Re: Help with a calculation
- To: mathgroup at smc.vnet.net
- Subject: [mg51816] Re: Help with a calculation
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 3 Nov 2004 01:23:32 -0500 (EST)
- Organization: The University of Western Australia
- References: <clnds5$o70$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <clnds5$o70$1 at smc.vnet.net>,
ss54 at york.ac.uk (Simone Severini) wrote:
> I' m here asking for some help with the following calculation:
>
> $\sum_{r=0}^{3}\operatorname{Re}\left( \left( \alpha_{3,r}^{\ast}%
> -\alpha_{4,r}^{\ast}\right) \left( \alpha_{2,r}e^{-i\phi_{r}}-\alpha
> _{1,r}e^{i\phi_{r}}\right) \right) =2\sqrt{2}$
>
> $\sum_{r=0}^{3}\alpha_{j,r}\alpha_{k,r}^{\ast}=\delta_{j,k}$ with
> $j,k=1,2,3,4$
>
> $\sum_{j=1}^{4}\left\vert \alpha_{j,r}\right\vert ^{2}=1$ for $r=0,1,2,3$
>
> Is Mathematica able to find solutions?
>
> In case of affirmative answer, how do I program Mathematica for this task?
Your question is a mathematical one, not really a Mathematica question.
Your last two equations are simply the requirement that the 4x4 matrix
A = Table[alpha[j,r], {j,4}, {r,0,3}]
is a unitary matrix of determinant one, i.e., an element of SU(4). The
dimension of SU(n) is n^2 - 1 and so a parameterization of SU(4) has 15
generators. (This is a significant reduction: writing the real and
imaginary part of alpha[j,r] = x[j,r] + I y[j,r], one has 32
non-independent parameters). See, e.g, "A parametrization of bipartite
systems based on SU(4) Euler angles" by T Tilma, M Byrd and E C G
Sudarshan, J. Phys. A: Math. Gen. 35 No 48 (6 December 2002) 10445-10465.
The first equation relates the values of phi[r], r = 0,1,2,3, but the
phi[r] are not determined by this equation.
See the Notebook appended below for a simpler example.
> Apologies, in the case my question is in some way naive.
And let me guess the application: some computation involving two qubit
density matrices?
Cheers,
Paul
(************** Content-type: application/mathematica **************
CreatedBy='Mathematica 5.0'
Mathematica-Compatible Notebook
This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.
To get the notebook into a Mathematica-compatible application, do
one of the following:
* Save the data starting with the line of stars above into a file
with a name ending in .nb, then open the file inside the
application;
* Copy the data starting with the line of stars above to the
clipboard, then use the Paste menu command inside the application.
Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode. Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).
NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.
For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
web: http://www.wolfram.com
email: info at wolfram.com
phone: +1-217-398-0700 (U.S.)
Notebook reader applications are available free of charge from
Wolfram Research.
*******************************************************************)
(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 10375, 324]*)
(*NotebookOutlinePosition[ 11013, 346]*)
(* CellTagsIndexPosition[ 10969, 342]*)
(*WindowFrame->Normal*)
Notebook[{
Cell[CellGroupData[{
Cell["SU(2) Example", "Section"],
Cell["Introduce the following definitions:", "Text"],
Cell[BoxData[
\(TraditionalForm\`\[Alpha]\_\(n_, r_\) :=
x\_\(n, r\) + \[ImaginaryI]\ y\_\(n, r\)\)], "Input"],
Cell[BoxData[
\(TraditionalForm\`\(x_\^*\) :=
x /. \[InvisibleSpace]Complex[a_, b_] \[RuleDelayed]
Complex[a, \(-b\)]\)], "Input"],
Cell[BoxData[
\(TraditionalForm\`x_\^\[Dagger] := \(x\^*\)\^T\)], "Input"],
Cell[TextData[{
"The general ",
Cell[BoxData[
\(TraditionalForm\`SU(2)\)]],
" matrix has the form ",
Cell[BoxData[
FormBox[
RowBox[{"(", GridBox[{
{"\[Alpha]", "\[Beta]"},
{\(-\(\[Beta]\^*\)\), \(\[Alpha]\^*\)}
}], ")"}], TraditionalForm]]],
" with the constraint ",
Cell[BoxData[
\(TraditionalForm\`\[LeftBracketingBar]\[Alpha]\
