▁and ▁mathemat icians ▁Nicolas ▁G is in , ▁Lane ▁P . ▁Hugh ston , ▁Richard ▁J oz sa ▁and ▁William ▁W oot ters , ▁though ▁much ▁of ▁it ▁was ▁established ▁dec ades ▁earlier ▁by ▁Er win ▁Sch r öd inger . ▁The ▁result ▁was ▁also ▁found ▁independently ▁by ▁Nicolas ▁Had j is av vas ▁building ▁upon ▁work ▁by ▁Ed ▁Jay nes , ▁while ▁a ▁significant ▁part ▁of ▁it ▁was ▁like wise ▁independently ▁discovered ▁by ▁N . ▁David ▁M erm in . ▁Thanks ▁to ▁its ▁complicated ▁history , ▁it ▁is ▁also ▁known ▁as ▁the ▁H J W ▁theorem ▁and ▁the ▁Sch r öd inger – H J W ▁theorem . ▁ ▁Pur ification ▁of ▁a ▁mixed ▁quantum ▁state ▁ ▁Consider ▁a ▁mixed ▁state ▁ ▁of ▁the ▁system ▁, ▁where ▁the ▁states ▁ ▁are ▁not ▁assumed ▁to ▁be ▁mut ually ▁orthogonal . ▁We ▁can ▁add ▁an ▁auxili ary ▁space ▁ ▁with ▁an ▁orth on ormal ▁basis ▁, ▁then ▁the ▁mixed ▁state ▁can ▁be ▁obtained ▁as ▁reduced ▁density ▁operator ▁from ▁the ▁pure ▁b ipart ite ▁state ▁▁▁▁ ▁More ▁precisely , ▁. ▁The ▁state ▁ ▁is ▁thus ▁called ▁the ▁pur ification ▁of ▁. ▁Since ▁the ▁auxili ary ▁space ▁and ▁the ▁basis ▁can ▁be ▁chosen ▁arbitr arily , ▁the ▁pur ification ▁of ▁a ▁mixed ▁state ▁is ▁not ▁unique ; ▁in ▁fact , ▁there ▁are ▁infinitely ▁many ▁pur ifications ▁of ▁a ▁given ▁mixed ▁state . ▁ ▁G H J W ▁theorem ▁▁ ▁Consider ▁a ▁mixed ▁quantum ▁state ▁ ▁with ▁two ▁different ▁realiz ations ▁as ▁ensemble ▁of ▁pure ▁states ▁as