Projective characterization of higher-order quantum transformations
Summarize this article with:
AbstractTransformations of transformations, also called higher-order transformations, is a natural concept in information processing, which has recently attracted significant interest in the study of quantum causal relations. In this work, a framework for characterizing higher-order quantum transformations which relies on the use of superoperator projectors is presented. More precisely, working with projectors in the Choi-Jamiołkowski picture is shown to provide a handy way of defining the characterization constraints on any class of higher-order transformations. The algebraic properties of these projectors are furthermore shown to obey rules similar to $\textit{multiplicative additive linear logic (MALL)}$, providing an intuitive way of comparing any two classes through their projectors. The main novelty of this work is the introduction to the algebra of the 'prec' connector. It is used for the characterization of maps that are no signaling from input to output or the other way around. This allows to assess the possible signaling structure of any transformation characterized within the projective framework. The properties of the prec are moreover shown to yield a normal form for projective expressions. This hints towards a general way to compare different classes of higher-order transformations.► BibTeX data@article{Hoffreumon2026projective, doi = {10.22331/q-2026-01-21-1978}, url = {https://doi.org/10.22331/q-2026-01-21-1978}, title = {Projective characterization of higher-order quantum transformations}, author = {Hoffreumon, Timoth{\'{e}}e and Oreshkov, Ognyan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1978}, month = jan, year = {2026} }► References [1] Karl Kraus. ``States, effects, and operations: Fundamental notions of quantum theory''. Volume 190 of Lecture notes in physics. Springer-Verlag Berlin Heidelberg. (1983). 1 edition. https://doi.org/10.1007/3-540-12732-1 [2] G. Chiribella, G. M. D’Ariano, and P. Perinotti. ``Transforming quantum operations: Quantum supermaps''. EPL (Europhysics Letters) 83, 30004 (2008). arXiv:0804.0180. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [3] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Physical Review A 80 (2009). arXiv:0904.4483. https://doi.org/10.1103/PhysRevA.80.022339 arXiv:0904.4483 [4] Paolo Perinotti. ``Causal structures and the classification of higher order quantum computations''. Tutorials, Schools, and Workshops in the Mathematical SciencesPage 103–127 (2017). arXiv:1612.05099. https://doi.org/10.1007/978-3-319-68655-4_7 arXiv:1612.05099 [5] Alessandro Bisio and Paolo Perinotti. ``Theoretical framework for higher-order quantum theory''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, 20180706 (2019). arXiv:1806.09554. https://doi.org/10.1098/rspa.2018.0706 arXiv:1806.09554 [6] Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Benoit Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013). arXiv:0912.0195. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [7] E. B. Davies and J. T. Lewis. ``An operational approach to quantum probability''. Communications in Mathematical Physics 17, 239–260 (1970). https://doi.org/10.1007/BF01647093 [8] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nature Communications 3, 1–13 (2012). arXiv:1105.4464. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464 [9] Aleks Kissinger and Sander Uijlen. ``A categorical semantics for causal structure''. Logical Methods in Computer Science Volume 15, Issue 3 (2019). arXiv:1701.04732. https://doi.org/10.23638/LMCS-15(3:15)2019 arXiv:1701.04732 [10] Mateus Araújo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and Časlav Brukner. ``Witnessing causal nonseparability''. New Journal of Physics 17, 102001 (2015). arXiv:1506.03776. https://doi.org/10.1088/1367-2630/17/10/102001 arXiv:1506.03776 [11] Timothée Hoffreumon and Ognyan Oreshkov. ``The multi-round process matrix''. Quantum 5, 384 (2021). arXiv:2005.04204. https://doi.org/10.22331/q-2021-01-20-384 arXiv:2005.04204 [12] Simon Milz, Jessica Bavaresco, and Giulio Chiribella. ``Resource theory of causal connection''. Quantum 6, 788 (2022). arXiv:2110.03233. https://doi.org/10.22331/q-2022-08-25-788 arXiv:2110.03233 [13] Simon Milz and Marco Túlio Quintino. ``Characterising transformations between quantum objects, of quantum properties, and transformations without a fixed causal order''. Quantum 8, 1415 (2024). arXiv:2305.01247. https://doi.org/10.22331/q-2024-07-17-1415 arXiv:2305.01247 [14] Jean-Yves Girard. ``Linear logic''.
