Alternative definition of secret sharing using entropy

I ame reading the paper "Secret-Sharing schemes: A survey" by Amos Beimel. Here there are two definitions of secret sharing. The first one states:\ A distribution scheme $\langle \Pi,\mu \rangle$ with domain of secrets $S$ is a secret-sharing scheme realizing an access structure $\mathscr{A}$ if the following holds:

• Correctness: the secret $s$ can be reconstructed by any authorized set of parties. That is, $\forall B \in \mathscr{A}$ there exists a $Recon_B$ algorithm such that $$ \Pr[Recon_B(\Pi(s,r)_B) = s] = 1 \qquad \forall s \in S $$ where $\Pi(s,r)_B$ denotes a restriction of $\Pi(s,r)$ to its $B$ entries.
• Perfect Privacy: every unauthorized set cannot learn anything about the secret from their shares. Formally, $\forall\,T \notin \mathscr{A}, \forall\,a,b \in S$ and for every vector of shares $\langle s_j\rangle_{P_j \in T}$: $$ \Pr[\,\Pi(a,r)_T = \langle s_j\rangle_{P_j \in T}\,] = \Pr[\,\Pi(b,r)_T = \langle s_j\rangle_{P_j \in T}\,]. $$

The second definition uses the notion of entropy and states: A distribution scheme $\langle \Pi,\mu \rangle$ is a secret-sharing scheme realizing an access structure $\mathscr{A}$ if the following conditions hold:

• Correctness: For every set $B \in \mathscr{A}$, \begin{equation*} H(S|S_B) = 0. \end{equation*}
• Perfect Privacy: For every unauthorized set $T \notin \mathscr{A}$, \begin{equation*} H(S|S_T) = H(S). \end{equation*}

What I am trying to do is prove that the two definitions are equivalent.