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Quantum Supermaps are Characterized by Locality

Quantum Science and Technology (arXiv overlay)

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Researchers Wilson, Chiribella, and Kissinger redefine quantum supermaps using a locality-based axiom, simplifying their characterization to sequential and parallel composition principles. The team introduces "locally-applicable transformations" in monoidal categories, offering a category-theoretic framework that generalizes supermaps to broader probabilistic theories beyond standard quantum mechanics. A key breakthrough shows these transformations correspond one-to-one with deterministic quantum supermaps, validated through intuitive diagrammatic proofs rather than complex algebraic methods. The framework extends to advanced structures like quantum switches and constrained channel spaces, proving robustness in scenarios with indefinite causal order or signaling limitations. This work bridges abstract category theory with practical quantum information, enabling clearer analysis of higher-order quantum processes and their operational constraints.

Quantum Supermaps are Characterized by Locality

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AbstractWe provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational probabilistic theories. We do so by providing a simple definition of $\textit{locally-applicable transformation}$ on a monoidal category. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to work for more general multiple-input supermaps such as the quantum switch and on arbitrary normal convex spaces of quantum channels such as those defined by satisfaction of signaling constraints.► BibTeX data@article{Wilson2026quantumsupermapsare, doi = {10.22331/q-2026-03-09-2013}, url = {https://doi.org/10.22331/q-2026-03-09-2013}, title = {Quantum {S}upermaps are {C}haracterized by {L}ocality}, author = {Wilson, Matt and Chiribella, Giulio and Kissinger, Aleks}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2013}, month = mar, year = {2026} }► References [1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83 no. 3, (7, 2008) 30004. https://doi.org/10.1209/0295-5075/83/30004 [2] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum circuit architecture,'' Physical Review Letters 101 no. 6, (8, 2008) 060401. https://doi.org/10.1103/PhysRevLett.101.060401 [3] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Physical Review A - Atomic, Molecular, and Optical Physics 88 no. 2, (8, 2013) 022318. https://doi.org/10.1103/PhysRevA.88.022318 [4] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Physical Review A - Atomic, Molecular, and Optical Physics 80 no. 2, (4, 2009). https://doi.org/10.1103/PhysRevA.80.022339 [5] G. Chiribella, A. Toigo, and V. Umanità, ``Normal completely positive maps on the space of quantum operations,'' Open Systems and Information Dynamics 20 no. 1, (12, 2010). https://doi.org/10.1142/S1230161213500030 [6] A. Bisio and P. Perinotti, ``Theoretical framework for higher-order quantum theory,'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 no. 2225, (5, 2019) 20180706. https://doi.org/10.1098/rspa.2018.0706 [7] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science 15 no. 3, (2019). https://doi.org/10.23638/LMCS-15(3:15)2019 [8] M. Wilson and G. Chiribella, ``A Mathematical Framework for Transformations of Physical Processes,'' arXiv:2204.04319 [quant-ph]. arXiv:2204.04319 [9] P. Perinotti, Causal Structures and the Classification of Higher Order Quantum Computations, pp. 103–127.

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AbstractWe provide a new characterisation of quantum supermaps in terms of an axiom that refers only to sequential and parallel composition. Consequently, we generalize quantum supermaps to arbitrary monoidal categories and operational probabilistic theories. We do so by providing a simple definition of $\textit{locally-applicable transformation}$ on a monoidal category. The definition can be rephrased in the language of category theory using the principle of naturality, and can be given an intuitive diagrammatic representation in terms of which all proofs are presented. In our main technical contribution, we use this diagrammatic representation to show that locally-applicable transformations on quantum channels are in one-to-one correspondence with deterministic quantum supermaps. This alternative characterization of quantum supermaps is proven to work for more general multiple-input supermaps such as the quantum switch and on arbitrary normal convex spaces of quantum channels such as those defined by satisfaction of signaling constraints.► BibTeX data@article{Wilson2026quantumsupermapsare, doi = {10.22331/q-2026-03-09-2013}, url = {https://doi.org/10.22331/q-2026-03-09-2013}, title = {Quantum {S}upermaps are {C}haracterized by {L}ocality}, author = {Wilson, Matt and Chiribella, Giulio and Kissinger, Aleks}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2013}, month = mar, year = {2026} }► References [1] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83 no. 3, (7, 2008) 30004. https://doi.org/10.1209/0295-5075/83/30004 [2] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum circuit architecture,'' Physical Review Letters 101 no. 6, (8, 2008) 060401. https://doi.org/10.1103/PhysRevLett.101.060401 [3] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Physical Review A - Atomic, Molecular, and Optical Physics 88 no. 2, (8, 2013) 022318. https://doi.org/10.1103/PhysRevA.88.022318 [4] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Physical Review A - Atomic, Molecular, and Optical Physics 80 no. 2, (4, 2009). https://doi.org/10.1103/PhysRevA.80.022339 [5] G. Chiribella, A. Toigo, and V. Umanità, ``Normal completely positive maps on the space of quantum operations,'' Open Systems and Information Dynamics 20 no. 1, (12, 2010). https://doi.org/10.1142/S1230161213500030 [6] A. Bisio and P. Perinotti, ``Theoretical framework for higher-order quantum theory,'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 no. 2225, (5, 2019) 20180706. https://doi.org/10.1098/rspa.2018.0706 [7] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science 15 no. 3, (2019). https://doi.org/10.23638/LMCS-15(3:15)2019 [8] M. Wilson and G. Chiribella, ``A Mathematical Framework for Transformations of Physical Processes,'' arXiv:2204.04319 [quant-ph]. arXiv:2204.04319 [9] P. Perinotti, Causal Structures and the Classification of Higher Order Quantum Computations, pp. 103–127.

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