A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category

J. Hamilton; C. J. Isham; J. Butterfield

Home/Archive/arXiv

arXiv · quant-ph/9911020

A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category

J. Hamilton·C. J. Isham·J. Butterfield

View original source ↗Read source document ↗Export metadata ↓

Abstract

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

Explore related subjects

quant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

J. Hamilton, C. J. Isham, J. Butterfield. 1999-11-05. A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category. https://arxiv.org/abs/quant-ph/9911020

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Entanglement-based quantum communication may prevent tracking the message sender

The possibility of attaining current position of the message sender without person's consent seriously compromises the secrecy of correspondence. Classical communication systems cannot guarantee the security of communication against unwanted location tracking, because the sender must broadcast a signal at the moment of sending the message. The source of the signal could be always located, at least in principle. We show that quantum communication enables sending the message with pre-distributed entangled quantum systems, local weak measurements, time synchronization between the sender and the receiver and classical communication by the message receiver alone, therefore, physically securing the sender's location.

quant-ph↗

arXiv2026

Fully Passive Quantum Conference Key Agreement

Quantum Conference Key Agreement (CKA) provides a secure method for multi-party communication. A recently developed interference-based prepare-and-measure quantum CKA possesses the advantages of measurement-device-independence, namely, being immune to side-channels from the detector side. Besides, it achieves good key rate performance, especially for high-loss channels, due to the use of single photon interference. Meanwhile, several fully passive QKD schemes have been proposed, which eliminate all side channels from the source modulation side. We extend the fully passive idea to an interference-based CKA, which has a high level of implementation security for many-user communication.

quant-ph↗

arXiv2026

Fast simulations of X-ray absorption spectroscopy for battery materials on a quantum computer

X-ray absorption spectroscopy (XAS) is a leading technique for understanding structural changes in advanced battery materials such as lithium-excess cathodes. However, extracting critical information like oxidation states from the experimental spectra requires expensive and time-consuming simulations. Building upon a recent proposal to simulate XAS using quantum computers, this work proposes a highly-optimized implementation of the time-domain algorithm for X-ray absorption. Among a host of improvements to Hamiltonian representation, circuit implementation, and measurement strategies, three optimizations are key to the efficiency of the algorithm. The first is the use of product formulas with the compressed double factorized form of the Hamiltonian. The second is recognizing that for spectroscopy applications, it is sufficient to control the error in the eigenvalues of the (approximate) Hamiltonian being implemented by the product formula, rather than the generic error on the full time evolution operator. Using perturbation theory to estimate this eigenvalue error, we find that significantly fewer Trotter steps are needed than expected from the time evolution error bound. The third is the choice of an optimized distribution of samples that takes advantage of the exponentially decaying Lorentzian kernel. Through constant factor resource estimates, we show that a challenging model Li$_4$Mn$_2$O cluster system with 18 spatial orbitals and 22 electrons in the active space can be simulated with 100 logical qubits and less than $4 \times 10^8$ Toffoli gates per circuit. Finally, the algorithm is implemented on a simulator, and the reconstructed spectrum is verified against a classical computational reference. The low cost of our algorithm makes it attractive to use on fault-tolerant quantum devices to accelerate the development and commercialization of high-capacity battery cathodes.

quant-ph↗