# Trace of a subsystem of a two qubit system

#### Haorong Wu

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.

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#### cpt_carrot

You might want to think again about the result you got for $\rho^{AB}$

#### Haorong Wu

You might want to think again about the result you got for $\rho^{AB}$
Hi, cpt_carrot. I still can't figure the mistake.

Here is my reasoning:

There are three possible states : $\left | 00 \right> , \left | 01 \right> , \left | 11 \right>$ all with probabilitities of $\frac 1 3$.

So $\rho \equiv \sum_i p_i \left | \psi _i \right > \left < \psi _i \right |=\frac { \left |00 \right > \left <00 \right |+ \left |01 \right > \left < 01\right |+\left | 11\right > \left < 11\right | } 3$.

Maybe I understand the definition of density operator in a wrong way.

Could you help me point the mistake? Thanks!

#### cpt_carrot

You need to include the cross terms in the outer product. The density matrix for your pure state $|\psi\rangle$ is $\rho= |\psi\rangle\langle\psi|$ which includes terms like $|00\rangle\langle 01|$

#### Haorong Wu

You need to include the cross terms in the outer product. The density matrix for your pure state $|\psi\rangle$ is $\rho= |\psi\rangle\langle\psi|$ which includes terms like $|00\rangle\langle 01|$
Oh, so $\rho \equiv \sum_i p_i \left | \psi _i \right > \left < \psi _i \right |$ should be applied to a collect of pure states.
I am going to redo the calculation again.
Thanks, cpt_carrot!

"Trace of a subsystem of a two qubit system"

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