1. The problem statement, all variables and given/known data Sakurai Modern Quantum Mechanics Revised Edition. Page 81. density matrix p = 3/4 [1 0; 0 0] + 1/4 [1/2 1/2; 1/2 1/2]. We leave it as an exercise to the reader the task of showing this ensemble can be decomposed in ways other than 3.4.24 2. Relevant equations 3.4.24 w( sz + ) = 0.75; w(sx +) = 0.25. 3. The attempt at a solution Various. First I found pure states SX+, SY+, SZ+,SX+, SX+, SY-, and SZ-. The fact that p = [7/8 1/8 ; 1/8 1/8 ] has no i's suggest that the weights for Sy+ = Sy-. a SX+ + b Sy+ + c Sz+ ... = [7/8 1/8 ; 1/8 1/8 ] can be but it always seems to lead to some weights which are negative. are negative weights allowed (I do not think so). Another attempt I tried was to rotate the basis states themselves to get more pure states. I was able to get the rotated density matrix equal to the linear combination of rotated basis states, but I could not get the density matrix [7/8 1/8 ; 1/8 1/8 ] to be a legitimate linear combination of the rotated basis states. I believe I used ideas that have not been developed by Sakurai at this point in this book. I may be making this problem harder than it is (I hope so ). Otherwise I tend to think the exercise is impossible. I tend to think Sakurai throws this out and does not assign it as a problem because it is more easy and straightforward. This has not been assigned to me as a problem. I am not taking a course in QM (not in the last decade). I am just interested I would appreciate the solution or any comments.