\[RightBracketingBar]\^2 + \[LeftBracketingBar]\[Beta]\
\[RightBracketingBar]\^2 \[LongEqual] 1\)]],
". Hence we set"
}], "Text"],
Cell[BoxData[
\(TraditionalForm\`\(\[Alpha] /: \[Alpha]\_\(2, 0\) = \(-\(\
\[Alpha]\_\(1, 1\)\%*\)\);\)\)], "Input"],
Cell["and", "Text"],
Cell[BoxData[
\(TraditionalForm\`\(\[Alpha] /: \[Alpha]\_\(2, 1\) = \
\(\[Alpha]\_\(1, 0\)\%*\);\)\)], "Input"],
Cell["Your second and third equations are", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(TraditionalForm\`Table[
ComplexExpand[\[Sum]\+\(r = 0\)\%1\( \[Alpha]\_\(j,
r\)\) \(\[Alpha]\_\(k, r\)\%*\)], {j, 2}, {k,
2}] \[Equal] IdentityMatrix[2]\)], "Input"],
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 + y\_\(1, 0\)\%2 +
y\_\(1, 1\)\%2\), "0"},
{
"0", \(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 +
y\_\(1, 0\)\%2 + y\_\(1, 1\)\%2\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
"\[LongEqual]",
RowBox[{"(", "\[NoBreak]", GridBox[{
{"1", "0"},
{"0", "1"}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}]}],
TraditionalForm]], "Output"]
}, Open ]],
Cell["and", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
RowBox[{"Table", "[",
RowBox[{
RowBox[{\(\[Sum]\+\(j = 1\)\%2\( \[Alpha]\_\(j,
r\)\) \(\[Alpha]\_\(j, r\)\%*\)\), "\[Equal]",
FormBox["1",
"TraditionalForm"]}], ",", \({r, 0, 1}\)}], "]"}],
"//",
"Simplify"}], "//", "Union"}], TraditionalForm]], "Input"],
Cell[BoxData[
\(TraditionalForm\`{x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 +
y\_\(1, 0\)\%2 + y\_\(1, 1\)\%2 \[LongEqual] 1}\)], "Output"]
}, Open ]],
Cell[TextData[{
"With the single constraint, ",
Cell[BoxData[
\(TraditionalForm\`x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 +
y\_\(1, 0\)\%2 + y\_\(1, 1\)\%2 \[LongEqual] 1\)]],
", we have 3 independent parameters as required for ",
Cell[BoxData[
\(TraditionalForm\`SU(2)\)]],
". Alternatively, your second and third equations are equivalent \
to"
}], "Text"],
Cell[CellGroupData[{
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(\[Alpha]\_\(1, 0\)\), \(\[Alpha]\_\(1, 1\)\)},
{\(\[Alpha]\_\(2, 0\)\), \(\[Alpha]\_\(2, 1\)\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}], ".",
SuperscriptBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(\[Alpha]\_\(1, 0\)\), \(\[Alpha]\_\(1, 1\)\)},
{\(\[Alpha]\_\(2, 0\)\), \(\[Alpha]\_\(2, 1\)\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
"\[Dagger]"]}], "//", "Simplify"}],
TraditionalForm]], "Input"],
Cell[BoxData[
FormBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 + y\_\(1, 0\)\%2 +
y\_\(1, 1\)\%2\), "0"},
{
"0", \(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 +
y\_\(1, 0\)\%2 + y\_\(1, 1\)\%2\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
TraditionalForm]], "Output"]
}, Open ]],
Cell["and", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SuperscriptBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(\[Alpha]\_\(1, 0\)\), \(\[Alpha]\_\(1, 1\)\)},
{\(\[Alpha]\_\(2, 0\)\), \(\[Alpha]\_\(2, 1\)\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
"\[Dagger]"], ".",
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(\[Alpha]\_\(1, 0\)\), \(\[Alpha]\_\(1, 1\)\)},
{\(\[Alpha]\_\(2, 0\)\), \(\[Alpha]\_\(2, 1\)\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}]}], "//",
"Simplify"}], TraditionalForm]], "Input"],
Cell[BoxData[
FormBox[
RowBox[{"(", "\[NoBreak]", GridBox[{
{\(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 + y\_\(1, 0\)\%2 +
y\_\(1, 1\)\%2\), "0"},
{
"0", \(x\_\(1, 0\)\%2 + x\_\(1, 1\)\%2 +
y\_\(1, 0\)\%2 + y\_\(1, 1\)\%2\)}
},
RowSpacings->1,
ColumnSpacings->1,
ColumnAlignments->{Left}], "\[NoBreak]", ")"}],
TraditionalForm]], "Output"]
}, Open ]],
Cell["\<\
Here is a simpler equation (involving just two complex \
parameters) of the type you are interested in:\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(TraditionalForm\`eqn = \[Sum]\+\(r = 0\)\%1\ \((\[Alpha]\_\(2, \
