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| related topics |
| {let, theorem, proof} |
| {bell, inequality, local} |
| {theory, mechanics, state} |
| {energy, state, states} |
| {observables, space, algebra} |
| {entanglement, phys, rev} |
| {group, space, representation} |
| {particle, mechanics, theory} |
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The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$< x^{3} -
x>$ and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram
Metod Saniga, Michel Planat, Milan Minarovjech
abstract: In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly
simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of
dimensions four and eight by making use of what has since been referred to as
the Mermin(-Peres) "magic square" and the Mermin pentagram, respectively. The
former is a $3 \times 3$ array of nine observables commuting pairwise in each
row and column and arranged so that their product properties contradict those
of the assigned eigenvalues. The latter is a set of ten observables arranged in
five groups of four lying along five edges of the pentagram and characterized
by similar contradiction. An interesting one-to-one correspondence between the
operators of the Mermin-Peres square and the points of the projective line over
the product ring ${\rm GF}(2) \otimes \rm{GF}(2)$ is established. Under this
mapping, the concept "mutually commuting" translates into "mutually distant"
and the distinguishing character of the third column's observables has its
counterpart in the distinguished properties of the coordinates of the
corresponding points, whose entries are both either zero-divisors, or units.
The ten operators of the Mermin pentagram answer to a specific subset of points
of the line over GF(2)[$x$]/$$. The situation here is, however, more
intricate as there are two different configurations that seem to serve equally
well our purpose. The first one comprises the three distinguished points of the
(sub)line over GF(2), their three "Jacobson" counterparts and the four points
whose both coordinates are zero-divisors; the other features the neighbourhood
of the point ($1, 0$) (or, equivalently, that of ($0, 1$)). Some other ring
lines that might be relevant for BKS proofs in higher dimensions are also
mentioned.
- oai_identifier:
- oai:arXiv.org:quant-ph/0603206
- categories:
- quant-ph math-ph math.MP
- comments:
- 6 pages, 5 figures
- doi:
- 10.1007/s11232-007-0049-5
- arxiv_id:
- quant-ph/0603206
- journal_ref:
- Theoretical and Mathematical Physics 151 (2007) 625-631
- created:
- 2006-03-23
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