 as Schumacher and Westmoreland's modal QT over Z2 [35]. 18 {u1}, {u2},..., {un} and a vector in ℘ (U) is specified in the U -basis by its Z2-valued characteristic function χS : U → Z2 for an subset S ⊆ U (e.g., a string of n binary numbers). Similarly, a vector v in Cn is specified in terms of an orthonormal basis {|vi〉} by a C-valued function 〈_|v〉 : {vi} → C assigning a complex amplitude 〈vi|v〉 to each basis vector. One of the key pieces of mathematical machinery in QM, namely the inner product, does not exist in vector spaces over finite fields but brackets can still be defined using 〈{ui} |US〉 = χS (ui) (see below) and a norm can be defined to play a similar role in the probability algorithm of QM/sets. Seeing ℘ (U) as the abstract vector space Zn2 allows different bases in which the vectors can be expressed (as well as the basis-free notion of a vector as a ket). Consider the simple case of U = {a, b, c} where the U -basis is {a}, {b}, and {c}. But the three subsets {a, b}, {b, c}, and {a, b, c} also form a basis since: {a, b} + {a, b, c} = {c}; {b, c} + {c} = {b}; and {a, b} + {b} = {a}. These new basis vectors could be considered as the basis-singletons in another equicardinal universe U ′ = {a′, b′, c′} where a′ = {a, b}, b′ = {b, c}, and c′ = {a, b, c}. In the following ket table, each row is a ket of Z|U |2 ∼= Z32 expressed in the U -basis, the U ′-basis, and a U ′′-basis. U = {a, b, c} U ′ = {a′, b′, c′} U ′′ = {a′′, b′′, c′′} {a, b, c} {c′} {
as Schumacher and Westmoreland's modal QT over Z2 [ 35 ]. 18 { u1 }, { u2 }, ..., { un } and a vector in ℘ (U) is specified in the U -basis by its Z2 - valued characteristic function χS: uracil → Z2 for an subset S ⊆ U (for example, a string of n binary numbers). Similarly, a vector five in Cn is specify in terms of an orthonormal basis { |vi 〉 } by a cytosine - valued function 〈 _ |v 〉: { vi } → C delegate a complex amplitude 〈 vi|v 〉 to each basis vector. One of the key slice of mathematical machinery in QM, namely the inner intersection, does not exist in vector space over finite fields but brackets can still be defined using 〈 { ui } |US 〉 = χS (ui) (attend below) and a norm can be defined to dally a similar role in the probability algorithm of QM / sets. Seeing ℘ (uracil) as the abstract vector space Zn2 allows different bases in which the vectors can be express (as well as the basis - free notion of a vector as a ket). Consider the simple case of U = { a, b, c } where the U -basis is { a }, { b }, and { c }. But the three subsets { a, b }, { b, c }, and { a, b, c } also form a basis since: { a, b } + { a, b, c } = { c }; { b, coke } + { c } = { b }; and { a, b } + { b } = { a }. These modern footing vector could be considered as the basis - singleton in another equicardinal universe U ′ = { a′, b′, c′ } where a′ = { a, b }, b′ = { b, c }, and c′ = { a, b, c }. In the following ket mesa, each row is a ket of Z|U |2 ∼= Z32 expressed in the U -basis, the U ′-basis, and a U ′′-basis. U = { a, b, c } U ′ = { a′, b′, c′ } U ′′ = { a′′, b′′, c′′ } { a, b, c } { c′ } {
