We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally nontransitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.

Topics
Lie algebras, Hermitian operator, Convex geometry, Operator theory, Topological groups, Representation theory, Functions and mappings, Hilbert space, Quantum information, Quantum measurement theory
1.
I. L.
 
Chuang
 and
M. A.
 
Nielsen
,
Quantum Information and Quantum Computation
(
Cambridge University Press
, Cambridge,
2000
).
Google Scholar
 
2.
A. S.
 
Holevo
,
J. Multivariate Anal.
https://doi.org/10.1016/0047-259X(73)90028-6
 
3
,
337
(
1973
).
Google Scholar
Crossref
Search ADS
 
3.
A.
 
Peres
,
Quantum Theory: Concepts and Methods
(
Kluwer Academic
, Dordrecht,
1993
), pp.
279
–
289
.
Google Scholar
 
4.
C. W.
 
Helstrom
,
Quantum Detection and Estimation Theory
(
Academic
, New York,
1976
).
Google Scholar
 
5.
A. S.
 
Holevo
,
Probabilistic and Statistical Aspects of Quantum Theory
(
North Holland
, Amsterdam,
1982
).
Google Scholar
 
6.
G. M.
 
D’Ariano
,
J. Math. Phys.
https://doi.org/10.1063/1.1777813
 
45
,
3620
(
2004
).
Google Scholar
Crossref
Search ADS
 
7.
G.
 
Chiribella
 and
G. M.
 
D’Ariano
,
J. Math. Phys.
https://doi.org/10.1063/1.1806262
 
45
,
4435
(
2004
).
Google Scholar
Crossref
Search ADS
 
8.
P.
Shor
 , in
Quantum Communication, Computing, and Measurement 3
, edited by
P.
Tombesi
 and
O.
Hirota
 (
Kluwer
, Dordrecht,
2001
);
Google Scholar
 
LANL e-print quant-ph/0009077.
9.
T.
 
Decker
, eprint quant-ph/0509122.
10.
The proof of this statement is the straightforward generalization of the corresponding proof for transitive group actions (see Ref. 5, pp.
166
–
169
).
11.
W. K.
 
Wootters
 and
B. D.
 
Fields
,
Ann. Phys. (N.Y.)
https://doi.org/10.1016/0003-4916(89)90322-9
 
191
,
363
(
1989
).
Google Scholar
Crossref
Search ADS
 
12.
K. R.
 
Parthasarathy
, eprint quant-ph/0408069.
13.
C. H.
 
Bennett
 and
G.
 
Brassard
, in
Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India
(
IEEE
, New York,
1984
), pp.
175
–
179
.
14.
If an operator O commutes with all the unitaries UpVq in the representation R(G̃)⁠, then it must commute at least with Up and Vq⁠. In other words, O must be diagonal both on the eigenvectors of Up and on the eigenvectors of Vq⁠. But these two bases are mutually unbiased, whence the only possibility is O proportional to the identity, i.e., the representation R(G̃) is irreducible.
15.
E. B.
 
Davies
,
IEEE Trans. Inf. Theory
https://doi.org/10.1109/TIT.1978.1055941
 
24
,
596
(
1978
).
Google Scholar
Crossref
Search ADS
 
© 2006 American Institute of Physics.
2006
American Institute of Physics
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