A paper I've been reading (1, just after eq. 5.1) claims that if $\rho$ is a quantum state on $n$ qubits, and $S(\rho) \le (1 - \epsilon)n$, then $$\|\rho - \tilde{I}\|_{\text{tr}} \ge \epsilon - \frac{1}{2^n},$$ where $S$ is the von Neumann entropy and $\tilde{I}$ is the maximally mixed state. The claim feels very believable, but I've tried to verify it using the Fannes inequality and only managed to derive weaker bounds. Is this claim true, and if so, how does one prove it?
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