Generalized Spectral Resolution & some of its applications

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1 Generlzed Spectrl Resoluton & some of ts pplctons Nchols Wheeler, Reed College Physcs Deprtment 27 Aprl 29 Introducton Fmlrly, f the n n mtrx H s complex hermtn (or, more prtculrly, rel symmetrc or mgnry ntsymmetrc) then the egenvlues λ,λ 2,,λ n re necessrly rel, nd mongst the egenvectors h, h 2,,h n those ssocted wth dstnct egenvlues re necessrly orthogonl: h h j f λ λ j If the spectrum s non-degenerte the egenvectors (whch we mght but won t ssume to hve been normlzed) provde n orthogonl bss n V n More generlly, we wrte (λ,δ ), (λ 2,δ 2 ),,(λ ν,δ ν ) where δ degenercy of λ : δ = n nd the λ re dstnct: ech such λ dentfes n egenspce V () δ -dmensonl subspce of V n Every element of V () s orthogonl to every element of V (β) ( β) The set of egenvectors s smlrly prttoned (h,,h δ ), (h δ+,,h δ+δ 2 ),,(h n δν,,h n ) where the vectors (h,,h δ ) spn V () nd cn be ssumed to hve been orthogonlzed ( by hnd ), etc Wth ths pprtus n hnd, we construct mtrces [ P h h t ] = (h, h) (h, h) h h h h2 h hn h 2 h h 2 h2 h 2 hn h n h h n h2 h n hn Here (h, h) h t h nd t sgnfes conjugted trnsposton

2 2 Generlzed spectrl resoluton whch project onto the egenrys : P h = h From the crcumstnce tht the h, h 2,,h n comprse utomtclly, else (n the cse of degenercy) by contrvnce n orthogonl bss n V n t follows tht the mtrces P, P 2,,P n comprse complete P = I () orthogonl set P P j = O : j (2) of projecton mtrces P 2 = P : ll (3) n terms of whch we hve the spectrl resoluton of H: H = λ P (2) In degenerte cses we cn lump the projectors onto the sme egenspce, wrtng H = λ P () wth P () P + P P δ, etc where P () = I P () P (β) = O : β From () nd (2) we obtn P 2 () = P () : ll H k = λ k P whch n the cse k = gves bck the completeness condton () For ll f(x) tht cn be expressed s weghted sums of powers we hve f(h) = f(λ P )= f(λ ) P (3) Relxton of the hermtcty ssumpton Let M be ny n n mtrx (no symmetry propertes ssumed) We proceed from the observton tht, whle M nd ts trnspose M T hve dentcl spectr, 2 they cn be expected to hve dstnct egenvectors We re led thus to dstngush rght egenvectors defned 2 So long s M remns unspeclzed we cn sy nothng bout the ny specl propertes of the egenvlues

3 Relxton of the hermtcty ssumpton 3 from left egenvectors, defned M = λ M T b = λ b equvlently b T M = λ b T Immedtely λ b b T M j = T j on the one hnd λ j b T j on the other from whch we conclude tht the sets, 2,, n nd b, b 2,,b n re borthogonl n the followng sense: b j f λ λ j Here b j mens tht (b, j ) b T j = Note tht n the precedng equtons we encounter the smple trnspose, not the conjugted trnspose And tht the / b dstncton dsppers when M s rel symmetrc: borthogonlty reduces then to smple orthogonlty Use the mterl now n hnd to defne b b 2 b n [ P b T ] (, b) = (, 2 b 2 b 2 2 b n b) n b n b 2 n b n nd observe tht P = P j = f λ λ j (by borthogonlty) b T P = b T b T P j = T f λ λ j (by borthogonlty) Moreover, P P = P (5) P P j = O f λ λ j (by borthogonlty) (52) nd f the spectrum s non-degenerte t s ssuredly the cse tht P = I (53) Fnlly, we hve ths unverslly vld generlzton of the fmlr spectrl resoluton formul (2): M = λ P (54) In the presence of spectrl degenerces we cn by contrvnce rrnge for equtons (5) to be vld (4)

4 4 Generlzed spectrl resoluton Frst pplcton: Hmltonn genertors of quntum gtes The controlled evoluton of the stte Ψ s ccomplshed by the cton of gtes, represented by untry mtrces the desgns of whch s reflect the bsc elements of Boolen logc The cton of such gtes s quntum dynmcl Ψ ψ t = e (/ħ) H t Ψ becomes Ugte t tme t = In quntum mechncs we re most commonly gven H, nd sked to construct U(t), but here we confront the nverse problem: we re gven U gte nd sked to construct the genertor H gte of tht mtrx Snce the untrty of U gte mples the hermtcty of H gte we cn solve the problem by ppel smply to (2), don t n ths nstnce need the generlty of (54) I llustrte the procedure by lookng to specfc exmple: The most common nstnce of the mportnt cnot ( controlled NOT ) gte s defned 3 U cnot = Mthemtc supples λ,λ 2,λ 3,λ 4 =,,, nd the (unnormlzed but orthogonl) egenvectors h =, h 2 = + from whch we obtn the projecton mtrces P =, P 2 = , h 3 =, h 4 =, So we hve P 3 =, P 3 = U cnot =( )P + (+)Q (6) 3 See N Dvd Mermn, Quntum Computer Scence: An Introducton (27), pge

