2,212 research outputs found
Abstract algebra, projective geometry and time encoding of quantum information
Algebraic geometrical concepts are playing an increasing role in quantum
applications such as coding, cryptography, tomography and computing. We point
out here the prominent role played by Galois fields viewed as cyclotomic
extensions of the integers modulo a prime characteristic . They can be used
to generate efficient cyclic encoding, for transmitting secrete quantum keys,
for quantum state recovery and for error correction in quantum computing.
Finite projective planes and their generalization are the geometric counterpart
to cyclotomic concepts, their coordinatization involves Galois fields, and they
have been used repetitively for enciphering and coding. Finally the characters
over Galois fields are fundamental for generating complete sets of mutually
unbiased bases, a generic concept of quantum information processing and quantum
entanglement. Gauss sums over Galois fields ensure minimum uncertainty under
such protocols. Some Galois rings which are cyclotomic extensions of the
integers modulo 4 are also becoming fashionable for their role in time encoding
and mutual unbiasedness.Comment: To appear in R. Buccheri, A.C. Elitzur and M. Saniga (eds.),
"Endophysics, Time, Quantum and the Subjective," World Scientific, Singapore.
16 page
On Some Dynamical Systems in Finite Fields and Residue Rings
We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning
- …