Among several tasks in Machine Learning, a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of this is the task of inferring the hidden Markov model underlying a given stochastic process. This is known as the positive realization problem (PRP), [L. Benvenuti and L. Farina, IEEE Trans. Autom. Control 49(5), 651–664 (2004)] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory. We consider the scenario where the latent variables are quantum (i.e., quantum states of a finite-dimensional system) and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument — if any — yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the hidden Markov model, or the iterated quantum instrument, is however devoid of any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The completely positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [M. Guţă, Phys. Rev. A 83(6), 062324 (2011); M. Guţă and N. Yamamoto, e-print arXiv:1303.3771(2013)].

1.
Accardi
,
L.
, “
Noncommutative Markov chains associated to a pressigned evolution: An application to the quantum theory of measurement
,”
Adv. Math.
29
(
2
),
226
243
(
1978
).
2.
Arenz
,
C.
,
Gualdi
,
G.
, and
Burgarth
,
D.
, “
Control of open quantum systems: Case study of the central spin model
,”
New J. Phys.
16
(
6
),
065023
(
2014
).
3.
Anderson
,
B. D. O.
, “
The realization problem for hidden Markov models
,”
Math. Control, Signals, Syst.
12
(
1
),
80
120
(
1999
).
4.
Benvenuti
,
L.
and
Farina
,
L.
, “
A tutorial on the positive realization problem
,”
IEEE Trans. Autom. Control
49
(
5
),
651
664
(
2004
).
5.
Blume-Kohout
,
R.
,
Gamble
,
J. K.
,
Nielsen
,
E.
,
Mizrahi
,
J.
,
Sterk
,
J. D.
, and
Maunz
,
P.
, “
Robust, self-consistent, closed-form tomography of quantum logic gates on a trapped ion qubit
,” e-print arXiv:1310.4492 [quant-ph] (
2013
).
6.
Barrett
,
J.
,
Linden
,
N.
,
Massar
,
S.
,
Pironio
,
S.
,
Popescu
,
S.
, and
Roberts
,
D.
, “
Nonlocal correlations as an information-theoretic resource
,”
Phys. Rev. A
71
(
2
),
022101
(
2005
).
7.
Blekherman
,
G.
,
Parrilo
,
P. A.
, and
Thomas
,
R. R.
,
Semidefinite Optimization and Convex Algebraic Geometry
(
SIAM
,
2012
).
8.
Burgarth
,
D.
and
Yuasa
,
K.
, “
Quantum system identification
,”
Phys. Rev. Lett.
108
(
8
),
080502
(
2012
).
9.
Burgarth
,
D.
and
Yuasa
,
K.
, “
Identifiability of open quantum systems
,”
Phys. Rev. A
89
,
030302
(
2014
); e-print arXiv:1401.5240 [quant-ph] (2014).
10.
Cybenko
,
G.
and
Crespi
,
V.
, “
Learning hidden Markov models using nonnegative matrix factorization
,”
IEEE Trans. Inf. Theory
57
(
6
),
3963
3970
(
2011
).
11.
Dharmadhikari
,
S. W.
, “
Sufficient conditions for a stationary process to be a function of a finite Markov chain
,”
Ann. Math. Stat.
34
(
3
),
1033
1041
(
1963
).
12.
Erickson
,
R. V.
, “
Functions of Markov chains
,”
Ann. Math. Stat.
41
(
3
),
843
850
(
1970
).
13.
Fiorini
,
S.
,
Massar
,
S.
,
Pokutta
,
S.
,
Tiwary
,
H. R.
, and
de Wolf
,
R.
, “
Linear vs. semidefinite extended formulations: Exponential separation and strong lower bounds
,” in
Proceedings of the 44th Symposium on Theory of Computing–STOC ’12
(
ACM
,
New York, NY, USA
,
2012
), pp.
95
106
.
14.
Fiorini
,
S.
,
Massar
,
S.
,
Patra
,
M. K.
, and
Tiwary
,
H. R.
, “
Generalized probabilistic theories and conic extensions of polytopes
,”
J. Phys. A: Math. Theor.
48
(
2
),
025302
(
2015
).
15.
Fannes
,
M.
,
Nachtergaele
,
B.
, and
Werner
,
R. F.
, “
Finitely correlated states on quantum spin chains
,”
Commun. Math. Phys. (1965-1997)
144
(
3
),
443
490
(
1992
).
16.
Farenick
,
D.
and
Paulsen
,
V. I.
, “
Operator system quotients of matrix algebras and their tensor products
,”
Math. Scand.
111
(
2
),
210
243
(
2012
).
17.
Guţă
,
M.
and
Kiukas
,
J.
, “
Equivalence classes and local asymptotic normality in system identification for quantum Markov chains
,”
Commun. Math. Phys.
335
(
3
),
1397
1428
(
2015
); e-print arXiv:1402.3535 [quant-ph, stat] (2014).
18.
Gouveia
,
J.
,
Parrilo
,
P. A.
, and
Thomas
,
R. R.
, “
Lifts of convex sets and cone factorizations
,”
Math. Oper. Res.