5 Applctons 5 wth Equton (6) cn now be wrtten Q P 2 + P 3 + P 4 = U cnot = e π P + e π P Q = e = e (/ħ) H cnot t t= 2 wth H cnot = π ħp Second pplcton: Contnuously nterpolted Mrkoff processes Let the elements of p p 2 P = p n be probbltes, wth p = Mrkoff processes hve the structure P k P k = M P k = M k P where the elements of M = m j re trnston probbltes nd elements of Pk = : ll k requres tht the columns of M sum to unty: m j = (ll j) Look, for exmple, to the cse M = The egenvlues re λ,λ 2,λ 3 =, 79, 79 It s chrcterstc of Mrkoff mtrces tht one of the egenvlues s unty, nd the others hve bsolute vlues tht re less thn one The negtve egenvlue s dmssble, but hs complex logrthm To vod the complex probbltes to whch we would be led when we construct the mtrx M t tht nterpoltes between the mtrces M k (k =,, 2,) I therefore dopt ths modfed defnton: M =

6 6 Generlzed spectrl resoluton The spectrum of ths M s the ssuredly non-negtve squre of the prevous spectrum (t reds, 624, 32 ) We construct the rght egenvectors, 2, 3, the left egenvectors b, b 2, b 3, nd from them ssemble P = P 2 = P 3 = T b (, b ) = T 2 b 2 = mtrx of undstngushed ppernce ( 2, b 2 ) 3 b 3 T ( 3, b 3 ) = dtto whch do n fct comprse complete set of orthogonl projecton mtrces We now hve M = P +(624) P 2 +(32) P 3 gvng M k = P +(624) k P 2 +(32) k P 3 (7) = P n the lmt k The mplcton s tht ll ntl probblty vectors P proceed symptotclly to the stte P = = sum of the elements of 48 Notce tht the elements of P re precsely the elements tht we see repeted n P re for redly understood resons precsely the elements tht we see repeted n the columns of P Returnng now to (7), we hve M = e log P + e log 624 P 2 + e log 32 P 3 47 P2 344 P3 = e gvng the nterpoltng mtrx M t = e L t wth L = log λ 2 P 2 + log λ 3 P 3 (8) In physcl pplctons the elements of M re subject to prncple of detled blncng: m j = m j The nlyss proceeds then not from (54) but from the more fmlr equton (2) It s found n such cses tht L s symmetrc, nd tht the elements n ts columns (rows) sum to zero And tht the clculton typclly proceeds L = M rther thn (s bove) M = L,

7 Applctons 7 wth the structure of L red drectly from (sy) the djcency mtrx of grph (s n Mtt Jemelt s thess (29)) Thrd pplcton: Proof of n elegnt dentty One frequently encounters rguments tht hnge on the dentty det M = e tr log M, e, log det M = tr log M proofs of whch usully pertn only to cses n whch M s equvlent to dgonl mtrx: M = S DS We re n poston now to construct proof whch s subject to no such lmtton For we hve M = λ P () = exp log λ P () where the λ re dstnct nd P () projects onto the δ -dmensonl egenspce V () Fmlrly, det M =(λ ) δ (λ 2 ) δ2 (λ ν ) δν But the trce of projecton mtrx s the dmenson of the spce onto whch t projects, so we hve tr log λ P () = δ log λ = log (λ ) δ (λ 2 ) δ2 (λ ν ) δν QED Concludng comments In ths short note my ntent hs been to mke more convenently vlble some of the mterl of whch I mde crtcl use n Mthemtc notebook ( New Mrkoff: Clsscl/Quntum Mrkoff Processes (22 Aprl 29)) wrtten n conjuncton wth Mtt Jemelt s thess, whch s concerned wth clsscl/quntum rndom wlks on grphs The hert of the note resdes n the generlzed spectrl representton (54) to whch my ttle refers I hve no doubt tht mthemtcns would consder (54) to be commonplce trvlty, but thnk t fr to sy tht (54) nd ts powerful mplctons re unfmlr to most physcsts though t ws few lnes n pper by physcst tht ntroduced me to ths topc 4 To sy the sme thng nother wy, one only seldom encounters references n the physcs lterture to borthogonlty, though t underles the entry of the recprocl lttce nto the sold stte physcs of perodc structures (crystls) 5 Ths hs probbly to do wth the fct tht the mtrces encountered n physcl pplctons re usully (nt)symmetrc or rottonl (n ether the Euclden or Lorentzn sense), (nt)hermtn or untry seldom symmetrc Or rectngulr, n whch context somethng very lke the present lne of rgument leds to the sngulr vlue decomposton (SVD) 4 I llude to remrks on pge 48 of Ellott W Montroll s Mrkoff chns, Wener ntegrls nd quntum theory, Comm Pure & Appl Mth 5, (952) 5 I hve explored spects of ths subject n Recrocl systems of nonorthogonl quntum sttes (998)