Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order

Simon Milz; Marco Túlio Quintino

doi:10.22331/q-2024-07-17-1415

Characterising transformations between quantum objects, ‘completeness’ of quantum properties, and transformations without a fixed causal order

Simon Milz^1,2,3,4 and Marco Túlio Quintino⁵

¹School of Physics, Trinity College Dublin, Dublin 2, Ireland
²Trinity Quantum Alliance, Unit 16, Trinity Technology and Enterprise Centre, Pearse Street, Dublin 2, D02YN67, Ireland
³Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
⁴Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
⁵Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

Published:	2024-07-17, volume 8, page 1415
Eprint:	arXiv:2305.01247v2
Doi:	https://doi.org/10.22331/q-2024-07-17-1415
Citation:	Quantum 8, 1415 (2024).

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Abstract

Many fundamental and key objects in quantum mechanics are linear mappings between particular affine/linear spaces. This structure includes basic quantum elements such as states, measurements, channels, instruments, non-signalling channels and channels with memory, and also higher-order operations such as superchannels, quantum combs, n-time processes, testers, and process matrices which may not respect a definite causal order. Deducing and characterising their structural properties in terms of linear and semidefinite constraints is not only of foundational relevance, but plays an important role in enabling the numerical optimisation over sets of quantum objects and allowing simpler connections between different concepts and objects. Here, we provide a general framework to deduce these properties in a direct and easy to use way. While primarily guided by practical quantum mechanical considerations, we also extend our analysis to mappings between $general$ linear/affine spaces and derive their properties, opening the possibility for analysing sets which are not explicitly forbidden by quantum theory, but are still not much explored. Together, these results yield versatile and readily applicable tools for all tasks that require the characterisation of linear transformations, in quantum mechanics and beyond. As an application of our methods, we discuss how the existence of indefinite causality naturally emerges in higher-order quantum transformations and provide a simple strategy for the characterisation of mappings that have to preserve properties in a 'complete' sense, i.e., when acting non-trivially only on parts of an input space.

Featured image: A pictorial illustration of an exemplary chain of higher-order operations. Channels describe transformations between states, superchannels describe transformation between channels, and supersuperchannels describe transformations between superchannels. The methods we provide allow for the characterization of all mappings in this chain, but more generally all mappings between arbitrary quantum objects.

Popular summary

In mathematics and computer science, a higher-order function is a function which may take a function as an input, and output another function. This idea conveys the concept of “transformations of transformations”, a concept which has also been proven to be fruitful for studying quantum theory and causality. In this case, quantum operations, such as quantum channels or unitary gates, are objects which may be subjected to transformations.
The crucial role played by transformations in quantum mechanics is already visible when considering its basic building blocks. To name but two pertinent examples, a quantum evolution — described by quantum channels — transforms quantum states at an earlier time to quantum states at a later time, while quantum measurements transform quantum states to outcome probabilities. Taking this role played by mappings between different sets of quantum objects as a starting point, one does not have to stop at quantum states, channels and measurements, but can, for example consider seemingly more exotic transformations, like those that map quantum channels onto quantum channels, or, say, pairs of quantum channels to quantum states.
Indeed, many of the resulting, so-called higher order transformations, like, e.g., quantum combs or quantum testers, turn out to be compatible with standard quantum mechanics and have found ample application in quantum information theory. Others, like, e.g., process matrices or the quantum time flip cannot necessarily be implemented within quantum mechanics when assuming a fixed causal order, but allow one to investigate the potential consequences of causal indefiniteness in quantum mechanics. Studying the hierarchy of transformations between sets of quantum objects thus promises a whole host of interesting higher-order transformations.
The present work rests agnostic of their respective physicality and offers a systematic way to derive the pertinent properties of transformations between arbitrary sets of quantum objects, thus providing the basic toolbox for this study.

► BibTeX data

@article{Milz2024characterising,
  doi = {10.22331/q-2024-07-17-1415},
  url = {https://doi.org/10.22331/q-2024-07-17-1415},
  title = {Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order},
  author = {Milz, Simon and Quintino, Marco T{\'{u}}lio},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1415},
  month = jul,
  year = {2024}
}

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[1] Michael Antesberger, Marco Túlio Quintino, Philip Walther, and Lee A. Rozema, "Higher-Order Process Matrix Tomography of a Passively-Stable Quantum Switch", PRX Quantum 5 1, 010325 (2024).

[2] Jessica Bavaresco, Patryk Lipka-Bartosik, Pavel Sekatski, and Mohammad Mehboudi, "Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations", Physical Review Research 6 2, 023305 (2024).

[3] Guilherme Zambon and Diogo O. Soares-Pinto, "Relations between Markovian and non-Markovian correlations in multitime quantum processes", Physical Review A 109 6, 062401 (2024).

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