38
(
2
),
248
264
(
2013
).
19.
Guţă
,
M.
, “
Fisher information and asymptotic normality in system identification for quantum Markov chains
,”
Phys. Rev. A
83
(
6
),
062324
(
2011
).
20.
Gu
,
M.
,
Wiesner
,
K.
,
Rieper
,
E.
, and
Vedral
,
V.
, “
Quantum mechanics can reduce the complexity of classical models
,”
Nat. Commun.
3
,
762
(
2012
).
21.
Guţă
,
M.
and
Yamamoto
,
N.
, “
Systems identification for passive linear quantum systems: The transfer function approach
,” in
2013 IEEE 52nd Annual Conference on Decision and Control (CDC)
(
IEEE
,
2013
), pp.
1930
1937
; e-print arXiv:1303.3771 (2013).
22.
Horodecki
,
M.
,
Horodecki
,
P.
, and
Horodecki
,
R.
, “
Separability of mixed states: Necessary and sufficient conditions
,”
Phys. Lett. A
223
(
1–2
),
1
8
(
1996
).
23.
Helton
,
J. W.
and
Nie
,
J.
, “
Sufficient and necessary conditions for semidefinite representability of convex hulls and sets
,”
SIAM J. Optim.
20
(
2
),
759
791
(
2009
).
24.
Holevo
,
A. S.
,
Statistical Structure of Quantum Theory
,
Lecture Notes in Physics Monographs
Vol.
M67
(
Springer Verlag
,
2001
).
25.
Ito
,
H.
,
Amari
,
S.-I.
, and
Kobayashi
,
K.
, “
Identifiability of hidden Markov processes and their minimum degrees of freedom
,”
Electron. Commun. Jpn. (Part III: Fundam. Electron. Sci.)
74
(
7
),
77
84
(
1991
).
26.
Kalman
,
R. E.
, “
Irreducible realizations and the degree of a rational matrix
,”
J. Soc. Ind. Appl. Math.
13
(
2
),
520
544
(
1965
).
27.
Kliesch
,
M.
,
Gross
,
D.
, and
Eisert
,
J.
, “
Matrix product operators and states: NP-hardness and undecidability
,”
Phys. Rev. Lett.
113
,
160503
(
2014
); e-print arXiv:1404.4466 [cond-mat, physics:math-ph, physics:quant-ph] (2014).
28.
Kavruk
,
A. S.
,
Paulsen
,
V. I.
,
Todorov
,
I. G.
, and
Tomforde
,
M.
, “
Quotients, exactness, and nuclearity in the operator system category
,”
Adv. Math.
235
,
321
360
(
2013
).
29.
Ludwig
,
G.
, “
Versuch einer axiomatischen grundlegung der quantenmechanik und allgemeinerer physikalischer theorien
,”
Z. Phys.
181
(
3
),
233
260
(
1964
).
30.
Ludwig
,
G.
, “
Attempt of an axiomatic foundation of quantum mechanics and more general theories, II
,”
Commun. Math. Phys.
4
(
5
),
331
348
(
1967
).
31.
Monras
,
A.
,
Beige
,
A.
, and
Wiesner
,
K.
, “
Hidden quantum Markov models and non-adaptive read-out of many-body states
,”
Appl. Math. Comput. Sci.
3
,
93
(
2011
).
32.
Paulsen
,
V. I.
,
Completely Bounded Maps and Operator Algebras
(
Cambridge University Press
,
2003
).
33.
Paulsen
,
V. I.
, personal communication (
2014
).
34.
Peres
,
A.
, “
Separability criterion for density matrices
,”
Phys. Rev. Lett.
77
,
1413
1415
(
1996
).
35.
Pfister
,
C.
, “
One simple postulate implies that every polytopic state space is classical
,” e-print arXiv:1203.5622 [quant-ph] (
2012
).
36.
Rudin
,
W.
,
Principles of Mathematical and Complex and Real Functional Analysis
(
Scottex Publishing
,
1966
).
37.
Sontag
,
E. D.
, “
On some questions of rationality and decidability
,”
J. Comput. Syst. Sci.
11
(
3
),
375
381
(
1975
).
38.
van den Hof
,
J. M.
and
van Schuppen
,
J. H.
, “
Positive matrix factorization via extremal polyhedral cones
,”
Linear Algebra Appl.
293
(
1-3
),
171
186
(
1999
).
39.
Vidyasagar
,
M.
, “
The complete realization problem for hidden Markov models: A survey and some new results
,”
Math. Control, Signals, Syst.
23
(
1-3
),
1
65
(
2011
).
40.
Wolf
,
M. M.
and
Perez-Garcia
,
D.
, “
Assessing quantum dimensionality from observable dynamics
,”
Phys. Rev. Lett.
102
(
19
),
190504
(
2009
).
41.
Werner
,
R. F.
and
Wolf
,
M. M.
, “
Bell inequalities and entanglement
,”
Quantum Inf. Comput.
1
(
3
),
1
25
(
2001
).
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