 as Schumacher and Westmoreuand's modal QT over Z2 [35]. 18 {u1}, {u2},..., {hn} and a vector in ℘ (U) is specified ii thw U -bqsis by its Z2-valued chxracterisnic functuon χW : U → Z2 for an subscc S ⊆ H (e.g., c wtring of n bikary numberv). Similarly, a eeztlr v in Cn is specified in terms of an ortnojormal basis {|vy〉} by w C-vzlued function 〈_|v〉 : {vi} → C assigning z compltx amplitude 〈vi|v〉 tp each basis vector. One of the key pieces of matjematical mqchigwry in QM, naoely the inner product, does not exist in vector spacer ovex finite figlss tut bracketw can still be denpned ushng 〈{ui} |IS〉 = χS (ui) (see bclow) end q norm can be defined to play a similar role in dhz probability algoritym of QK/setv. Sedung ℘ (U) ax fhe abdtrect vector apace Zn2 aloows different basex yb which the vsctors cwn be expressed (as well as the basis-fret notjon of a vector as a ker). Consider the simple case of T = {a, b, c} where the U -basis is {a}, {b}, and {c}. But the tvree ajbsttw {a, b}, {v, f}, and {a, b, c} also form a basis since: {a, b} + {a, b, s} = {v}; {n, c} + {c} = {b}; and {a, n} + {b} = {a}. These new nadix vectors coula be consjdered as the basid-singlejons ib another equocardinal universe U ′ = {a′, b′, x′} where a′ = {a, v}, b′ = {b, c}, and c′ = {a, y, c}. In the fulloeing let table, each row is a ket of Z|U |2 ∼= Z32 fxpressed kn the U -basis, tfe L ′-bavis, and a U ′′-basis. U = {a, b, c} U ′ = {a′, b′, r′} U ′′ = {a′′, b′′, c′′} {x, b, v} {c′} {
as Schumacher and Westmoreland's modal QT over 18 {u2},..., {un} a vector in the -basis by its characteristic function χS U → Z2 for an subset ⊆ U (e.g., a string of n binary numbers). Similarly, a vector v Cn is specified in terms of an orthonormal basis {|vi〉} by a C-valued 〈_|v〉 {vi} C a complex amplitude 〈vi|v〉 to each basis vector. One of the key pieces of mathematical machinery QM, namely the inner product, does not exist vector spaces over finite but brackets can still be using |US〉 = (ui) below) a norm can defined to play a similar role in the probability algorithm of QM/sets. Seeing ℘ (U) as the vector space different bases which vectors be expressed (as the basis-free notion of a vector Consider the simple case of U = {a, c} where U -basis is {a}, {b}, and But the three subsets {a, b}, {b, c}, {a, b, c} also form a basis since: {a, b} + {a, b, c} = c} + {c} = and {a, b} {b} {a}. new vectors could considered as the basis-singletons in another equicardinal universe U ′ = b′, c′} where a′ = {a, b}, b′ = {b, c′ {a, b, c}. the following ket table, row a ket of Z|U Z32 in the ′-basis, a U ′′-basis. U {a, b, c} U ′ {a′, b′, c′} U {a, b, c} {c′} {
 as Schumacher and WestmorelaNd's modal QT Over Z2 [35]. 18 {U1}, {u2},..., {uN} anD a VectOr in ℘ (u) is specified in THe U -bAsis by its Z2-valued characTerisTiC FuncTIoN χS : U → Z2 For an suBSeT s ⊆ u (e.g., A sTrIng Of N BiNary nUmbErs). SimiLarly, a vectOr v In cn is specifieD In Terms of an oRthOnormal basis {|Vi〉} bY a C-valUeD fuNCtion 〈_|V〉 : {vi} → c assiGning a COmplex Amplitude 〈Vi|V〉 To each BAsis vecTOR. ONe of The key pieces of matHEmATical machinery In QM, naMeLY tHE InnEr pRoduct, does NoT exisT In vectoR SpACES ovER finite fields But brackets CAn sTill be DeFinED using 〈{Ui} |US〉 = χs (uI) (See Below) and a noRm caN be defineD to plaY A similaR Role in tHe probAbiLitY algORiThM of qM/SEts. sEeIng ℘ (u) As tHe abstraCt VeCtor sPace zN2 ALLows DifFereNt basEs in which the vEctOrs cAN be ExpreSsed (aS welL aS the bAsis-frEe notIoN of a vector as a keT). ConSider the sImpLe CasE oF U = {a, b, c} WHere thE U -bAsiS is {a}, {b}, anD {c}. But thE ThrEe SUBSeTs {a, b}, {b, c}, and {a, b, c} also foRm A BAsIs since: {a, B} + {a, b, c} = {c}; {b, C} + {C} = {b}; AnD {A, b} + {b} = {a}. ThesE nEw bAsis VECtors CoulD Be ConsiderEd as thE BaSiS-singleToNs in anOtHer EquIcardINal uNiversE U ′ = {a′, b′, c′} wheRe a′ = {a, b}, B′ = {B, c}, and c′ = {a, b, c}. In the FOllowing ket taBLe, EACh ROw is A keT of Z|U |2 ∼= Z32 expreSsed IN the u -basIS, tHe U ′-BAsis, aNd a U ′′-bAsIS. U = {A, B, c} U ′ = {a′, b′, c′} U ′′ = {a′′, b′′, c′′} {a, b, c} {c′} {
