  We present a systematic analysis of X-ray archival data of all the 29 quasars
(QSOs) at $z$ > 5.5 observed so far with Chandra, XMM-Newton and Swift-XRT,
including the most-distant quasar ever discovered, ULAS J1120+0641 ($z$ =
7.08). This study allows us to place constraints on the mean spectral
properties of the primordial population of luminous Type 1 (unobscured)
quasars. Eighteen quasars are detected in the X-ray band, and we provide
spectral-fitting results for their X-ray properties, while for the others we
provide upper limits to their soft (0.5-2.0 keV) X-ray flux. We measured the
power-law photon index and derived an upper limit to the column density for the
five quasars (J1306+0356, J0100+2802, J1030+0524, J1148+5251, J1120+0641) with
the best spectra (> 30 net counts in the 0.5-7.0 keV energy range) and find
that they are consistent with values from the literature and lower-redshift
quasars. By stacking the spectra of ten quasars detected by Chandra in the
redshift range 5.7 $\le$ $z$ $\le$ 6.1 we find a mean X-ray power-law photon
index of $\Gamma = 1.92_{-0.27}^{+0.28}$ and a neutral intrinsic absorption
column density of $N_H \le 10^{23}$ cm$^{-2}$. These results suggest that the
X-ray spectral properties of luminous quasars have not evolved up to $z$
$\approx$ 6. We also derived the optical-X-ray spectral slopes ($\alpha_{ox}$)
of our sample and combined them with those of previous works, confirming that
$\alpha_{ox}$ strongly correlates with UV monochromatic luminosity at 2500 \AA
. These results strengthen the non-evolutionary scenario for the spectral
properties of luminous active galactic nuclei (AGN).
