What does it mean to hold a diversified portfolio?

Imagine we have a portfolio of equal positions in different securities? Is this diversified? Probably not. What if these securities were all bonds with small volatilities, and one equity position with a much higher volatility - that clearly would not count as properly diversified. So individual security variances need to be considered when thinking of diversification. What about correlation? This too should be included. The best measure of diversification probably considers the contribution of the pieces to the overall variance of the portfolio. But a complication here is that variance is not additive - so instead we often talk of factor variances, calculated based on some form of Principal Component Analysis (PCA) - basically computing the eigenvectors of the original covariance matrix in order to produce orthogonal risk vectors (whose variances are then additive). Having PCA risk factor variances then allows us to estimate the cumulative contribution of different securities to aggregate variance starting with the greatest factor contribution. Much like a Pareto Analysis, we ask how different is the actual contribution to perfectly diversified case - a straight line, and this gives us a robust measure of diversification.