Quantum convolutional channels and multiparameter families of 2-unitary matrices
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AbstractMany alternative approaches to construct quantum channels with large entangling capacity were proposed in the past decade, resulting in multiple isolated gates. In this work, we put forward a novel one, inspired by convolution, which provides greater freedom of nonlocal parameters. Although quantum counterparts of convolution have been shown not to exist for pure states, several attempts with various degrees of rigorousness have been proposed for mixed states. In this work, we follow the approach based on coherifications of multi-stochastic operations and demonstrate a surprising connection to gates with high entangling power. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. Furthermore, we establish new, continuous classes of bipartite 2-unitary matrices of dimension $d^2$ for $d = 7$ and $d = 9$, with $2$ and $4$ free nonlocal parameters beyond simple phasing of matrix elements, corresponding to perfect tensors of rank $4$ or 4-partite absolutely maximally entangled states.Featured image: Classical convolution can be viewed as an operation that takes two probability distributions as input and produces a new distribution as output. We introduce quantum convolutional channels as coherifications of tristochastic tensors, which can be realised, via the Stinespring representation, as partial traces of particular unitary channels. In particular, the admissible unitary channels include new continuous families of 2‑unitary matrices with maximal entangling power. A visual representation of an exemplary family for local dimension $7$ is presented in the upper-right corner.Popular summaryQuantum entanglement is one of the central resources of quantum information science, and understanding how to generate it efficiently and universally is important for both fundamental physics and practical applications. To address this problem, we introduce a new method for constructing quantum channels inspired by the idea of convolution. Although a direct quantum analogue of convolution does not exist for pure states, we show that a related construction can be implemented for mixed-state dynamics through parametrised unitary evolution followed by partial tracing. This approach provides a flexible framework for building quantum operations with a rich nonlocal structure. A key result of our study is the identification of a link between these convolutional channels and quantum gates with maximal entangling power, known as 2-unitary gates. These gates are also connected to highly entangled objects such as perfect tensors and absolutely maximally entangled states, which are relevant to quantum error correction, quantum communication, and holographic models. We derive conditions under which our convolutional construction yields maximally entangling channels, and we use it to obtain new continuous families of bipartite 2-unitary matrices in dimensions $7^2$ and $9^2$, with genuinely free nonlocal parameters. These results provide new examples of highly entangling quantum operations and suggest a broader pathway for their systematic discovery.► BibTeX data@article{Bistron2026quantum, doi = {10.22331/q-2026-04-16-2072}, url = {https://doi.org/10.22331/q-2026-04-16-2072}, title = {Quantum convolutional channels and multiparameter families of 2-unitary matrices}, author = {Bistro{\'{n}}, Rafa{\l{}} and Czartowski, Jakub and {\.{Z}}yczkowski, Karol}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2072}, month = apr, year = {2026} }► References [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] P. Zanardi, C. Zalka, and L. Faoro. ``Entangling power of quantum evolutions''. Phys. Rev. A 62, 030301 (2000). https://doi.org/10.1103/PhysRevA.62.030301 [3] L. Clarisse, S. Ghosh, S. Severini, and A. Sudbery. ``Entangling power of permutations''. Phys. Rev. A 72, 012314 (2005). https://doi.org/10.1103/PhysRevA.72.012314 [4] A. A. Rico. ``Absolutely maximally entangled states in small system sizes''. Master Thesis, University of Innsbruck (2020). [5] G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan, and K. Życzkowski. ``Absolutely maximally entangled pure states of multipartite quantum systems''. Rep. Prog. Phys. (2026, in press). https://doi.org/10.48550/arXiv.2508.04777 [6] S. Aravinda, S. A. Rather, and A. Lakshminarayan. ``From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy''. Phys. Rev. Res. 3, 043034 (2021). https://doi.org/10.1103/PhysRevResearch.3.043034 [7] I. S. Reed and G. Solomon. ``Polynomial codes over certain finite fields''. Journal of the Society for Industrial and Applied Mathematics 8, 300–304 (1960). https://doi.org/10.1137/0108018 [8] A. J. Scott. ``Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions''. Phys. Rev. A 69, 052330 (2004). https://doi.org/10.1103/PhysRevA.69.052330 [9] J. I. Latorre and G. Sierra. ``Holographic codes'' (2015). [10] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill. ``Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence''. J.
High Energy Phys. 6, 1029–8479 (2015). https://doi.org/10.1007/JHEP06(2015)149 [11] W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo. ``Absolute maximal entanglement and quantum secret sharing''. Phys. Rev. A 86, 052335 (2012). https://doi.org/10.1103/PhysRevA.86.052335 [12] B. Bertini, P. Kos, and T. Prosen. ``Exact correlation functions for dual-unitary lattice models in $1+1$ dimensions''. Phys. Rev. Lett. 123, 210601 (2019). https://doi.org/10.1103/PhysRevLett.123.210601 [13] J. Harper. ``Perfect tensor hyperthreads''. J.
