A Novel Stabilizer-based Entanglement Distillation Protocol for Qudits
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AbstractEntanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension $d$, we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.Popular summaryEntanglement is a remarkable feature of quantum mechanics, allowing particles to share correlations beyond classical physics. These correlations underpin quantum communication, computing, and sensing, but in practice entanglement is fragile: noise quickly degrades it. To counter this, we use entanglement distillation, which converts several weak or noisy states into fewer, less noisy, and thereby more usable ones. In this study, we introduce a new distillation method for qudits—quantum systems with two or more levels. Using stabilizer codes, a framework from quantum error correction, we establish a systematic way to describe how stabilizer-based protocols act on imperfect states. Building on this foundation, we propose a two-copy protocol that applies to all states in prime dimensions and achieves the maximum possible improvement in each round for noise-affected Bell states. Our simulations show that the method outperforms established protocols, especially for high levels of noise. By clarifying the link between input states, stabilizers, and outcomes, we advance efficient distillation strategies and strengthen the foundation for robust quantum networks and scalable quantum technology.► BibTeX data@article{Popp2025novelstabilizer, doi = {10.22331/q-2025-12-15-1945}, url = {https://doi.org/10.22331/q-2025-12-15-1945}, title = {A {N}ovel {S}tabilizer-based {E}ntanglement {D}istillation {P}rotocol for {Q}udits}, author = {Popp, Christopher and Sutter, Tobias C. and Hiesmayr, Beatrix C.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1945}, month = dec, year = {2025} }► References [1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels'' Physical Review Letters 70, 1895-1899 (1993) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.70.1895 [2] Charles H. Bennettand Stephen J. Wiesner ``Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states'' Physical Review Letters 69, 2881–2884 (1992) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.69.2881 [3] R. F. Werner ``All teleportation and dense coding schemes'' Journal of Physics A: Mathematical and General 34, 7081 (2001). https://doi.org/10.1088/0305-4470/34/35/332 [4] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, ``Measurement-based quantum computation'' Nature Physics 5, 19–26 (2009) Publisher: Nature Publishing Group. https://doi.org/10.1038/nphys1157 https://www.nature.com/articles/nphys1157 [5] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki, ``Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?'' Physical Review Letters 80, 5239–5242 (1998). https://doi.org/10.1103/PhysRevLett.80.5239 [6] Beatrix C. Hiesmayr, Christopher Popp, and Tobias C. Sutter, ``Bipartite bound entanglement'' International Journal of Quantum Information 23, 2530003 (2025) Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0219749925300037 [7] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters, ``Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels'' Physical Review Letters 76, 722–725 (1996). https://doi.org/10.1103/PhysRevLett.76.722 [8] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher, ``Concentrating partial entanglement by local operations'' Physical Review A 53, 2046–2052 (1996). https://doi.org/10.1103/PhysRevA.53.2046 [9] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, and Anna Sanpera, ``Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels'' Physical Review Letters 77, 2818–2821 (1996) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.77.2818 [10] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, ``Measurement-based logical qubit entanglement purification'' Physical Review A 105, 062418 (2022) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.105.062418 [11] Lan Zhou, Wei Zhong, and Yu-Bo Sheng, ``Purification of the residual entanglement'' Optics Express 28, 2291–2301 (2020) Publisher: Optica Publishing Group. https://doi.org/10.1364/OE.383499 https://opg.optica.org/oe/abstract.cfm?uri=oe-28-2-2291 [12] Gernot Alber, Aldo Delgado, Nicolas Gisin, and Igor Jex, ``Efficient bipartite quantum state purification in arbitrary dimensional Hilbert spaces'' Journal of Physics A: Mathematical and General 34, 8821 (2001). https://doi.org/10.1088/0305-4470/34/42/307 [13] Michał Horodeckiand Paweł Horodecki ``Reduction criterion of separability and limits for a class of distillation protocols'' Physical Review A 59, 4206–4216 (1999) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.59.4206 [14] Karl Gerd H. Vollbrechtand Michael M. Wolf ``Efficient distillation beyond qubits'' Physical Review A 67, 012303 (2003). https://doi.org/10.1103/PhysRevA.67.012303 [15] Jeroen Dehaene, Maarten Van Den Nest, Bart De Moor, and Frank Verstraete, ``Local permutations of products of Bell states and entanglement distillation'' Physical Review A 67, 022310 (2003). https://doi.org/10.1103/PhysRevA.67.022310 [16] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' (1997) arXiv:quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 http://arxiv.org/abs/quant-ph/9705052 [17] A. Ashikhminand E. Knill ``Nonbinary quantum stabilizer codes'' IEEE Transactions on Information Theory 47, 3065–3072 (2001). https://doi.org/10.1109/18.959288 https://ieeexplore.ieee.org/document/959288 [18] Mark M. Wilde ``Quantum Coding with Entanglement'' (2008) arXiv:0806.4214 [quant-ph]. https://doi.org/10.48550/arXiv.0806.4214 http://arxiv.org/abs/0806.4214 [19] W. Dürand H.-J. Briegel ``Entanglement Purification for Quantum Computation'' Physical Review Letters 90, 067901 (2003). https://doi.org/10.1103/PhysRevLett.90.067901 [20] W. Dürand H. J. Briegel ``Entanglement purification and quantum error correction'' Reports on Progress in Physics 70, 1381 (2007). https://doi.org/10.1088/0034-4885/70/8/R03 [21] Ryutaroh Matsumoto ``Conversion of a general quantum stabilizer code to an entanglement distillation protocol'' Journal of Physics A: Mathematical and General 36, 8113–8127 (2003) arXiv:quant-ph/0209091. https://doi.org/10.1088/0305-4470/36/29/316 http://arxiv.org/abs/quant-ph/0209091 [22] Ryutaroh Matsumoto ``Breeding protocols are advantageous for finite-length entanglement distillation'' (2024) arXiv:2401.02265 [quant-ph]. https://doi.org/10.48550/arXiv.2401.02265 http://arxiv.org/abs/2401.02265 [23] Shun Watanabe, Ryutaroh Matsumoto, and Tomohiko Uyematsu, ``Improvement of stabilizer-based entanglement distillation protocols by encoding operators'' Journal of Physics A: Mathematical and General 39, 4273 (2006). https://doi.org/10.1088/0305-4470/39/16/013 [24] J. Miguel-Ramiroand W. Dür ``Efficient entanglement purification protocols for d -level systems'' Physical Review A 98, 042309 (2018). https://doi.org/10.1103/PhysRevA.98.042309 [25] E. Knill ``Non-binary unitary error bases and quantum codes'' report (1996). https://doi.org/10.2172/373768 http://www.osti.gov/servlets/purl/373768-BfsVyz/webviewable/ [26] E.M. Rains ``Nonbinary quantum codes'' IEEE Transactions on Information Theory 45, 1827–1832 (1999) Conference Name: IEEE Transactions on Information Theory. https://doi.org/10.1109/18.782103 https://ieeexplore.ieee.org/document/782103 [27] Bernhard Baumgartner, Beatrix Hiesmayr, and Heide Narnhofer, ``A special simplex in the state space for entangled qudits'' Journal of Physics A: Mathematical and Theoretical 40, 7919–7938 (2007) arXiv:quant-ph/0610100. https://doi.org/10.1088/1751-8113/40/28/S03 http://arxiv.org/abs/quant-ph/0610100 [28] Christopher Poppand Beatrix C. Hiesmayr ``Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts'' Scientific Reports 13, 2037 (2023) Number: 1 Publisher: Nature Publishing Group. https://doi.org/10.1038/s41598-023-29211-w https://www.nature.com/articles/s41598-023-29211-w [29] Christopher Poppand Beatrix C. Hiesmayr ``Special features of the Weyl–Heisenberg Bell basis imply unusual entanglement structure of Bell-diagonal states'' New Journal of Physics 26, 013039 (2024) Publisher: IOP Publishing. https://doi.org/10.1088/1367-2630/ad1d0e [30] William Karush ``Minima of functions of several variables with inequalities as side conditions'' thesis (1939) OCLC: 43268508. https://catalog.lib.uchicago.edu/vufind/Record/4111654 [31] Harold W. Kuhnand Albert W. Tucker ``Nonlinear Programming'' Springer (2014). https://doi.org/10.1007/978-3-0348-0439-4_11 [32] Christopher Popp, Tobias C. Sutter, and Beatrix C. Hiesmayr, ``Low-fidelity entanglement distillation with FIMAX'' International Journal of Quantum Information 23, 2550017 (2025) Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0219749925500170 [33] Christopher Popp ``BellDiagonalQudits: A package for entanglement analyses of mixed maximally entangled qudits'' Journal of Open Source Software 8, 4924 (2023). https://doi.org/10.21105/joss.04924 [34] H. Bombinand M. A. Martin-Delgado ``Entanglement distillation protocols and number theory'' Physical Review A 72, 032313 (2005). https://doi.org/10.1103/PhysRevA.72.032313 [35] Erik Hostens, Jeroen Dehaene, and Bart De Moor, ``The equivalence of two approaches to the design of entanglement distillation protocols'' (2004) arXiv:quant-ph/0406017. https://doi.org/10.48550/arXiv.quant-ph/0406017 http://arxiv.org/abs/quant-ph/0406017 [36] Sumeet Khatriand Mark M. Wilde ``Principles of Quantum Communication Theory: A Modern Approach'' (2024) arXiv:2011.04672 [cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.2011.04672 http://arxiv.org/abs/2011.04672Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-15 09:48:46: Could not fetch cited-by data for 10.22331/q-2025-12-15-1945 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 09:48:53: 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. AbstractEntanglement distillation, the process of converting weakly entangled states into maximally entangled ones using Local Operations and Classical Communication (LOCC), is pivotal for robust entanglement-assisted quantum information processing in error-prone environments. A construction based on stabilizer codes offers an effective method for designing such protocols. By analytically investigating the effective action of stabilizer protocols for systems of prime dimension $d$, we establish a standard form for the output states of recurrent stabilizer-based distillation. This links the properties of input states, stabilizers, and encodings to the properties of the protocol. Based on those insights, we present a novel two-copy distillation protocol, applicable to all bipartite states in prime dimension, that maximizes the fidelity increase per iteration for Bell-diagonal states. The power of this framework and the protocol is demonstrated through numerical investigations, which provide evidence for superior performance in terms of efficiency and distillability of low-fidelity states compared to other well-established recurrence protocols. By elucidating the interplay between states, errors, and protocols, our contribution advances the systematic development of highly effective distillation protocols, enhancing our understanding of distillability.Popular summaryEntanglement is a remarkable feature of quantum mechanics, allowing particles to share correlations beyond classical physics. These correlations underpin quantum communication, computing, and sensing, but in practice entanglement is fragile: noise quickly degrades it. To counter this, we use entanglement distillation, which converts several weak or noisy states into fewer, less noisy, and thereby more usable ones. In this study, we introduce a new distillation method for qudits—quantum systems with two or more levels. Using stabilizer codes, a framework from quantum error correction, we establish a systematic way to describe how stabilizer-based protocols act on imperfect states. Building on this foundation, we propose a two-copy protocol that applies to all states in prime dimensions and achieves the maximum possible improvement in each round for noise-affected Bell states. Our simulations show that the method outperforms established protocols, especially for high levels of noise. By clarifying the link between input states, stabilizers, and outcomes, we advance efficient distillation strategies and strengthen the foundation for robust quantum networks and scalable quantum technology.