This article is cited in 9 scientific papers (total in 9 papers)
Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: The Mermin "magic" square/pentagram
M. Sanigaa, M. Planatb, M. Minarovjecha
a Astronomical Institute, Slovak Academy of Sciences
b CNRS — Institut FEMTO-ST, Département LPMO
In 1993, Mermin gave surprisingly simple proofs of the Bell–Kochen–Specker
(BKS) theorem in Hilbert spaces of dimensions four and eight
respectively using what has since been called the Mermin–Peres ‘`magic"
square and the Mermin pentagram. The former is a $3\times 3$ array of nine
observables commuting pairwise in each row and column and arranged such 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 a similar
contradiction. We establish a one-to-one correspondence between the operators
of the Mermin–Peres square and the points of the projective line over
the product ring $GF(2)\otimes GF(2)$. Under this map, the concept mutually
commuting transforms 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 either both zero divisors or both units. The ten operators
of the Mermin pentagram correspond to a specific subset of points of the line
over $GF(2)[x]/\langle x^3-x\rangle$. But the situation in this case is more
intricate because there are two different configurations that seem to serve
our purpose equally well. 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 configuration features the neighborhood of the point $(1,0)$
(or, equivalently, that of $(0,1)$). We also mention some other ring
lines that might be relevant to BKS proofs in higher dimensions.
projective ring line, neighbor relation, distant relation, Mermin's square, Mermin's pentagram, quantum entanglement
PDF file (400 kB)
Theoretical and Mathematical Physics, 2007, 151:2, 625–631
M. Saniga, M. Planat, M. Minarovjech, “Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: The Mermin "magic" square/pentagram”, TMF, 151:2 (2007), 219–227; Theoret. and Math. Phys., 151:2 (2007), 625–631
Citation in format AMSBIB
\by M.~Saniga, M.~Planat, M.~Minarovjech
\paper Projective line over the~finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: The~Mermin ``magic" square/pentagram
\jour Theoret. and Math. Phys.
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This publication is cited in the following articles:
Michel Planat, Metod Saniga, Maurice R. Kibler, “Quantum Entanglement and Projective Ring Geometry”, SIGMA, 2 (2006), 066, 14 pp.
M. Saniga, M. Planat, “Projective line over the finite quotient ring $GF(2)[x]/\langle x^3-x\rangle$ and quantum entanglement: Theoretical background”, Theoret. and Math. Phys., 151:1 (2007), 474–481
Metod Saniga, Michel Planat, Petr Pracna, Hans Havlicek, “The Veldkamp Space of Two-Qubits”, SIGMA, 3 (2007), 075, 7 pp.
Havlicek, H, “Projective ring line of a specific qudit”, Journal of Physics A-Mathematical and Theoretical, 40:43 (2007), F943
M. Saniga, M. Planat, P. Pracna, “Projective ring line encompassing two-qubits”, Theoret. and Math. Phys., 155:3 (2008), 905–913
Planat M, Baboin AC, Saniga M, “Multi-line geometry of qubit-qutrit and higher-order Pauli operators”, International Journal of Theoretical Physics, 47:4 (2008), 1127–1135
Planat M, Saniga M, “On the Pauli graphs on N-qudits”, Quantum Information & Computation, 8:1–2 (2008), 127–146
Hans Havlicek, Boris Odehnal, Metod Saniga, “Factor-Group-Generated Polar Spaces and (Multi-)Qudits”, SIGMA, 5 (2009), 096, 15 pp.
Green R.M. Saniga M., “The Veldkamp Space of the Smallest Slim Dense Near Hexagon”, Int. J. Geom. Methods Mod. Phys., 10:2 (2013), 1250082
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