Conditional probabilities in quantum systems which have both initial and final boundary conditions are commonly evaluated using the Aharonov–Bergmann–Lebowitz rule. In this short note, we present a seemingly disturbing paradox that appears when applying the rule to systems with slightly broken degeneracies. In these cases, we encounter a singular limit—the probability “jumps” when going from perfect degeneracy to negligibly broken one. We trace the origin of the paradox and solve it from both traditional and modern perspectives in order to highlight the physics behind it: the necessity to take into account the finite resolution of the measuring device. As a practical example, we study the application of the rule to the Zeeman effect. The analysis presented here may stress the general need to first consider the governing physical principles before heading to the mathematical formalism, in particular, when exploring puzzling quantum phenomena.

1.
Yakir
Aharonov
and
Daniel
Rohrlich
,
Quantum Paradoxes: Quantum Theory for the Perplexed
(
Wiley-VCH
,
Weinheim
,
2008
).
2.
Yakir
Aharonov
,
Alonso
Botero
,
Sandu
Popescu
,
Benni
Reznik
, and
Jeff
Tollaksen
, “
Revisiting Hardy's paradox: Counterfactual statements, real measurements, entanglement and weak values
,”
Phys. Lett. A
301
(
3–4
),
130
138
(
2002
).
3.
Yakir
Aharonov
,
Eliahu
Cohen
,
Ariel
Landau
, and
Avshalom C.
Elitzur
, “
The case of the disappearing (and re-appearing) particle
,”
Sci. Rep.
7
(
531
),
1
9
(
2017
).
4.
Yakir
Aharonov
,
Eliahu
Cohen
,
Avshalom C.
Elitzur
, and
Lee
Smolin
, “
Interaction-free effects between distant atoms
,”
Found. Phys.
48
(
1
),
1
16
(
2018
).
5.
Yakir
Aharonov
,
Eliahu
Cohen
,
Avishy
Carmi
, and
Avshalom C.
Elitzur
, “
Extraordinary interactions between light and matter determined by anomalous weak values
,”
Proc. R. Soc. A
474
(
2215
),
20180030
(
2018
).
6.
Christopher A.
Fuchs
, “
Quantum mechanics as quantum information (and only a little more)
,” e-print arXiv:quant-ph/0205039.
7.
Carlton M.
Caves
,
Christopher A.
Fuchs
, and
Ruediger
Schack
, “
Quantum probabilities as Bayesian probabilities
,”
Phys. Rev. A
65
,
022305
(
2002
).
8.
Rob
Clifton
,
Jeffrey
Bub
, and
Hans
Halvorson
, “
Characterizing quantum theory in terms of information-theoretic constraints
,”
Found. Phys.
33
(
11
),
1561
1591
(
2003
).
9.
Jeffrey
Bub
, “
Quantum mechanics is about quantum information
,”
Found. Phys.
35
(
4
),
541
560
(
2005
).
10.
Christopher A.
Fuchs
,
N.
David Mermin
, and
Ruediger
Schack
, “
An introduction to QBism with an application to the locality of quantum mechanics
,”
Am. J. Phys.
82
(
8
),
749
754
(
2014
).
11.
Yakir
Aharonov
and
Lev
Vaidman
, “
Complete description of a quantum system at a given time
,”
J. Phys. A: Math. Gen.
24
(
10
),
2315
2328
(
1991
).
12.
Ruth E.
Kastner
, “
The three-box ‘paradox’ and other reasons to reject the counterfactual usage of the ABL rule
,”
Found. Phys.
29
(
6
),
851
863
(
1999
).
13.
Ulrich
Mohrhoff
, “
What quantum mechanics is trying to tell us
,”
Am. J. Phys.
68
(
8
),
728
745
(
2000
).
14.
Ruth E.
Kastner
, “
Comment on ‘What quantum mechanics is trying to tell us,’ by Ulrich Mohrhoff, Am. J. Phys. 68(8), 728–745 (2000)]
,”
Am. J. Phys.
69
(
8
),
860
863
(
2001
).
15.
Ulrich
Mohrhoff
, “
Objective probabilities, quantum counterfactuals, and the ABL rule—A response to R.E. Kastner
,”
Am. J. Phys.
69
(
8
),
864
873
(
2001
).
16.
Matthew S.
Leifer
and
Robert W.
Spekkens
, “
Pre-and post-selection paradoxes and contextuality in quantum mechanics
,”
Phys. Rev. Lett.
95
,
200405
(
2005
).
17.
Dmitri I.
Sokolovski
,
Irene P.
Gimenez
, and
R.
Sala Mayato
, “
Path integrals, the ABL rule and the three-box paradox
,”
Phys. Lett. A
372
(
44
),
6578
6583
(
2008
).
18.
Eliahu
Cohen
and
Marcin
Nowakowski
, “
Comment on measurements without probabilities in the final state proposal
,”
Phys. Rev. D
97
,
088501
(
2018
).
19.
Michael V.
Berry
, “
Singular limits
,”
Phys. Today
55
(
5
),
10
11
(
2002
).
20.
Yakir
Aharonov
,
Peter G.
Bergmann
, and
Joel L.
Lebowitz
, “
Time symmetry in the quantum process of measurement
,”
Phys. Rev.
134
,
B1410
B1416
(
1964
).
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