- #1

Peter Morgan

Gold Member

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If we open the box and measure only whether the cat is alive, using the projection operator ##\hat A=\left(\begin{array}{cc}1 & 0\\0 & 0\end{array}\right)##, we can't tell, because for example for two different density matrices, a mixture ##\hat M_\alpha=\left(\begin{array}{cc}\alpha & 0\\0 & 1{-}\alpha\end{array}\right)## or a pure state ##\hat S_\alpha=\left(\begin{array}{cc}\alpha\!\! & \!\!\!\!\sqrt{\alpha(1{-}\alpha)}\!\\ \!\!\sqrt{\alpha(1{-}\alpha)}\!\!\!\! & \!\!1{-}\alpha\end{array}\right)##, we obtain precisely the same probability, ##\alpha##, that the cat is alive. To tell whether the physicist is lying, we have to use other observables, such as what I'll call the Lewis Carroll operator, ##\hat C=\left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right)##, which takes a live cat and kills it and takes a dead cat and resuscitates it. With this operator as well as ##\hat A##, we can construct a different projection operator, ##\hat P_{\!+}=\frac{1}{2}\left(\begin{array}{cc}1 & 1\\1 & 1\end{array}\right)## and measure a probability such as ##\mathsf{Tr}\left[\hat P_{\!+}\hat\rho\right]##, where ##\hat\rho## is whatever the state is, and discriminate at least between ##\hat M_\alpha## and ##\hat S_\alpha## (to differentiate between all possible states requires more projection operators, but the above is enough for the discussion that follows).

Quantum Mechanics, however, tells us that measurements in Classical Mechanics are

*always*mutually commutative, and, indeed, that's perhaps

*the*fundamental difference between QM and CM. In that case, CM measurements cannot tell whether the alleged QM preparation of a superposition is what the quantum preparer says it is or not. If CM accepts that all its measurements are and must be mutually commutative, a classical physicist can just say, "Huh, it's just a mixture, which I understand well enough", you're just lying that it's a superposition. In fact, however, the Lewis Carroll operator is well-enough-defined, as above, for a classical physicist, it's just not easy to implement for a classical cat. QM is just making a straw man for itself.

If the classical physicist is

*not*prevented from using the Lewis Carroll and similar operators, then they can tell whether the state is a pure state, and they can confirm the quantum physicist's claims about the state, but with that expansion of what a classical physicist can do, to what I call CM

_{+}in my paper Unary Classical Mechanics, a quantum physicist is much less different from a classical physicist than has usually been asserted.

So: what flaws are there in this argument, what subtleties do people feel should be developed, and is this argument in the literature already?