Information and successive measurments

[Higgsono]

Suppose you measure the spin of an electron along some axis, we get the result +1 with probability 0.5 and -1 with probability 0.5. The average information we receive is 1 bit of information. Now if we rotate our apparatus 90 degrees and make a new measurement, we will as before get the value +1 with probability 0.5 and -1 with probability 0.5. But now we lost the information about the spin along the direction of our first measurement. We lost 1 bit of information but gain 1 new bit of information.

So, I have a question: Does there exist a pair of noncommuting observables A,B such that if we measure A first and gain some amount of information $I_{A}$ and then make a measurement of B and gain some information $I_{B}$. Is is possible that the new information we gained is less then the information we lose from our previous measurement. That is is it possible that $I_{B} - I_{A} < 0$
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[azntoon]

Yes, it is possible. Consider a two-dimensional system with the two non-commuting observables A and B. Suppose that A has eigenvalues a1 and a2, with corresponding probabilities of p1 and p2, and B has eigenvalues b1 and b2, with probabilities q1 and q2. Then if we measure A first, the average information gained is I_A = -p1log(p1) - p2log(p2). If we then measure B, the average information gained is I_B = -q1log(q1) - q2log(q2). It is possible that the difference between I_B and I_A is negative, i.e., that I_B - I_A < 0. For example, consider a system with A having eigenvalues a1 and a2 with probabilities p1 = 0.5 and p2 = 0.5, and B having eigenvalues b1 and b2 with probabilities q1 = 0.7 and q2 = 0.3. Then I_A = 1 bit and I_B = 0.88 bits, so I_B - I_A = -0.12 bits.