bibkey: renes2004sic authors: J. M. Renes, R. Blume-Kohout, A. J. Scott, C. M. Caves year: 2004 title: “Symmetric Informationally Complete Quantum Measurements” doi: 10.1063/1.1737053 url: https://arxiv.org/abs/quant-ph/0310075v1 claim: “A set of n ≥ C(t+d−1, d−1) unit vectors in C^d is a spherical t-design iff its t-th frame potential Σ_{j,k}|⟨φ_j|φ_k⟩|^{2t} equals n² t!(d−1)!/(t+d−1)!; consequently every SIC-POVM is a 2-design, and every 2-design with d² elements is a SIC-POVM.” strata_touched:

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Renes, Blume-Kohout, Scott and Caves, symmetric informationally complete measurements

J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, Symmetric Informationally Complete Quantum Measurements, J. Math. Phys. 45, 2171 (2004); arXiv:quant-ph/0310075v1 (13 October 2003).

Verified locator

DOI: 10.1063/1.1737053. Preprint: https://arxiv.org/abs/quant-ph/0310075v1 (the only arXiv version). The TeX source sicpovmarchive.tex supplies §2 (frames and spherical $t$-designs, Theorem 2 and the SIC-POVM consequence).

Source statements

Spherical design (§2): “A \emph{spherical $t$-design/} is a set of $n$ normalized vectors $\{|{\varphi }_k⟩\in {\mathbb{S}}^d\}$ such that the average value of any $t$-th order polynomial $f_t(\psi )$ over the set $\{|{\varphi }_k⟩\}$ is equal to the average of $f_t(\psi )$ over \emph{all/} normalized vectors $|\psi ⟩$.”

Theorem 2: “A set of normalized vectors $\{|{\varphi }_k⟩\in {\mathbb{S}}^d{\}}_{k=1}^n$ with $n\ge \left(\genfrac{}{}{0ex}{}{t+d-1}{d-1}\right)$ forms a spherical $t$-design if and only if $\mathrm{Tr}[S_t^2]={\sum }_{j,k}|⟨{\varphi }_j|{\varphi }_k⟩{|}^{2t}=\frac{n^{2}t!(d-1)!}{(t+d-1)!}$. Furthermore, this value is the global minimum of $\mathrm{Tr}[S_t^2]$.”

Consequence (§2): “Immediately we can infer that every SIC-POVM is a 2-design since Tr[S_2^2]=\sum_{j,k}|\bracket{\phi_j}{\phi_k}|^4=2d^3/(d+1), the required value for a 2-design.”

Scope

For $d=2$ and $n=4$ the 2-design property is the operator identity ${\sum }_k({\psi }_k{\psi }_k^{†}{)}^{\otimes 2}=\frac{2}{3}(I+\mathrm{SWAP})$ on ${\mathbb{C}}^2\otimes {\mathbb{C}}^2$; the repository uses it in this form.