r\)\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \[Phi]\_r\) - \
\[Alpha]\_\(1, r\)\ \[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\_r\))\) \
\[Equal] 2 \@ 2\)], "Input"],
Cell[BoxData[
\(TraditionalForm\`\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ \
\[Phi]\_1\)\ \((x\_\(1, 0\) - \[ImaginaryI]\ y\_\(1, 0\))\) - \
\[ExponentialE]\^\(\[ImaginaryI]\ \[Phi]\_0\)\ \((x\_\(1, 0\) + \
\[ImaginaryI]\ y\_\(1, 0\))\) + \[ExponentialE]\^\(\(-\[ImaginaryI]\)\
\ \[Phi]\_0\)\ \((\[ImaginaryI]\ y\_\(1, 1\) -
x\_\(1, 1\))\) - \[ExponentialE]\^\(\[ImaginaryI]\ \
\[Phi]\_1\)\ \((x\_\(1, 1\) + \[ImaginaryI]\ y\_\(1, 1\))\) \
\[LongEqual] 2\ \@2\)], "Output"]
}, Open ]],
Cell["Here are the real and imaginary parts of this equation:", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(TraditionalForm\`\(Collect[
ComplexExpand /@ \(Re /@ #\), {cos(_), sin(_)}] &\) /@
eqn\)], "Input"],
Cell[BoxData[
\(TraditionalForm\`\(cos(\[Phi]\_0)\)\ \((\(-x\_\(1, 0\)\) -
x\_\(1, 1\))\) + \(cos(\[Phi]\_1)\)\ \((x\_\(1, 0\) -
x\_\(1, 1\))\) + \(sin(\[Phi]\_1)\)\ \((y\_\(1, 1\) -
y\_\(1, 0\))\) + \(sin(\[Phi]\_0)\)\ \((y\_\(1, 0\) +
y\_\(1, 1\))\) \[LongEqual] 2\ \@2\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\(TraditionalForm\`\(Collect[
ComplexExpand /@ \(Im /@ #\), {cos(_), sin(_)}] &\) /@
eqn\)], "Input"],
Cell[BoxData[
\(TraditionalForm\`\(sin(\[Phi]\_1)\)\ \((\(-x\_\(1, 0\)\) -
x\_\(1, 1\))\) + \(sin(\[Phi]\_0)\)\ \((x\_\(1, 1\) -
x\_\(1, 0\))\) + \(cos(\[Phi]\_1)\)\ \((\(-y\_\(1, \
0\)\) - y\_\(1, 1\))\) + \(cos(\[Phi]\_0)\)\ \((y\_\(1, 1\) -
y\_\(1, 0\))\) \[LongEqual] 0\)], "Output"]
}, Open ]],
Cell[TextData[{
"Clearly it is possible to solve this pair of equations for ",
Cell[BoxData[
\(TraditionalForm\`\[Phi]\_0\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`\[Phi]\_1\)]],
" (since ",
Cell[BoxData[
\(TraditionalForm\`\(sin\^2\)(\[Phi]\_r) + \
\(cos\^2\)(\[Phi]\_r) \[LongEqual] 1\)]],
") in terms of the parameters ",
Cell[BoxData[
\(TraditionalForm\`x\_\(j, r\)\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`y\_\(j, r\)\)]],
"."
}], "Text"]
}, Open ]]
},
FrontEndVersion->"5.0 for Macintosh",
ScreenRectangle->{{0, 1436}, {0, 878}},
WindowSize->{520, 740},
WindowMargins->{{100, Automatic}, {12, Automatic}}
]
(*******************************************************************
Cached data follows. If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of the file. The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[1776, 53, 32, 0, 69, "Section"],
Cell[1811, 55, 52, 0, 32, "Text"],
Cell[1866, 57, 121, 2, 29, "Input"],
Cell[1990, 61, 152, 3, 28, "Input"],
Cell[2145, 66, 78, 1, 29, "Input"],
Cell[2226, 69, 541, 17, 43, "Text"],
Cell[2770, 88, 120, 2, 30, "Input"],
Cell[2893, 92, 19, 0, 32, "Text"],
Cell[2915, 94, 115, 2, 30, "Input"],
Cell[3033, 98, 51, 0, 32, "Text"],
Cell[CellGroupData[{
Cell[3109, 102, 223, 4, 52, "Input"],
Cell[3335, 108, 756, 21, 51, "Output"]
}, Open ]],
Cell[4106, 132, 19, 0, 32, "Text"],
Cell[CellGroupData[{
Cell[4150, 136, 453, 12, 54, "Input"],
Cell[4606, 150, 144, 2, 30, "Output"]
}, Open ]],
Cell[4765, 155, 381, 10, 68, "Text"],
Cell[CellGroupData[{
Cell[5171, 169, 832, 20, 46, "Input"],
Cell[6006, 191, 461, 12, 51, "Output"]
}, Open ]],
Cell[6482, 206, 19, 0, 32, "Text"],
Cell[CellGroupData[{
Cell[6526, 210, 834, 20, 46, "Input"],
Cell[7363, 232, 461, 12, 51, "Output"]
}, Open ]],
Cell[7839, 247, 127, 3, 50, "Text"],
Cell[CellGroupData[{
Cell[7991, 254, 244, 4, 52, "Input"],
Cell[8238, 260, 489, 8, 49, "Output"]
}, Open ]],
Cell[8742, 271, 71, 0, 32, "Text"],
Cell[CellGroupData[{
Cell[8838, 275, 136, 3, 28, "Input"],
Cell[8977, 280, 349, 5, 48, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[9363, 290, 136, 3, 28, "Input"],
Cell[9502, 295, 336, 5, 46, "Output"]
}, Open ]],
Cell[9853, 303, 506, 18, 50, "Text"]
}, Open ]]
}
]
*)
(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul