Entropy, a term used in Physics to quantify the degree of randomness in a complex system, is shown to be relevant for portfolio diversification. The link between entropy and diversification lies in the notion of uncertainty. We introduce the concept of available diversification in an investment universe and of diversification curves. We build a framework for assembling a fully diversified risk parity-like portfolio with a fundamental-based high-conviction strategy, through a constrained entropy-maximisation process by which a portion of potential portfolio return is swapped for extra diversification. The main results of this study are:• mean-variance optimised portfolios are highly concentrated and scarcely related to the asset return assumptions;• few basis points of expected returns can be converted into a huge amount of extra diversification that making the portfolio allocation more robust to parameter uncertainty;• on a more conceptual ground, we investigate the relationship between portfolio risk and diversification concluding that they should be managed distinctly.The empirical analysis presented in this work shows that entropy is a useful means to alleviate the lack of diversification of portfolios on the efficient frontier.