1 Introduction
2 Quantum states and POVMs
3 Symmetric, resolving and highly symmetric sets in metric spaces
4 Symmetric, informationally complete and highly symmetric normalized rank-1 POVMs
-
\(\varPi \) is a symmetric POVM \(\Leftrightarrow \) S is symmetric in \(({{\mathcal {P}}}\left( {\mathbb {C}}^{d}\right) ,D_{FS})\);
-
\(\varPi \) is a highly symmetric POVM (HS-POVM) \(\Leftrightarrow \) S is highly symmetric in \(({\mathcal {P}}\left( {\mathbb {C}}^{d}\right) ,D_{FS})\).
5 Classification of highly symmetric POVMs in dimension 2
Convex hull of the orbit | Cardinality of the orbit | Group | Stabilizer |
---|---|---|---|
Digon | 2 |
\(D_{\infty h}\)
|
\(C_{\infty v}\)
|
Regular n-gon (\(n\ge 3\)) |
n
|
\(D_{nh}\)
|
\(C_{2v}\)
|
Tetrahedron | 4 |
\(T_{d}\)
|
\(C_{3v}\)
|
Octahedron | 6 |
\(O_{h}\)
|
\(C_{4v}\)
|
Cube | 8 |
\(O_{h}\)
|
\(C_{3v}\)
|
Cuboctahedron | 12 |
\(O_{h}\)
|
\(C_{2v}\)
|
Icosahedron | 12 |
\(I_{h}\)
|
\(C_{5v}\)
|
Dodecahedron | 20 |
\(I_{h}\)
|
\(C_{3v}\)
|
Icosidodecahedron | 30 |
\(I_{h}\)
|
\(C_{2v}\)
|
6 Entropy and relative entropy of measurement
6.1 Definition
6.2 Relation to informational power
6.3 Relation to entropic uncertainty principles
6.4 Relation to Wehrl entropy minimization
6.5 Relation to quantum dynamical entropy
6.6 Entropy in the Bloch representation
7 Local extrema of entropy for symmetric POVMs in dimension 2
8 Global minima of entropy for highly symmetric POVMs in dimension 2
8.1 The minimization method based on the Hermite interpolation
Kv
|
\(\left| Kv\right| \)
|
K
|
\(K_{v}\)
|
\(n_{a}(v)\)
|
\(n_{s}(v)\)
|
n(v) |
\(\deg p_{v}\le \)
|
---|---|---|---|---|---|---|---|
Regular n-gon (n-even) |
n
|
\(C_{n}\)
|
\(C_{1}\)
|
\(n-2\)
| 2 |
\(n/2+1\)
|
\(n-1\)
|
Regular n-gon (n-odd) |
n
|
\(C_{n}\)
|
\(C_{1}\)
|
\(n-1\)
| 1 |
\(n/2+1/2\)
|
\(n-1\)
|
Tetrahedron | 4 |
T
|
\(C_{3}\)
| 0 | 2 | 2 | 2 |
Octahedron | 6 |
O
|
\(C_{4}\)
| 0 | 3 | 3 | 3 |
Cube | 8 |
O
|
\(C_{3}\)
| 0 | 4 | 4 | 5 |
Cuboctahedron | 12 |
O
|
\(C_{2}\)
| 4 | 3 | 5 | 7 |
Icosahedron | 12 |
I
|
\(C_{5}\)
| 0 | 4 | 4 | 5 |
Dodecahedron | 20 |
I
|
\(C_{3}\)
| 4 | 4 | 6 | 9 |
Icosidodecahedron | 30 |
I
|
\(C_{2}\)
| 14 | 2 | 9 | 15 |
8.2 Group-invariant polynomials
Group | Primary invariants |
---|---|
\(D_{nh}\)
|
\(z^{2},\rho ,\gamma _{n}\)
|
\(T_{d}\)
|
\(I_{2},I_{3},I_{4}\)
|
\(O_{h}\)
|
\(I_{2},I_{4},I_{6}\)
|
\(I_{h}\)
|
\(I_{2},I_{6}^{\prime },I_{10}\)
|
8.3 The main theorem
Gv
|
\(\left| Gv\right| \)
|
T
|
---|---|---|
Regular n-gon (n even) |
n
|
\(\big \{\cos \left( \frac{2\pi j}{n}\right) :j=1,\ldots ,n\big \}\)
|
Regular n-gon (n odd) |
n
|
\(\big \{-\cos \left( \frac{2\pi j}{n}\right) :j=1,\ldots ,n\big \}\)
|
Tetrahedron | 4 |
\(\big \{ -1,\frac{1}{3}\big \} \)
|
Octahedron | 6 |
\(\big \{ -1,0,1\big \} \)
|
Cube | 8 |
\(\big \{-1,-\frac{1}{3},\frac{1}{3},1\big \} \)
|
Cuboctahedron | 12 |
\(\big \{ -1,-\frac{1}{2},0,\frac{1}{2},1\big \} \)
|
Icosahedron | 12 |
\(\big \{-1,-\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5} },1\big \} \)
|
Dodecahedron | 20 |
\(\big \{-1,-\frac{\sqrt{5}}{3},-\frac{1}{3},\frac{1}{3},\frac{\sqrt{5}}{3},1\big \}\)
|
Icosidodecahedron | 30 |
\(\big \{ -1,-\frac{\tau }{2},-\frac{1}{2},-\frac{1}{2\tau },0,\frac{1}{2\tau },\frac{1}{2},\frac{\tau }{2},1\big \}\)
|
9 Informational power and the average value of relative entropy
Convex hull of the orbit | Informational power |
---|---|
Digon | 0.69315 |
Regular n-gon (\(n\rightarrow \infty \)) | 0.30685 |
Tetrahedron | 0.28768 |
Octahedron | 0.23105 |
Cube | 0.21576 |
Cuboctahedron | 0.20273 |
Icosahedron | 0.20189 |
Dodecahedron | 0.19686 |
Icosidodecahedron | 0.19486 |
Average value of relative entropy | 0.19315 |