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TMF, 2007, Volume 151, Number 2, Pages 219–227 (Mi tmf6041)  

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

Abstract: 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.

Keywords: projective ring line, neighbor relation, distant relation, Mermin's square, Mermin's pentagram, quantum entanglement


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English version:
Theoretical and Mathematical Physics, 2007, 151:2, 625–631

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Received: 21.07.2006

Citation: 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 TMF
\yr 2007
\vol 151
\issue 2
\pages 219--227
\jour Theoret. and Math. Phys.
\yr 2007
\vol 151
\issue 2
\pages 625--631

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    This publication is cited in the following articles:
    1. Michel Planat, Metod Saniga, Maurice R. Kibler, “Quantum Entanglement and Projective Ring Geometry”, SIGMA, 2 (2006), 066, 14 pp.  mathnet  crossref  mathscinet  zmath
    2. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Metod Saniga, Michel Planat, Petr Pracna, Hans Havlicek, “The Veldkamp Space of Two-Qubits”, SIGMA, 3 (2007), 075, 7 pp.  mathnet  crossref  mathscinet  zmath
    4. Havlicek, H, “Projective ring line of a specific qudit”, Journal of Physics A-Mathematical and Theoretical, 40:43 (2007), F943  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. M. Saniga, M. Planat, P. Pracna, “Projective ring line encompassing two-qubits”, Theoret. and Math. Phys., 155:3 (2008), 905–913  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. 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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Planat M, Saniga M, “On the Pauli graphs on N-qudits”, Quantum Information & Computation, 8:1–2 (2008), 127–146  mathscinet  zmath  adsnasa  isi
    8. Hans Havlicek, Boris Odehnal, Metod Saniga, “Factor-Group-Generated Polar Spaces and (Multi-)Qudits”, SIGMA, 5 (2009), 096, 15 pp.  mathnet  crossref  mathscinet
    9. 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  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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