Let us consider a closed quantum system \(S\) consisting of two subsystems \(S_1\) and \(S_2\). The Hilbert space \(\mathcal{H}\) of system \(S\) is a tensor product of Hilbert spaces \(\mathcal{H}_1\) and \(\mathcal{H}_2\) of subsystems \(S_1\) and \(S_2\), respectively, i.e., \[\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2 \,.\] Let \(\{ | a \rangle \}\) and \(\{ | b \rangle \}\) be complete orthonormal bases in \(\mathcal{H}_1\) and \(\mathcal{H}_2\), respectively, so that \[\langle a | a' \rangle = \delta_{aa'} \,, \qquad \langle b | b' \rangle = \delta_{bb'} \,,\] and any state \(| \Psi \rangle\) of \(S\) can be written as \[| \Psi \rangle = \sum_{a,b} \Psi_{ab} | a b \rangle \,, \qquad | a b \rangle \equiv | a \rangle \otimes | b \rangle \,.\] We take \(| \Psi \rangle\) to be normalized to one: \[\langle \Psi | \Psi \rangle = \sum_{a,b} |\Psi_{ab}|^2 = 1 \,.\] For notational simplicity, we treat sets \(\{ | a \rangle \}\) and \(\{ | b \rangle \}\) as being discrete; a generalization to the case of continuous (or mixed) basis sets is straightforward. Also, indexes \(a\) and \(b\) may in general denote sets of quantum numbers.

Let us also consider an observable \(A\) of the subsystem \(S_1\). In the full Hilbert space \(H\), the observable is describe by the operator
\[A = A_1 \otimes I_2 \,,\]
where \(A_1\) is the corresponding operator in \(\mathcal{H}_1\), and \(I_2\) is the identity operator in \(\mathcal{H}_2\). The expectation value of the observable is given by
\[\langle A \rangle \equiv \langle \Psi | A | \Psi \rangle = \sum_{a,a',b,b'} \Psi_{ab}^* \Psi_{a'b'} \langle a b | A| a' b' \rangle \,,\]
or
\[\langle A \rangle = \sum_{a,a',b} \Psi_{ab}^* \Psi_{a'b} \langle a | A_1 | a' \rangle = \sum_{a,a'} \sigma_{a'a} \langle a | A_1 | a' \rangle \,,\]
where
\[\sigma_{a'a} = \sum_b \Psi_{ab}^* \Psi_{a'b} \,.\]
This expectation value can be expressed in terms of a (reduced) *density operator* of the subsystem \(S_1\), defined as
\[\rho_1 = \sum_{a,a'} \sigma_{a'a} | a' \rangle \langle a | \,.\]
The density operator is normalized to one:
\[\operatorname{Tr} \rho_1 = \sum_a \sigma_{aa} = \sum_{a,b} |\Psi_{ab}|^2 = 1 \,.\]
In terms of \(\rho_1\), the expectation value of \(A\) can be written as
\[\langle A \rangle = \operatorname{Tr} \rho_1 A_1 \,.\]
Indeed,
\[\operatorname{Tr} \rho_1 A_1 = \sum_{a'} \langle a' | \rho_1 A_1 | a' \rangle = \sum_{a,a'} \langle a' | \rho_1 | a \rangle \langle a | A_1 | a' \rangle = \sum_{a,a'} \sigma_{a'a} \langle a | A_1 | a' \rangle = \langle A \rangle \,.\]

The density operator has the following properties:

*All eigenvalues of \(\rho_1\) are real and lie inside the interval \([0,1]\).*

**Proof:**Since \[\sigma_{aa'} = \sum_b \Psi_{a'b}^* \Psi_{ab} = \left( \sum_b \Psi_{ab}^* \Psi_{a'b} \right)^* = \sigma_{a'a}^* \,,\] the density operator is Hermitian, \(\rho_1^{\dagger} = \rho_1\). This implies that all eigenvalues of \(\rho_1\) are real. The fact that all eigenvalues of \(\rho_1\) lie between 0 and 1 can be established as follows. For any state \(| \varphi \rangle\) in \(\mathcal{H}_1\), we have \[\langle \varphi | \rho_1 | \varphi \rangle = \sum_{aa'} \sigma_{a'a} \langle \varphi | a' \rangle \langle a | \varphi \rangle = \sum_{a,a',b} \langle \Psi | a b \rangle \langle a | \varphi \rangle \langle \varphi | a' \rangle \langle a' b | \Psi \rangle \,,\] or \[\langle \varphi | \rho_1 | \varphi \rangle = \langle \Psi | P_{\varphi} | \Psi \rangle \,, \qquad P_{\varphi} = | \varphi \rangle \langle \varphi | \otimes I_2 \,.\] Here, \(P_{\varphi}\) is the projection operator that projects any state of subsystem \(S_1\) onto \(| \varphi \rangle\), and its expectation value must lie in the interval \([0,1]\).*The density matrix satisfies*\[\operatorname{Tr} \rho_1^2 \le 1 \,.\]

**Proof:**It follows from the previous property that \(\rho_1\) can be diagonalized: \[\rho_1 = \sum_{n} p_n | n \rangle \langle n | \,,\] where \(\{ | n \rangle \}\) is an orthonormal basis in \(\mathcal{H}_1\), and eigenvalues \(p_n\) are such that \(0 \le p_n \le 1\) and \[\operatorname{Tr} \rho_1 = \sum_n p_n = 1 \,.\] Then, \[\operatorname{Tr} \rho_1^2 = \sum_n p_n^2 \le 1 \,.\]

The subsystem \(S_1\) is in a *mixed* state if \(\operatorname{Tr} \rho_1^2 < 1\), and in a *pure* state if \(\operatorname{Tr} \rho_1^2 = 1\). The latter is the case if and only if the state \(| \Psi \rangle\) of the whole system \(S\) is a product state, \(| \Psi \rangle = | \psi_1 \rangle \otimes | \psi_2 \rangle\) with \(| \psi_1 \rangle \in \mathcal{H}_1\) and \(| \psi_2 \rangle \in \mathcal{H}_2\).