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 10
 Problem Statement

From Nielsen's QC&QI, in page 109 (The schmidt decomposition), it reads that:
As an example, consider the state of two qubits, ##\left( \left  00 \right> +\left  01 \right> +\left  11 \right> \right) / \sqrt 3##. This has no obvious symmetry property, yet if you calculate ##tr \left ( {\left( \rho ^A \right )}^2 \right )## and ##tr \left ( {\left( \rho ^B \right )}^2 \right )## you will discover that they have the same value, ##\frac 7 9## in each case.
 Relevant Equations

The density operator for a system is ## \rho \equiv \sum_i p_i \left \psi _i \right> \left < \psi _i \right ##.
Also, ## tr \left( A \left  \psi \right > \left < \psi \right  \right) =\left < \psi \left A \right  \psi \right > ##
Consider the first qubit (subsystem A):
First, the density operator for the system AB is ## \rho ^{AB} =\frac { \left 00 \right > \left <00 \right + \left 01 \right > \left < 01\right +\left  11\right > \left < 11\right  } 3 ##.
Then, the reduced density operator of subsystem A is ## \rho ^A = \frac { \left 0 \right > \left <0 \right + \left 0 \right > \left < 0\right +\left  1\right > \left < 1\right  } 3 =\frac { 2\left 0 \right > \left <0 \right +\left  1\right > \left < 1\right  } 3 ##.
Thus, ## \left ( \rho ^A \right ) ^2=\frac { 4\left 0 \right > \left <0 \right +\left  1\right > \left < 1\right  } 9##.
So, ## tr \left ( {\left( \rho ^A \right )}^2 \right ) =\frac 5 9##.
I overchecked the procedure several times, but I can't see where am I wrong.
Thanks for reading.
First, the density operator for the system AB is ## \rho ^{AB} =\frac { \left 00 \right > \left <00 \right + \left 01 \right > \left < 01\right +\left  11\right > \left < 11\right  } 3 ##.
Then, the reduced density operator of subsystem A is ## \rho ^A = \frac { \left 0 \right > \left <0 \right + \left 0 \right > \left < 0\right +\left  1\right > \left < 1\right  } 3 =\frac { 2\left 0 \right > \left <0 \right +\left  1\right > \left < 1\right  } 3 ##.
Thus, ## \left ( \rho ^A \right ) ^2=\frac { 4\left 0 \right > \left <0 \right +\left  1\right > \left < 1\right  } 9##.
So, ## tr \left ( {\left( \rho ^A \right )}^2 \right ) =\frac 5 9##.
I overchecked the procedure several times, but I can't see where am I wrong.
Thanks for reading.