A uniform matrix product state defined on a tripartite system of spins, denoted by ABC, is shown to be an approximate quantum Markov chain when the size of subsystem B, denoted |B|, is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to exp(−q(|B| − K) + 2K ln |B|), with q and K computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate q. We show the improved value of q to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.

1.

The QCMI may be contrasted with the information mutually shared by regions A and C and described by the quantum mutual information (QMI) I(A: C) ≔ S(ρA) + S(ρC) − S(ρAC).

2.
E. H.
Lieb
and
M. B.
Ruskai
, “
A fundamental property of quantum-mechanical entropy
,”
Phys. Rev. Lett.
30
,
434
436
(
1973
).
3.
E. H.
Lieb
and
M. B.
Ruskai
, “
Proof of the strong subadditivity of quantum-mechanical entropy
,”
J. Math. Phys.
14
,
1938
1941
(
1973
).
4.
P.
Hayden
,
R.
Jozsa
,
D.
Petz
, and
A.
Winter
, “
Structure of states which satisfy strong subadditivity of quantum entropy with equality
,”
Commun. Math. Phys.
246
,
359
374
(
2004
).
5.
D.
Petz
, “
Sufficient subalgebras and the relative entropy of states of a von Neumann algebra
,”
Commun. Math. Phys.
105
,
123
131
(
1986
).
6.
D.
Petz
, “
Sufficiency of channels over von Neumann algebras
,”
Q. J. Math.
39
,
97
108
(
1988
).
7.
O.
Fawzi
and
R.
Renner
, “
Quantum conditional mutual information and approximate Markov chains
,”
Commun. Math. Phys.
340
,
575
611
(
2015
).
8.
D.
Sutter
,
O.
Fawzi
, and
R.
Renner
, “
Universal recovery map for approximate Markov chains
,”
Proc. R. Soc. A
472
,
20150623
(
2016
).
9.
M.
Junge
,
R.
Renner
,
D.
Sutter
,
M. M.
Wilde
, and
A.
Winter
, “
Universal recovery maps and approximate sufficiency of quantum relative entropy
,”
Ann. Henri Poincaré
19
,
2955
2978
(
2018
).
10.
D.
Sutter
,
Approximate Quantum Markov Chains
(
Springer
,
Cham
,
2018
).
11.
S. T.
Flammia
,
J.
Haah
,
M. J.
Kastoryano
, and
I. H.
Kim
, “
Limits on the storage of quantum information in a volume of space
,”
Quantum
1
,
4
(
2017
).
12.
K.
Kato
and
F. G. S. L.
Brandão
, “
Quantum approximate Markov chains are thermal
,”
Commun. Math. Phys.
370
,
117
149
(
2019
).
13.
F. G. S. L.
Brandão
and
M. J.
Kastoryano
, “
Finite correlation length implies efficient preparation of quantum thermal states
,”
Commun. Math. Phys.
365
,
1
16
(
2019
).
14.
B.
Swingle
and
J.
McGreevy
, “
Mixed s-sourcery: Building many-body states using bubbles of nothing
,”
Phys. Rev. B
94
,
155125
(
2016
).
15.
P.
Hayden
and
G.
Penington
, “
Approximate quantum error correction revisited: Introducing the alpha-bit
,”
Commun. Math. Phys.
374
,
369
432
(
2020
).
16.
A.
Klümper
,
A.
Schadschneider
, and
J.
Zittartz
, “
Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets
,”
Europhys. Lett.
24
,
293
297
(
1993
).
17.
G.
Vidal
, “
Efficient classical simulation of slightly entangled quantum computations
,”
Phys. Rev. Lett.
91
,
147902
(
2003
).
18.
D.
Pérez-García
,
F.
Verstraete
,
M. M.
Wolf
, and
J. I.
Cirac
, “
Matrix product state representations
,”
Quantum Inf. Comput.
7
,
401
430
(
2007
).
19.
F.
Verstraete
,
V.
Murg
, and
J.
Cirac
, “
Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
,”
Adv. Phys.
57
,
143
224
(
2008
).
20.
M.
Fannes
,
B.
Nachtergaele
, and
R. F.
Werner
, “
Finitely correlated states on quantum spin chains
,”
Commun. Math. Phys.
144
,
443
490
(
1992
).
21.
M.
Fannes
,
B.
Nachtergaele
, and
R. F.
Werner
, “
Finitely correlated pure states
,”
J. Funct. Anal.
120
,
511
534
(
1994
).
22.
S. R.
White
, “
Density matrix formulation for quantum renormalization groups
,”
Phys. Rev. Lett.
69
,
2863
2866
(
1992
).
23.
F.
Verstraete
,
J. J.
García-Ripoll
, and
J. I.
Cirac
, “
Matrix product density operators: Simulation of finite-temperature and dissipative systems
,”
Phys. Rev. Lett.
93
,
207204
(
2004
).
24.
T.
Kuwahara
,
K.
Kato
, and
F. G. S. L.
Brandão
, “
Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
,”
Phys. Rev. Lett.
124
,
220601
(
2020
).
25.
T.
Kuwahara
,
A. M.
Alhambra
, and
A.
Anshu
, “
Improved thermal area law and quasilinear time algorithm for quantum Gibbs states
,”
Phys. Rev. X
11
,
011047
(
2021
).
26.
C.
Schön
,
K.
Hammerer
,
M. M.
Wolf
,
J. I.
Cirac
, and
E.
Solano
, “
Sequential generation of matrix-product states in cavity QED
,”
Phys. Rev. A
75
,
032311
(
2007
).
27.
L.
Zhou
,
S.
Choi
, and
M. D.
Lukin
, “
Symmetry-protected dissipative preparation of matrix product states
,”
Phys. Rev. A
104
,
032418
(
2021
).
28.
M. M.
Wolf
,
F.
Verstraete
,
M. B.
Hastings
, and
J. I.
Cirac
, “
Area laws in quantum systems: Mutual information and correlations
,”
Phys. Rev. Lett.
100
,
070502
(
2008
).
29.
F. G. S. L.
Brandão
and
M.
Horodecki
, “
Exponential decay of correlations implies area law
,”
Commun. Math. Phys.
333
,
761
798
(
2015
).
30.
J.
Guth Jarkovský
,
A.
Molnár
,
N.
Schuch
, and
J. I.
Cirac
, “
Efficient description of many-body systems with matrix product density operators
,”
PRX Quantum
1
,
010304
(
2020
).
31.
I.
Kim
, “
Markovian matrix product density operators: Efficient computation of global entropy
,” arXiv:1709.07828 [quant-ph] (
2017
).
32.
C.-F.
Chen
,
K.
Kato
, and
F. G. S. L.
Brandão
, “
Matrix product density operators: When do they have a local parent Hamiltonian?
,” arXiv:2010.14682 [quant-ph] (
2021
).
33.
P.
Svetlichnyy
and
T. A. B.
Kennedy
, “
Decay of quantum conditional mutual information for purely generated finitely correlated states
,”
J. Math. Phys.
63
,
072201
(
2022
).
34.
R.
Alicki
and
M.
Fannes
,
J. Phys. A: Math. Gen.
37
,
L55
L57
(
2004
).
35.
A.
Winter
, “
Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints
,”
Commun. Math. Phys.
347
,
291
313
(
2016
).
36.
W. F.
Stinespring
, “
Positive functions on C*-algebras
,”
Proc. Am. Math. Soc.
6
,
211
216
(
1955
).
37.
D.
Kretschmann
,
D.
Schlingemann
, and
R. F.
Werner
, “
The information-disturbance tradeoff and the continuity of Stinespring’s representation
,”
IEEE Trans. Inf. Theory
54
,
1708
1717
(
2008
).
38.
M.-D.
Choi
, “
Completely positive linear maps on complex matrices
,”
Linear Algebra Appl.
10
,
285
290
(
1975
).
39.
O.
Szehr
,
D.
Reeb
, and
M. M.
Wolf
, “
Spectral convergence bounds for classical and quantum Markov processes
,”
Commun. Math. Phys.
333
,
565
595
(
2015
).
40.
M. B.
Ruskai
, “
Inequalities for quantum entropy: A review with conditions for equality
,”
J. Math. Phys.
43
,
4358
4375
(
2002
).
41.

