The Exercise

A quantum system $Q$ with a $d$-dimensional state space is prepared in a state $\rho$ and sent through a noisy process, a trace-preserving quantum operation $\mathcal{E}(\rho) = \sum_i E_i \rho E_i^\dagger$ whose operation elements $E_1, \dots, E_n$ satisfy the completeness relation $$\sum_{i=1}^{n} E_i^\dagger E_i = I .$$(1) Two numbers are attached to the pair $\rho$ and $\mathcal{E}$: the entanglement fidelity $F(\rho, \mathcal{E})$, the fidelity between $\rho$ and the state the operation leaves behind, each purified with a reference system, and the entropy exchange $S(\rho, \mathcal{E})$. Theorem 12.9, the quantum analogue of Fano’s inequality, bounds the entropy exchange from above by a function of the entanglement fidelity and of the dimension $d$.

States and Entropy

The state space of $Q$ is a complex Hilbert space $\mathcal{H}$ of dimension $d$. A state $\rho$ of $Q$ is a density operator on $\mathcal{H}$: a positive operator on $\mathcal{H}$ of unit trace.^✦

The Shannon entropy of a probability distribution $p = (p_i)$ on a finite set is $$H(p) = -\sum_i p_i \log p_i ,$$(2)^✦ with the natural logarithm, so that entropies are measured in nats.

The von Neumann entropy $S(T)$ of a Hermitian operator $T$ on an inner product space of finite dimension $m$ is the Shannon entropy of its eigenvalues $\lambda_1, \dots, \lambda_m$, counted with multiplicity: $$S(T) = H(\lambda_1, \dots, \lambda_m) = -\sum_{i=1}^{m} \lambda_i \log \lambda_i .$$(3) For an operator that is not Hermitian, $S(T)$ is zero.

The Entropy Exchange

For operators $\rho$ and $E_1, \dots, E_n$ on $\mathcal{H}$, the matrix $W$ is the $n \times n$ complex matrix with entries $$W_{jk} = \operatorname{tr}\bigl(E_j \rho E_k^\dagger\bigr), \qquad j, k = 1, \dots, n .$$(4)^✦ The entropy exchange $S(\rho, \mathcal{E})$ of $\rho$ under the operation with elements $E_1, \dots, E_n$ is the von Neumann entropy of $W$, taken as an operator on $\mathbb{C}^n$: $$S(\rho, \mathcal{E}) = S(W) .$$(5)^✦

The Claim

The bound enters through the binary entropy $h(p) = H(p, 1-p)$, the Shannon entropy (2) of the two-outcome distribution $(p, 1-p)$, evaluated at the entanglement fidelity.

What Is Checked

The claim is stated over the density operators on a complex Hilbert space of finite dimension $d$ and over every finite family of operators obeying (1). Its left side is the entropy exchange (5), built from the matrix $W$ of (4) and the von Neumann entropy (3), which rests in turn on the Shannon entropy (2).

About these constructions the library proves the inequality (6) in every finite dimension and for any finite number of operation elements.