## What is the definition of Bell state on a n-qubit system?

6

Question 1: The bell state for a 2-qubit system has been defined in Neilsen and Chuang's book as the set of maximally entangled states spanned by $$\{|00\rangle + |11\rangle, |00\rangle - |11\rangle, |01\rangle + |10\rangle, |01\rangle- |10\rangle \}$$. What is the higher dimensional definition of a Bell state for an n-qubits system?

Question 2: More specificity, Consider the toric code on a $$L\times L$$ lattice. There are $$2L^2$$ qubits on it. Consider the minimal case when $$L=3$$, ie. we have $$18$$ qubits. How does the code space look like? Is it some sort of "bell-state" for these qubits? Is it possible to explicitly write the code space for this as in the case of say, a repetition code.

Thanks!

1

Does this GHZ article help?

– Mark S – 2019-06-08T02:16:06.210

I see, but what about other linear combinations? Its still not clear and well-motivated to me.The GHZ state is just one state whereas there are more than one bell state. Is the toric code space a GHZ state? – John Jacob – 2019-06-08T10:41:38.840

1I don't think of a"Bell state" (singular) as the set of the four enumerated states - a pair of qubits are in a Bell state (singular) if they are in one of the four enumerated Bell states (plural). Similarly it might be ok to think of a qubit being in a superposition of $\vert 010\rangle+\vert 101\rangle$ as being in a "GHZ" state. – Mark S – 2019-06-08T12:27:07.210

1) There are 4 Bell states, namely the ones you listed divided by $$\sqrt{2}$$. There is no "the bell state". The Bell states are only defined for 2 qubits, so there is no "higher dimensional definition of a Bell state". One of the key features of the Bell states is that they're maximally entangled. If this is what you'd like in a higher dimensional analog of the Bell states then you'll want the GHZ states as Mark S suggests. An analysis coming to this conclusion can be found here: https://arxiv.org/pdf/quant-ph/9804045
For the toric code the stabilizers are strings of $$X$$ and $$I$$ or $$Z$$ and $$I$$ derived from the topology of a torus. It would be a tedious but straight forward exercise to explicitly enumerate all the stabilizers for the 3x3 lattice then find the elements spanning their mutual +1 eigenspace (hint: there's 4 of them encoding $$|00\rangle$$, $$|01\rangle$$, $$|10\rangle$$ and $$|11\rangle$$)