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MathGroup Archive 2004

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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

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-- 
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


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