By Andrew Soltis
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Additional info for Winning with the Ruy Lopez Exchange Variation (Subtitle: Fischer's Weapon)
Some explicit eigenstates of Bk The interest of the quantization Bk lies in the fact that its spectrum and eigenstates can be analytically computed. Vol. 1. 15) and the periodicity of the Fourier transform) is that this operator is periodic, with period 2k (when D = 2) or 4k (when D ≥ 3): D = 2 =⇒ ∀k ≥ 1, (Bk )2k = I2k D ≥ 2 =⇒ ∀k ≥ 1, (Bk )4k = IDk . 1) where Π is the “parity operator” on C , which sends e to e¯, with +¯ ≡ 0 mod D. log h| As we noticed above, k = |log D is the Ehrenfest time of the system, so the above periodicity can be compared with the “short quantum periods” of the quantum cat map [5, 12], which allowed one to construct eigenstates with a partial localization on some periodic orbits.
Indeed, the following “Entropic Uncertainty Principle”, ﬁrst conjectured 56 N. Anantharaman and S. Nonnenmacher Ann. Henri Poincar´e by Kraus  and proven in , directly provides the desired lower bound for h(w). 1 (Entropic uncertainty principle ). For any M ∈ N∗ , let U be a def unitary M × M matrix and c(U ) = supi,j |Uij |. Then, for any normalized state ψ ∈ CM , one has h(ψ) + h(U ψ) ≥ −2 log c(U ) , where the entropy is deﬁned as h(ψ) = − i |ψi |2 log |ψi |2 . 2, see Section 5) is outlined in the Appendix.
We will use here a diﬀerent quantization, based on the Walsh–Fourier transform : this choice makes the quantum model amenable to an analytic treatment. The map and its quantization will be described in more detail in Sections 2–3. The localization in phase space of an eigenfunction ψ will be analyzed using its Walsh–Husimi measure W Hψ , which is a probability measure on the torus, associated with the state ψ . For any sequence of eigenfunctions (ψ ) →0 of the quantized map, one can extract a subsequence of W Hψ j j →0 which weakly converges towards a probability measure µ.
Winning with the Ruy Lopez Exchange Variation (Subtitle: Fischer's Weapon) by Andrew Soltis