Two quantities quantifying uncertainty relations are examined. Busch and Pearson [J. Math. Phys.48, 082103 (2007)] investigated the limitation on joint localizability and joint measurement of position and momentum by introducing overall width and error bar width. In this paper, we show a simple relationship between these quantities for finite-dimensional systems. Our result indicates that if there is a bound on joint localizability, it is possible to obtain a similar bound on joint measurability. For finite-dimensional systems, uncertainty relations for a pair of general projection-valued measures are obtained as by-products.

1.
H. P.
Robertson
,
Phys. Rev.
34
,
163
(
1929
).
2.
D.
Deutsch
,
Phys. Rev. Lett.
50
,
631
(
1983
).
3.
H.
Maassen
and
J. B. M.
Uffink
,
Phys. Rev. Lett.
60
,
1103
(
1988
).
4.
M.
Krishna
and
K. R.
Parthasarathy
,
Sankhya, Ser. A
64
,
842
(
2002
).
5.
T.
Miyadera
and
H.
Imai
,
Phys. Rev. A
76
,
062108
(
2007
).
6.
W.
Heisenberg
,
Z. Phys.
43
,
172
(
1927
).
7.
D. M.
Appleby
,
Int. J. Theor. Phys.
37
,
1491
(
1998
).
9.
R. F.
Werner
,
Quantum Inf. Comput.
4
,
546
(
2004
).
10.
P.
Busch
and
D. B.
Pearson
,
J. Math. Phys.
48
,
082103
(
2007
).
11.
T.
Miyadera
and
H.
Imai
,
Phys. Rev. A
78
,
052119
(
2008
).
12.
P.
Busch
,
T.
Heinonen
, and
P.
Lahti
,
Phys. Rep.
452
,
155
(
2007
).
13.
That is, d: Ω × Ω → [0, ∞) satisfies (i) symmetry: d(x, y) = d(y, x) for all x, y ∈ Ω, (ii) positive definiteness: d(x, y) = 0 if and only if x = y, and (iii) triangular inequality: d(x, y) + d(y, z) ⩾ d(x, z) for all x, y, z ∈ Ω.
14.
M.
Keyl
,
D.
Schlingemann
, and
R. F.
Werner
,
Quantum Inf. Comput.
3
,
281
(
2003
).
15.
A.
Dvurečenskij
and
S.
Pulmannová
,
New Trends in Quantum Structures
(
Kluwer
,
Dordrecht
,
2000
).
You do not currently have access to this content.