# What is the relation between a Fock state and a qudit?

What is the relation between a Fock state and a qudit? Is the Fock state $$|n\rangle, n=0, 1, 2, ...$$ a qudit having $$d=\infty$$?

• Some approaches to QC approximate a qudit by taking the first d Fock states and ignoring the rest. Mathematically that's not very clean, since the other states are still there and the system can in principle drift into them, but experimentally that can still be used to do QC. Aug 25 '21 at 17:49

## 1 Answer

I would say it is not exactly the same.

A qudit of size $$d$$ has a finite number of states. A Fock state can have discretely infinite in principle.

The goal of having multiple qudits, let's say $$N$$, is to make a larger space of size $$d^N$$. Rarely you consider product of Fock spaces (but it can be done). Usually, you define a Fock space that includes all configurations.

Fock spaces can be fermionic or bosonic (the ladder operators have defined commutation relations). The qudits are usually neither fermions nor bosons (check how weird is the commutation algebra of qutrits).

If you have a chain of qudits, there are mappings to converting them to Fock states and viceversa. But without a mapping I would not force them to be equivalent.

You can also have continuous Fock spaces, but it would be weird to have continuous qudits?

Any way if you want to think about $$d=\infty$$ qudits, consider that you can always write a Fock space to describe not only your $$\infty$$-qdit but also a chain of $$\infty$$ number of $$\infty$$-qdit. The Fock space is just more general.

• The commutation algebra you link to is that of $su(3)$; I guess you can call it the algebra of qutrits but there are probably subtleties. If they're exactly the same, then "quarks" might be a better name for "qutrit". Do you see $su(2)$ as the algebra of qubits? Aug 25 '21 at 17:53
• @unknown is it not? Why are the subtleties? Aug 25 '21 at 18:32
• For one, $su(3)$ is 8 dimensional (the link defines 8 matrices that can serve as basis for it). Exponentiating these gives you $SU(3)$. QC uses $U(3)$... Aug 25 '21 at 19:26
• @unknown sure but working with $U(n)$ or $SU(n)$ is equivalent for quantum computation (same projective unitary group). Aug 25 '21 at 21:21
• @unknown exactly, check this quantumcomputing.stackexchange.com/a/17763/15775 Aug 25 '21 at 21:28