Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras

• URL: http://arxiv.org/abs/2207.05832v2
• Date: Wed, 15 Nov 2023 11:44:24 GMT
• Title: Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras
• Authors: Sam Staton (University of Oxford), Ned Summers (University of Oxford)
• Abstract summary: We show that quantum de Finetti construction has a universal property as a categorical limit. This allows us to pass canonically between categorical treatments of finite dimensional quantum theory and the infinite dimensional. We also show that the same categorical analysis also justifies a continuous de Finetti theorem for classical probability.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: De Finetti theorems tell us that if we expect the likelihood of outcomes to be independent of their order, then these sequences of outcomes could be equivalently generated by drawing an experiment at random from a distribution, and repeating it over and over. In particular, the quantum de Finetti theorem says that exchangeable sequences of quantum states are always represented by distributions over a single state produced over and over. The main result of this paper is that this quantum de Finetti construction has a universal property as a categorical limit. This allows us to pass canonically between categorical treatments of finite dimensional quantum theory and the infinite dimensional. The treatment here is through understanding properties of (co)limits with respect to the contravariant functor which takes a C*-algebra describing a physical system to its convex, compact space of states, and through discussion of the Radon probability monad. We also show that the same categorical analysis also justifies a continuous de Finetti theorem for classical probability.

Related papers

• Heading towards an Algebraic Heisenberg Cut [0.0]
  We show that early signs of macroscopic behaviour appear before infinity. This lays the grounds for justifying the inclusion in quantum physics of the ITP formalism.
  arXiv  Detail & Related papers  (2024-12-21T10:39:40Z)
• A unified approach to quantum de Finetti theorems and SoS rounding via geometric quantization [0.0]
  We study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem. We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function.
  arXiv  Detail & Related papers  (2024-11-06T17:09:28Z)
• A non-hermitean momentum operator for the particle in a box [49.1574468325115]
  We show how to construct the corresponding hermitean Hamiltonian for the infinite as well as concrete example. The resulting Hilbert space can be decomposed into a physical and unphysical subspace.
  arXiv  Detail & Related papers  (2024-03-20T12:51:58Z)
• Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics [0.0]
  We prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions. We discuss the relation of our results with the quantum comb formalism.
  arXiv  Detail & Related papers  (2024-02-02T08:40:03Z)
• Dilation theorem via Schr\"odingerisation, with applications to the quantum simulation of differential equations [29.171574903651283]
  Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. In this study, we demonstrate the viability of the recently devised Schr"odingerisation approach.
  arXiv  Detail & Related papers  (2023-09-28T08:55:43Z)
• Connecting classical finite exchangeability to quantum theory [45.76759085727843]
  Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. It is well known that both theorems do not hold for finitely exchangeable sequences.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Resolving game theoretical dilemmas with quantum states [0.0]
  We present a new framework for creating a quantum version of a classical game. Using Fine's theorem, we re-express both the player payoffs and their strategies in terms of a set of marginals. We then consider particular quantum states that can potentially resolve dilemmas inherent in classical games.
  arXiv  Detail & Related papers  (2023-04-07T11:58:58Z)
• Quantum dissipation and the virial theorem [22.1682776279474]
  We study the celebrated virial theorem for dissipative systems, both classical and quantum. The non-Markovian nature of the quantum noise leads to novel bath-induced terms in the virial theorem. We also consider the case of an electrical circuit with thermal noise and analyze the role of non-Markovian noise in the context of the virial theorem.
  arXiv  Detail & Related papers  (2023-02-23T13:28:11Z)
• Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality [69.62715388742298]
  We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs) We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT.
  arXiv  Detail & Related papers  (2022-03-08T14:47:39Z)
• Stochastic approximate state conversion for entanglement and general quantum resource theories [41.94295877935867]
  An important problem in any quantum resource theory is to determine how quantum states can be converted into each other. Very few results have been presented on the intermediate regime between probabilistic and approximate transformations. We show that these bounds imply an upper bound on the rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels.
  arXiv  Detail & Related papers  (2021-11-24T17:29:43Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.