Theoretical Computer Science 50, 1–101 (1987). https://doi.org/10.1016/0304-3975(87)90045-4 [15] Andrej Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Reports on Mathematical Physics 3, 275–278 (1972). https://doi.org/10.1016/0034-4877(72)90011-0 [16] Man-Duen Choi. ``Positive linear maps on complex matrices''. Linear Algebra and its Applications 10, 285–290 (1975). https://doi.org/10.1016/0024-3795(75)90075-0 [17] David Beckman, Daniel Gottesman, Michael A. Nielsen, and John Preskill. ``Causal and localizable quantum operations''. Physical Review A 64 (2001). arXiv:quant-ph/0102043. https://doi.org/10.1103/physreva.64.052309 arXiv:quant-ph/0102043 [18] Marco Piani, Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Properties of quantum nonsignaling boxes''. Physical Review A 74 (2006). arXiv:quant-ph/0505110. https://doi.org/10.1103/physreva.74.012305 arXiv:quant-ph/0505110 [19] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''.
Cambridge University Press. (2009). 2 edition. https://doi.org/10.1017/cbo9780511976667 [20] Giulio Chiribella, Giacomo M. D'Ariano, and Paolo Perinotti. ``Memory effects in quantum channel discrimination''.
Physical Review Lett. 101, 180501 (2008). arXiv:0803.3237. https://doi.org/10.1103/PhysRevLett.101.180501 arXiv:0803.3237 [21] Sally Shrapnel, Fabio Costa, and Gerard Milburn. ``Updating the Born rule''. New Journal of Physics 20 (2018). arXiv:1702.01845. https://doi.org/10.1088/1367-2630/aabe12 arXiv:1702.01845 [22] Man-Duen Choi and Edward G. Effros. ``Injectivity and operator spaces''. Journal of Functional Analysis 24, 156 – 209 (1977). https://doi.org/10.1016/0022-1236(77)90052-0 [23] Fumio Hiai and Dénes Petz. ``Introduction to matrix analysis and applications''. Universitext. Springer, Cham. (2014). https://doi.org/10.1007/978-3-319-04150-6 [24] Esteban Castro-Ruiz, Flaminia Giacomini, and Časlav Brukner. ``Dynamics of Quantum Causal Structures''. Physical Review X 8, 1–18 (2018). arXiv:1710.03139. https://doi.org/10.1103/PhysRevX.8.011047 arXiv:1710.03139 [25] Esteban Castro-Ruiz (2019). private communication. [26] Will Simmons and Aleks Kissinger. ``Higher-order causal theories are models of bv-logic''.
In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 80:1–80:14. Dagstuhl, Germany (2022). Schloss Dagstuhl – Leibniz-Zentrum für Informatik. arXiv:2205.11219. https://doi.org/10.4230/LIPIcs.MFCS.2022.80 arXiv:2205.11219 [27] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Informational derivation of quantum theory''. Physical Review A 84 (2011). arXiv:1011.6451. https://doi.org/10.1103/physreva.84.012311 arXiv:1011.6451 [28] Ognyan Oreshkov and Christina Giarmatzi. ``Causal and causally separable processes''. New Journal of Physics 18, 093020 (2016). arXiv:1506.05449. https://doi.org/10.1088/1367-2630/18/9/093020 arXiv:1506.05449 [29] Julian Wechs, Alastair A Abbott, and Cyril Branciard. ``On the definition and characterisation of multipartite causal (non)separability''. New Journal of Physics 21, 013027 (2019). arXiv:1807.10557. https://doi.org/10.1088/1367-2630/aaf352 arXiv:1807.10557 [30] Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019). arXiv:1801.07594. https://doi.org/10.22331/q-2019-12-02-206 arXiv:1801.07594 [31] Julian Wechs, Cyril Branciard, and Ognyan Oreshkov. ``Existence of processes violating causal inequalities on time-delocalised subsystems''. Nature Communications 14, 1471 (2023). arXiv:2201.11832. https://doi.org/10.1038/s41467-023-36893-3 arXiv:2201.11832 [32] Steven Roman. ``Advanced linear algebra''. Springer. (2008). 3 edition. https://doi.org/10.1007/978-0-387-72831-5 [33] Robert Piziak, Patrick L. Odell, and R. Hahn. ``Constructing projections on sums and intersections''. Computers & Mathematics with Applications 37, 67–74 (1999). https://doi.org/10.1016/S0898-1221(98)00242-9 [34] Tomoyuki Morimae. ``The process matrix framework for a single-party system'' (2014). arXiv:1408.1464. arXiv:1408.1464Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-21 10:10:10: Could not fetch cited-by data for 10.22331/q-2026-01-21-1978 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-21 10:10:10: 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. AbstractTransformations of transformations, also called higher-order transformations, is a natural concept in information processing, which has recently attracted significant interest in the study of quantum causal relations. In this work, a framework for characterizing higher-order quantum transformations which relies on the use of superoperator projectors is presented. More precisely, working with projectors in the Choi-Jamiołkowski picture is shown to provide a handy way of defining the characterization constraints on any class of higher-order transformations. The algebraic properties of these projectors are furthermore shown to obey rules similar to $\textit{multiplicative additive linear logic (MALL)}$, providing an intuitive way of comparing any two classes through their projectors. The main novelty of this work is the introduction to the algebra of the 'prec' connector. It is used for the characterization of maps that are no signaling from input to output or the other way around. This allows to assess the possible signaling structure of any transformation characterized within the projective framework. The properties of the prec are moreover shown to yield a normal form for projective expressions. This hints towards a general way to compare different classes of higher-order transformations.► BibTeX data@article{Hoffreumon2026projective, doi = {10.22331/q-2026-01-21-1978}, url = {https://doi.org/10.22331/q-2026-01-21-1978}, title = {Projective characterization of higher-order quantum transformations}, author = {Hoffreumon, Timoth{\'{e}}e and Oreshkov, Ognyan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1978}, month = jan, year = {2026} }► References [1] Karl Kraus. ``States, effects, and operations: Fundamental notions of quantum theory''. Volume 190 of Lecture notes in physics. Springer-Verlag Berlin Heidelberg. (1983). 1 edition. https://doi.org/10.1007/3-540-12732-1 [2] G. Chiribella, G. M. D’Ariano, and P. Perinotti. ``Transforming quantum operations: Quantum supermaps''. EPL (Europhysics Letters) 83, 30004 (2008). arXiv:0804.0180. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [3] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Physical Review A 80 (2009). arXiv:0904.4483. https://doi.org/10.1103/PhysRevA.80.022339 arXiv:0904.4483 [4] Paolo Perinotti. ``Causal structures and the classification of higher order quantum computations''. Tutorials, Schools, and Workshops in the Mathematical SciencesPage 103–127 (2017). arXiv:1612.05099. https://doi.org/10.1007/978-3-319-68655-4_7 arXiv:1612.05099 [5] Alessandro Bisio and Paolo Perinotti. ``Theoretical framework for higher-order quantum theory''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, 20180706 (2019). arXiv:1806.09554. https://doi.org/10.1098/rspa.2018.0706 arXiv:1806.09554 [6] Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Benoit Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013). arXiv:0912.0195. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [7] E. B. Davies and J. T. Lewis. ``An operational approach to quantum probability''. Communications in Mathematical Physics 17, 239–260 (1970). https://doi.org/10.1007/BF01647093 [8] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nature Communications 3, 1–13 (2012). arXiv:1105.4464. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464 [9] Aleks Kissinger and Sander Uijlen. ``A categorical semantics for causal structure''. Logical Methods in Computer Science Volume 15, Issue 3 (2019). arXiv:1701.04732. https://doi.org/10.23638/LMCS-15(3:15)2019 arXiv:1701.04732 [10] Mateus Araújo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and Časlav Brukner. ``Witnessing causal nonseparability''. New Journal of Physics 17, 102001 (2015). arXiv:1506.03776. https://doi.org/10.1088/1367-2630/17/10/102001 arXiv:1506.03776 [11] Timothée Hoffreumon and Ognyan Oreshkov. ``The multi-round process matrix''. Quantum 5, 384 (2021). arXiv:2005.04204. https://doi.org/10.22331/q-2021-01-20-384 arXiv:2005.04204 [12] Simon Milz, Jessica Bavaresco, and Giulio Chiribella. ``Resource theory of causal connection''. Quantum 6, 788 (2022). arXiv:2110.03233. https://doi.org/10.22331/q-2022-08-25-788 arXiv:2110.03233 [13] Simon Milz and Marco Túlio Quintino. ``Characterising transformations between quantum objects, of quantum properties, and transformations without a fixed causal order''. Quantum 8, 1415 (2024). arXiv:2305.01247. https://doi.org/10.22331/q-2024-07-17-1415 arXiv:2305.01247 [14] Jean-Yves Girard. ``Linear logic''.
