A statement found for example in (Ambainis Emerson 2007) is
Quantum $t$-designs are related to $t$-designs of vectors on the unit sphere in $\mathbb{R}^N$, called spherical $t$-designs.
[...]
A spherical $t$-design in $\mathbb{R}^N$ can be transformed into a $(t/2,t/2)$-design in $\mathbb{C}^{N/2}$.
I'm trying to understand more precisely the relation between complex-projective $t$-designs (which I think is what they mean with "quantum $t$-designs") and spherical $t$-designs. What does the above-mentioned "transformation" look like exactly?
My issue is that if I take a complex-projective $t$-design, defined as a set of vectors allowing to compute averages of polynomials in the states of degree up to $t$ (in a sense clarified below), how does that correspond to a spherical (real) $t/2$ design? How do you go from the complex vectors (technically, complex projective rays) forming the complex-projective design, to a set of real unit vectors?
Can I just take real and imaginary parts and be done with it? As in, map each normalised $\psi\in\mathbb{C}^{N}$ into $(\operatorname{Re}(\psi),\operatorname{Im}(\psi))\in\mathbb{R}^{2N}$? What's a good way to show that this procedure always maps a complex-projective $t$-design on $\mathbb{CP}^{N-1}$ into a spherical $(t/2)$-design on $S^{2N-1}\subset \mathbb{R}^{2N}$?
What also confuses me a little bit in this relation is that complex-projective designs are defined based on the behaviour of averages of polynomials that have degree $t$ in two different types of arguments (those $\psi_i$ inputs and those taking $\bar\psi_i$ inputs). On the other hand, spherical designs allow any polynomial up to a given degree. So if I convert complex vectors into larger real vectors, to have a spherical design I need to have the correct values averaging polynomials that don't have an obvious counterpart in the complex-projective case.
Some background info and context
Definition of spherical designs — A spherical t-design on $S^d$, as defined e.g. in (Bajnok 1998), is a set of points $X\subset S^d$ such that $$ \frac{1}{|X|}\sum_{\boldsymbol x\in X} p_t(\boldsymbol x)=\int_{S^d}d\mu(\boldsymbol x) p_t(\boldsymbol x), $$ for all real polynomials $p_t\in\mathbb{R}[X_1,...,X_{d+1}]$ of degree at most $t$.
Definition of complex-projective designs — On the other hand, a complex-projective t-design, following for example (Ambainis Emerson 2007), is a set of states $\{|\psi_i\rangle\}_{i=1}^{|X|}=X\subset\mathbb{CP}^d$ such that $$\frac1{|X|} \sum_{i=1}^{|X|} p_{t,t}(\psi_i) = \int_\psi p_{t,t}(\psi) d\psi,$$ where the integral in the RHS is over the Haar measure on the unit sphere in $\mathbb{C}^d$, and $p_{t,t}(X_1,...,X_{d+1},Y_1,...,Y_{d+1})$ is a polynomial of degree at most $t$ in both $X_i$ and $Y_i$ arguments, and we defined $p_{t,t}(\psi)\equiv p_{t,t}(\psi_1,...,\psi_{d+1},\psi_1^*,...,\psi_{d+1}^*)$, given $|\psi\rangle\equiv\sum_{i=1}^{d+1} \psi_i|i\rangle$. I'm assuming in these definitions we are defining the expansion coefficients $\psi_i$ by taking a representative of the complex-projective rays $|\psi\rangle\in\mathbb{CP}^d$ with unit norm, otherwise I'm not sure how the expressions are well-defined.
Example of spherical design — In the $d=1$ case, spherical $t$-designs are (all and only) the vertices of regular $n$-gons with $n\ge t+1$. As a more explicit toy example, $X=\{(1,0),(0,1),(-1,0),(0,-1)\}$ are the vertices of a square, and can be readily verified to give a spherical 3-design: they allow to compute averages of polynomials of degree 3, $$ \frac14(1^2 + 0^2+(-1)^2+0^2) = \frac12 = \int_0^{2\pi} \frac{d\theta}{2\pi} \cos^2(\theta), \qquad(p_2(x,y)\equiv x^2) \\ \frac14(1\cdot 0 + 0\cdot 1+(-1)\cdot0+0\cdot(-1)) = 0 = \int_0^{2\pi} \frac{d\theta}{2\pi} \cos(\theta)\sin(\theta), \qquad(p_2(x,y)\equiv x y), \\ \frac14(1^3 + 0^3 + (-1)^3 + 0^3) = 0 = \int_0^{2\pi} \frac{d\theta}{2\pi} \cos^3(\theta), \qquad(p_3(x,y)\equiv x^3) $$ but they give the wrong averages for polynomials of degree 4: $$\frac12=\frac14(1^4 + 0^4 + (-1)^4 + 0^4) \neq \int_0^{2\pi} \frac{d\theta}{2\pi} \cos^4(\theta) = \frac38, \qquad(p_4(x,y)\equiv x^4).$$
Example of complex-projective design — A standard example of a complex-projective 2-design on $\mathbb{CP}^1$ is a SIC-POVM. More specifically, the four pure states $$|\psi_1\rangle=|0\rangle,\\ |\psi_2\rangle = \frac{1}{\sqrt3}(|0\rangle+\sqrt2|1\rangle), \qquad |\psi_3\rangle = \frac{1}{\sqrt3}(|0\rangle+\sqrt2 e^{2\pi i/3}|1\rangle),\\ |\psi_4\rangle = \frac{1}{\sqrt3}(|0\rangle+\sqrt2 e^{4\pi i/3}|1\rangle), $$ form a complex-projective 2-design. And we can verify this computing for example the average of $|\psi_0|^{2k}$ on these states: $$ \frac14(1^k + 3/3^k) = \begin{cases}\frac12, & k=1, \\ \frac13, & k=2, \\ \frac{5}{18}, & k=3.\end{cases} $$ These values are to be compared with the uniform (with respect to the Fubini-Study metric) averages over the whole of $\mathbb{CP}^1$, which give (assuming I'm parametrising things correctly): $$\int_{\mathbb{CP}^1 }\mathrm d\mu_{\rm FS}([\psi])|\psi_0|^{2k} = \int_0^{\pi/2} \mathrm d\theta \sin(2\theta) \cos^{2k}(\theta) = \frac{1}{k+1}.$$ Clearly, the two averages are equal for $k=1,2$, but not for $k=3$, confirming the set being a 2-design, but not a 3-design (of course, this isn't a proof, I'm just using these calculations as sanity checks and ways to show explicitly what the design condition entails).
