Universal Properties in Quantum Theory

Huot, Mathieu; Staton, Sam

doi:10.4204/eptcs.287.12

Cited by 12 publications

(19 citation statements)

References 23 publications

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“…These are more syntactic than the present work, which can be an advantage, but an advantage of the present work is that it is arguably more canonical through its categorical nature. We highlight in particular the recent extention of ZX with hiding [7], which indeed makes an explicit connection to the universal properties considered in [21].…”

Section: Quantum Programming Languagesmentioning

confidence: 86%

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Quantum channels as a categorical completion

Huot

Staton

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories.First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries.Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologicallyenriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.• The multiplication of numbers corresponds to spatial juxtaposition of systems. For example, given two bijections on a bit, f, g : 2 → 2, we have a bijection arXiv:1904.09600v2 [cs.LO]

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Section: Quantum Programming Languagesmentioning

confidence: 86%

“…In [21] we presented a similar paradigm for the restricted version of quantum channels between matrix algebras. We proved that those quantum channels are the affine completion of the category of isometries, both seen as monoidal categories.…”

Section: Admit Hidingmentioning

confidence: 99%

Quantum channels as a categorical completion

Huot

Staton

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…Proof. Since Isometry is a trivial restriction category, CPTP L(Isometry) Aux(Isometry) by [10,Corollary 7]. Also, CPTP is already well-pointed, so Ext(CPTP) CPTP.…”

Section: Quantum Channels and Classical Functions As Completionsmentioning

confidence: 99%

“…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%

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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Universal Properties in Quantum Theory

Cited by 12 publications

References 23 publications

Quantum channels as a categorical completion

Quantum channels as a categorical completion

Bennett and Stinespring, Together at Last

Quantum Information Effects

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