Quantum: why linear combination of vectors (superposition) is described as "both at the same time"?

Question

I want to get a better understanding of quantum phenomena and out world in general. Before long I've thought of Schrödinger cat as being both alive and dead (or spin both up and down). Now after some reading of math, it looks to me it is not technically correct, as the cat/the spin is in a linear combination of vectors in Hilbert space (which is just another vector in that space). In my understanding linear combination of not the same as being both base vectors at the same time and the "naive" viewpoint of the cat being both alive and dead is oversimplification and misleading. Why is it commonly called "both at the same time"?

Answer 1

“Both at the same time” is actually bad language, especially as this is a basis-dependent statement. For instance, the eigenstate $\vert \uparrow\rangle_x$ of $\sigma_x$ is one state. If you make a measurement along $\hat x$, you get one outcome but if you make measurements along $\hat z$ you can get two outcomes. It's at best imprecise but colourful language to suggest the spin is in both $\hat z$ direction at the same time because there can be two outcomes when measuring in that basis.

Now, one can rotate a magnetic field gradient so that it is intuitive that we can measure one state in several bases, but it is non-sensical to imagine this for the case of the cat: the basis vectors |dead$\rangle$ and |alive$\rangle$ cannot be “rotated” to another position (in space or otherwise) in any sense that is to be understood rationally. So we are left with the imprecise but colourful language, which is now etched in popular culture.

Answer 2

In quantum theory the evolution of a measurable quantity is described by a linear operator called and observable. The eigenvalues of that observable represent the possible outcomes of a measurement of that observable. In general the evolution of an observable can't be described in terms of what is happening to just one of the possible values. Rather, the results of experiments in general depend on what is happening to all of its possible values.

This isn't just a mathematical artefact. In single particle interference experiments, the results of the experiment depend on what is happening along all of the paths the particle could go down. The only known explanation of such an experiment involves the existence of many versions of the particle that interfere at the end of the experiment to produce the outcome, see "The Fabric of Reality" by David Deutsch, Chapter 2.

Now, if there are multiple versions of every particle in your body and every particle in a cat's body then there should be multiple versions of both you and the cat. But you don't see multiple versions of a cat or a person or anything else you can see with the naked eye. The standard response to this issue is to say that somehow those multiple versions don't exist and to ignore the issue of having an account of what's happening in reality.

Quantum theory explains why you only see one version of the cat without collapse. When you copy information out of a system that suppresses quantum interference: this is called decoherence:

https://arxiv.org/abs/1911.06282

The different versions of the system evolve autonomously and form layers each of which looks approximately like the universe as described by classical physics:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is commonly called the many worlds interpretation (MWI) of quantum theory but it is just an implication of the theory when it is taken seriously as a description of reality. Reality as described by the MWI isn't just a collection of different versions of every object because the isolation between different layers is never perfect. An electron in your body has position and momentum observables that are narrowly peaked in position and momentum on the scale of everyday life, but it isn't a point particle at a single location. So to talk of multiple versions of a cat is a simplification but not one that matters much in everyday life.

Answer 3

Suppose you're travelling with a bearing of 45 degrees to North. We call this direction "Northeast". Are you travelling North? Some might say yes. Are you travelling East? Some might say yes. So are you travelling both North and East at the same time? Some might say yes.

So your travel vector is a superposition of the North and East vectors so we might say you are travelling both North and East at the same time.

It's not so strange to call the superposition vector the cat being both dead and alive at the same time. The strange thing, in my opinion, is that physical states are represented by state vectors in a Hilbert space. But once we accept that physical states ARE represented by state vectors in Hilbert space, it's not so strange to call superpositions "both at the same time".

Though I do agree with comments that this feature of quantum mechanics starts to stretch the philosophical boundaries of human thought and language.