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Could not fetch ADS cited-by data during last attempt 2026-04-16 08:09:38: 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. AbstractMany alternative approaches to construct quantum channels with large entangling capacity were proposed in the past decade, resulting in multiple isolated gates. In this work, we put forward a novel one, inspired by convolution, which provides greater freedom of nonlocal parameters. Although quantum counterparts of convolution have been shown not to exist for pure states, several attempts with various degrees of rigorousness have been proposed for mixed states. In this work, we follow the approach based on coherifications of multi-stochastic operations and demonstrate a surprising connection to gates with high entangling power. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. Furthermore, we establish new, continuous classes of bipartite 2-unitary matrices of dimension $d^2$ for $d = 7$ and $d = 9$, with $2$ and $4$ free nonlocal parameters beyond simple phasing of matrix elements, corresponding to perfect tensors of rank $4$ or 4-partite absolutely maximally entangled states.Featured image: Classical convolution can be viewed as an operation that takes two probability distributions as input and produces a new distribution as output. We introduce quantum convolutional channels as coherifications of tristochastic tensors, which can be realised, via the Stinespring representation, as partial traces of particular unitary channels. In particular, the admissible unitary channels include new continuous families of 2‑unitary matrices with maximal entangling power. A visual representation of an exemplary family for local dimension $7$ is presented in the upper-right corner.Popular summaryQuantum entanglement is one of the central resources of quantum information science, and understanding how to generate it efficiently and universally is important for both fundamental physics and practical applications. To address this problem, we introduce a new method for constructing quantum channels inspired by the idea of convolution. Although a direct quantum analogue of convolution does not exist for pure states, we show that a related construction can be implemented for mixed-state dynamics through parametrised unitary evolution followed by partial tracing. This approach provides a flexible framework for building quantum operations with a rich nonlocal structure. A key result of our study is the identification of a link between these convolutional channels and quantum gates with maximal entangling power, known as 2-unitary gates. These gates are also connected to highly entangled objects such as perfect tensors and absolutely maximally entangled states, which are relevant to quantum error correction, quantum communication, and holographic models. We derive conditions under which our convolutional construction yields maximally entangling channels, and we use it to obtain new continuous families of bipartite 2-unitary matrices in dimensions $7^2$ and $9^2$, with genuinely free nonlocal parameters. These results provide new examples of highly entangling quantum operations and suggest a broader pathway for their systematic discovery.► BibTeX data@article{Bistron2026quantum, doi = {10.22331/q-2026-04-16-2072}, url = {https://doi.org/10.22331/q-2026-04-16-2072}, title = {Quantum convolutional channels and multiparameter families of 2-unitary matrices}, author = {Bistro{\'{n}}, Rafa{\l{}} and Czartowski, Jakub and {\.{Z}}yczkowski, Karol}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2072}, month = apr, year = {2026} }► References [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009). https://doi.org/10.1103/RevModPhys.81.865 [2] P. Zanardi, C. Zalka, and L. Faoro. ``Entangling power of quantum evolutions''. Phys. Rev. A 62, 030301 (2000). https://doi.org/10.1103/PhysRevA.62.030301 [3] L. Clarisse, S. Ghosh, S. Severini, and A. Sudbery. ``Entangling power of permutations''. Phys. Rev. A 72, 012314 (2005). https://doi.org/10.1103/PhysRevA.72.012314 [4] A. A. Rico. ``Absolutely maximally entangled states in small system sizes''. Master Thesis, University of Innsbruck (2020). [5] G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan, and K. Życzkowski. ``Absolutely maximally entangled pure states of multipartite quantum systems''. Rep. Prog. Phys. (2026, in press). https://doi.org/10.48550/arXiv.2508.04777 [6] S. Aravinda, S. A. Rather, and A. Lakshminarayan. ``From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy''. Phys. Rev. Res. 3, 043034 (2021). https://doi.org/10.1103/PhysRevResearch.3.043034 [7] I. S. Reed and G. Solomon. ``Polynomial codes over certain finite fields''. Journal of the Society for Industrial and Applied Mathematics 8, 300–304 (1960). https://doi.org/10.1137/0108018 [8] A. J. Scott. ``Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions''. Phys. Rev. A 69, 052330 (2004). https://doi.org/10.1103/PhysRevA.69.052330 [9] J. I. Latorre and G. Sierra. ``Holographic codes'' (2015). [10] F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill. ``Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence''. J.
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