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 $\Ket{\phi^+}=\frac{1}{\sqrt{2}}\left(\Ket{00}+\Ket{11}\right)$.


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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.


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