the localization of $\CC_\Lambda^t[x_1,\ldots, x_n]$ at the monoid generated by ${x_1,\ldots, x_n}$, which is an Ore set. The quantum torus is clearly contained in the skew-field of fractions $\mathcal{F}_\Lambda$ of $\CC_\Lambda^t[x_1,\ldots, x_n]$, and the Laurent monomials define a lattice $L\subset  H_\Lambda^t\subset
\mathcal{F}_\Lambda$ isomorphic to $\ZZ^n$. Denote for each ${\bf e}=(e_1,\ldots e_n)\in \ZZ^n$ by $x^{\bf e}$ the monomial $x_1^{e_1}\ldots x_n^{e_n}$.

We need the notion of a toric frame in order to define the quantum cluster algebra.

A toric frame in $ \mathfrak F$ is a mapping $M:\ZZ^n\to  \mathfrak F-\{0\}$ of the form $$M(c)=\phi(X^{\eta(c)})\ ,$$ where $\phi$ is a $\QQ(\frac{1}{2})$-algebra automorphism of $\mathfrak F$ and $\eta: \ZZ^n\to L$ an isomorphism of lattices.

Since a toric frame $M$ is determined uniquely by the images of the standard basis vectors $\phi(X^{\eta(e_1)})$,…, $\phi(X^{\eta(e_n)})$ of $\ZZ^n$, we can associate to each toric frame a skew commutative $n\times n$-integer matrix $\Lambda_M$. We can now define the quantized version of a seed.

[@bz-qclust Definition 4.5] A quantum seed is a pair $(M,B)$ where

-   $M$ is a toric frame in $\mathfrak F$.

-   $B$ is a $n\times m$-integer matrix with rows labeled by $[1,m]$ and columns labeled by $[1,n]$.

-   The pair $(B,\Lambda_M)$ is compatible.

Now we define 