Taking the log moves nonzero complex numbers to complex numbers, so it makes exactly as much sense as expecting to observe complex odds.
Odds are the ratios of outcomes. How might you get complex outcomes like that?
Well, if the outcomes are quantum states, you could get something like 0.5 |a> + 0.5(1+i) |b> − 0.5|c> in which case the odds of each outcome are 1:2:1 if you make them decohere right then. But let’s suppose you don’t decohere them and work with the amplitudes.
What would taking the logarithm do to these things?
If your states are energy eigenstates, taking the log has the interesting effect of extracting the time dependence, which would manifest as an oscillation between various cases of interference between the states.
If the states are angular momentum eigenstates, taking the log would extract the angle dependence of the outcome. What this means depends more on your context than the first case. Also, you’re not going to be able to predict the time dependence from this (the combined state might already be an energy eigenstate, which would help).
And so on, with complementary pairs of observables. If they’re eigenstates of one observable operator in a pair of complementary observables, this will encode some information about the dependence on the complementary operator.
If your choice of states don’t imply (as far as you can tell) a complementary operator, you’re out of luck.
In short:
The real part of your deciban figure would be the log of the amplitude. That acts pretty much like you’re used to it acting for normal real quantities. Just throw in a correction factor of 2 to use the squared amplitude instead of the amplitude.
To make any use of that imaginary part, you’d need to know a lot of details about the exact definitions of your states and do a lot of math. If you don’t, then you just throw out the imaginary part of that log and use the real part.
Taking the log moves nonzero complex numbers to complex numbers, so it makes exactly as much sense as expecting to observe complex odds.
Odds are the ratios of outcomes. How might you get complex outcomes like that?
Well, if the outcomes are quantum states, you could get something like 0.5 |a> + 0.5(1+i) |b> − 0.5|c> in which case the odds of each outcome are 1:2:1 if you make them decohere right then. But let’s suppose you don’t decohere them and work with the amplitudes.
What would taking the logarithm do to these things?
If your states are energy eigenstates, taking the log has the interesting effect of extracting the time dependence, which would manifest as an oscillation between various cases of interference between the states.
If the states are angular momentum eigenstates, taking the log would extract the angle dependence of the outcome. What this means depends more on your context than the first case. Also, you’re not going to be able to predict the time dependence from this (the combined state might already be an energy eigenstate, which would help).
And so on, with complementary pairs of observables. If they’re eigenstates of one observable operator in a pair of complementary observables, this will encode some information about the dependence on the complementary operator.
If your choice of states don’t imply (as far as you can tell) a complementary operator, you’re out of luck.
In short:
The real part of your deciban figure would be the log of the amplitude. That acts pretty much like you’re used to it acting for normal real quantities. Just throw in a correction factor of 2 to use the squared amplitude instead of the amplitude.
To make any use of that imaginary part, you’d need to know a lot of details about the exact definitions of your states and do a lot of math. If you don’t, then you just throw out the imaginary part of that log and use the real part.