The Exercise

Exercise 2.77 concerns a quantum system made of three parts $A$, $B$ and $C$. A pure state of two parts can always be written, by the Schmidt decomposition, as a sum of products of one state from each part, with real coefficients and with the states of each part orthonormal. For three parts this fails: some pure states are no such sum with an orthonormal family in each part. Three qubits give an example, built below from the basis state $|0\rangle$ of the first and the Bell state $|\Phi^+\rangle$ of the other two.

Three Qubits

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space. A pure state $|\psi\rangle$ of $S$ is a vector of unit norm in the state space of $S$.^✦ Two systems $S$ and $T$ make up the composite system $S \otimes T$, the quantum system whose state space is the tensor product of the state spaces of $S$ and $T$.^✦ For pure states $|\psi\rangle$ of $S$ and $|\varphi\rangle$ of $T$, the product $|\psi\rangle \otimes |\varphi\rangle$ is the pure state of $S \otimes T$ whose vector is the tensor product of the two vectors.^✦

The qubit, written $\mathbb{C}^2$, is the quantum system of two levels, whose state space is $\mathbb{C}^2$.^✦ The basis states $|0\rangle$ and $|1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦ On two qubits we write $|00\rangle$ for $|0\rangle \otimes |0\rangle$ and $|11\rangle$ for $|1\rangle \otimes |1\rangle$. The Bell state $|\Phi^+\rangle$ is the pure state of two qubits whose vector is $|00\rangle + |11\rangle$ divided by its norm: $$|\Phi^+\rangle = \frac{1}{\sqrt{2}}\bigl(|00\rangle + |11\rangle\bigr).$$(1)^✦

The state of the example lives on a qubit and a pair of qubits, the composite system $\mathbb{C}^2 \otimes (\mathbb{C}^2 \otimes \mathbb{C}^2)$. It is the product of the basis state $|0\rangle$ of the single qubit with the Bell state of the pair: $$|0\rangle \otimes |\Phi^+\rangle = \frac{1}{\sqrt{2}}\, |0\rangle \otimes \bigl(|00\rangle + |11\rangle\bigr).$$(2)^✦

The Triorthogonal Decomposition

Let $A$, $B$ and $C$ be quantum systems. A triorthogonal decomposition of a pure state $|\psi\rangle$ of the composite system $A \otimes (B \otimes C)$ is an expansion $$|\psi\rangle = \sum_{i=1}^{n} \lambda_i\, |a_i\rangle \otimes \bigl(|b_i\rangle \otimes |c_i\rangle\bigr)$$(3)^✦ with some number $n$ of terms, real coefficients $\lambda_1, \dots, \lambda_n$, and orthonormal families $|a_1\rangle, \dots, |a_n\rangle$ in the state space of $A$, $|b_1\rangle, \dots, |b_n\rangle$ in that of $B$ and $|c_1\rangle, \dots, |c_n\rangle$ in that of $C$.

The Claim

What Is Checked

The claim is stated over the state (2), the product of the basis state $|0\rangle$ of one qubit with the Bell state (1) of two more, and over the triorthogonal decomposition (3) of a pure state of three systems.

About these constructions the library proves that the product of two pure states is again a unit vector, since the norm of a product vector is the product of the norms, and that $|00\rangle + |11\rangle$ has norm $\sqrt{2}$, so that $|\Phi^+\rangle$ is a unit vector.