SIC-POVM

A symmetric, informationally complete, positive operator valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement," utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a connection has been discovered with Hilbert's twelfth problem.

Definition

Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used throughout this article to represent elements in a Hilbert space.

A POVM over a $d$-dimensional Hilbert space ${\mathcal {H}}$ is a set of $m$ positive semidefinite operators $\{F_{i}\}$ on the Hilbert space that sum to the identity:$$\sum _{i=1}^{m}F_{i}=I.$$If a POVM consists of at least $d^{2}$ operators which span ${\mathcal {L}}({\mathcal {H}})$, it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly $d^{2}$ elements are called minimal. A set of $d^{2}$ rank-1 projectors $\{\Pi _{i}\}$ which have equal pairwise Hilbert–Schmidt inner products,$$\mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}},$$defines a minimal IC-POVM $F_{i}={\frac {1}{d}}\Pi _{i}$ called a SIC-POVM.

Properties

Symmetry

The condition that the projectors $\{\Pi _{i}\}$ defined above have equal pairwise inner products actually fixes the value of this constant. Recall that ${\frac {1}{d}}\sum _{i}\Pi _{i}=I$ and set $\mathrm {Tr} (\Pi _{i}\Pi _{j})=c$. Then$${\begin{aligned}d&=\mathrm {Tr} (I^{2})\\&=\displaystyle {\frac {1}{d^{2}}}\sum _{i,j}\mathrm {Tr} (\Pi _{i}\Pi _{j})\\&=\displaystyle {\frac {1}{d^{2}}}\left(d^{2}+cd^{2}(d^{2}-1)\right)\end{aligned}}$$implies that $c={\frac {1}{d+1}}$. Thus,$$\mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}}.$$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 ${\mathcal {B}}({\mathcal {H}})\rightarrow {\mathcal {B}}({\mathcal {H}})$. This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map

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

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

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

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:

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

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

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

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

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

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

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

Finding SIC sets

Group covariance

General group covariance

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

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

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 $|\phi \rangle$ such that

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

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

The case of Z_d × Z_d

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

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

and the shift operator as

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

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

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

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

Zauner's conjecture

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

More specifically,

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

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

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

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

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

Partial results

Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is $d=1,\dots ,21,24,28,30,31,35,37,39,43,48$. Furthermore, using the Heisenberg group covariance on $\mathbb {Z} _{d}\times \mathbb {Z} _{d}$, numerical solutions have been found for all integers up through $d=151$.

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

Relation to spherical t-designs

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

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

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

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

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

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

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

Relation to MUBs

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

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

This seems similar in nature to the symmetric property of SIC-POVMs. In fact, the problem of finding a SIC-POVM is precisely the problem of finding equiangular lines in $\mathbb {C} ^{d}$; whereas mutually unbiased bases are analogous to affine spaces. In fact it can be shown that the geometric analogy of finding a "complete set of $N+1$ mutually unbiased bases is identical to the geometric structure analogous to a SIC-POVM ". It is important to note that the equivalence of these problems is in the strict sense of an abstract geometry, and since the space on which each of these geometric analogues differs, there's no guarantee that a solution on one space will directly correlate with the other.

An example of where this analogous relation has yet to necessarily produce results is the case of 6-dimensional Hilbert space, in which a SIC-POVM has been analytically computed using mathematical software, but no complete mutually unbiased bases has yet been discovered.

See also

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