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

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"?

• It's simply because there's no better phrase available in common English. You're completely right that this is a serious limitation of popular science. Commented Jul 9 at 3:07
• Agreed. Human language was built around things "being" a certain way because the objects that we observe in day-to-day life are only one way. In quantum mechanics, the whole idea of "being" breaks down into probabilities of having certain properties, with each combination of quantized properties being a possible state and a particle being defined as a linear combination of those possible states. It isn't a limitation of understanding, just that human language never had to deal with the idea of quantum mechanics, and now that we do, we're left with "being in two places at once" and the like. Commented Jul 9 at 3:34
• It's not "both at the same time" but "neither until it is measured". There is nothing particularly special going on here. Think about the properties of dice. Are dice that are still rolling in all six possible outcome states at the same time? No. Rolling dice can simply not be described as being in any of their outcome states. The same is happening here, except that we are dealing with quantum mechanics instead of probability theory, but the implications of not knowing what the outcome of a future measurement will be are the same. Commented Jul 9 at 8:13
• Related Commented Jul 9 at 8:48
• For much hilarity check out Wigner's Friend en.wikipedia.org/wiki/Wigner%27s_friend , which is Schrödinger's Cat on steroids.
– AnoE
Commented Jul 9 at 13:43

“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.

• "the basis vectors |dead⟩ and |alive⟩ cannot be “rotated” to another position (in space or otherwise) in any sense that is to be understood rationally." I seem to have no trouble at all understanding rotations of a two dimensional Hilbert space. Does this make me irrational? Commented Jul 9 at 4:17
• @WillO since the phase factor is complex in general, I guess this means that you are not real :O Commented Jul 9 at 6:36
• the problem as you know is not mathematical: there is no problem mathematically but students or professors cannot half-dead/half-alive: this is colloquial. We are in practice restricted to measurement in only one basis: the one where the outcomes are dead or alive. This is why the superposition appears absurd. Commented Jul 9 at 11:35
• I think this comic expresses it quite well... "Superposition doesn't mean 'and,' but it also doesn't mean 'or.' ... You should think of it as a new ontologial category: a way of combining things that doesn't really map onto any classical concept." Commented Jul 9 at 18:50

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.

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.

• Others would say you are in a state of moving Northeast, which is neither moving north nor moving east, since a defining feature of moving north is that you are neither moving east nor west. Of course, in this example it is inconsequential, but it becomes very confusing for, say, the position basis to say "the particle is at every point in space", especially since position eigenstates are not physical. Commented Jul 9 at 18:24
• @BioPhysicist but this analogy stretches. If your bearing is instead 1 degree to North then we might say you're mostly travelling North, but a little bit East. So for the particle's position state we could say the particle is everywhere, but it is mostly within some imaginary coordinate box. Commented Jul 9 at 18:57
• While the box is closed, do you think the cat knows s/he is in superposition (I mean senses something extra-ordinary)? Or it ceases to be a cat... Or more like MWI with two branches each forming its own history? Or the question makes no sense? Commented Jul 10 at 14:09
• @Martian2020 your comment really needs to be asked as a separate question. But I'll say that I don't think the cat can tell it's in a superposition. Commented Jul 10 at 16:29
• @Martian2020 : You are asking what happens when (or if) the cat makes a measurement of the observable "Am I either definitely alive or definitely dead?". Presumably <alive> and <dead> are both in the "yes" eigenspace of this observable. It follows that any superposition of <alive> and <dead> is in that eigenspace. So the answer the cat comes to is always "yes". Commented Jul 14 at 14:27