D(p)(p.),where the maximum is over all POVM p and are distributions of outcomes obtained by applying [Ek}on p and o,respectively. This implies that the trace distance between p and o is the largest difference one can tell by a measurement. โ D is a metric.Triangle inequality holds. Quantum operations don't increase trace distance.(One cannot make two states more distinguishable by operating on them.)D((p),())D(p,) โ Strong convexity:D(iPiPi,โiqioi)โคD(p,q)+โipiD(pi,oi) ใfidelity:Fo,o)=trpaไธ It's symmetric:F(p,o)=F(a,p). If p and o commute,then F(p,a)=F((p),()) โ F(p,o)=1iffp=0. F(p,o)=0 iff p and o haveorthogonal supports Special case of pure state(s):F(),p)=)F())=) F(UpUt,UoUt)=F(p,a). F(p,a)=max{):),lo)purify p,o,respectively} F(p,o)=max{l):l)purifies p},for any fixed purification lo)of a. F(p)(.).where the maximum is over all POVM Ex p and q are distributions ofoutcomes obtained by applying [Ek}on p and o,respectively โ F((p),ฮฆ(a)โฅF(p,o). โ FCiPiP1,โiqio)โฅโiPiqiF(P,o) A basic relation between the two distance measures is as follows. 1-F(p,o)โคD(p,o)โคV1-F(0,G)2 Note There are a couple of excellent references for quantum information.Part III of [NC0O]is still very good.[Will3]is a new book with emphasis on quantum Shannon theory.[Wat11] contains more other stuff and it is somewhat closer to computer science perspectives.โผ ๐ท(๐, ๐) = max {๐ธ๐} ๐ท(๐,๐), where the maximum is over all POVM {๐ธ๐ }, ๐ and ๐ are distributions of outcomes obtained by applying {๐ธ๐ } on ๐ and ๐, respectively. This implies that the trace distance between ๐ and ๐ is the largest difference one can tell by a measurement. โผ ๐ท is a metric. Triangle inequality holds. โผ Quantum operations donโt increase trace distance. (One cannot make two states more distinguishable by operating on them.) ๐ท(ฮฆ(๐),ฮฆ(๐)) โค ๐ท(๐,๐). โผ Strong convexity: ๐ท(โ๐ ๐๐๐๐ ,โ๐ ๐๐๐๐ ) โค ๐ท(๐, ๐) + โ ๐๐๐ท(๐๐ ๐ ,๐๐). โซ fidelity: ๐น(๐, ๐) = ๐ก๐โโ๐๐โ๐ โผ Itโs symmetric: ๐น(๐,๐) = ๐น(๐,๐). โผ If ๐ and ๐ commute, then ๐น(๐, ๐) = ๐น(๐(๐), ๐(๐)). โผ ๐น(๐, ๐) = 1 iff ๐ = ๐. โผ ๐น(๐, ๐) = 0 iff ๐ and ๐ have orthogonal supports. โผ Special case of pure state(s): ๐น(|๐โช,๐) = โโฉ๐|๐|๐โช, ๐น(|๐โช,|๐โช) = |โฉ๐|๐โช|. โผ ๐น(๐๐๐โ ,๐๐๐โ ) = ๐น(๐, ๐). โผ ๐น(๐, ๐) = max{|โฉ๐|๐โช|:|๐โช,|๐โช purify ๐,๐, respectively} โผ ๐น(๐, ๐) = max{|โฉ๐|๐โช|:|๐โช purifies ๐}, for any fixed purification |๐โช of ๐. โผ ๐น(๐, ๐) = min {๐ธ๐} ๐น(๐,๐), where the maximum is over all POVM {๐ธ๐ }, ๐ and ๐ are distributions of outcomes obtained by applying {๐ธ๐ } on ๐ and ๐, respectively. โผ ๐น(ฮฆ(๐),ฮฆ(๐)) โฅ ๐น(๐, ๐). โผ ๐น(โ๐ ๐๐๐๐ ,โ๐ ๐๐๐๐ ) โฅ โ โ๐๐๐๐๐น(๐๐ ๐ ,๐๐). A basic relation between the two distance measures is as follows. 1 โ ๐น(๐, ๐) โค ๐ท(๐,๐) โค โ1 โ ๐น(๐,๐) 2 . Note There are a couple of excellent references for quantum information. Part III of [NC00] is still very good. [Wil13] is a new book with emphasis on quantum Shannon theory. [Wat11] contains more other stuff and it is somewhat closer to computer science perspectives