendible product bases, Phys. Rev. A 81, 062330 (2010). https://doi.org/10.1103/PhysRevA.81.062330 [21] Ł. Skowronek, Three-by-three bound entanglement with general unextendible product bases, J. Math. Phys. 52, 122202 (2011). https://doi.org/10.1063/1.3663836 [22] S. Agrawal1, S. Halder, and M. Banik, Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition, Phys. Rev. A 99, 032335 (2019). https://doi.org/10.1103/PhysRevA.99.032335 [23] M. Waegell and J. Dressel, Benchmarks of nonclassicality for qubit arrays, npj Quantum Inf 5, 66 (2019). https://doi.org/10.1038/s41534-019-0181-8 [24] O. Makuta and R. Augusiak, Self-testing maximally-dimensional genuinely entangled subspaces within the stabilizer formalism, New J. Phys. 23, 043042 (2020). https://doi.org/10.1088/1367-2630/abee40 [25] O. Makuta, B. Kuzaka, and R. Augusiak, Fully non-positive-partial-transpose genuinely entangled subspaces, Quantum 7, 915 (2023). https://doi.org/10.22331/q-2023-02-09-915 [26] K. V. Antipin, Construction of genuinely entangled subspaces and the associated bounds on entanglement measures for mixed states, J. Phys. A: Math. Theor. 54, 505303 (2021). https://doi.org/10.1103/PhysRevA.98.012313 [27] K. V. Antipin, Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties, Phys. Lett. A 445, 128248 (2022). https://doi.org/10.1016/j.physleta.2022.128248 [28] B. Lovitz and N. Johnston, Entangled subspaces and generic local state discrimination with pre-shared entanglement, Quantum 6, 760 (2022). https://doi.org/10.22331/q-2022-07-07-760 [29] G. Gour and N. R. Wallach, Entanglement of subspaces and error-correcting codes, Phys. Rev. A 76, 042309 (2007). https://doi.org/10.1103/PhysRevA.76.042309 [30] F. Huber and M. Grassl, Quantum Codes of Maximal Distance and Highly Entangled Subspaces, Quantum 4, 284 (2020). https://doi.org/10.22331/q-2020-06-18-284 [31] A. H. Shenoy and R. Srikanth, Maximally nonlocal subspaces, J. Phys. A: Math. Theor. 52, 095302 (2019). https://doi.org/10.1088/1751-8121/ab0046 [32] J. Harris, Algebraic Geometry: A First Course, (Graduate Texts in Mathematics, Vol. 133) (Springer New York, NY, 1992). https://doi.org/10.1007/978-1-4757-2189-8. https://doi.org/10.1007/978-1-4757-2189-8 [33] J. M. Landsberg, Tensors: Geometry and Applications, (Graduate Studies in Mathematics Vol. 128) (American Mathematical Society 2012). https://doi.org/10.1090/gsm/128. https://doi.org/10.1090/gsm/128 [34] M. Gharahi, S. Mancini, and G. Ottaviani, Fine-structure classification of multiqubit entanglement by algebraic geometry, Phys. Rev. Research 2, 043003 (2020). https://doi.org/10.1103/PhysRevResearch.2.043003 [35] M. Gharahi and S. Mancini, Algebraic-geometric characterization of tripartite entanglement, Phys. Rev. A 104, 042402 (2021). https://doi.org/10.1103/PhysRevA.104.042402 [36] M. Gharahi, $\ell$-Multilinear Ranks of Multipartite Quantum States via Tensor Flattening: A Mathematica Codebase, Zenodo (2025). https://doi.org/10.5281/zenodo.15299720. https://doi.org/10.5281/zenodo.15299720 [37] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going Beyond Bell's Theorem, (in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos) (Springer, Dordrecht, 1989). pp. 69-72. https://doi.org/10.1007/978-94-017-0849-4_10. https://doi.org/10.1007/978-94-017-0849-4_10 [38] N. R. Wallach, An unentangled Gleason’s theorem, Contemp. Math. 305, 291 (2002). https://doi.org/10.1090/conm/305 [39] T. Cubitt, A. Montanaro, and A. Winter, On the dimension of subspaces with bounded Schmidt rank, J. Math. Phys. 49, 022107 (2008). https://doi.org/10.1063/1.2862998 [40] I. R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, (Springer Berlin, Heidelberg, 2013). https://doi.org/10.1007/978-3-642-37956-7. https://doi.org/10.1007/978-3-642-37956-7 [41] R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99 (1954). https://doi.org/10.1103/PhysRev.93.99 [42] J. J. Sylvester, On the principles of the calculus of forms, Cambridge and Dublin Mathematical Journal VII, 52 (1852). [43] M. Gharahi, Classifying entanglement by algebraic geometry, Int. J. Quant. Inf. 22, 2350047 (2024). https://doi.org/10.1142/S0219749923500478 [44] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, (Lecture Notes in Mathematics, Vol. 1721) (Springer Berlin, Heidelberg, 1999). https://doi.org/10.1007/BFb0093426. https://doi.org/10.1007/BFb0093426 [45] P. Aluffi and C. Faber, Linear orbits of $d$-tuples of points in $\mathbb{P}^1$, J. Reine Angew. Math. 445, 205 (1993). https://eudml.org/doc/153581 [46] N. Linden, S. Popescu, and J. A. Smolin, Entanglement of Superpositions, Phys. Rev. Lett. 97, 100502 (2006). https://doi.org/10.1103/PhysRevLett.97.100502 [47] Z. Ma, Z. Chen, and S.-M. Fei, Genuine multipartite entanglement of superpositions, Phys. Rev. A 90, 032307 (2014). https://doi.org/10.1103/PhysRevA.90.032307 [48] R. A. Horn and C. R. Johnson, Matrix Analysis, (Cambridge University Press, Cambridge, 1985). https://doi.org/10.1017/CBO9780511810817. https://doi.org/10.1017/CBO9780511810817Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-12-28 15:38:49). Could not fetch ADS cited-by data during last attempt 2025-12-28 15:38:50: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe analyze the performance of quantum stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. We introduce a novel approach that combines the knowledge of WE with a logical operator analysis, allowing the derivation of the exact asymptotic error rate for short codes. For example, on a depolarizing channel with physical error rate $\rho \to 0$, the logical error rate $\rho_\mathrm{L}$ is asymptotically $\rho_\mathrm{L} \approx 16 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \approx 16.3 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \approx 18.7 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \approx 149.3 \rho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $\rho_\mathrm{L} \approx 1215 \rho^4$ and $\rho_\mathrm{L} \approx 663 \rho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively. Finally, we extend our analysis to include realistic, noisy syndrome extraction circuits by modeling error propagation throughout gadgets. This enables estimation of logical error rates under faulty measurements. The performance analysis serves as a design tool for developing fault-tolerant quantum systems by guiding the selection of quantum codes based on their error correction capability. Additionally, it offers a novel perspective on quantum degeneracy, showing it represents the fraction of non-correctable error patterns shared by multiple logical operators.Featured image: Example of errors leading to a logical operator of weight $w=4$ for the $[[ 13,1,3 ]]$ surface code.Popular summaryThe exploitation of the unique features of quantum mechanics has opened new perspectives on how we can sense, process, and communicate information. From an engineering point of view, there are many challenges to solve, calling for both theoretical and experimental research studies. The aim is to progress towards the already known possible applications of quantum information technologies, as well as those currently still unforeseen, that will arise when practical implementations become available. One of the main challenges is how to deal with the noise caused by unwanted interaction of the quantum information with the environment. Quantum error correcting codes, where a redundant representation of quantum states protects from certain types of errors, are therefore of paramount importance for quantum computation, quantum memories, and quantum communication systems. In this paper we provide an analytical evaluation of the performance of stabilizer codes, like quantum low-density parity-check (QLDPC) codes and surface codes. We propose a framework for the performance investigation of stabilizer codes by means of the quantum MacWilliams identities. Moreover, we develop a logical operator analysis leading to exact expressions for the logical error