Lie algebraic invariants in quantum linear optics
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AbstractQuantum linear optics without post-selection is not powerful enough to produce any quantum state from a given input state. This limits its utility since some applications require entangled resources that are difficult to prepare. Thus, a deeper understanding of linear optical state preparation is needed. In this work, we give a recipe to derive conserved quantities in the evolution of arbitrary states along any possible passive linear interferometer. One example of such an invariant is the spectrum of a density matrix mapped onto the Lie algebra of passive linear optical Hamiltonians. These invariants give necessary conditions for exact state preparation: if the input and output states have different invariants, it is impossible to design a passive linear interferometer that evolves one into the other. Moreover, we provide a lower bound to the distance between an output and target state based on the distance between their invariants. This gives a necessary condition for approximate or heralded state preparations. Therefore, the invariants allow us to narrow the search when trying to prepare useful entangled states, like NOON states, from easy-to-prepare states, like Fock states. We conclude that future exact and approximate state preparation methods will need to consider the necessary conditions given by our invariants to weed out impossible linear optical evolutions.Featured image: For a given state $\rho$, there is a subset of states in Fock space that we can reach with linear optics (LO) and it is contained in the set of states sharing the same invariant.Popular summaryPhysics studies change. But change would be nothing without constancy. In fact, quantities that remain constant under a physical process can reveal a lot about the underlying physics. In this work, we study conserved quantities in circuits of light. These circuits, called photonic circuits, consist of a network of crystals and mirrors where beams of light split, bounce and mix. When the light is extremely faint, the quantum properties of light start to emerge. Instead of beams, we find photons, indivisible packets of light, bouncing through the circuit. However, quantum light has some weird properties. Photons can be in a superposition and they can interfere in ways classical light cannot. This quantum interference results in conservation laws that photons inside a circuit must respect. In our work, we found some of these conserved quantities using the mathematics of photonic circuits. These invariants help explain why generating the entangled states needed for quantum computing is so hard with photonic circuits. But they also give us a clearer understanding of what photonic circuits could do, opening the door for new quantum technologies based on these circuits of light.► BibTeX data@article{VParellada2026liealgebraic, doi = {10.22331/q-2026-06-12-2132}, url = {https://doi.org/10.22331/q-2026-06-12-2132}, title = {Lie algebraic invariants in quantum linear optics}, author = {V. Parellada, Pablo and Gimeno i Garcia, Vicent and Moyano-Fern{\'{a}}ndez, Julio Jos{\'{e}} and Garcia-Escartin, Juan Carlos}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2132}, month = jun, year = {2026} }► References [1] C. K. Hong, Z. Y. Ou, and L. Mandel, ``Measurement of subpicosecond time intervals between two photons by interference'' Physical Review Letters 59, 2044-2046 (1987). https://doi.org/10.1103/PhysRevLett.59.2044 [2] Marco Barbieri ``Optical Quantum Metrology'' PRX Quantum 3, 010202 (2022). https://doi.org/10.1103/PRXQuantum.3.010202 [3] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, ``Long-distance quantum communication with atomic ensembles and linear optics'' Nature 414, 413–418 (2001). https://doi.org/10.1038/35106500 [4] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn, ``Linear optical quantum computing with photonic qubits'' Reviews of Modern Physics 79, 135–174 (2007). https://doi.org/10.1103/RevModPhys.79.135 [5] N. J. Cerf, C. Adami, and P. G. 