Fig. 1: White-noise tolerance of our Schmidt-number witness.
From: High-dimensional entanglement witnessed by correlations in arbitrary bases
The upper bounds of the white-noise ratio, \({p}_{c,m}^{(k)}\) in eq. (10), for witnessing Schmidt number k + 1 versus \({\epsilon }_{\min }:= 1/d-{c}_{\min }\) in local dimension d = 5, where we set \({c}_{\max }=1-(d-1){c}_{\min }\). When \({\epsilon }_{\min }=0\) (i.e., \({c}_{\min }=1/d\) for MUBs), \({p}_{c,m}^{(k)}\) reaches its maximum, (m − 1)(d − k)/[m(d − 1)]. When m = d + 1 = 6, it coincides with \({p}_{\,\text{iso}\,}^{(k)}\), the maximum white-noise ratio for \({\rho }_{A\,B}^{{\rm{iso}}}\) having Schmidt number k + 1. The noise tolerance of the witness is higher for larger m or smaller k but reduces as \({\epsilon }_{\min }\) increases, and eventually, when \({\epsilon }_{\min }\ge 1/5-(7-\sqrt{17})/20\approx 0.0562\), we cannot witness non-trivial Schmidt numbers of \({\rho }_{AB}^{{\rm{iso}}}\).