The benefit of finding eigenvalues of ρn by projecting ρn onto its support versus direct diagonalization is revealed by comparing how the memory required to implement the two schemes scales with n. Direct diagonalization requires knowledge of all elements of ρn, that is Odsn real numbers. Projecting ρn onto its support requires storing and multiplying 2n copies of M, n copies each of W and W and 2 copies of Σ12 [refer to (57)], that is, O2ndsdM2+2dM4=On real numbers. Therefore, the projection method is exponentially more efficient than direct diagonalization.

42.
H.
Cartan
,
Differential Calculus
(
Houghton Mifflin Co
,
1971
).
43.
C. E.
González-Guillén
,
M.
Junge
, and
I.
Nechita
, “
On the spectral gap of random quantum channels
,” arXiv:1811.08847 (
2018
).
44.
C.
Lancien
and
D.
Pérez-García
, “
Correlation length in random MPS and PEPS
,”
Ann. Henri Poincaré
23
,
141
222
(
2021
).
45.
R.
Kukulski
,
I.
Nechita
,
L.
Pawela
,
Z.
Puchała
, and
K.
Życzkowski
, “
Generating random quantum channels
,”
J. Math. Phys.
62
,
062201
(
2021
).
46.
W.
Bruzda
,
M.
Smaczyński
,
V.
Cappellini
,
H.-J.
Sommers
, and
K.
Życzkowski
, “
Universality of spectra for interacting quantum chaotic systems
,”
Phys. Rev. E
81
,
066209
(
2010
).
47.
M. B.
Hastings
, “
Random unitaries give quantum expanders
,”
Phys. Rev. A
76
,
032315
(
2007
).
48.
J.
Watrous
,
The Theory of Quantum Information
(
Cambridge University Press
,
2018
).
49.
J.
Schwinger
, “
Unitary operator bases
,”
Proc. Natl. Acad. Sci. U. S. A.
46
,
570
579
(
1960
).
50.
J.
Siewert
, “
On orthogonal bases in the Hilbert–Schmidt space of matrices
,”
J. Phys. Commun.
6
,
055014
(
2022
).
You do not currently have access to this content.