Quantum mechanics says that physical observables are self-adjoint operators. Is this correspondence a bijection, ie can we realize any such operator as a physical observable? There are obvious practical concerns with physically realizing certain contrived operators. But are there any theoretical limitations (eg locality) to measuring in an arbitrary basis? Suppose we place an electron in a box with two compartments, left and right, and insert a barrier between the compartments, so that the electron is in a superposition of being in either. When we open the box and measure the electron's position by simply looking, we find the electron either in the L state or in the R state. This is because the "simply looking" measurement process corresponds to the "position" operator, which has eigenstates L and R, and these eigenstates are the possible results of the measurement. Of course, "a superposition of left and right" is one of many ways to describe the electron's state before measurement. We can choose another basis consisting of the eigenstates of another operator. I can easily write an abstract operator with (L+R)/√2 and (L-R)/√2 as eigenstates, but can I observe the electron in one of these states? With some effort, yes. We can position a lens at the opening of each compartment and direct the two signals (light from the electron) to a system (such as a one-way mirror) that combines the signals. We can do this such that, due to constructive interference, (L+R)√2 yields a signal while, due to destructive interference, (L-R)/√2 does not yield a signal. We have measured in the new eigenbasis. That's great. But now consider Schrodinger's Cat. Can I design an experiment (however contrived) to measure in the (Alive±Dead)/√2 basis? Why, in the theoretical sense, is this measurement extremely difficult (if not impossible)? My guess is that it has something to do with the locality of physical interactions. Somehow (?), locality gives preference to the position basis (eg L and R). Since it is impossible to fundamentally overcome this preference, we must instead convert (via contrived experiments) arbitrary states into position states (eg (L+R)/√2 becomes "YES, I observe a signal through the lens"). Am I on the right track with locality and the preferred basis?