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Often in quantum computing the idea of quantum **superposition** is introduced well before the concept of **entanglement**. I suspect this may be because our conception of (classical) computing privileges *bits*, and hence we also privilege *qubits* in a Hilbert space of dimension $d=2$. It's easy enough to consider a single qubit in superposition, but transitioning to entanglement requires a plurality of such particles.

Or does it?

For example, suppose we lived in a world that privileged qudits, with $d=4$; e.g. four-level quantum systems as opposed to two-level qubits. We can think of our system (say, a particle-in-a-box or a harmonic-oscillator or what-have-you); our qudit could be in any superposition of $\{\vert 0\rangle,\vert 1\rangle,\vert 2\rangle,\vert 3\rangle\}$.

We can think of a particle in a superposition of $\vert \Psi\rangle=\frac{1}{\sqrt{2}}\vert 0\rangle\pm\vert 3\rangle$, or $\vert\Phi\rangle=\frac{1}{\sqrt{2}}\vert 1\rangle\pm\vert 2\rangle$.

Now if we envision our (single) qudit instead as two virtual qubits, with a mapping/isomorphism such as:

$$\vert 0\rangle_{qudit}=\vert 00\rangle_{qubit}$$ $$\vert 1\rangle_{qudit}=\vert 01\rangle_{qubit}$$ $$\vert 2\rangle_{qudit}=\vert 10\rangle_{qubit}$$ $$\vert 3\rangle_{qudit}=\vert 11\rangle_{qubit},$$

then we can see that both $\vert \Psi\rangle$ and $\vert \Phi\rangle$ are the Bell states, e.g. are entangled.

This works nicely for $d=4$ or any other power of $2$. But would it work for any other dimension, such as $d=3$?

Can we decompose a qutrit that is in superposition into smaller components, and ask whether the qutrit thusly is in some sense entangled?

Thanks! There's another question that I'll have to consider how to formalize, along the lines of "what are the requirements for such a factorization to be meaningful?". If we had a harmonic oscillator that can be in a superposition of one of $n$ eigenstates for some

non-prime$n=p\times q$, then this can be factored non-trivially into into the tensor product of two different Hilbert spaces of dimension $p$ and $q$. Can we then think of these two separate Hilbert spaces as somehow separate "particles" or separate "qudits?" – Mark S – 2020-10-09T13:55:41.5001I guess the question is what IS a separate "particle" in the first place? Maybe some kind of system which you can consider to be isolated in a certain limit? In this case, your example would not really meet the criteria. But one could maybe think about these "subsystems" as "logical" qudits which your big system is simulating. – Markus Heinrich – 2020-10-12T09:09:21.200