{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 
0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 
0 1 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "# This spread sheet
 computes the answers to the example problem\n# of Homework 1.\n\n#  P
lease do not use the Maple worksheet QC6_Lib.mws for this example\n#  \+
This worksheet works without it.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(linalg):\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warni
ng, the protected names norm and trace have been redefined and unprote
cted\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "# We wish to obs
erve the state psi with respect to the observable A\n\npsi:=evalm( (1/
sqrt(3)) * matrix(4,1,[1,I,0,-1]) );\nA:=matrix(4,4, [0, 0,1,-I,\n    \+
            0, 0,I,-1,\n                1,-I,0, 0,\n                I,
-1,0, 0  ]  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiG-%'matrixG6#7
&7#,$*$-%%sqrtG6#\"\"$\"\"\"#F0F/7#*&^#F1F0F,F07#\"\"!7#,$F+#!\"\"F/" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"!F*\"\"\"^
#!\"\"7&F*F*^#F+F-7&F+F,F*F*7&F/F-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 275 "# We now compute the two eingenvals of A along with \+
their eigenvectors\n\nList:=[eigenvects(A)];\na1:=op(1, op(1,List) );
\nv1:=op(1, op(3, op(1,List) )  );\nv2:=op(2, op(3, op(1,List) )  );\n
\na2:=op(1, op(2,List) );\nv3:=op(1, op(3, op(2,List) )  );\nv4:=op(2,
 op(3, op(2,List) )  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ListG7$7
%*$-%%sqrtG6#\"\"#\"\"\"F+<$-%'vectorG6#7&\"\"!,$F'!\"\"^#F,F,-F/6#7&F
,F5F'F27%F3F+<$-F/6#7&F2F'F5F,-F/6#7&F,F5F3F2" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#a1G*$-%%sqrtG6#\"\"#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v1G-%'vectorG6#7&\"\"\"^#F)*$-%%sqrtG6#\"\"#F)\"\"!
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G-%'vectorG6#7&\"\"!,$*$-%%sq
rtG6#\"\"#\"\"\"!\"\"^#F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G,
$*$-%%sqrtG6#\"\"#\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v3
G-%'vectorG6#7&\"\"!*$-%%sqrtG6#\"\"#\"\"\"^#F/F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#v4G-%'vectorG6#7&\"\"\"^#F),$*$-%%sqrtG6#\"\"#F)!\"
\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "# We now you th
e GramSchmidt to create orthogonal bases for the two eigenspaces V+ an
d V-.\n\nVectorSet1:=GramSchmidt(\{v1,v2\});\nww1:=op(1,VectorSet1);  \+
 ww2:=op(2,VectorSet1);\nVectorSet2:=GramSchmidt(\{v3,v4\});\nww3:=op(
1,VectorSet2);   ww4:=op(2,VectorSet2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%+VectorSet1G<$7&\"\"\"^#F'*$-%%sqrtG6#\"\"#F'\"\"!7&*&^##!\"\"
F-F'F*F',$F)F2F.F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ww1G7&\"\"\"^
#F&*$-%%sqrtG6#\"\"#F&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ww2G
7&*&^##!\"\"\"\"#\"\"\"-%%sqrtG6#F*F+,$*$F,F+F(\"\"!F+" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%+VectorSet2G<$7&\"\"!*$-%%sqrtG6#\"\"#\"\"\"^#F-
F-7&F-F',$F(#!\"\"F,*&^#F1F-F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$ww3G7&\"\"!*$-%%sqrtG6#\"\"#\"\"\"^#F,F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$ww4G7&\"\"\"\"\"!,$*$-%%sqrtG6#\"\"#F&#!\"\"F-*&^#F.
F&F*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "# We now normali
ze the eigenkets\n\nfor i from 1 to 4 do\n   w||i:=convert(evalm( (1/n
orm(ww||i, 2)) * ww||i ), matrix):\nod;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#w1G-%'matrixG6#7&7##\"\"\"\"\"#7#^#F*7#,$*$-%%sqrtG6#F,F+F*7#
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w2G-%'matrixG6#7&7#^##!\"
\"\"\"#7#F+7#\"\"!7#,$*$-%%sqrtG6#F-\"\"\"#F7F-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#w3G-%'matrixG6#7&7#\"\"!7#,$*$-%%sqrtG6#\"\"#\"\"\"#
F2F17#^#F37#F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w4G-%'matrixG6#7&
7#,$*$-%%sqrtG6#\"\"#\"\"\"#F0F/7#\"\"!7##!\"\"F/7#^#F5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "# Thus, the eigenvectors for the e
igenvalue a1 are w1 and w2.\n# They form an orthonormal basis of the e
igenspace  V+\n# Also, the eigenvectors for the eigenvalue a2 are w3 a
nd w4.\n# They form an orthonormal basis of the eigenspace  V- \n# Mor
eover, all w1 & w2 are each mutually orthogonal to w3 & w4\n# because \+
eigenvectors corresponding to distinct eigenvectors\n# must be orthogo
