Symmetry edit

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

Superoperator edit

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 edit

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 edit

General group covariance edit

A SIC-POVM ${\displaystyle P}$  is said to be group covariant if there exists a group ${\displaystyle G}$  with a ${\displaystyle d^{2}}$ -dimensional unitary representation such that

• ${\displaystyle \forall |\psi \rangle \langle \psi |\in P,\quad \forall U_{g}\in G,\quad U_{g}|\psi \rangle \in P}$ 
• ${\displaystyle \forall |\psi \rangle \langle \psi |,|\phi \rangle \langle \phi |\in P,\quad \exists U_{g}\in G,\quad U_{g}|\phi \rangle =|\psi \rangle }$ 

The search for SIC-POVMs can be greatly simplified by exploiting the property of group covariance. Indeed, the problem is reduced to finding a normalized fiducial vector ${\displaystyle |\phi \rangle }$  such that

${\displaystyle |\langle \phi |U_{g}|\phi \rangle |^{2}={\frac {1}{d+1}}\ \forall g\neq id}$ .

The SIC-POVM is then the set generated by the group action of ${\displaystyle U_{g}}$  on ${\displaystyle |\phi \rangle }$ .

The case of Z_d × Z_d edit

So far, most SIC-POVM's have been found by considering group covariance under ${\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}}$ .^[5] To construct the unitary representation, we map ${\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}}$  to ${\displaystyle U(d)}$ , the group of unitary operators on d-dimensions. Several operators must first be introduced. Let ${\displaystyle |e_{i}\rangle }$  be a basis for ${\displaystyle {\mathcal {H}}}$ , then the phase operator is

${\displaystyle T|e_{i}\rangle =\omega ^{i}|e_{i}\rangle }$  where ${\displaystyle \omega =e^{\frac {2\pi i}{d}}}$  is a root of unity

and the shift operator as

${\displaystyle S|e_{i}\rangle =|e_{i+1{\pmod {d}}}\rangle }$ 

Combining these two operators yields the Weyl operator ${\displaystyle W(p,q)=S^{p}T^{q}}$  which generates the Heisenberg-Weyl group. This is a unitary operator since

${\displaystyle {\begin{aligned}W(p,q)W^{\dagger }(p,q)&=S^{p}T^{q}T^{-q}S^{-p}\\&=Id\end{aligned}}}$ 

It can be checked that the mapping ${\displaystyle (p,q)\in \mathbb {Z} _{d}\times \mathbb {Z} _{d}\rightarrow W(p,q)}$  is a projective unitary representation. It also satisfies all of the properties for group covariance,^[6] and is useful for numerical calculation of SIC sets.

Zauner's conjecture edit

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 edit

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.^[a] 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 ${\displaystyle d=2208}$ .^[b]

A spherical t-design is a set of vectors ${\displaystyle S=\left\{|\phi _{k}\rangle :|\phi _{k}\rangle \in \mathbb {S} ^{d}\right\}}$  on the d-dimensional generalized hypersphere, such that the average value of any ${\displaystyle t^{th}}$ -order polynomial ${\displaystyle f_{t}(\psi )}$  over ${\displaystyle S}$  is equal to the average of ${\displaystyle f_{t}(\psi )}$  over all normalized vectors ${\displaystyle |\psi \rangle }$ . Defining ${\displaystyle {\mathcal {H}}_{t}=\displaystyle \bigotimes _{i=1}^{t}{\mathcal {H}}}$  as the t-fold tensor product of the Hilbert spaces, and

${\displaystyle S_{t}=\displaystyle \sum _{k=1}^{n}|\Phi _{k}^{t}\rangle \langle \Phi _{k}^{t}|,\quad |\Phi _{k}^{t}\rangle =|\phi _{k}\rangle ^{\otimes t}}$ 

as the t-fold tensor product frame operator, it can be shown that^[8] a set of normalized vectors ${\displaystyle \left\{|\phi _{k}\rangle \in \mathbb {S} ^{d}\right\}_{k=1}^{n}}$  with ${\displaystyle n\geq {t+d-1 \choose d-1}}$  forms a spherical t-design if and only if

${\displaystyle \displaystyle \mathrm {Tr} \left[S_{t}^{2}\right]=\sum _{j,k}\left|\langle \phi _{j}|\phi _{k}\rangle \right|^{2t}={\frac {n^{2}t!(d-1)!}{(t+d-1)!}}}$ 

It then immediately follows that every SIC-POVM is a 2-design, since

${\displaystyle \mathrm {Tr} (S_{2}^{2})=\displaystyle \sum _{j,k}|\langle \phi _{j}|\phi _{k}\rangle |^{4}={\frac {2d^{3}}{d+1}}}$ 

which is precisely the necessary value that satisfies the above theorem.

In a d-dimensional Hilbert space, two distinct bases ${\displaystyle \left\{|\psi _{i}\rangle \right\},\left\{|\phi _{j}\rangle \right\}}$  are said to be mutually unbiased if

${\displaystyle \displaystyle |\langle \psi _{i}|\phi _{j}\rangle |^{2}={\frac {1}{d}},\quad \forall i,j}$ 

This seems similar in nature to the symmetric property of SIC-POVMs. Wootters points out that a complete set of ${\displaystyle d+1}$  unbiased bases yields a geometric structure known as a finite projective plane, while a SIC-POVM (in any dimension that is a prime power) yields a finite affine plane, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged. In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.^[17]

In dimension ${\displaystyle d=3}$ , the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.^[18] The 9 vectors of the SIC-POVM, together with the 12 vectors of the mutually unbiased bases, form a set that can be used in a Kochen–Specker proof.^[19] However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete set of mutually unbiased bases has yet been discovered, and it is widely believed that no such set exists.^[20][21]

• Measurement in quantum mechanics
• Mutually unbiased bases
• POVM
• QBism