4 research outputs found
Contextuality in foundations and quantum computation
Contextuality is a key concept in quantum theory. We reveal just how important it is by demonstrating that quantum theory builds on contextuality in a fundamental
way: a number of key theorems in quantum foundations can be given a unifi ed presentation in the topos approach to quantum theory, which is based on contextuality as the common underlying principle. We review existing results and complement them by providing contextual reformulations for Stinespring's and Bell's theorem.
Both have a number of consequences that go far beyond the evident confirmation of the unifying character of contextuality in quantum theory. Complete positivity of
quantum channels is already encoded in contexts, nonlocality arises from a notion of composition of contexts, and quantum states can be singled out among more general non-signalling correlations over the composite context structure by a notion of time orientation in subsystems, thus solving a much discussed open problem in quantum information theory. We also discuss nonlocal correlations under the generalisation to orthomodular lattices and provide generalised Bell inequalities in this setting. The dominant role of contextuality in quantum foundations further supports a recent hypothesis in quantum computation, which identifi es contextuality as the resource for the supposed quantum advantage over classical computers. In particular, within the architecture of measurement-based quantum computation, the resource
character of nonlocality and contextuality exhibits rather clearly. We study contextuality in this framework and generalise the strong link between contextuality and computation observed in the qubit case to qudit systems. More precisely, we provide new proofs of contextuality as well as a universal implementation of computation in this setting, while emphasising the crucial role played by phase
relations between measurement eigenstates. Finally, we suggest a fine-grained measure for contextuality in the form of the number of qubits required for implementation in the non-adaptive, deterministic case.Open Acces
Contextuality and the fundamental theorems of quantum mechanics
Contextuality is a key feature of quantum mechanics, as was first brought to
light by Bohr and later realised more technically by Kochen and Specker. Isham
and Butterfield put contextuality at the heart of their topos-based formalism
and gave a reformulation of the Kochen-Specker theorem in the language of
presheaves. Here, we broaden this perspective considerably (partly drawing on
existing, but scattered results) and show that apart from the Kochen-Specker
theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate
fundamentally to contextuality. We provide reformulations of the theorems using
the language of presheaves over contexts and give general versions valid for
von Neumann algebras. This shows that a very substantial part of the structure
of quantum theory is encoded by contextuality.Comment: v2: minor revisions, added definition of Bell presheaf, adjustment of
Bell's theorem in contextual for
Bipartite entanglement and the arrow of time
We provide a new perspective on the close relationship between entanglement
and time. Our main focus is on bipartite entanglement, where this connection is
foreshadowed both in the positive partial transpose criterion due to Peres [A.
Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of quantum
within more general non-signalling bipartite correlations [M. Frembs and A.
D\"oring, arXiv:2204.11471]. Extracting the relevant common features, we
identify a necessary and sufficient condition for bipartite entanglement in
terms of a compatibility condition with respect to time orientations in local
observable algebras, which express the dynamics in the respective subsystems.
We discuss the relevance of the latter in the broader context of von Neumann
algebras and the thermodynamical notion of time naturally arising within the
latter.Comment: 16 page