► BibTeX data@article{Popp2025novelstabilizer, doi = {10.22331/q-2025-12-15-1945}, url = {https://doi.org/10.22331/q-2025-12-15-1945}, title = {A {N}ovel {S}tabilizer-based {E}ntanglement {D}istillation {P}rotocol for {Q}udits}, author = {Popp, Christopher and Sutter, Tobias C. and Hiesmayr, Beatrix C.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1945}, month = dec, year = {2025} }► References [1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels'' Physical Review Letters 70, 1895-1899 (1993) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.70.1895 [2] Charles H. Bennettand Stephen J. Wiesner ``Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states'' Physical Review Letters 69, 2881–2884 (1992) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.69.2881 [3] R. F. Werner ``All teleportation and dense coding schemes'' Journal of Physics A: Mathematical and General 34, 7081 (2001). https://doi.org/10.1088/0305-4470/34/35/332 [4] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, ``Measurement-based quantum computation'' Nature Physics 5, 19–26 (2009) Publisher: Nature Publishing Group. https://doi.org/10.1038/nphys1157 https://www.nature.com/articles/nphys1157 [5] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki, ``Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?'' Physical Review Letters 80, 5239–5242 (1998). https://doi.org/10.1103/PhysRevLett.80.5239 [6] Beatrix C. Hiesmayr, Christopher Popp, and Tobias C. Sutter, ``Bipartite bound entanglement'' International Journal of Quantum Information 23, 2530003 (2025) Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0219749925300037 [7] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters, ``Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels'' Physical Review Letters 76, 722–725 (1996). https://doi.org/10.1103/PhysRevLett.76.722 [8] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, and Benjamin Schumacher, ``Concentrating partial entanglement by local operations'' Physical Review A 53, 2046–2052 (1996). https://doi.org/10.1103/PhysRevA.53.2046 [9] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, and Anna Sanpera, ``Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels'' Physical Review Letters 77, 2818–2821 (1996) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevLett.77.2818 [10] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, ``Measurement-based logical qubit entanglement purification'' Physical Review A 105, 062418 (2022) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.105.062418 [11] Lan Zhou, Wei Zhong, and Yu-Bo Sheng, ``Purification of the residual entanglement'' Optics Express 28, 2291–2301 (2020) Publisher: Optica Publishing Group. https://doi.org/10.1364/OE.383499 https://opg.optica.org/oe/abstract.cfm?uri=oe-28-2-2291 [12] Gernot Alber, Aldo Delgado, Nicolas Gisin, and Igor Jex, ``Efficient bipartite quantum state purification in arbitrary dimensional Hilbert spaces'' Journal of Physics A: Mathematical and General 34, 8821 (2001). https://doi.org/10.1088/0305-4470/34/42/307 [13] Michał Horodeckiand Paweł Horodecki ``Reduction criterion of separability and limits for a class of distillation protocols'' Physical Review A 59, 4206–4216 (1999) Publisher: American Physical Society. https://doi.org/10.1103/PhysRevA.59.4206 [14] Karl Gerd H. Vollbrechtand Michael M. Wolf ``Efficient distillation beyond qubits'' Physical Review A 67, 012303 (2003). https://doi.org/10.1103/PhysRevA.67.012303 [15] Jeroen Dehaene, Maarten Van Den Nest, Bart De Moor, and Frank Verstraete, ``Local permutations of products of Bell states and entanglement distillation'' Physical Review A 67, 022310 (2003). https://doi.org/10.1103/PhysRevA.67.022310 [16] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' (1997) arXiv:quant-ph/9705052. https://doi.org/10.48550/arXiv.quant-ph/9705052 http://arxiv.org/abs/quant-ph/9705052 [17] A. Ashikhminand E. Knill ``Nonbinary quantum stabilizer codes'' IEEE Transactions on Information Theory 47, 3065–3072 (2001). https://doi.org/10.1109/18.959288 https://ieeexplore.ieee.org/document/959288 [18] Mark M. Wilde ``Quantum Coding with Entanglement'' (2008) arXiv:0806.4214 [quant-ph]. https://doi.org/10.48550/arXiv.0806.4214 http://arxiv.org/abs/0806.4214 [19] W. Dürand H.