▁can ▁be ▁described ▁by : ▁ ▁where ▁ ▁is ▁the ▁deleting ▁operation ▁which ▁is ▁not ▁necessarily ▁unit ary ▁( but ▁a ▁linear ▁operator ), ▁ ▁is ▁the ▁unknown ▁quantum ▁ ▁state , ▁ ▁is ▁the ▁blank ▁state , ▁ ▁is ▁the ▁initial ▁state ▁of ▁ ▁the ▁deleting ▁machine ▁and ▁ ▁is ▁the ▁final ▁state ▁of ▁the ▁machine . ▁ ▁It ▁may ▁be ▁noted ▁that ▁classical ▁bits ▁can ▁be ▁copied ▁and ▁deleted , ▁as ▁can ▁qu bits ▁in ▁orthogonal ▁states . ▁For ▁example , ▁if ▁we ▁have ▁two ▁identical ▁qu bits ▁ ▁and ▁ ▁then ▁we ▁can ▁transform ▁to ▁ ▁and ▁. ▁In ▁this ▁case ▁we ▁have ▁deleted ▁the ▁second ▁copy . ▁However , ▁it ▁follows ▁from ▁linear ity ▁of ▁quantum ▁theory ▁that ▁there ▁is ▁no ▁ ▁that ▁can ▁perform ▁the ▁deleting ▁operation ▁for ▁any ▁arbitrary ▁state ▁. ▁ ▁F ormal ▁statement ▁of ▁the ▁no - de let ing ▁theorem ▁▁ ▁Let ▁ ▁be ▁an ▁unknown ▁quantum ▁state ▁in ▁some ▁Hil bert ▁space ▁( and ▁let ▁other ▁states ▁have ▁their ▁usual ▁meaning ). ▁Then , ▁ ▁there ▁is ▁no ▁linear ▁is ometric ▁transformation ▁such ▁that ▁ ▁, ▁with ▁the ▁final ▁state ▁of ▁the ▁anc illa ▁being ▁independent ▁of ▁. ▁ ▁Proof ▁▁ ▁The ▁theorem ▁holds ▁for ▁quantum ▁states ▁in ▁a ▁Hil bert ▁space ▁of ▁any ▁dimension . ▁For ▁simplicity , ▁ ▁consider ▁the ▁deleting ▁transformation ▁for ▁two ▁identical ▁qu bits . ▁If ▁two ▁qu bits ▁are ▁in ▁orthogonal ▁states , ▁then ▁delet ion ▁requires ▁that ▁ ▁, ▁. ▁ ▁Let ▁ ▁be ▁the ▁state