Possible typo in the paper "Graph states for quantum secret sharing"

Continuing from my last question that I posted about a paper on Graph States. I have another doubt regarding a possible typo error in the paper. Here it goes. The authors define a 'labeled state' as $$|G_{\vec{l}}\rangle=\bigotimes_{i}X_i^{l_{i1}}Z_i^{l_{i2}}|G'\rangle$$where G is a graph state. Then they define an 'encoded graph state' as $$|G_{\vec{l}_{*2}}\rangle=\bigotimes_{i}Z_i^{l_{i2}}|G'\rangle$$ and say that the encoded graph state is the labeled graph state with $$l_{1i}=0, \forall i$$.

Here $$\vec{l}_{i*}=(l_{i1},l_{i2})$$ for the i-th vertex, $$\vec{l}_{*j}=(l_{1j},l_{2j},....l_{nj})$$ for the $$j$$ th bit over all the $$n$$ vertices, and $$\vec{l}=(\vec{l}_{1*}, \vec{l}_{2*},........\vec{l}_{n*}$$, each $$l_{ij}\in \{0,1\}$$.

My question is shouldn't the condition be $$l_{i1}=0, \forall i$$, because only then the X gate is removed. Can somebody check?

The link for the paper is https://journals.aps.org/pra/abstract/10.1103/PhysRevA.78.042309

• Can you specify what is $\vec{l}$? And what are the superscripts $l_{i1}$ and $l_{i2}$? Is $\vec{l}$ a $n \times 2$ 'vector'? Is it some $n \times n$ matrix? If so, is it Hermitian/Symmetric? Because then clearly $l_{1i} = l_{i1}$. Also, is there an open-access version of the paper?
– JSdJ
Jun 23 at 10:52
• Here is the open access version arxiv.org/abs/0808.1532 Jun 23 at 11:36
• Why is there a downvote on this question? Jun 23 at 16:20