We extend the notion of Gacs quantum algorithmic entropy, originally formulated for finitely many qubits, to infinite dimensional quantum spin chains and investigate the relation of this extension with two quantum dynamical entropies that have been proposed in recent years.

1.
R. J.
Solomonoff
, “
A formal theory of inductive inference
,”
Inform. Contr.
7
,
1
22
and 224–254 (
1964
).
2.
A. N.
Kolmogorov
, “
Three approaches to the quantitative definition of information
,”
Prob. Inf. Transm.
1
,
1
7
(
1965
).
3.
J. G.
Chaitin
, “
On the length of programs for computing finite binary sequences
,”
AMS Notices
13
,
547
569
(
1966
).
4.
R. G.
Downey
and
D. R.
Hirschfeldt
,
Algorithmic Randomness and Complexity
(
Springer
,
2010
).
5.
M.
Ohya
and
I.
Volovich
,
Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano-and-Bio-Systems
(
Springer
,
2011
).
6.
A.
Nies
,
Computability and Randomness
(
Oxford University Press
,
2009
).
7.
M.
Li
and
P. M. B.
Vitányi
,
An Introduction to Kolmogorov Complexity and Its Applications
(
Springer
,
New York
,
2008
).
8.
C.
Calude
,
Information and Randomness. An Algorithmic Perspective
(
Springer
,
Berlin
,
2002
).
9.
A.
Berthiaume
,
W.
van Dam
, and
S.
Laplante
, “
Quantum Kolmogorov complexity
,”
J. Comput. System. Sci.
63
,
201
221
(
2001
).
10.
P. M. B.
Vitányi
, “
Quantum Kolmogorov complexity based on classical descriptions
,”
IEEE Trans. Inf. Th.
47
,
2464
2479
(
2001
).
11.
C. E.
Mora
and
H. J.
Briegel
, “
Algorithmic complexity and entanglement of quantum states
,”
Phys. Rev. Lett.
95
,
200503
200507
(
2005
).
12.
P.
Gacs
, “
Quantum algorithmic entropy
,”
J. Phys. A
34
,
6859
6880
(
2001
).
13.
A. A.
Brudno
, “
Entropy and the complexity of the trajectories of a dynamical system
,”
Trans. Moscow Math. Soc.
2
,
127
151
(
1983
).
14.
A.
Connes
,
H.
Narnhofer
, and
W.
Thirring
, “
Dynamical entropy of C*-algebras and von Neumann algebras
,”
Commun. Math. Phys.
112
,
691
719
(
1987
).
15.
R.
Alicki
and
M.
Fannes
, “
Defining quantum dynamical entropy
,”
Lett. Math. Phys.
32
,
75
82
(
1994
).
16.
R.
Alicki
and
M.
Fannes
,
Quantum Dynamical Systems
(
Oxford University Press
,
2001
).
17.
W.
Slomczynski
and
K.
Zyczkowski
, “
Quantum chaos: An entropy approach
,”
J. Math. Phys.
35
,
5674
5701
(
1994
).
18.
D.
Voiculescu
, “
Dynamical approximation entropies and topological entropy in operator algebras
,”
Commun. Math. Phys.
170
,
249
281
(
1995
).
19.
F.
Benatti
, “
Quantum dynamical entropy and Gacs algorithmic entropy
,”
Entropy
14
,
1259
1273
(
2012
).
20.
M.
Davis
,
R.
Sigal
, and
E. J.
Weyuker
,
Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science
(
Academic Press
,
San Diego
,
1994
).
21.
F.
Strocchi
,
Symmetry Breaking
,
Lec. Notes Phys.
(
Springer
,
2005
), Vol.
643
.
22.
F.
Benatti
,
Dynamics, Information and Complexity in Quantum Systems
,
Theoretical and Mathematical Physics
(
Springer
,
2009
).
23.
J.-L.
Sauvageot
and
J.-P
Thouvenot
, “
Une nouvelle définition de lentropie dynamique des systèmes non commutatifs
,”
Commun. Math. Phys.
145
,
411
423
(
1992
).
24.
An instance of such an injection is the map ι ○ ι defined in Sec. IV.
You do not currently have access to this content.