# Constructing a maximally entangled basis of a bipartite system $\mathcal{H}=\mathbb{C}^d\otimes\mathbb{C}^d$, starting from the reference state?

$$\newcommand{\Ket}[1]{\left|#1\right>}$$

Given $$\Ket{\phi}=\frac{1}{\sqrt{d}}\Sigma_{i=1}^d\Ket{i}_A\otimes\Ket{i}_B$$, where $$d=\textrm{dim}(\mathcal{H}_A)=\textrm{dim}(\mathcal{H}_B)$$, how does one construct a maximally entangled basis of $$\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$$, using local unitaries?

I am looking for the higher dimensional equivalent of constructing the Bell basis for a pair of qubits, using Pauli matrices in one subsystem (i.e. $$\mathbb{1}\otimes\sigma_i$$) and the Bell state $$\Ket{\phi^+}=\frac{1}{\sqrt{2}}\left(\Ket{00}+\Ket{11}\right)$$.

What you need is a set of unitaries $$U_k$$ such that $$\mathrm{tr}[U_kU_\ell^\dagger]=\delta_{k\ell}$$, and then build $$(U_k\otimes I)|\phi\rangle$$. Unitarity ensures that you get maximally entangled states, and the orthogonality condition ensures that they are orthogonal.
For instance, you can use the set of unitaries generated by $$X=\sum_{x=1}^d |x+1\rangle\langle x|$$ and $$Z=\sum_{x=1}^d e^{2\pi i x/d}|x\rangle\langle x|$$, i.e. $$U_{(nm)}=X^nZ^m$$, with a double index $$k=(nm)$$. It is easy to see that they satisfy the orthogonality condition.