Symmetry

Consider an arbitrary set of rank-1 projectors ${\displaystyle (\Pi _{i})_{i=1}^{d^{2}}}$ such that ${\displaystyle F_{i}=\Pi _{i}/d}$ is a POVM, and thus ${\displaystyle {\frac {1}{d}}\sum _{i}\Pi _{i}=I}$. Asking the projectors to have equal pairwise inner products, ${\displaystyle \mathrm {Tr} (\Pi _{i}\Pi _{j})=c}$ for all ${\displaystyle i\neq j}$, fixes the value of ${\displaystyle c}$. To see this, observe that

$${\displaystyle {\begin{aligned}d&=\mathrm {Tr} (I^{2})\\&={\frac {1}{d^{2}}}\sum _{i,j}\mathrm {Tr} (\Pi _{i}\Pi _{j})\\&={\frac {1}{d^{2}}}\left(d^{2}+cd^{2}(d^{2}-1)\right)\end{aligned}}}$$

implies that ${\displaystyle c={\frac {1}{d+1}}}$. Thus,

$${\displaystyle \mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}}.}$$

This property is what makes SIC-POVMs symmetric; with respect to the Hilbert–Schmidt inner product, any pair of elements is equivalent to any other pair.

Superoperator

In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map ${\displaystyle {\mathcal {L}}({\mathcal {H}})\rightarrow {\mathcal {L}}({\mathcal {H}})}$. This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map

${\displaystyle {\begin{aligned}{\mathcal {G}}:{\mathcal {L}}({\mathcal {H}})&\rightarrow {\mathcal {L}}({\mathcal {H}})\\A&\mapsto \displaystyle \sum _{\alpha }|\psi _{\alpha }\rangle \langle \psi _{\alpha }|A|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\end{aligned}}}$

This operator acts on a SIC-POVM element in a way very similar to identity, in that

${\displaystyle {\begin{aligned}{\mathcal {G}}(\Pi _{\beta })&=\displaystyle \sum _{\alpha }\Pi _{\alpha }\left|\langle \psi _{\alpha }|\psi _{\beta }\rangle \right|^{2}\\&=\displaystyle \Pi _{\beta }+{\frac {1}{d+1}}\sum _{\alpha \neq \beta }\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\Pi _{\beta }+{\frac {1}{d+1}}\Pi _{\beta }+{\frac {1}{d+1}}\sum _{\alpha \neq \beta }\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\Pi _{\beta }+{\frac {d}{d+1}}\sum _{\alpha }{\frac {1}{d}}\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\left(\Pi _{\beta }+I\right)\end{aligned}}}$

But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:

${\displaystyle G={\frac {d}{d+1}}\left({\mathcal {I}}+I\right)}$ where ${\displaystyle I(A)=A{\text{ and }}{\mathcal {I}}(A)=\mathrm {Tr} (A)I}$

From here, the left inverse can be calculated^[4] to be ${\displaystyle G^{-1}={\frac {1}{d}}\left[\left(d+1\right)I-{\mathcal {I}}\right]}$, and so with the knowledge that

${\displaystyle I=G^{-1}G={\frac {1}{d}}\sum _{\alpha }\left[(d+1)\Pi _{\alpha }\odot \Pi _{\alpha }-I\odot \Pi _{\alpha }\right]}$,

an expression for a state ${\displaystyle \rho }$ can be created in terms of a quasi-probability distribution, as follows:

${\displaystyle {\begin{aligned}\rho =I|\rho )&=\displaystyle \sum _{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]{\frac {(\Pi _{\alpha }|\rho )}{d}}\\&=\displaystyle \sum _{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]{\frac {\mathrm {Tr} (\Pi _{\alpha }\rho )}{d}}\\&=\displaystyle \sum _{\alpha }p_{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]\quad {\text{ where }}p_{\alpha }=\mathrm {Tr} (\Pi _{\alpha }\rho )/d\\&=\displaystyle -I+(d+1)\sum _{\alpha }p_{\alpha }|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\\&=\displaystyle \sum _{\alpha }\left[(d+1)p_{\alpha }-{\frac {1}{d}}\right]|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\end{aligned}}}$

where ${\displaystyle |\rho )}$ is the Dirac notation for the density operator viewed in the Hilbert space ${\displaystyle {\mathcal {L}}({\mathcal {H}})}$. This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state ${\displaystyle \rho }$ is given by

${\displaystyle (d+1)p_{\alpha }-{\frac {1}{d}}}$

Simplest example

For ${\displaystyle d=2}$ the equations that define the SIC-POVM can be solved by hand, yielding the vectors

$${\displaystyle {\begin{aligned}|\psi _{1}\rangle &=|0\rangle \\|\psi _{2}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}|1\rangle \\|\psi _{3}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}e^{i{\frac {2\pi }{3}}}|1\rangle \\|\psi _{4}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}e^{i{\frac {4\pi }{3}}}|1\rangle ,\end{aligned}}}$$

which form the vertices of a regular tetrahedron in the Bloch sphere. The projectors that define the SIC-POVM are given by ${\displaystyle \Pi _{i}=|\psi _{i}\rangle \langle \psi _{i}|}$, and the elements of the SIC-POVM are thus ${\displaystyle F_{i}=\Pi _{i}/2=|\psi _{i}\rangle \!\langle \psi _{i}|/2}$.

For higher dimensions this is not feasible, necessitating the use of a more sophisticated approach.

Group covariance

Zauner's conjecture

Given some of the useful properties of SIC-POVMs, it would be useful if it were positively known whether such sets could be constructed in a Hilbert space of arbitrary dimension. Originally proposed in the dissertation of Zauner,^[7] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.

More specifically,

For every dimension ${\displaystyle d\geq 2}$ there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator ${\displaystyle E_{0}}$ under the Weyl–Heisenberg group ${\displaystyle H_{d}}$. What is more, ${\displaystyle E_{0}}$ commutes with an element T of the Jacobi group ${\displaystyle J_{d}=H_{d}\rtimes SL(2,\mathbb {Z} _{d})}$. The action of T on ${\displaystyle H_{d}}$ modulo the center has order three.

Utilizing the notion of group covariance on ${\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}}$, this can be restated as ^[8]

For any dimension ${\displaystyle d\in \mathbb {N} }$, let ${\displaystyle \left\{k\right\}_{k=0}^{d-1}}$ be an orthonormal basis for ${\displaystyle \mathbb {C} ^{d}}$, and define

${\displaystyle \displaystyle \omega =e^{\frac {2\pi i}{d}},\quad \quad D_{j,k}=\omega ^{\frac {jk}{2}}\sum _{m=0}^{d-1}\omega ^{jm}|k+m{\pmod {d}}\rangle \langle m|}$

Then ${\displaystyle \exists |\phi \rangle \in \mathbb {C} ^{d}}$ such that the set ${\displaystyle \left\{D_{j,k}|\phi \rangle \right\}_{j,k=1}^{d}}$ is a SIC-POVM.

Partial results

The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,^[6] but is an ongoing field of research in the quantum information community.

Exact expressions for SIC sets have been found for Hilbert spaces of all dimensions from ${\displaystyle d=2}$ through ${\displaystyle d=53}$ inclusive, and in some higher dimensions as large as ${\displaystyle d=5779}$, for 115 values of ${\displaystyle d}$ in all.^{[lower-alpha 1]} Furthermore, using the Heisenberg group covariance on ${\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}}$, numerical solutions have been found for all integers up through ${\displaystyle d=193}$, and in some larger dimensions up to ${\displaystyle d=2208}$.^{[lower-alpha 2]}