nal.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "# We can now ver
ify that the spectral decomposition theorem is true,\n# which of cours
e it is.\n\nProj1:=evalm( w1 &* htranspose(evalm(w1)) );\nProj2:=evalm
( w2 &* htranspose(evalm(w2)) );\nProj3:=evalm( w3 &* htranspose(evalm
(w3)) );\nProj4:=evalm( w4 &* htranspose(evalm(w4)) );\n\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&Proj1G-%'matrixG6#7&7&#\"\"\"\"\"%^##!\"
\"F,,$*$-%%sqrtG6#\"\"#F+F*\"\"!7&^#F*F**&F8F+F2F+F67&F0*&F-F+F2F+#F+F
5F67&F6F6F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Proj2G-%'matrixG6#
7&7&#\"\"\"\"\"%^#F*\"\"!*&^##!\"\"F,F+-%%sqrtG6#\"\"#F+7&F0F*F.,$*$F3
F+F17&F.F.F.F.7&*&F-F+F3F+F8F.#F+F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%&Proj3G-%'matrixG6#7&7&\"\"!F*F*F*7&F*#\"\"\"\"\"#*&^##!\"\"\"\"%F-
-%%sqrtG6#F.F-,$*$F4F-#F-F37&F**&^#F9F-F4F-F9F<7&F*F7F0F9" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&Proj4G-%'matrixG6#7&7&#\"\"\"\"\"#\"\"!,$*$-
%%sqrtG6#F,F+#!\"\"\"\"%*&^##F+F5F+F0F+7&F-F-F-F-7&F.F-F8^#F37&*&F;F+F
0F+F-F7F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "# So the Proj
ection operators for the eigenspace V+ and V- are respectively\nProjPl
us:= evalm(Proj1 + Proj2);\nProjMinus:=evalm(Proj3 + Proj4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)ProjPlusG-%'matrixG6#7&7&#\"\"\"\"\"#\"\"
!,$*$-%%sqrtG6#F,F+#F+\"\"%*&^##!\"\"F4F+F0F+7&F-F**&^#F3F+F0F+,$F/F77
&F.F5F*F-7&F:F<F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ProjMinusG-%
'matrixG6#7&7&#\"\"\"\"\"#\"\"!,$*$-%%sqrtG6#F,F+#!\"\"\"\"%*&^##F+F5F
+F0F+7&F-F**&^#F3F+F0F+,$F/F87&F.F6F*F-7&F:F<F-F*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 182 "# Since by the spectral decomposition theor
em, A = a1* ProjPlus + a2*ProjMinus,\n# we should get A with the follo
wing computation\n\nevalm(  evalm(a1*ProjPlus) + evalm(a2*ProjMinus)  \+
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"!F(\"\"\"^#!
\"\"7&F(F(^#F)F+7&F)F*F(F(7&F-F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 274 "# We now compute the coefficients of the expansion o
f psi in\n# the eigenbasis w1, w2, w3, w4\n\nalpha||1:=evalm(htranspos
e(w1) &* psi)[1,1];\nalpha||2:=evalm(htranspose(w2) &* psi)[1,1];\nalp
ha||3:=evalm(htranspose(w3) &* psi)[1,1];\nalpha||4:=evalm(htranspose(
w4) &* psi)[1,1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha1G,$*$-%
%sqrtG6#\"\"$\"\"\"#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha2G
,$*&-%%sqrtG6#\"\"#\"\"\"-F(6#\"\"$F+#!\"\"\"\"'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'alpha3G,&*(^##\"\"\"\"\"'F)-%%sqrtG6#\"\"#F)-F,6#\"
\"$F)F)*&#F)F*F)*$F/F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'al
pha4G,&*&-%%sqrtG6#\"\"#\"\"\"-F(6#\"\"$F+#F+\"\"'*&^##!\"\"F0F+F,F+F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "# We now verify that t
hese coefficients are correct.\n\nmap(simplify, evalm(evalm(alpha1*w1 \+
+ alpha2*w2) + evalm(alpha3*w3 + alpha4*w4)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7&7#,$*$-%%sqrtG6#\"\"$\"\"\"#F.F-7#*&^#F/F
.F*F.7#\"\"!7#,$F)#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
172 "# Thus, the probability that eigenvalue a1 is read\n# as a result
 of the measurement is:\n\nProbPlus:= abs(alpha1)^2 + abs(alpha2)^2;\n
ProbMinus:=abs(alpha3)^2 + abs(alpha4)^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)ProbPlusG#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%*ProbMinusG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 326 "# The respective resulting states are:\n\nStatePlus:= map(sim
plify, \n          evalm( (1/sqrt(abs(alpha1)^2 + abs(alpha2)^2)) * ev
alm(alpha1*w1 + alpha2*w2) )\n                       );\nStateMinus:=m
ap(simplify, \n          evalm( (1/sqrt(abs(alpha3)^2 + abs(alpha4)^2)
) * evalm(alpha3*w3 + alpha4*w4) )\n                       );\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*StatePlusG-%'matrixG6#7&7#,&*&-%%sq
rtG6#\"\"#\"\"\"-F-6#\"\"$F0#F0\"\"'*&^#F4F0F1F0F07#,$*(F,F0F1F0,&^#F/
F0*$F,F0F0F0#F0\"#77#,$*$F1F0#F0F37#,$F+#!\"\"F5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%+StateMinusG-%'matrixG6#7&7#,&*&-%%sqrtG6#\"\"#\"\"\"
-F-6#\"\"$F0#F0\"\"'*&^##!\"\"F5F0F1F0F07#,&*(^#F4F0F,F0F1F0F0*&#F0F5F
0*$F1F0F0F97#,$F@#F9F37#,$F+F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "14 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }