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We can operate on the cat/box/sensor/particle system with a quantum operator if we like. And, if I may be a bit informal with it, the system after interaction might be $|\text{cat}_{after}\rangle=a|\text{alive}\rangle+b|\text{dead}\rangle+c|\text{wierd}\rangle$, where $a$ $b$ and $c$ are just real numbers. The $|\text{alive}\rangle$ handles the cases which are handled intuitively as having an alive cat, $|\text{dead}\rangle$ handles the cases which are handled intuitively as having a dead cat, and $|\text{wierd}\rangle$ lumps together all of the really wonky cases where quantum mechanics says one thing where our intuition says another. One of the great things about the bra-ket notation that physicists like to use is I can use it to correctly capture a system, even when using really oddball states like "wierd."

So now we come back to the detector. This detector could have been any system really. There's more interesting things to throw into a box with a cat, but the experiment calls for a detector. And, hand-waving emphatically, one aspect of a good detector in physics land is that it minimizes the probability of any weird things happening. Using the above equation, we try to design sensors in such a way that, for any interaction one may wish to do with the system (opening the box, or any quantum operator), the constant $c$ in $c|\text{wierd}\rangle$ is vanishingly small ($c\approx 0$). A sensor which doesn't have this property is a pretty poor sensor, and I would no longer be comfortable with the intuitive idea that it "detects" the radioactive isotope decaying.

Now you can construct more interesting experiments with things other than nice clean detectors. And you can start to see quantum effects at the macroscopic level. There's an entire approach to QM around studying "decoherence" which handles this in a statistically rigorous way and does a good job predicting the results of more odd systems that permit more $|\text{wierd}\rangle$ through by design. For example, there's a whole approach of using "weak measurements" which are measurements designed to not disturb "weirdness" that was already happening in the experiment. But in this case we can comfortably say the detector "collapsed" the wave form. And, approaching the topic through the idea of decoherence, we can even show why that term is valid: we intentionally designed the detector to "collapse" the weird part of the waveform into a vanishingly small part.

We can operate on the cat/box/sensor/particle system with a quantum operator if we like. And, if I may be a bit informal with it, the system after interaction might be $|\text{cat}_{after}\rangle=a|\text{alive}\rangle+b|\text{dead}\rangle+c|\text{weird}\rangle$, where $a$ $b$ and $c$ are just real numbers. The $|\text{alive}\rangle$ handles the cases which are handled intuitively as having an alive cat, $|\text{dead}\rangle$ handles the cases which are handled intuitively as having a dead cat, and $|\text{weird}\rangle$ lumps together all of the really wonky cases where quantum mechanics says one thing where our intuition says another. One of the great things about the bra-ket notation that physicists like to use is I can use it to correctly capture a system, even when using really oddball states like "weird."

So now we come back to the detector. This detector could have been any system really. There's more interesting things to throw into a box with a cat, but the experiment calls for a detector. And, hand-waving emphatically, one aspect of a good detector in physics land is that it minimizes the probability of any weird things happening. Using the above equation, we try to design sensors in such a way that, for any interaction one may wish to do with the system (opening the box, or any quantum operator), the constant $c$ in $c|\text{weird}\rangle$ is vanishingly small ($c\approx 0$). A sensor which doesn't have this property is a pretty poor sensor, and I would no longer be comfortable with the intuitive idea that it "detects" the radioactive isotope decaying.

Now you can construct more interesting experiments with things other than nice clean detectors. And you can start to see quantum effects at the macroscopic level. There's an entire approach to QM around studying "decoherence" which handles this in a statistically rigorous way and does a good job predicting the results of more odd systems that permit more $|\text{weird}\rangle$ through by design. For example, there's a whole approach of using "weak measurements" which are measurements designed to not disturb "weirdness" that was already happening in the experiment. But in this case we can comfortably say the detector "collapsed" the wave form. And, approaching the topic through the idea of decoherence, we can even show why that term is valid: we intentionally designed the detector to "collapse" the weird part of the waveform into a vanishingly small part.

More to the point of Schrödinger's cat, these new rules obey a principle known as "superposition." In Aaron Steven's answer, he was very careful to point out that the cat exists in exactly one state at all times. There's a good reason he was so careful there. When we write something like $|\text{cat}_{initial}\rangle=|\text{alive}\rangle$ or $|\text{cat}_{final}\rangle=\frac{1}{\sqrt{2}}\left(|\text{alive}\rangle+|\text{dead}\rangle\right)$, we are describing the one and only state that the cat is in. However, by the rules of superposition, we can figure out the state the cat will be in by looking at each branch of an addition, one at a time, and then add them up later. This is convenient for you and I, because we are much more comfortable thinking through what happens to an "alive" cat or a "dead" cat, rather than trying to handle some complex mathematical equations that links both. The fact that QM wavefunctions have this superposition property lets us do this rigorously.*

More to the point of Schrödinger's cat, these new rules obey a principle known as "superposition." In Aaron Steven's answer, he was very careful to point out that the cat exists in exactly one state at all times. There's a good reason he was so careful there. When we write something like $|\text{cat}_{initial}\rangle=|\text{alive}\rangle$ or $|\text{cat}_{final}\rangle=\frac{1}{\sqrt{2}}\left(|\text{alive}\rangle+|\text{dead}\rangle\right)$, we are describing the one and only state that the cat is in. However, by the rules of superposition (which all quantum systems obey), we can figure out the state the cat will be in by looking at each branch of an addition, one at a time, and then add them up later (Formally, we can say that for any linear operation $f$ on the system $(f(x_1+x_2) = f(x_1) + f(x_2)$). This is convenient for you and I, because we are much more comfortable thinking through what happens to an "alive" cat or a "dead" cat, rather than trying to handle some complex mathematical equations that links both. The fact that QM wavefunctions have this superposition property lets us do this rigorously.*

^{*. As a perhaps useful aside, the decomposition itself isn't all that important. This could have been $|\text{cat}\rangle=a\left(|\text{male}\rangle+|a\rangle female\right)$, describing what happened to the cat if it was male or the cat if it was female. The math would actually end up right either way. However, by selecting states which are convenient to the human doing the math (alive and dead), it becomes easier to leverage the superposition principle to actually start picking away at the problem, rather than merely developing new bases.}

^{*. As a perhaps useful aside, the decomposition itself isn't all that important. This could have been $|\text{cat}\rangle=a|\text{male}\rangle+b|female\rangle$, describing what happened to the cat if it was male or the cat if it was female. The math would actually end up right either way. However, by selecting states which are convenient to the human doing the math (alive and dead), it becomes easier to leverage the superposition principle to actually start picking away at the problem, rather than merely developing new bases.}