A comonadic view of simulation and quantum resources

Abramsky, Samson; Barbosa, Rui Soares; Karvonen, Martti; Mansfield, Shane

doi:10.1109/lics.2019.8785677

Cited by 18 publications

(41 citation statements)

References 29 publications

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“…Over the past decade, the sheaf-theoretic framework has evolved from its seminal form [9] in a number of directions. The core framework itself has grown through the addition of extended operational semantics [10], All-vs-Nothing arguments [5], logical characterisation of no-signalling polytopes [6], contextual fraction [7], an extension to continuous variable models [15], and the development of a categorical structure on empirical models [4,8,48,49]. Deep connections have been discovered with cohomology [1,11,14,22,23], logic [3,13,51,52,55,58,59] and other frameworks, including database theory [2], Spekkens's framework for contextuality [70], effect algebras [69] and non-commutative geometry [33].…”

Section: Literature Reviewmentioning

confidence: 99%

The Sheaf-Theoretic Structure of Definite Causality

Gogioso,

Pinzani

2021

Preprint

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Section: Literature Reviewmentioning

confidence: 99%

The Sheaf-Theoretic Structure of Definite Causality

Gogioso,

Pinzani

2021

Preprint

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“…To overcome such issues, we work in the general approach to contextuality initiated in [31], and later extended in [32] to capture building some correlations from others using such probabilistic and adaptive means, resulting in a resource theory for contextuality.…”

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confidence: 99%

“…No catalysis for contextuality We begin by briefly reviewing the resource theory of contextuality as defined in [32]. To start, we formalize the idea of a measurement scenario S: we imagine a situation where there is a finite set X S of measurements available, each measurement x ∈ X S giving rise to outcomes in some finite set O S,x .…”

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confidence: 99%

“…The most general formulation of this idea would allow a measurement x in T to be simulated by a probabilistic and adaptive procedure, that first measures (depending on some classical randomness) something in S, then based on the outcome (and possibly further classical randomness) chooses what to measure next (if anything) and so on. This idea is formalized carefully in [32] in two stages, the first one adding adaptivity and the second adding randomness.…”

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confidence: 99%

“…Then one can define adaptive (but still deterministic) transformations S → T between scenarios as deterministic transformations MP(S) → T as in Figure 2. In [32] we show that the assignment S → MP(S) defines a comonad on the category of scenarios: this abstract language is not needed here, but can be thought of as guaranteeing that one has a well-behaved way of composing MP(S) → T with MP(T ) → U to obtain a map MP(S) → U. Intuitively, the composite is obtained by first interpreting each measurement in U as a measurement protocol over T , and each measurement in that measurement protocol as an measurement protocol over S, and then "flattening" the resulting measurement protocol of measurement protocols (i.e.…”

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confidence: 99%

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Neither Contextuality nor Nonlocality Admits Catalysts

Karvonen

2021

Preprint

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We show that the resource theory of contextuality does not admit catalysts. As a corollary, we observe that the same holds for non-locality. This adds a further example to the list of "anomalies of entanglement", showing that non-locality and entanglement behave differently as resources. We also show that catalysis remains impossible even if instead of classical randomness we allow some more powerful behaviors to be used freely in the free transformations of the resource theory. Introduction. Contextuality [1] and non-locality [2, 3] play a prominent role in a wide variety of applications of quantum mechanics, with non-locality being used for example in quantum key-distribution [4], certified randomness [5] and randomness expansion [6].Similarly, contextuality powers quantum computation in some computational models [7][8][9][10][11][12] and even increases expressive power in quantum machine learning [13]. Consequently, it is vital to understand how non-locality and contextuality behave as resources.In this Letter, we show that neither contextuality nor non-locality admit catalysts: that is, there are no correlations that can be used to enable an otherwise impossible conversion between correlations and still recovered afterwards. To state this slightly more precisely, let us write d, e, f . . . for various correlations (whether classical or not), d ⊗ e for having independent instances of d and e, and d e (read as "d simulates e") for the existence of a conversion d → e. Then our results state that, in suitably formalized resource theories of contextuality and non-locality, whenever d ⊗ e d ⊗ f , then e f already. This gives a strong indication that contextuality (and non-locality) are resources that get spent when you use them: there is no way of using a correlation d to achieve a task you could not do otherwise while keeping d intact. As entanglement theory famously allows for catalysts [14],this can be seen as yet another "anomaly of non-locality" [15] and thus further testament to the fact that non-locality and entanglement are different resources. [16] We do this by working in precisely defined resource theories of contextuality and nonlocality. These are not strictly speaking quantum resource theories [17], but resource theories in a more general sense [18, 19], as we allow resources such as PR-boxes [20] that are not quantum realizable. The kinds of conversions between correlations we have in mind capture the intuitive idea of using one system to simulate another one, and have been studied in earlier literature [21][22][23][24][25][26][27][28][29]. These roughly correspond to the local operations and shared randomness-paradigm ((LOSR)) or to wirings and prior-to-input classical communication

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Categorical composable cryptography

Broadbent

Karvonen

2022

Lecture Notes in Computer Science

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We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in cryptography. Our model is able to incorporate computational security, set-up assumptions and various attack models such as colluding or independently acting subsets of adversaries in a modular, flexible fashion. We conclude by using string diagrams to rederive the security of the one-time pad and no-go results concerning the limits of bipartite and tripartite cryptography, ruling out e.g., composable commitments and broadcasting.

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A comonadic view of simulation and quantum resources

Cited by 18 publications

References 29 publications

The Sheaf-Theoretic Structure of Definite Causality

The Sheaf-Theoretic Structure of Definite Causality

Neither Contextuality nor Nonlocality Admits Catalysts

Categorical composable cryptography

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