The Exercise

Two quantum systems $A$ and $B$ are in a joint state $\rho$ , a mixture of mutually orthogonal pure states $|\psi_1\rangle, \dots, |\psi_n\rangle$ of the pair with positive weights $\lambda_1, \dots, \lambda_n$ summing to one. The component $|\psi_i\rangle$ alone would leave $A$ in the reduced state $\rho^A_i$ and $B$ in the reduced state $\rho^B_i$ . The triangle inequality of Araki and Lieb bounds the joint entropy $S(A,B)$ of $\rho$ from below by $S(B) - S(A)$ , the difference between the entropies of its reduced states $\rho^B$ and $\rho^A$ . The exercise asks what the $\rho^A_i$ and the $\rho^B_i$ must satisfy for $\rho$ to reach that bound.

States of Two Systems

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ Two systems $A$ and $B$ form the composite system $A \otimes B$ , whose state space is the tensor product of the state spaces of $A$ and $B$ .^✦

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$ .^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$ ^✦; it determines the state $|\psi\rangle\langle\psi|$ of $S$ , the projector onto $|\psi\rangle$ .^✦ Given states $\rho_1, \dots, \rho_n$ of $S$ and weights $\lambda_i \geq 0$ with $\sum_i \lambda_i = 1$ , the mixture of the $\rho_i$ is the state $\sum_{i=1}^{n} \lambda_i \rho_i$ (1)^✦ of $S$ .

For a state $\rho$ of $A \otimes B$ , the reduced state of $A$ is the partial trace $\rho^A = \operatorname{tr}_B \rho$ over $B$ , a state of $A$ , and the reduced state of $B$ is the partial trace $\rho^B = \operatorname{tr}_A \rho$ over $A$ , a state of $B$ .^✦

The support $\operatorname{supp}\rho$ of a state $\rho$ of $S$ is its range, the subspace spanned by the eigenvectors of $\rho$ with nonzero eigenvalue; the support of $|\psi\rangle\langle\psi|$ is the line through $|\psi\rangle$ .^✦ States $\rho_1, \dots, \rho_n$ of $S$ have orthogonal supports when $\operatorname{supp}\rho_i \perp \operatorname{supp}\rho_j$ for every $i \neq j$ .^✦

The von Neumann entropy of a state $\rho$ is $S(\rho) = -\operatorname{tr}(\rho \ln \rho),$ (2)^✦ the Shannon entropy of the eigenvalues of $\rho$ , in nats.

The Claim

Theorem 1. Let $A$ and $B$ be quantum systems and let $|\psi_1\rangle, \dots, |\psi_n\rangle$ be pure states of $A \otimes B$ whose projectors $|\psi_i\rangle\langle\psi_i|$ have orthogonal supports. Let $\rho = \sum_i \lambda_i |\psi_i\rangle\langle\psi_i|$ be their mixture with positive weights $\lambda_i$ summing to one, and for each $i$ let $\rho^A_i = \operatorname{tr}_B |\psi_i\rangle\langle\psi_i|$ and $\rho^B_i = \operatorname{tr}_A |\psi_i\rangle\langle\psi_i|$ . Then $S(A,B) = S(B) - S(A) \iff \rho^A_i = \rho^A_j \ \text{ and } \ \operatorname{supp}\rho^B_i \perp \operatorname{supp}\rho^B_j \ \text{ for all } i \neq j .$ (3)^✦

Two Classically Correlated Qubits

States $\rho_1, \dots, \rho_n$ of $S$ have a common eigenbasis when some orthonormal basis of the state space of $S$ consists of eigenvectors of every $\rho_i$ .^✦

The qubit is the quantum system whose state space is $\mathbb{C}^2$ ^✦; its basis states $|0\rangle$ and $|1\rangle$ are the pure states whose vectors are the two standard basis vectors of $\mathbb{C}^2$ .^✦

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.^✦

On two qubits $A$ and $B$ , the products of equal basis states are $|ii\rangle = |i\rangle \otimes |i\rangle, \qquad i = 0, 1,$ (4)^✦ and the even mixture of their projectors is the classically correlated state $\rho = \tfrac{1}{2}\bigl(|00\rangle\langle 00| + |11\rangle\langle 11|\bigr)$ (5)^✦ of the pair. Each $|ii\rangle$ leaves both qubits in the state $|i\rangle\langle i|$ .

Theorem 2. The state $\rho$ of two qubits in (5) is a counterexample to the weaker condition: the reduced states $\rho^A_i = \operatorname{tr}_B |ii\rangle\langle ii|$ of the first qubit have a common eigenbasis and the reduced states $\rho^B_i = \operatorname{tr}_A |ii\rangle\langle ii|$ of the second have orthogonal supports, yet $S(A,B) \neq S(B) - S(A).$ (6)^✦

By the first theorem equality would need $\rho^A_0 = \rho^A_1$ , and $|0\rangle\langle 0| \neq |1\rangle\langle 1|$ .

What Is Checked

Both theorems are stated over states of a composite system $A \otimes B$ and of its two factors: pure states and their projectors, the mixture (1), the two partial traces, supports and the entropy (2). The second is stated also over the qubit, its basis states, their products (4) and the common eigenbasis.

About these constructions the library proves that the projector onto a unit vector is a density operator, and that a mixture of states with nonnegative weights summing to one is again a state. It proves that each partial trace of a state of $A \otimes B$ is a state of the remaining factor, and that the product of two pure states is a unit vector, since the norm of a product vector is the product of the norms.