We study uncertainty relations for pairs of conjugate variables like number and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of number and angle outputs, respectively. For each type of observable, we discuss two natural choices of metric and discuss the resulting optimal bounds with both numerical and analytical methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.

1.
E.
Kennard
, “
Zur quantenmechanik einfacher bewegungstypen
,”
Z. Phys.
44
,
326
352
(
1927
).
2.
H.
Robertson
, “
The uncertainty principle
,”
Phys. Rev.
34
,
163
164
(
1929
).
3.
W.
Heisenberg
, “
Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik
,”
Z. Phys.
43
,
172
198
(
1927
).
4.
R.
Werner
, “
The uncertainty relation for joint measurement of position and momentum
,”
Quantum Inf. Comput.
4
,
546
562
(
2004
); e-print arXiv:quant-ph/0405184.
5.
P.
Busch
and
D.
Pearson
, “
Universal joint-measurement uncertainty relation for error bars
,”
J. Math. Phys.
48
,
082103
(
2007
).
6.
P.
Busch
,
P.
Lahti
, and
R.
Werner
, “
Proof of Heisenberg’s error-disturbance relation
,”
Phys. Rev. Lett.
111
,
160405
(
2013
).
7.
P.
Busch
,
P.
Lahti
, and
R.
Werner
, “
Measurement uncertainty relations
,”
J. Math. Phys.
55
,
042111
(
2014
).
8.
J.
Biniok
,
P.
Busch
, and
J.
Kiukas
, “
Uncertainty in the context of multislit interferometry
,”
Phys. Rev. A
90
,
022115
(
2014
).
9.
S.
Tanimura
, “
Complementarity and the nature of uncertainty relations in Einstein–Bohr recoiling slit experiment
,”
Quanta
4
,
1
(
2015
).
10.
D.
Judge
and
J.
Lewis
, “
On the commutator [Lz, φ]
,”
Phys. Lett.
5
,
190
(
1963
).
11.
K.
Kraus
, “
Remark on the uncertainty between angle and angular momentum
,”
Z. Phys.
188
,
374
(
1965
).
12.
D.
Judge
, “
On the uncertainty relation for Lz and φ
,”
Phys. Lett.
5
,
189
(
1963
).
13.
A.
Evett
and
H.
Mahmoud
, “
Uncertainty relation for angle variables
,”
Il Nuovo Cimento
38
,
295
(
1965
).
14.
L.
Schotsmans
and
P. V.
Leuven
, “
Numerical evaluation of the uncertainty relation for angular variables
,”
Il Nuovo Cimento
39
,
776
(
1965
).
15.
C.
Villani
,
Optimal Transport: Old and New
(
Springer
,
2009
).
16.
L.
Dammeier
,
R.
Schwonnek
, and
R.
Werner
, “
Uncertainty relations for angular momentum
,”
New J. Phys.
17
,
093046
(
2015
).
17.
P.
Lévy
, “
L’ addition des variables aléatoires définies sur une circonférence
,”
Bull. Soc. Math. France
67
,
1
41
(
1939
).
18.
R.
von Mises
, “
Über die ‘ganzzahligkeit’ der atomgewichte und verwandte fragen
,”
Phys. Z.
19
,
490
(
1918
).
19.
E.
Breitenberger
, “
Uncertainty measures and uncertainty relations for angle observables
,”
Found. Phys.
15
,
353
(
1985
).
20.
M. B. M. A.
Alonso
, “
Mapping-based width measures and uncertainty relations for periodic functions
,”
Signal Process.
84
,
2425
(
2004
).
21.
R.
Werner
, “
Quantum harmonic analysis on phase space
,”
J. Math. Phys.
25
,
1404
(
1984
).
22.
N.
Dunford
and
J.
Schwartz
,
Linear Operators, Part I: General Theory
(
Wiley
,
1957
).
23.
M.
Abramowitz
and
I.
Stegun
,
Handbook of Mathematical Functions
(
Dover Publications
,
1965
).
24.
J.
Řeháček
,
Z.
Bouchal
,
R.
Čelechovský
,
Z.
Hradil
, and
L.
Sánchez-Soto
, “
Experimental test of uncertainty relations for quantum mechanics on a circle
,”
Phys. Rev. A
77
,
032110
(
2008
).
25.
M.
Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators
(
Academic Press
,
1978
).
26.
D.
Judge
, “
On the uncertainty relation for angle variables
,”
Il Nuovo Cimento
31
,
332
(
1964
).
27.
M.
Bouten
,
N.
Maene
, and
P. V.
Leuven
, “
On an uncertainty relation for angular variables
,”
Il Nuovo Cimento
37
,
1119
(
1965
).
28.
M.
Nieto
, “
Quantum phase and quantum phase operators: Some physics and some history
,”
Phys. Scr.
T48
,
5
(
1993
).
29.
M. N. P.
Carruthers
, “
Phase and angle variables in quantum mechanics
,”
Rev. Mod. Phys.
40
,
411
(
1968
).
30.
R.
Jackiw
, “
Minimum uncertainty product, number-phase uncertainty product, and coherent states
,”
J. Math. Phys.
9
,
339
(
1968
).
31.
P.
Lahti
,
J.-P.
Pellonpää
, and
J.
Schultz
, “
Number and phase: Complementarity and joint measurement uncertainties
,”
J. Phys. A: Math. Theor.
50
,
375301
(
2017
).
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