How can one compute the alpha decay of a systematic trading strategy?
The short answer (which represents one way of surely many ways to do it) is to watch the t-stat of a performance metric such as information coefficient vanish over time. IC is the correlation of predicted expected returns from your alpha strategy to the underlying benchmark.
Look at the expected returns your alpha strategy predicted over the past N time intervals and see how those predicted returns were correlated with the benchmark. This is the IC.
The crux of course is that you need to test if the IC is statistically different from zero or is just random noise. You would do this by computing the t-stat over time and watch it decay as the strategy you managed to build over hundreds of hours of research is unceremoniously drained of edge.
You can come up with many specific answers depending on the application, such as high frequency vs. low frequency, or cross-sectional (e.g. single stock equity relative value) vs. time-series strategies (e.g. trading E-mini S&P 500) (strimp099's suggestion of using information coefficient is a good suggestion for equity relative value). However, a general answer that is likely to at least give you some indication for any strategy is to look at the time series of returns over two adjacent periods and perform a t-test for difference in means. If the later period has a statistically significantly lower mean return, then the alpha has likely decayed. Measuring the precise degree of decay is going to be nearly impossible for anything but a relatively high frequency strategy with a very long time sample.
The blog post http://www.portfolioprobe.com/2011/11/30/alpha-decay-in-portfolios/ tells of one possible method:
Generate two sets of random portfolios. Both sets satisfy your portfolio constraints, and one set is additionally constrained to have high expected returns. You can now look at the distribution of the difference in returns for various time frames.