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Knill ``Quantum gates using linear optics and postselection'' Physical Review A 66, 052306 (2002). https://doi.org/10.1103/PhysRevA.66.052306 [58] Stasja Stanisic, Noah Linden, Ashley Montanaro, and Peter S. Turner, ``Generating entanglement with linear optics'' Physical Review A 96, 043861 (2017). https://doi.org/10.1103/PhysRevA.96.043861 [59] Michael Ragone, Paolo Braccia, Quynh T. Nguyen, Louis Schatzki, Patrick J. Coles, Frederic Sauvage, Martin Larocca, and M. Cerezo, ``Representation Theory for Geometric Quantum Machine Learning'' (2023) arXiv:2210.07980 [quant-ph, stat]. https://doi.org/10.48550/arXiv.2210.07980Cited byCould not fetch Crossref cited-by data during last attempt 2026-06-12 10:40:52: Could not fetch cited-by data for 10.22331/q-2026-06-12-2132 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-06-12 10:40:53: Cannot retrieve data from ADS due to rate limitations.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. AbstractQuantum linear optics without post-selection is not powerful enough to produce any quantum state from a given input state. This limits its utility since some applications require entangled resources that are difficult to prepare. Thus, a deeper understanding of linear optical state preparation is needed. In this work, we give a recipe to derive conserved quantities in the evolution of arbitrary states along any possible passive linear interferometer. One example of such an invariant is the spectrum of a density matrix mapped onto the Lie algebra of passive linear optical Hamiltonians. These invariants give necessary conditions for exact state preparation: if the input and output states have different invariants, it is impossible to design a passive linear interferometer that evolves one into the other. Moreover, we provide a lower bound to the distance between an output and target state based on the distance between their invariants. This gives a necessary condition for approximate or heralded state preparations. Therefore, the invariants allow us to narrow the search when trying to prepare useful entangled states, like NOON states, from easy-to-prepare states, like Fock states. We conclude that future exact and approximate state preparation methods will need to consider the necessary conditions given by our invariants to weed out impossible linear optical evolutions.Featured image: For a given state $\rho$, there is a subset of states in Fock space that we can reach with linear optics (LO) and it is contained in the set of states sharing the same invariant.Popular summaryPhysics studies change. But change would be nothing without constancy. In fact, quantities that remain constant under a physical process can reveal a lot about the underlying physics. In this work, we study conserved quantities in circuits of light. These circuits, called photonic circuits, consist of a network of crystals and mirrors where beams of light split, bounce and mix. When the light is extremely faint, the quantum properties of light start to emerge. Instead of beams, we find photons, indivisible packets of light, bouncing through the circuit. However, quantum light has some weird properties. Photons can be in a superposition and they can interfere in ways classical light cannot. This quantum interference results in conservation laws that photons inside a circuit must respect. In our work, we found some of these conserved quantities using the mathematics of photonic circuits. These invariants help explain why generating the entangled states needed for quantum computing is so hard with photonic circuits. But they also give us a clearer understanding of what photonic circuits could do, opening the door for new quantum technologies based on these circuits of light.