How can an outcome be Pareto efficient but not Kaldor-Hicks efficient? An policy is a Pareto improvement if it makes some people better off and no one worse off.  And a policy is a Kaldor-Hicks improvement if it can be turned into a Pareto improvement by redistributing money.  Every Pareto improvement is a Kaldor-Hicks improvement, but not every Kaldor-Hicks improvement is a Pareto improvement.  

Now an outcome is called Pareto efficient if there are no possible Pareto improvements of it, and Kaldor-Hicks efficient if there are no Kaldor-Hicks improvement of it.  Now clearly every Kaldor-Hicks efficient outcome is Pareto-efficient, because if a Pareto improvement were possible it would also be a Kaldor-Hicks improvement.  But my question is, why is the converse not true?

Here is my argument for why the converse is true.  Suppose that X is a Pareto-efficient outcome which is not Kaldor-Hicks efficient.  Then there exists a Kaldor-Hicks improvement which can be turned into a Pareto improvement by a redistribution of money.  But then that Kaldor-Hicks improvement together with that redistribution is a Pareto improvement, so there exists a possible Pareto improvement of X, contradicting the fact that X was Pareto efficient.  Where am I going wrong?
Example. Each month, God gives Adam $60$ apples and Eve $40$ (for a total of $100$ apples). Let's write this allocation as $X=(60,40)$.

The Devil now comes along and offers to increase their total monthly allotment of apples to $101$, but on the condition that the allocation must be $Y=(59,42)$.

Observe:


$X$ is Pareto efficient but not Kaldor-Hicks efficient.
$Y$ is both Pareto efficient and Kaldor-Hicks efficient.
$Y$ is a Kaldor-Hicks, but not a Pareto improvement over $X$.
Through imaginary redistribution, we could get from $Y=(59,42)$ to $Z=(60,41)$. And $Z$ would then be a Pareto improvement over $X$. But the problem is that the option of making the redistribution $Y\rightarrow Z$ doesn't actually exist. And so, while $X \rightarrow Z$ would indeed be a Pareto improvement, the problem is that it does not exist and is not a possibility. It is merely something we or Adam and Eve may have imagined.




So let's see where you went wrong in your argument:


  But then that Kaldor-Hicks improvement together with that redistribution is a Pareto improvement


This is correct. Using the above example, $X \rightarrow Y$ is the Kaldor-Hicks improvement, $Y \rightarrow Z$ is the imaginary redistribution, and $X \rightarrow Z$ is the Pareto improvement.


  so there exists a possible Pareto improvement of X


The error is here. The redistribution $Y\rightarrow Z$ is imaginary and does not exist. So, the reallocation $X \rightarrow Z$ is also imaginary and does not exist.

(In this case the redistribution  $Y\rightarrow Z$ isn't possible due to arbitrary constraints set by the Devil. But in the real world there will be other constraints.)

Edit: Another example

Suppose you and I are neighbors. I really like practicing the drums at 9 am on a Saturday. You really, really, like sleeping until 10 am on Saturdays, which is, of course, significantly disrupted by my drumming. Suppose I value drumming at \$10, and you value sleeping at \$20. What are the efficient outcomes?

The clear Kaldor-Hicks efficient outcome is for me to not drum. Sure, I lose (the equivalent utility of \$10 to be precise), but I lose less than you gain. If desired, you could compensate my loss, and still be better off than if I drum.

The set of Pareto efficient outcomes is bigger. It's Pareto efficient for me to either not drum (letting you sleep), or drum away at 9 am!

Of course, we could come to a mutually agreeable resolution where I don't drum and you pay me some amount between \$10 and \$20. However this adds another layer of complexity that we didn't include: that enforceable agreements and transfers are possible (and costless). In the real world, where there are many other pragmatic issues (such as enforcement, positive transaction costs, lack of information about who the "gainers" and "losers" are, etc.), these payments might not be possible, and hence, we're reduced to considering the initial problem.