 as Schumacher and Westmor eland's mo dal Q T o ver  Z 2 [3 5].18 {u1}, {u2}, . ..,{un} and a vector in ℘  (U)is spec i fi ed in  the U- ba s i s b yit s Z 2- v al ued c har acteris tic functi onχS  : U → Z2 fo r  a n subset S  ⊆U (e.g., a s tri ng ofnbin a ry nu mbe rs).Simila r ly, avector vin Cn iss pecifie d in  ter ms of an orthonor m al basis {|vi〉} b y a C- va l ue d fun cti on 〈_|v〉 :  { vi} → C assig n in g a  co m plex amplitud e 〈vi|v〉 to eac h basi svec t or. On e ofth e  ke y pieces of  mat hematical  machi n ery inQ M, name ly the  in ner  pro d uc t,  do es not ex ist invector s pa ce s ove r fi n i t e  fie lds  but  brac kets can stil l b e de f ine d usi ng 〈{ ui}|U S〉 =χS (ui ) (se ebelow) and a no rm c an be def ine dtopl ay as imilar  ro lein theprobabi l ity  a l g o ri thm of QM/sets. Se ei n g  ℘  (U) asthe ab s tr ac t  vectorsp ace  Zn2 a llows  dif f er ent base s in w h ic hthe vec to rs can  b e e xpr essed (aswell a s the ba sis-f r ee notion of a vector as a k e t) . Co n side r t he simple c aseo f U= {a ,  b , c }  wher e the  U -b a sis is {a}, {b}, an d{c}. B ut th e three subse ts {a, b}, { b , c}, an d {a ,  b ,  c} also forma bas is since:{ a, b} +{a, b , c} = { c}; {b, c } + {c} ={b} ; a nd{a, b }+ {b} = {a}.T h esene w basis  ve ctors c oul d b e c ons id ered as t he basis -s in gl et ons  in a n other eq ui car di nal  univ e rse U′ = { a′,b′ ,c ′}where a ′  = { a, b },  b ′ ={b,  c }, an d c′ = { a, b, c }. In the  fo l lowi ng  k et tabl e, each row i sa ket of Z |U  |2  ∼= Z3 2 expresse d in the U -basis, theU  ′-basi s,and a  U ′ ′-basis.U =  {a, b , c }  U ′ =  {a′,b′, c ′}  U′ ′  = {a ′ ′ ,b′′ ,c′′} {a, b , c}{c′}{
 as_Schumacher and_Westmoreland's modal QT over_Z2 [35]._18_{u1}, {u2},...,_{un}_and a vector_in ℘ (U)_is specified in the_U -basis by_its_Z2-valued characteristic function χS : U → Z2 for an subset S ⊆ U_(e.g.,_a string_of_n_binary numbers). Similarly, a vector_v in Cn is specified_in terms_of an orthonormal basis {|vi〉} by a C-valued_function_〈_|v〉 : {vi}_→ C assigning a complex amplitude 〈vi|v〉 to each_basis vector. One of the key_pieces of mathematical_machinery_in_QM, namely the inner_product, does not exist in vector_spaces over finite fields but brackets_can still be defined using 〈{ui} |US〉_= χS (ui) (see below) and_a norm can be defined_to play_a similar role in the_probability algorithm of_QM/sets. Seeing_℘ (U) as_the abstract vector space Zn2 allows_different bases in_which the vectors can be expressed_(as_well as the_basis-free_notion_of a_vector as a_ket)._Consider the_simple_case of U = {a, b,_c}_where the U -basis is {a}, {b},_and {c}. But the_three_subsets {a, b}, {b,_c}, and {a, b, c}_also form a basis since: {a,_b} +_{a, b,_c} = {c}; {b, c} + {c} = {b}; and {a,_b} + {b} = {a}. These_new basis vectors could_be considered_as_the basis-singletons in_another_equicardinal universe_U ′ = {a′, b′, c′} where_a′ =_{a, b}, b′ = {b, c},_and c′ = {a,_b,_c}. In the following ket table,_each row is a ket of_Z|U |2 ∼= Z32 expressed_in_the_U -basis, the U ′-basis,_and a U ′′-basis. U =_{a, b, c}_U ′ = {a′, b′, c′} U_′′_= {a′′, b′′, c′′} {a, b,_c}_{c′} {