4 research outputs found
Extreme nonlocality with one photon
Quantum nonlocality is typically assigned to systems of two or more well
separated particles, but nonlocality can also exist in systems consisting of
just a single particle, when one considers the subsystems to be distant spatial
field modes. Single particle nonlocality has been confirmed experimentally via
a bipartite Bell inequality. In this paper, we introduce an N-party Hardy-like
proof of impossibility of local elements of reality and a Bell inequality for
local realistic theories for a single particle superposed symmetrical over N
spatial field modes (i.e. a N qubit W state). We show that, in the limit of
large N, the Hardy-like proof effectively becomes an all-versus nothing (or
GHZ-like) proof, and the quantum-classical gap of the Bell inequality tends to
be same of the one in a three-particle GHZ experiment. We detail how to test
the nonlocality in realistic systems.Comment: 11 single column pages, 2 figures; v3 now includes a Bell inequality
in addition to the results in the previous versio
Locality for quantum systems on graphs depends on the number field
Adapting a definition of Aaronson and Ambainis [Theory Comput. 1 (2005),
47--79], we call a quantum dynamics on a digraph "saturated Z-local" if the
nonzero transition amplitudes specifying the unitary evolution are in exact
correspondence with the directed edges (including loops) of the digraph. This
idea appears recurrently in a variety of contexts including angular momentum,
quantum chaos, and combinatorial matrix theory. Complete characterization of
the digraph properties that allow such a process to exist is a long-standing
open question that can also be formulated in terms of minimum rank problems. We
prove that saturated Z-local dynamics involving complex amplitudes occur on a
proper superset of the digraphs that allow restriction to the real numbers or,
even further, the rationals. Consequently, among these fields, complex numbers
guarantee the largest possible choice of topologies supporting a discrete
quantum evolution. A similar construction separates complex numbers from the
skew field of quaternions. The result proposes a concrete ground for
distinguishing between complex and quaternionic quantum mechanics.Comment: 9 page
The Sheaf-Theoretic Structure Of Non-Locality and Contextuality
We use the mathematical language of sheaf theory to give a unified treatment
of non-locality and contextuality, in a setting which generalizes the familiar
probability tables used in non-locality theory to arbitrary measurement covers;
this includes Kochen-Specker configurations and more. We show that
contextuality, and non-locality as a special case, correspond exactly to
obstructions to the existence of global sections. We describe a linear
algebraic approach to computing these obstructions, which allows a systematic
treatment of arguments for non-locality and contextuality. We distinguish a
proper hierarchy of strengths of no-go theorems, and show that three leading
examples --- due to Bell, Hardy, and Greenberger, Horne and Zeilinger,
respectively --- occupy successively higher levels of this hierarchy. A general
correspondence is shown between the existence of local hidden-variable
realizations using negative probabilities, and no-signalling; this is based on
a result showing that the linear subspaces generated by the non-contextual and
no-signalling models, over an arbitrary measurement cover, coincide. Maximal
non-locality is generalized to maximal contextuality, and characterized in
purely qualitative terms, as the non-existence of global sections in the
support. A general setting is developed for Kochen-Specker type results, as
generic, model-independent proofs of maximal contextuality, and a new
combinatorial condition is given, which generalizes the `parity proofs'
commonly found in the literature. We also show how our abstract setting can be
represented in quantum mechanics. This leads to a strengthening of the usual
no-signalling theorem, which shows that quantum mechanics obeys no-signalling
for arbitrary families of commuting observables, not just those represented on
different factors of a tensor product.Comment: 33 pages. Extensively revised, new results included. Published in New
Journal of Physic