What is the Reduced Density Matrix?

The reduced matrix is defined as the partial trace of the density matrix.

Be $A$, $B$ finite dimensional Hilbert spaces and be $T$ $\in$ $L(A \otimes B)$ (Linear operators on $A \otimes B$), then the partial trace of T is defined as $\rm{Tr}_B [T]$ in $L(A)$ is defined by

\begin{equation} \langle a | \rm{Tr}_B [T]| b \rangle = \sum_n \langle a | \langle n | T| n\rangle | b \rangle \end{equation}

where $| n \rangle$ is an orthonormal basis in $B$, $|a\rangle$ and $|b\rangle$ are vectors in $A$.

Finally, note that the reduced matrix isn't the correct way of the describing a quantum state, is just a way to describe it as seen by looking just at a subsystem. This usually involves ignoring part of the information of the state and therefore the reduced density matrix of a pure state may be a mixed state. This is spectacular for the Bell states, as their reduced matrix is $\rm{Id}/2$, the most disordered state.

The reduced density matrix can be found by taking the trace over the subspaces of the Hilbert space that represent systems you're not interested in. For the Bell state the density matrix of the whole system is $$\tfrac{1}{2}(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)\\ = \tfrac{1}{2} (|00\rangle\langle 00|+|00\rangle\langle 11|+|11\rangle\langle 00|+|11\rangle\langle 11|)\\ \tfrac{1}{2}(|0\rangle\langle 0|\otimes|0\rangle\langle 0|+|0\rangle\langle 1|\otimes|0\rangle\langle 1|+|1\rangle\langle 0|\otimes|1\rangle\langle 0|+|1\rangle\langle 1|\otimes|1\rangle\langle 1|)$$.

So to get the reduced density matrix for the first qubit you take the trace over the Hilbert space for the second qubit. You take all the terms that have $|0\rangle\langle 0|$ or $|1\rangle\langle 1|$ for the second qubit, throw out the rest and then just take the parts of those terms that refer to the first qubit. This gives $$\rho_1 = \tfrac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|),$$ and $tr(\rho_1^2)<1$.

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For more detail and rigour, you should read "Quantum Computation and Quantum Information" by Nielsen and Chuang.