Complete extension: the non-signaling analog of quantum purification

Marek Winczewski1,2, Tamoghna Das1,3, John H. Selby1, Karol Horodecki4,1, Paweł Horodecki1,5,6, Łukasz Pankowski7, Marco Piani8,9, and Ravishankar Ramanathan10

1International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
2Institute of Theoretical Physics and Astrophysics and National Quantum Information Centre in Gdańsk, University of Gdańsk, 80–952 Gdańsk, Poland
3Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
4Institute of Informatics and National Quantum Information Centre in Gdańsk, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80–952 Gdańsk, Poland
5Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80–233 Gdańsk, Poland
6National Quantum Information Centre, University of Gdańsk, ul. Jana Bażyńskiego 8, 80-309 Gdańsk, Poland
7VOICELAB.AI, Al. Grunwaldzka 135A; 80-264 Gdańsk, Poland,
8evolutionQ Inc., Waterloo, Ontario, N2L 3L3, Canada
9SUPA and Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK
10Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong

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Abstract

Deriving quantum mechanics from information-theoretic postulates is a recent research direction taken, in part, with the view of finding a beyond-quantum theory; once the postulates are clear, we can consider modifications to them. A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.

Quantum mechanics can be derived from a few information-theoretic postulates. However, if there exists a theory of Nature with more explanatory power than quantum mechanics, it must not satisfy one of these. We study the consequences of replacing a particular postulate with a less restrictive one. Namely, we replace the purification postulate (PP) with a less restrictive complete extension postulate (CEP). This new postulate requires, for all states, the existence of extensions from which any other extension can be generated, i.e., complete extensions. We do so by studying the properties of theories satisfying CEP in the framework of the so-called generalized probabilistic theories (GPTs). First, we show that PP can not hold in any discrete convex theory. Second, we show that the mentioned replacement does not trivialize important cryptographic tasks (i.e., bit commitment). Second, we show that CEP, contrary to PP, may not exclude (post-quantum) theories that hyper-decohere to quantum mechanics. As a case study, we construct non-signaling complete extensions in the theory of non-signaling behaviors. We also show how the structure of the famous Popescu-Rohrlich behavior (PR box) arises as a non-signaling complete extension of the maximally mixed behavior. Finally, we proved that classical probability theory, quantum theory, and theory of non-signaling behaviors satisfy CEP. Therefore, we say that complete extensions are non-signaling analogs of quantum-mechanical purifications. In this way, our findings might establish a step towards finding a more fundamental theory of Nature.

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Cited by

[1] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classical theories with entanglement", Physical Review A 101 4, 042118 (2020).

[2] Marek Winczewski, Tamoghna Das, and Karol Horodecki, "Limitations on device independent key secure against non signaling adversary via the squashed non-locality", arXiv:1903.12154, (2019).

[3] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classicality without local discriminability: Decoupling entanglement and complementarity", Physical Review A 102 5, 052216 (2020).

[4] Manik Banik, Sutapa Saha, Tamal Guha, Sristy Agrawal, Some Sankar Bhattacharya, Arup Roy, and A. S. Majumdar, "Constraining the state space in any physical theory with the principle of information symmetry", Physical Review A 100 6, 060101 (2019).

[5] Marek Winczewski, "Fundamental Limitations within the Selected Cryptographic Scenarios and Supra-Quantum Theories", arXiv:2311.08211, (2023).

[6] Marek Winczewski, Tamoghna Das, and Karol Horodecki, "Limitations on a device-independent key secure against a nonsignaling adversary via squashed nonlocality", Physical Review A 106 5, 052612 (2022).

[7] Manik Banik, Sutapa Saha, Tamal Guha, Sristy Agrawal, Some Sankar Bhattacharya, Arup Roy, and A. S. Majumdar, "The principle of information symmetry constrains the state-space in any physical theory", arXiv:1905.09413, (2019).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-26 03:30:01). The list may be incomplete as not all publishers provide suitable and complete citation data.

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