-J. Briegel ``Entanglement Purification for Quantum Computation'' Physical Review Letters 90, 067901 (2003). https://doi.org/10.1103/PhysRevLett.90.067901 [20] W. Dürand H. J. Briegel ``Entanglement purification and quantum error correction'' Reports on Progress in Physics 70, 1381 (2007). https://doi.org/10.1088/0034-4885/70/8/R03 [21] Ryutaroh Matsumoto ``Conversion of a general quantum stabilizer code to an entanglement distillation protocol'' Journal of Physics A: Mathematical and General 36, 8113–8127 (2003) arXiv:quant-ph/0209091. https://doi.org/10.1088/0305-4470/36/29/316 http://arxiv.org/abs/quant-ph/0209091 [22] Ryutaroh Matsumoto ``Breeding protocols are advantageous for finite-length entanglement distillation'' (2024) arXiv:2401.02265 [quant-ph]. https://doi.org/10.48550/arXiv.2401.02265 http://arxiv.org/abs/2401.02265 [23] Shun Watanabe, Ryutaroh Matsumoto, and Tomohiko Uyematsu, ``Improvement of stabilizer-based entanglement distillation protocols by encoding operators'' Journal of Physics A: Mathematical and General 39, 4273 (2006). https://doi.org/10.1088/0305-4470/39/16/013 [24] J. Miguel-Ramiroand W. Dür ``Efficient entanglement purification protocols for d -level systems'' Physical Review A 98, 042309 (2018). https://doi.org/10.1103/PhysRevA.98.042309 [25] E. Knill ``Non-binary unitary error bases and quantum codes'' report (1996). https://doi.org/10.2172/373768 http://www.osti.gov/servlets/purl/373768-BfsVyz/webviewable/ [26] E.M. Rains ``Nonbinary quantum codes'' IEEE Transactions on Information Theory 45, 1827–1832 (1999) Conference Name: IEEE Transactions on Information Theory. https://doi.org/10.1109/18.782103 https://ieeexplore.ieee.org/document/782103 [27] Bernhard Baumgartner, Beatrix Hiesmayr, and Heide Narnhofer, ``A special simplex in the state space for entangled qudits'' Journal of Physics A: Mathematical and Theoretical 40, 7919–7938 (2007) arXiv:quant-ph/0610100. https://doi.org/10.1088/1751-8113/40/28/S03 http://arxiv.org/abs/quant-ph/0610100 [28] Christopher Poppand Beatrix C. Hiesmayr ``Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts'' Scientific Reports 13, 2037 (2023) Number: 1 Publisher: Nature Publishing Group. https://doi.org/10.1038/s41598-023-29211-w https://www.nature.com/articles/s41598-023-29211-w [29] Christopher Poppand Beatrix C. Hiesmayr ``Special features of the Weyl–Heisenberg Bell basis imply unusual entanglement structure of Bell-diagonal states'' New Journal of Physics 26, 013039 (2024) Publisher: IOP Publishing. https://doi.org/10.1088/1367-2630/ad1d0e [30] William Karush ``Minima of functions of several variables with inequalities as side conditions'' thesis (1939) OCLC: 43268508. https://catalog.lib.uchicago.edu/vufind/Record/4111654 [31] Harold W. Kuhnand Albert W. Tucker ``Nonlinear Programming'' Springer (2014). https://doi.org/10.1007/978-3-0348-0439-4_11 [32] Christopher Popp, Tobias C. Sutter, and Beatrix C. Hiesmayr, ``Low-fidelity entanglement distillation with FIMAX'' International Journal of Quantum Information 23, 2550017 (2025) Publisher: World Scientific Publishing Co. https://doi.org/10.1142/S0219749925500170 [33] Christopher Popp ``BellDiagonalQudits: A package for entanglement analyses of mixed maximally entangled qudits'' Journal of Open Source Software 8, 4924 (2023). https://doi.org/10.21105/joss.04924 [34] H. Bombinand M. A. Martin-Delgado ``Entanglement distillation protocols and number theory'' Physical Review A 72, 032313 (2005). https://doi.org/10.1103/PhysRevA.72.032313 [35] Erik Hostens, Jeroen Dehaene, and Bart De Moor, ``The equivalence of two approaches to the design of entanglement distillation protocols'' (2004) arXiv:quant-ph/0406017. https://doi.org/10.48550/arXiv.quant-ph/0406017 http://arxiv.org/abs/quant-ph/0406017 [36] Sumeet Khatriand Mark M. Wilde ``Principles of Quantum Communication Theory: A Modern Approach'' (2024) arXiv:2011.04672 [cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph]. https://doi.org/10.48550/arXiv.2011.04672 http://arxiv.org/abs/2011.04672Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-15 09:48:46: Could not fetch cited-by data for 10.22331/q-2025-12-15-1945 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-15 09:48:53: 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.