Combining standard errors

For each individual I have an estimate for A and a standard error P associated with that. A measures the sum of two factors X and Y that I don't know the separate values of.

A = X + Y

For each individual I also have an estimate for B and a standard error Q associated with that. B measures the difference between the same two factors X and Y.

B = X - Y

(I don't know if it makes a difference, but P and Q are highly correlated. In other words, if one individual has a high SE for their A estimate, they also have a high SE for their B estimate.)

I want to use A and B to estimate X and Y for each individual. I can do so using these equations:

X = (A + B)/2
Y = (A - B)/2

My question is what are the standard errors for X and Y? The sum of P and Q?