rates, assuming complete decoders (decoders that always attempt to correct the error). Specifically, we analyze minimum weight (MW) decoding, which finds the lowest weight error consistent with the syndrome. The analysis is conducted for both symmetric and asymmetric models of quantum channel errors. In practical quantum systems, however, syndrome extraction is a critical yet error-prone component of quantum error correction. Measurements are inherently noisy and typically require repetition to ensure reliability. Additionally, faults during extraction can propagate, causing high weight correlated errors. Thus, we introduce a framework that models the full syndrome extraction process, incorporating gate-specific noise and measurement imperfections. This enables the estimation of logical error rates under realistic circuit-level noise assumptions. The key contributions of the paper can be summarized as follows: – we derive the weight enumerator L(z) for the undetectable errors of arbitrary stabilizer codes via MacWilliams identities; – we derive theoretical upper bounds for the error correction capability of CSS stabilizer codes; – we derive closed form expressions for the coefficients of L(z) which significantly impact the performance of surface codes for any code distance; – we derive the exact performance of stabilizer codes under MW decoding, including surface codes under MWPM decoding, over symmetric and asymmetric channels; – we introduce a novel perspective on quantum degeneracy, analyzing its influence on the error correction capability of a quantum code.► BibTeX data@article{Forlivesi2025performanceanalysis, doi = {10.22331/q-2025-12-16-1950}, url = {https://doi.org/10.22331/q-2025-12-16-1950}, title = {Performance {A}nalysis of {Q}uantum {CSS} {E}rror-{C}orrecting {C}odes via {M}ac{W}illiams {I}dentities}, author = {Forlivesi, Diego and Valentini, Lorenzo and Chiani, Marco}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1950}, month = dec, year = {2025} }► References [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases and Bound Entanglement, Phys. Rev. Lett. 82, 5385 (1999). https://doi.org/10.1103/PhysRevLett.82.5385 [3] K. R. Parthasarathy, On the maximal dimension of a completely entangled subspace for finite level quantum systems, Proc. Math. Sci. 114, 365 (2004). https://doi.org/10.1007/BF02829441 [4] B. V. R. Bhat, A completely entangled subspace of maximal dimension, Int. J. Quantum Inf. 4, 325 (2006). https://doi.org/10.1142/S0219749906001797 [5] J. Walgate and A. J. Scott, Generic local distinguishability and completely entangled subspaces, J. Phys. A: Math. Theor. 41, 375305 (2008). https://doi.org/10.1088/1751-8113/41/37/375305 [6] R. Augusiak, J. Tura, and M. Lewenstein, A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces, J. Phys. A: Math. Theor. 44, 212001 (2011). https://doi.org/10.1088/1751-8113/44/21/212001 [7] N. Johnston, Non-positive-partial-transpose subspaces can be as large as any entangled subspace, Phys. Rev. A 87, 064302 (2013). https://doi.org/10.1103/PhysRevA.87.064302 [8] R. Sengupta, Arvind, and A. I. Singh, Entanglement properties of positive operators with ranges in completely entangled subspaces, Phys. Rev. A 90, 062323 (2014). https://doi.org/10.1103/PhysRevA.90.062323 [9] M. Brannan and B. Collins, Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups, Commun. Math. Phys. 358, 1007 (2018). https://doi.org/10.1007/s00220-017-3023-6 [10] N. Alon and L. Lovász, Unextendible Product Bases, J. Combinat. Theor. Series A 95, 169 (2001). https://doi.org/10.1006/jcta.2000.3122 [11] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement, Commun. Math. Phys. 238, 379 (2003). https://doi.org/10.1007/s00220-003-0877-6 [12] A. O. Pittenger, Unextendible product bases and the construction of inseparable states, Linear Algebra Appl. 359, 235 (2003). https://doi.org/10.1016/S0024-3795(02)00423-8 [13] N. Johnston, The structure of qubit unextendible product bases, J. Phys. A: Math. Theor. 