For a given target system and apparatus described by quantum theory, the so-called quantum no-programming theorem indicates that a family of states called programs in the apparatus with a fixed unitary operation on total system programs distinct unitary dynamics to the target system only if the initial programs are orthogonal to each other. The current study aims at revealing whether a similar behavior can be observed in generalized probabilistic theories (GPTs). Generalizing the programming scheme to GPTs, we derive a similar theorem to the quantum no-programming theorem. We, furthermore, demonstrate that programming of reversible dynamics is closely related to a curious structure named a quasi-classical structure on the state space. Programming of irreversible dynamics, i.e., channels, in GPTs is also investigated.

1.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information: 10th Anniversary Edition
(
Cambridge University Press
,
2010
).
2.
M. A.
Nielsen
and
I. L.
Chuang
,
Phys. Rev. Lett.
79
,
321
(
1997
).
3.
Y.
Yang
,
R.
Renner
, and
G.
Chiribella
,
Phys. Rev. Lett.
125
,
210501
(
2020
).
4.
T.
Heinosaari
,
T.
Miyadera
, and
M.
Tukiainen
,
Quantum Inf. Process.
16
,
85
(
2017
).
5.
L.
Hardy
, arXiv:quant-ph/0101012 (
2001
).
6.
H.
Barnum
,
J.
Barrett
,
M.
Leifer
, and
A.
Wilce
, arXiv:quant-ph/0611295 (
2006
).
7.
H.
Barnum
,
J.
Barrett
,
M.
Leifer
, and
A.
Wilce
,
Phys. Rev. Lett.
99
,
240501
(
2007
).
8.
J.
Barrett
,
Phys. Rev. A
75
,
032304
(
2007
).
9.
G.
Chiribella
,
G. M.
D’Ariano
, and
P.
Perinotti
,
Phys. Rev. A
81
,
062348
(
2010
).
10.
G.
Chiribella
,
G. M.
D’Ariano
, and
P.
Perinotti
,
Phys. Rev. A
84
,
012311
(
2011
).
11.
L.
Masanes
and
M. P.
Müller
,
New J. Phys.
13
,
063001
(
2011
).
12.
H.
Barnum
,
J.
Barrett
,
M.
Leifer
, and
A.
Wilce
, in
Proceedings of Symposia in Applied Mathematics
(
American Mathematical Society
,
2012
), Vol. 71, pp.
25
48
.
13.
L.
Lami
, “
Non-classical correlations in quantum mechanics and beyond
,” Ph.D. thesis,
Universitat Autònoma de Barcelona
,
2017
.
14.
M.
Plávala
, arXiv:2103.07469 [quant-ph] (
2021
).
15.
R.
Takakura
, “
Convexity and uncertainty in operational quantum foundations
,” Ph.D. thesis,
Kyoto University
,
2022
.
16.
P.
Janotta
,
C.
Gogolin
,
J.
Barrett
, and
N.
Brunner
,
New J. Phys.
13
,
063024
(
2011
).
17.

For subset A of a vector space, its convex hull conv(A), affine hull aff(A), and linear span span(A) are given by conv(A){i=1nλiaiaiA,λi[0,1],iλi=1,n: finite}, aff(A){i=1nλiaiaiA,λiR,iλi=1,n: finite}, and span(A){i=1nλiaiaiA,λiR,n: finite}, respectively.

18.
P.
Janotta
and
R.
Lal
,
Phys. Rev. A
87
,
052131
(
2013
).
19.
N.
Brunner
,
M.
Kaplan
,
A.
Leverrier
, and
P.
Skrzypczyk
,
New J. Phys.
16
,
123050
(
2014
).
20.
P.
Busch
,
P. J.
Lahti
,
J.-P.
Pellonpää
, and
K.
Ylinen
,
Quantum Measurement
, Theoretical and Mathematical Physics (
Springer International Publishing
,
2016
).
21.
T.
Heinosaari
and
M.
Ziman
,
The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement
(
Cambridge University Press
,
Cambridge
,
2011
).
22.
G.
Aubrun
,
L.
Lami
,
C.
Palazuelos
, and
M.
Plávala
,
Geom. Funct. Anal.
31
,
181
(
2021
).
23.
C.
Zander
and
A. R.
Plastino
,
Europhys. Lett.
86
,
18004
(
2009
).
24.
G.
Kimura
,
K.
Nuida
, and
H.
Imai
,
Rep. Math. Phys.
66
,
175
(
2010
).
25.
I.
Namioka
and
R. R.
Phelps
,
Pac. J. Math.
31
,
469
(
1969
).
26.
R.
Takakura
and
T.
Miyadera
,
J. Math. Phys.
61
,
082203
(
2020
).
27.
Z.
Wu
,
C.
Zhu
, and
X.
Zhang
,
Cent. Eur. J. Phys.
11
,
317
(
2013
).
28.
T. M.
Cover
and
J. A.
Thomas
,
Elements of Information Theory
,
2nd ed.
(
John Wiley & Sons, Inc.
,
Hoboken, NJ
,
2006
).
29.
G.
Chiribella
,
Int. J. Software Inf.
8
,
209
(
2014
); available at http://www.ijsi.org/ijsi/article/abstract/i191.
30.
J. B.
Conway
,
A Course in Functional Analysis
,
1st ed.
, Graduate Texts in Mathematics Vol. 96 (
Springer-Verlag
,
New York
,
1985
).
31.
J.
Bae
and
L.-C.
Kwek
,
J. Phys. A: Math. Theor.
48
,
083001
(
2015
).
32.
G.
Kimura
,
T.
Miyadera
, and
H.
Imai
,
Phys. Rev. A
79
,
062306
(
2009
).
33.
J.
Bae
,
D.-G.
Kim
, and
L.-C.
Kwek
,
Entropy
18
,
39
(
2016
).
34.
R.
Takagi
and
B.
Regula
,
Phys. Rev. X
9
,
031053
(
2019
).
35.
Y.
Yoshida
,
H.
Arai
, and
M.
Hayashi
,
Phys. Rev. Lett.
125
,
150402
(
2020
).
36.
S.
Popescu
and
D.
Rohrlich
,
Found. Phys.
24
,
379
(
1994
).
37.
A.
Jenčová
and
M.
Plávala
,
Phys. Rev. A
96
,
022113
(
2017
).
38.
P.
Busch
,
Int. J. Theor. Phys.
30
,
1217
(
1991
).
39.
M.
Singer
and
W.
Stulpe
,
J. Math. Phys.
33
,
131
(
1992
).
40.
R. T.
Rockafellar
,
Convex Analysis
, Princeton Mathematical Series Vol. 28 (
Princeton University Press
,
1970
).
You do not currently have access to this content.