Quantum channels as a categorical completion

Huot, Mathieu; Staton, Sam

doi:10.1109/lics.2019.8785700

Cited by 10 publications

(7 citation statements)

References 39 publications

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“…Hermiticity is, however, a simpler property, allowing direct proof of completeness using normal forms. The restatement of the proof in terms of purification is left for future works generalising the notion of purification for Hermiticity-preserving operators, paving the way toward a categorical characterisation of HP via a universal property as done for CP in [29]. Such uniform characterisation would allow us to extend completeness to other graphical languages like ZX-and ZH-calculi more naturally than by direct translation of the present axioms, and could even prove useful for similar issues in quantum circuits, which are not compact close.…”

Section: Discussionmentioning

confidence: 99%

Complete Graphical Language for Hermiticity-Preserving Superoperators

Carette¹,

Timothée²,

Larroque³

et al. 2023

Preprint

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Universal and complete graphical languages have been successfully designed for pure state quantum mechanics, corresponding to linear maps between Hilbert spaces, and mixed states quantum mechanics, corresponding to completely positive superoperators. In this paper, we go one step further and present a universal and complete graphical language for Hermiticity-preserving superoperators. Such a language opens the possibility of diagrammatic compositional investigations of antilinear transformations featured in various physical situations, such as the Choi-Jamio lkowski isomorphism, spin-flip, or entanglement witnesses. Our construction relies on an extension of the ZW-calculus exhibiting a normal form for Hermitian matrices.for any two states |ψ and |φ . In 1931, Wigner proved that the symmetries of quantum theory should be either unitary or antiunitary (see the appendix to Chapter 20 of [48]), meaning that the general form of T in Eq. ( 1) iswhere U is a linear operator respecting U † U = I = U U † , and |ψ is the complex conjugation of |ψ . This abstract point of view turned out to be extremely useful for all aspects of quantum theory. On the one hand, the study of symmetries has been a very fruitful method for simplifying problems encountered at all scales, from solid-state physics to high-energy physics, with notable uses in atomic physics and quantum information sciences. On the other hand, symmetries have been a guiding principle for the construction of theories such as the standard model and, by extension, Yang-Mills theory (see e.g. [45,46]).So far, physicists and quantum computer scientists have been mainly concerned with unitary transformations, the ubiquitous ingredient in the study of dynamics, and little attention has been devoted to antiunitary transformations. There is a clear reason why: unitary transformations represent what can be realised and tested in a (closed) lab. In contrast, one year after his theorem, Wigner showed that antiunitaries are typically involved in the case of a time reversal [47] (see, for example, §11.4.2 of [41] for a source in English), something experimentalists cannot achieve (at least within the current theoretical framework of physics). Still, some quantum computer scientists have started investigating the possible computational advantages processes involving time reversal could provide. We mention in particular the complexity-theoretic study of [1] and the categorical semantics of [39]. However, little has been done on the computational power of antiunitaries in general, although some results suggest possible advantages in quantum information [8,22,36], and that they may be simulable [5,19,40]. Such investigations are actually made difficult by the lack of a clearly defined computational framework because it would require going beyond the usual mathematics underlying quantum circuits. Rigorously put, antiuniatries are associated with positive but not completely positive mappings. Hence, they are outside the framework of quantum computing with open systems, and are assume...

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Section: Discussionmentioning

confidence: 99%

Complete Graphical Language for Hermiticity-Preserving Superoperators

Carette¹,

Timothée²,

Larroque³

et al. 2023

Preprint

View full text Add to dashboard Cite

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“…Despite the similarity of their statements, these categorical completions are surprisingly dissimilar. The universal construction of completely positive trace-preserving map from isometries and unitaries is due to Huot and Staton [10,11]. A different categorical approach to Stinespring's dilation theorem as a universal construction is given by Westerbaan and Westerbaan [18].…”

Section: Reversible Dynamics On Open Systemsmentioning

confidence: 99%

“…We can now show that Pfn and CPTP arise as completions of the inverse categories PInj and Unitary. The quantum case relies on Huot and Staton's characterisation of Isometry as a completion of Unitary [11] making initial the unit of the direct sum. We consider PInj and Unitary as inverse rig categories, using the Inp-construction to make the unit of the direct sum initial, and then the Aux-construction to make the tensor unit terminal.…”

Section: Cofree Reversible Foundationsmentioning

confidence: 99%

Bennett and Stinespring, Together at Last

Heunen,

Kaarsgaard

2021

Preprint

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We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the well-pointed restriction affine completion of a monoidal restriction category. This categorical completion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennett's method. Moreover, in these two cases, we show how our construction can be 'undone' by a further universal construction. This shows how both mixed quantum theory and classical computation rest on entirely reversible foundations.

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“…This work provides the missing origin story, by showing that measurement-as-an-effect arises through a sequence of arrow constructions that can be applied (and given precise meaning) to any rig groupoid. Thus our categorical constructions eliminate the need for involved functional-analytic semantics using operator algebras [8,32,30], and is much more general than earlier work specific to Hilbert spaces [16,17] and restriction categories [13].…”

Section: Introductionmentioning

confidence: 97%

Quantum Information Effects

Heunen,

Kaarsgaard

2021

Preprint

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We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

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Quantum channels as a categorical completion

Cited by 10 publications

References 39 publications

Complete Graphical Language for Hermiticity-Preserving Superoperators

Complete Graphical Language for Hermiticity-Preserving Superoperators

Bennett and Stinespring, Together at Last

Quantum Information Effects

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