Entangled Subspaces through Algebraic Geometry
Summarize this article with:
AbstractWe propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.► BibTeX data@article{Gharahi2025entangledsubspaces, doi = {10.22331/q-2025-12-15-1947}, url = {https://doi.org/10.22331/q-2025-12-15-1947}, title = {Entangled {S}ubspaces through {A}lgebraic {G}eometry}, author = {Gharahi, Masoud and Mancini, Stefano}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1947}, 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 byCould not fetch Crossref cited-by data during last attempt 2025-12-15 11:29:50: Could not fetch cited-by data for 10.22331/q-2025-12-15-1947 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 11:29:51: 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 propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.► BibTeX data@article{Gharahi2025entangledsubspaces, doi = {10.22331/q-2025-12-15-1947}, url = {https://doi.org/10.22331/q-2025-12-15-1947}, title = {Entangled {S}ubspaces through {A}lgebraic {G}eometry}, author = {Gharahi, Masoud and Mancini, Stefano}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1947}, 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 byCould not fetch Crossref cited-by data during last attempt 2025-12-15 11:29:50: Could not fetch cited-by data for 10.22331/q-2025-12-15-1947 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 11:29:51: 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.