In the context of quantum mechanics and quantum information theory, symmetric, informationally complete, positive operator-valued measures (SIC-POVMs) are a particular type of generalized measurement (POVM). SIC-POVMs are particularly notable thanks to their defining features of (1) being informationally complete; (2)having the minimal number of outcomes compatible with informational completeness, and (3) being highly symmetric. In this context, informational completeness is the property of a POVM of allowing to fully reconstruct input states from measurement data.
The properties of SIC-POVMs make them an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. SIC-POVMs have several applications in the context of quantum state tomography[1] and quantum cryptography,[2] and a possible connection has been discovered with Hilbert's twelfth problem.[3]
A POVM over a
d
l{H}
m
\left\{Fi
m | |
\right\} | |
i=1 |
d2
l{L}(l{H})
d2
d2
\left\{\Pii
d2 | |
\right\} | |
i=1 |
Fi=
1 | |
d |
\Pii
Consider an arbitrary set of rank-1 projectors
(\Pii)
d2 | |
i=1 |
Fi=\Pii/d
1 | |
d |
\sumi\Pii=I
Tr(\Pii\Pij)=c
i ≠ j
c
c=
1 | |
d+1 |
In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map
l{L}(l{H}) → l{L}(l{H})
\begin{align}l{G}:l{L}(l{H})& → l{L}(l{H})\\ A&\mapsto\displaystyle\sum\alpha|\psi\alpha\rangle\langle\psi\alpha|A|\psi\alpha\rangle\langle\psi\alpha|\end{align}
This operator acts on a SIC-POVM element in a way very similar to identity, in that
\begin{align}l{G}(\Pi\beta)&=\displaystyle\sum\alpha\Pi\alpha\left|\langle\psi\alpha|\psi\beta\rangle\right|2\\ &=\displaystyle\Pi\beta+
1 | |
d+1 |
\sum\alpha\Pi\alpha\\ &=\displaystyle
d | |
d+1 |
\Pi\beta+
1 | |
d+1 |
\Pi\beta+
1 | |
d+1 |
\sum\alpha\Pi\alpha\\ &=\displaystyle
d | |
d+1 |
\Pi\beta+
d | |
d+1 |
\sum\alpha
1 | |
d |
\Pi\alpha\\ &=\displaystyle
d | |
d+1 |
\left(\Pi\beta+I\right)\end{align}
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=
d | |
d+1 |
\left(l{I}+I\right)
I(A)=Aandl{I}(A)=Tr(A)I
G-1=
1d | |
\left[ |
\left(d+1\right)I-l{I}\right]
I=G-1G=
1d | |
\sum |
\alpha\left[(d+1)\Pi\alpha\odot\Pi\alpha-I\odot\Pi\alpha\right]
\rho
\begin{align}\rho=I|\rho)&=\displaystyle\sum\alpha\left[(d+1)\Pi\alpha-I\right]
(\Pi\alpha|\rho) | |
d |
\\ &=\displaystyle\sum\alpha\left[(d+1)\Pi\alpha-I\right]
Tr(\Pi\alpha\rho) | |
d |
\\ &=\displaystyle\sum\alphap\alpha\left[(d+1)\Pi\alpha-I\right] wherep\alpha=Tr(\Pi\alpha\rho)/d\\ &=\displaystyle-I+(d+1)\sum\alphap\alpha|\psi\alpha\rangle\langle\psi\alpha|\\ &=\displaystyle\sum\alpha\left[(d+1)p\alpha-
1d | |
\right] |
|\psi\alpha\rangle\langle\psi\alpha| \end{align}
|\rho)
l{L}(l{H})
\rho
(d+1)p\alpha-
1d | |
For
d=2
which form the vertices of a regular tetrahedron in the Bloch sphere. The projectors that define the SIC-POVM are given by
\Pii=|\psii\rangle\langle\psii|
Fi=\Pii/2=|\psii\rangle\langle\psii|/2
For higher dimensions this is not feasible, necessitating the use of a more sophisticated approach.
A SIC-POVM
P
G
d2
\forall|\psi\rangle\langle\psi|\inP, \forallUg\inG, Ug|\psi\rangle\inP
\forall|\psi\rangle\langle\psi|,|\phi\rangle\langle\phi|\inP, \existsUg\inG, Ug|\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
|\langle\phi|Ug|\phi\rangle|2=
1 | |
d+1 |
\forallg ≠ id
Ug
|\phi\rangle
So far, most SIC-POVM's have been found by considering group covariance under
Zd x Zd
Zd x Zd
U(d)
|ei\rangle
l{H}
T|ei\rangle=\omegai|ei\rangle
\omega=
| ||||
e |
S|ei\rangle=|ei+1
Combining these two operators yields the Weyl operator
W(p,q)=SpTq
\begin{align}W(p,q)W\dagger(p,q)&=SpTqT-qS-p\\ &=Id\end{align}
(p,q)\inZd x Zd → W(p,q)
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,[5] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.
More specifically,
For every dimensionthere exists a SIC-POVM whose elements are the orbit of a positive rank-one operatord\geq2
under the Weyl - Heisenberg groupE0
. What is more,Hd
commutes with an element T of the Jacobi groupE0
. The action of T onJd=Hd\rtimesSL(2,Zd)
modulo the center has order three.Hd
Utilizing the notion of group covariance on
Zd x Zd
For any dimension, letd\inN
be an orthonormal basis for\left\{k
d-1 \right\} k=0 , and defineCd
Then\displaystyle\omega=
2\pii d e , Dj,k=
jk 2 \omega
d-1 \sum m=0 \omegajm|k+m\pmod{d}\rangle\langlem|
such that the set\exists|\phi\rangle\inCd
is a SIC-POVM.\left\{Dj,k|\phi\rangle
d \right\} j,k=1
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
d=2
d=53
d=5779
d
Zd x Zd
d=193
d=2208
A spherical t-design is a set of vectors
S=\left\{|\phik\rangle:|\phik\rangle\inSd\right\}
tth
ft(\psi)
S
ft(\psi)
|\psi\rangle
l{H}t=\displaystyle
t | |
otimes | |
i=1 |
l{H}
St=\displaystyle
n | |
\sum | |
k=1 |
|
t | |
\Phi | |
k |
\rangle\langle
t | |
\Phi | |
k |
|,
t\rangle | |
|\Phi | |
k |
=
⊗ t | |
|\phi | |
k\rangle |
\left\{|\phik\rangle\inSd
n | |
\right\} | |
k=1 |
n\geq{t+d-1\choosed-1}
\displaystyleTr\left[
2 | |
S | |
t |
\right]=\sumj,k\left|\langle\phij|\phik\rangle\right|2t=
n2t!(d-1)! | |
(t+d-1)! |
It then immediately follows that every SIC-POVM is a 2-design, since
2 | |
Tr(S | |
2) |
=\displaystyle\sumj,k|\langle\phij|\phik\rangle|4=
2d3 | |
d+1 |
In a d-dimensional Hilbert space, two distinct bases
\left\{|\psii\rangle\right\},\left\{|\phij\rangle\right\}
\displaystyle|\langle\psii|\phij\rangle|2=
1 | |
d |
, \foralli,j
d+1
In dimension
d=3