► BibTeX data@article{VParellada2026liealgebraic, doi = {10.22331/q-2026-06-12-2132}, url = {https://doi.org/10.22331/q-2026-06-12-2132}, title = {Lie algebraic invariants in quantum linear optics}, author = {V. Parellada, Pablo and Gimeno i Garcia, Vicent and Moyano-Fern{\'{a}}ndez, Julio Jos{\'{e}} and Garcia-Escartin, Juan Carlos}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2132}, month = jun, year = {2026} }► References [1] C. K. Hong, Z. Y. Ou, and L. Mandel, ``Measurement of subpicosecond time intervals between two photons by interference'' Physical Review Letters 59, 2044-2046 (1987). https://doi.org/10.1103/PhysRevLett.59.2044 [2] Marco Barbieri ``Optical Quantum Metrology'' PRX Quantum 3, 010202 (2022). https://doi.org/10.1103/PRXQuantum.3.010202 [3] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, ``Long-distance quantum communication with atomic ensembles and linear optics'' Nature 414, 413–418 (2001). https://doi.org/10.1038/35106500 [4] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn, ``Linear optical quantum computing with photonic qubits'' Reviews of Modern Physics 79, 135–174 (2007). https://doi.org/10.1103/RevModPhys.79.135 [5] N. J. Cerf, C. Adami, and P. G. Kwiat, ``Optical simulation of quantum logic'' Physical Review A 57, R1477–R1480 (1998). https://doi.org/10.1103/PhysRevA.57.R1477 [6] Julio José Moyano-Fernándezand Juan Carlos Garcia-Escartin ``Linear optics only allows every possible quantum operation for one photon or one port'' Optics Communications 382, 237–240 (2017). https://doi.org/10.1016/j.optcom.2016.07.085 [7] E. Knill, R. Laflamme, and G. J. Milburn, ``A scheme for efficient quantum computation with linear optics'' Nature 409, 46–52 (2001). https://doi.org/10.1038/35051009 [8] Shunya Konno, Warit Asavanant, Fumiya Hanamura, Hironari Nagayoshi, Kosuke Fukui, Atsushi Sakaguchi, Ryuhoh Ide, Fumihiro China, Masahiro Yabuno, Shigehito Miki, Hirotaka Terai, Kan Takase, Mamoru Endo, Petr Marek, Radim Filip, Peter Van Loock, and Akira Furusawa, ``Logical states for fault-tolerant quantum computation with propagating light'' Science 383, 289–293 (2024). https://doi.org/10.1126/science.adk7560 [9] Scott Aaronsonand Alex Arkhipov ``The Computational Complexity of Linear Optics'' Theory of Computing 9, 143–252 (2013). https://doi.org/10.1145/1993636.1993682 [10] Daniel J. Brod, Ernesto F. Galvão, Andrea Crespi, Roberto Osellame, Nicolò Spagnolo, and Fabio Sciarrino, ``Photonic implementation of boson sampling: a review'' Advanced Photonics 1, 034001 (2019). https://doi.org/10.1117/1.AP.1.3.034001 [11] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien, ``A variational eigenvalue solver on a photonic quantum processor'' Nature communications 5, 1–7 (2014). https://doi.org/10.1038/ncomms5213 [12] Ali W Elshaari, Wolfram Pernice, Kartik Srinivasan, Oliver Benson, and Val Zwiller, ``Hybrid integrated quantum photonic circuits'' Nature Photonics 14, 285–298 (2020). https://doi.org/10.1038/s41566-020-0609-x [13] Wim Bogaerts, Daniel Pérez, José Capmany, David A. B. Miller, Joyce Poon, Dirk Englund, Francesco Morichetti, and Andrea Melloni, ``Programmable photonic circuits'' Nature 586, 207–216 (2020). https://doi.org/10.1038/s41586-020-2764-0 [14] Caterina Taballione, Reinier van der Meer, Henk J Snijders, Peter Hooijschuur, Jörn P Epping, Michiel de Goede, Ben Kassenberg, Pim Venderbosch, Chris Toebes, Hans van den Vlekkert, Pepijn W H Pinkse, and Jelmer J Renema, ``A universal fully reconfigurable 12-mode quantum photonic processor'' Materials for Quantum Technology 1, 035002 (2021). https://doi.org/10.1088/2633-4356/ac168c [15] Francesco Hoch, Simone Piacentini, Taira Giordani, Zhen-Nan Tian, Mariagrazia Iuliano, Chiara Esposito, Anita Camillini, Gonzalo Carvacho, Francesco Ceccarelli, Nicoló Spagnolo, Andrea Crespi, Fabio Sciarrino, and Roberto Osellame, ``Reconfigurable continuously-coupled 3D photonic circuit for Boson Sampling experiments'' npj Quantum Information 8, 1–7 (2022). https://doi.org/10.1038/s41534-022-00568-6 [16] Caterina Taballione, Malaquias Correa Anguita, Michiel de Goede, Pim Venderbosch, Ben Kassenberg, Henk Snijders, Narasimhan Kannan, Ward L. Vleeshouwers, Devin Smith, Jörn P. Epping, Reinier van der Meer, Pepijn W. H. Pinkse, Hans van den Vlekkert, and Jelmer J. Renema, ``20-Mode Universal Quantum Photonic Processor'' Quantum 7, 1071 (2023). https://doi.org/10.22331/q-2023-08-01-1071 [17] Piotr Migdał, Javier Rodríguez-Laguna, Michał Oszmaniec, and Maciej Lewenstein, ``Multiphoton states related via linear optics'' Phys. Rev. A 89, 062329 (2014). https://doi.org/10.1103/PhysRevA.89.062329 [18] Y. S. Li, B. Zeng, X. S. Liu, and G. L. Long, ``Entanglement in a two-identical-particle system'' Phys. Rev. A 64, 054302 (2001). https://doi.org/10.1103/PhysRevA.64.054302 [19] Pablo V. 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