Zur Geometrie der Zahlen

Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes

“…Similar to the one-mode case, it is possible to increase the correctable radius of displacement in N -mode case, by choosing a 2N -dimensional symplectic lattice allowing more efficient sphere packing than the square lattice. It is known that there exists a 2N -dimensional lattice in Euclidean space allowing d min ≥ N/(πe) [65] and a stronger statement was proven in [66] that the same holds also for symplectic lattices. Choosing such a lattice to define the GKP code, one can correct all displacement error within radius r ≤ N/(2ed) .…”

A bound, and a conjecture, on the maximum lattice‐packing density of a superball

“…Since GL(k, R) includes scalings by arbitrarily small factors, s^0+ IxcZ 1 LeR 1 exp(-/(x))dV = integral lim s k Xvcz* ex P (~f( sx )) limit of Riemann sum and so the density bound given by Theorem 2, for any superball, is at least like the Minkowski-Hlawka bound See [7], and also [12], [6] and [1], regarding the Minkowski-Hlawka bound.…”

“…Z" n B y G)dy. Hence v=0 I exp(-i//(Ax))= e" v card(Z"nB^A v = 0and we haveL R » exp (-^(Ac))dV_ f" =0 e-y Vol (B.A'Qdy X«z" exp (-^(AJC))J7 =o e" v card (Z" n B y A^G)dy ' Therefore there must be some y > 0 such that f xeR -exp(-iKAr))dy_, Vol (B^-'G) Ix e z" ex P (-MA*)) * card (Z" n B y A" 1 G)" In(7), put B = B v , and apply the inequality above, arriving at log, 8,(G) liminf6 s* -1 +lim inf log 2 M(i/>). (8)Restricting A to the block-diagonal subgroup GL(k, R)0GL(/, R)©.. ,eGL(m, R), we get, for some CeGL(k, R), De GL(/, R),..., EeGL(m,R), the result exp eX P [ex P (-/(Qc))^VL«' ex P (-g(Dx))dV X xeZ -exp(-g(Dx)) "" l xeZ -exp (-h(Ex)) So M(i/0" 3= M(/) fc M(g)'.…”

The lifting construction: A general solution for the fat strut problem

“…The conjecture was proved by Hlawka in 1944 [6] (see also [4]). The Minkowski-Hlawka theorem is proved by an averaging over a general class of lattices.…”