47, 424034 (2014). https://doi.org/10.1088/1751-8113/47/42/424034 [14] M. Demianowicz and R. Augusiak, From unextendible product bases to genuinely entangled subspaces, Phys. Rev. A 98, 012313 (2018). https://doi.org/10.1103/PhysRevA.98.012313 [15] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000). https://doi.org/10.1103/PhysRevA.62.062314 [16] P. Hayden, D. W. Leung, and A. Winter, Aspects of Generic Entanglement, Commun. Math. Phys. 265, 95 (2004). https://doi.org/10.1007/s00220-006-1535-6 [17] P. Hayden, Entanglement in Random Subspaces, AIP Conf. Proc. 734, 226 (2004). https://doi.org/10.1063/1.1834421 [18] M. Demianowicz and R. Augusiak, An approach to constructing genuinely entangled subspaces of maximal dimension, Quantum Inf. Process. 19, 199 (2020). https://doi.org/10.1007/s11128-020-02688-4 [19] M. Demianowicz, Universal construction of genuinely entangled subspaces of any size, Quantum 6, 854 (2022). https://doi.org/10.22331/q-2022-11-10-854 [20] J. M. Leinaas, J. Myrheim, and P. O. Sollid, Low-rank extremal positive-partial-transpose states and unextendible product bases, Phys. Rev. A 81, 062330 (2010). https://doi.org/10.1103/PhysRevA.81.062330 [21] Ł. Skowronek, Three-by-three bound entanglement with general unextendible product bases, J. Math. Phys. 52, 122202 (2011). https://doi.org/10.1063/1.3663836 [22] S. Agrawal1, S. Halder, and M. Banik, Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition, Phys. Rev. A 99, 032335 (2019). https://doi.org/10.1103/PhysRevA.99.032335 [23] M. Waegell and J. Dressel, Benchmarks of nonclassicality for qubit arrays, npj Quantum Inf 5, 66 (2019). https://doi.org/10.1038/s41534-019-0181-8 [24] O. Makuta and R. Augusiak, Self-testing maximally-dimensional genuinely entangled subspaces within the stabilizer formalism, New J. Phys. 23, 043042 (2020). https://doi.org/10.1088/1367-2630/abee40 [25] O. Makuta, B. Kuzaka, and R. Augusiak, Fully non-positive-partial-transpose genuinely entangled subspaces, Quantum 7, 915 (2023). https://doi.org/10.22331/q-2023-02-09-915 [26] K. V. Antipin, Construction of genuinely entangled subspaces and the associated bounds on entanglement measures for mixed states, J. Phys. A: Math. Theor. 54, 505303 (2021). https://doi.org/10.1103/PhysRevA.98.012313 [27] K. V. Antipin, Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties, Phys. Lett. A 445, 128248 (2022). https://doi.org/10.1016/j.physleta.2022.128248 [28] B. Lovitz and N. Johnston, Entangled subspaces and generic local state discrimination with pre-shared entanglement, Quantum 6, 760 (2022). https://doi.org/10.22331/q-2022-07-07-760 [29] G. Gour and N. R. Wallach, Entanglement of subspaces and error-correcting codes, Phys. Rev. A 76, 042309 (2007). https://doi.org/10.1103/PhysRevA.76.042309 [30] F. Huber and M. Grassl, Quantum Codes of Maximal Distance and Highly Entangled Subspaces, Quantum 4, 284 (2020). https://doi.org/10.22331/q-2020-06-18-284 [31] A. H. Shenoy and R. Srikanth, Maximally nonlocal subspaces, J. Phys. A: Math. Theor. 52, 095302 (2019). https://doi.org/10.1088/1751-8121/ab0046 [32] J. 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Gharahi, $\ell$-Multilinear Ranks of Multipartite Quantum States via Tensor Flattening: A Mathematica Codebase, Zenodo (2025). https://doi.org/10.5281/zenodo.15299720. https://doi.org/10.5281/zenodo.15299720 [37] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going Beyond Bell's Theorem, (in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos) (Springer, Dordrecht, 1989). pp. 69-72. https://doi.org/10.1007/978-94-017-0849-4_10. https://doi.org/10.1007/978-94-017-0849-4_10 [38] N. R. Wallach, An unentangled Gleason’s theorem, Contemp. Math. 305, 291 (2002). https://doi.org/10.1090/conm/305 [39] T. Cubitt, A. Montanaro, and A. Winter, On the dimension of subspaces with bounded Schmidt rank, J. Math. Phys. 49, 022107 (2008). https://doi.org/10.1063/1.2862998 [40] I. R. 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