Can Time be Measured in Bits?

Every living thing follows along a set path. And if you could see your path or channel, then you could see into the future, right? Like err… that’s a form of time travel. – Donnie Darko

Time is nature’s way to keep everything from happening all at once. – John Archibald Wheeler

The past exists only as recorded in the present.

What is time? This question, in various forms, has been pondered by physicists and philosophers for eons. No doubt, various advances in our understanding of time have been made (the relativity of time for different observers, the symmetry breaking of time invariance as demostrated directly by K^0-K^0-bar experiments, etc.) but there are still enough troubling aspects of time for a good theoretician to get lost in. For example, the role of time in quantum theory (and in particular in a theory of quantum gravity) is one question which has consumed the soul of more than a few physicists.
In physics, a question which often bothers theoreticians is the origin of an arrow of time. The problem roughly is that we have underlying laws which are time symmetric, yet the universe seems to pick out a particular direction for the evolution in time. One explanation for the arrow of time is that it comes from thermodynamics. If we start with a universe which has a very low entropy, then the forward march of time can be marked by the upward increase in entropy. If we had started in a universe with maximal entropy, presumibly there would be no advance of time. But if the increase in entropy corresponds to an increase in the forward direction of time, does this mean that we can measure time in the same units of entropy? Can we measure time, then, in bits?
At first sight, this seems wrong. Take, for example a reversible computer. This computer acts according to reversible rules and so the entropy of the computer does not increase, even though, time is increasing as the computer runs a program. But maybe there is a way out of this puzzle. One possibility is that it is impossible to construct a truely reversible computer. This might seem silly, since we think the laws of physics are reversible, and so we can think about some physical system as enacting a reversible computation. But it’s not clear to me that robust computation is possible with a totally reversible system (more specifically without some effective irreversibility, such as cold ancilla bits which are discarded.)
Another possibility is that it might be true that a reversible computer can be constructed, but that it is impossible to construct a clock without irreversible evolution. I.e. to see the evolution of a reversible computer with respect to time, we need a clock around. Here things get rather tricky. Can’t I can think of a simple reversible two state system which simply cycles between the two states as a clock? I don’t think so. The reason is that a clock isn’t really just a system which counts, but it’s really a way in which we callibrate the basic units of time. So I use a cesium atom as a clock by using it to calibrate what a second is. Thus I run an experiment which performs measurements on the cesium clock which gives me a basic calibration upon which all clocks can be run. But why can’t this callibration be made totally reversible? I’m not sure, but it’s a good homework problem. I suspect that the callibration experiement cannot be made reversible (whenever I try the simple methods to make it reversible, I run into “effective” irreversibilities.)
So it seems that we can measure time in bits, or at least thermodynamic time in bits. What about other arrows of time (such as the arrow of time arrising from K^0-K^0-bar experiments or a cosmological arrow of time?) It would be fun to try and design a K^0-K^0-bar experiment which acts as a clock. And what of the relationship between time being measured in bits and the holographic principle, where surface areas are measured in bits?
See I told you a theoretical physicist could lose his soul thinking about time.