According to Quantum Mechanics, Schrödinger’s cat is in a superposition state of $\frac{1}{\sqrt{2}}(\left|A\right> + \left|D\right>)$, where $\left|A\right>$ and $\left|D\right>$ correspond to alive and dead state respectively. An external observer can make a measurement $M_1$ to obtain a result of the cat being either dead or alive. Mathematically, $M_1$ is a self-adjoint operator $M_1=\left|A\right>\left<A\right| - \left|D\right>\left<D\right|$ with eigen vector $\left|A\right>$ and $\left|D\right>$. After the measurement, the cat is in a state of either $\left|A\right>$ or $\left|D\right>$. It seems that theoretically we can perform a different kind of measurement $M_2=\left|U\right>\left<U\right| - \left|V\right>\left<V\right|$ where $\left|U\right>=\frac{1}{\sqrt{2}}(\left|D\right>+\left|A\right>)$ and $\left|U\right>=\frac{1}{\sqrt{2}}(\left|D\right>-\left|A\right>)$. After $M_2$, the cat will be in either state $\left|U\right>$ or $\left|V\right>$. Now suppose we first do measurement $M_1$ and get the result that the cat is dead. Now we subsequently perform measurement $M_2$ and will get result either $\left|U\right>$ or $\left|V\right>$ with equal probability because $\left|\left<U|D\right>\right|=\left|\left<V|D\right>\right| = \frac{1}{\sqrt{2}}$. Until this step, everything seems plausible. But what if after measurement $M_2$, we perform measurement $M_1$ again. What will be the result? Because now the cat is now in state $\left|U\right>$ or $\left|V\right>$ and $\left|\left<D|U\right>\right|=\left|\left<A|U\right>\right|=\left|\left<D|V\right>\right|=\left|\left<A|V\right>\right|$, the result of this additional $M_1$ will be either dead or alive with equal probability. However, from the first measurement of $M_1$, we know the cat is already dead, how can we have equal probability of obtaining result of dead and alive from the second $M_1$? And actually, after the initial measurement $M_1$, we can repeat $M_2$ and $M_1$ for indefinite number of times until we see the cat is alive. So we can effectively revive the cat by measuring $M_2$ and $M_1$ many times. This seems to be implausible.
Edit:
I need to emphasize that here cat is only an example. And I admit that it's nearly impossible to get a cat in pure quantum state. But that does not make this question invalid. We can replace the cat with a microscopic object such as a molecule which can probably be isolated into a pure state.
My current thinking is that $M_2$ is the problem. Perhaps there is some theory about whether it's possible to measure $M_2$. I see several possibilities:
- It's physically absolutely impossible to measure $M_2$. If it's this, I want to know the reason behind it.
- It's nearly impossible to measure $M_2$. This might involves some thermodynamic argument. I want the argument to be concrete instead just saying something like the process is not reversible (in a thermodynamic sense). Basically, I want to know that given a mathematical construction of a measurement, how can we know whether it's thermodynamically possible or not.
- It's actually possible to measure $M_2$ for some objects. This will be very interesting and I want to understand how we can physically measure $M_2$.