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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!

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    $\begingroup$ Does this GHZ article help? $\endgroup$ – Mark S Jun 8 at 2:16
  • $\begingroup$ 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? $\endgroup$ – John Jacob Jun 8 at 10:41
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    $\begingroup$ I 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. $\endgroup$ – Mark S Jun 8 at 12:27
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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

2) The code space for the toric code on a 3x3 lattice consists of 4 different, long, complex superpositions of kets which each have 18 elements in them. I doubt anyone is willing to write them out for you. However it wouldn't be too difficult for a computer algebra system to crunch them.

Beyond being discussed in entry level texts for clarity, the actual codespace of an error correcting code is rarely ever used. The codespace is uniquely determined as the mutual +1 eigenspace of the code's stabilizers. In fact the stabilizers encode everything about the code so it's common to think of them as the code itself. Compared to the codespace, the stabilizers are much cleaner, more concise and obey nice algebraic properties which makes them preferable to use.

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 $|11\rangle$)

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