Theoretical Computer Science 50, 1–101 (1987). https://doi.org/10.1016/0304-3975(87)90045-4 [15] Andrej Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Reports on Mathematical Physics 3, 275–278 (1972). https://doi.org/10.1016/0034-4877(72)90011-0 [16] Man-Duen Choi. ``Positive linear maps on complex matrices''. Linear Algebra and its Applications 10, 285–290 (1975). https://doi.org/10.1016/0024-3795(75)90075-0 [17] David Beckman, Daniel Gottesman, Michael A. Nielsen, and John Preskill. ``Causal and localizable quantum operations''. Physical Review A 64 (2001). arXiv:quant-ph/0102043. https://doi.org/10.1103/physreva.64.052309 arXiv:quant-ph/0102043 [18] Marco Piani, Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Properties of quantum nonsignaling boxes''. Physical Review A 74 (2006). arXiv:quant-ph/0505110. https://doi.org/10.1103/physreva.74.012305 arXiv:quant-ph/0505110 [19] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''.
Cambridge University Press. (2009). 2 edition. https://doi.org/10.1017/cbo9780511976667 [20] Giulio Chiribella, Giacomo M. D'Ariano, and Paolo Perinotti. ``Memory effects in quantum channel discrimination''.
Physical Review Lett. 101, 180501 (2008). arXiv:0803.3237. https://doi.org/10.1103/PhysRevLett.101.180501 arXiv:0803.3237 [21] Sally Shrapnel, Fabio Costa, and Gerard Milburn. ``Updating the Born rule''. New Journal of Physics 20 (2018). arXiv:1702.01845. https://doi.org/10.1088/1367-2630/aabe12 arXiv:1702.01845 [22] Man-Duen Choi and Edward G. Effros. ``Injectivity and operator spaces''. Journal of Functional Analysis 24, 156 – 209 (1977). https://doi.org/10.1016/0022-1236(77)90052-0 [23] Fumio Hiai and Dénes Petz. ``Introduction to matrix analysis and applications''. Universitext. Springer, Cham. (2014). https://doi.org/10.1007/978-3-319-04150-6 [24] Esteban Castro-Ruiz, Flaminia Giacomini, and Časlav Brukner. ``Dynamics of Quantum Causal Structures''. Physical Review X 8, 1–18 (2018). arXiv:1710.03139. https://doi.org/10.1103/PhysRevX.8.011047 arXiv:1710.03139 [25] Esteban Castro-Ruiz (2019). private communication. [26] Will Simmons and Aleks Kissinger. ``Higher-order causal theories are models of bv-logic''.
In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 80:1–80:14. Dagstuhl, Germany (2022). Schloss Dagstuhl – Leibniz-Zentrum für Informatik. arXiv:2205.11219. https://doi.org/10.4230/LIPIcs.MFCS.2022.80 arXiv:2205.11219 [27] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Informational derivation of quantum theory''. Physical Review A 84 (2011). arXiv:1011.6451. https://doi.org/10.1103/physreva.84.012311 arXiv:1011.6451 [28] Ognyan Oreshkov and Christina Giarmatzi. ``Causal and causally separable processes''. New Journal of Physics 18, 093020 (2016). arXiv:1506.05449. https://doi.org/10.1088/1367-2630/18/9/093020 arXiv:1506.05449 [29] Julian Wechs, Alastair A Abbott, and Cyril Branciard. ``On the definition and characterisation of multipartite causal (non)separability''. New Journal of Physics 21, 013027 (2019). arXiv:1807.10557. https://doi.org/10.1088/1367-2630/aaf352 arXiv:1807.10557 [30] Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019). arXiv:1801.07594. https://doi.org/10.22331/q-2019-12-02-206 arXiv:1801.07594 [31] Julian Wechs, Cyril Branciard, and Ognyan Oreshkov. ``Existence of processes violating causal inequalities on time-delocalised subsystems''. Nature Communications 14, 1471 (2023). arXiv:2201.11832. https://doi.org/10.1038/s41467-023-36893-3 arXiv:2201.11832 [32] Steven Roman. ``Advanced linear algebra''. Springer. (2008). 3 edition. https://doi.org/10.1007/978-0-387-72831-5 [33] Robert Piziak, Patrick L. Odell, and R. Hahn. ``Constructing projections on sums and intersections''. Computers & Mathematics with Applications 37, 67–74 (1999). https://doi.org/10.1016/S0898-1221(98)00242-9 [34] Tomoyuki Morimae. ``The process matrix framework for a single-party system'' (2014). arXiv:1408.1464. arXiv:1408.1464Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-21 10:10:10: Could not fetch cited-by data for 10.22331/q-2026-01-21-1978 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-21 10:10:10: 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.