We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.

1.
C. H.
Bennett
,
D. P.
DiVincenzo
,
T.
Mor
,
P. W.
Shor
,
J. A.
Smolin
, and
B. M.
Terhal
, “
Unextendible product bases and bound entanglement
,”
Phys. Rev. Lett.
82
,
5385
5388
(
1999
).
2.
C. H.
Bennett
,
D. P.
DiVincenzo
, and
J. A.
Smolin
, “
Capacities of quantum erasure channels
,”
Phys. Rev. Lett.
78
(
16
),
3217
(
1997
).
3.
M.
Berta
,
M.
Christandl
,
F. G.
Brandao
, and
S.
Wehner
, “
Entanglement cost of quantum channels
,” in
2012 IEEE International Symposium on Information Theory Proceedings (ISIT)
(
IEEE
,
2012
), pp.
900
904
.
4.
R.
Bhatia
, “
Matrix analysis
,” in
Graduate Texts in Mathematics
(
Springer
,
New York
,
1997
).
5.
D.
Bruß
,
D. P.
DiVincenzo
,
A.
Ekert
,
C. A.
Fuchs
,
C.
Macchiavello
, and
J. A.
Smolin
, “
Optimal universal and state-dependent quantum cloning
,”
Phys. Rev. A
57
(
4
),
2368
(
1998
).
6.
E.
Chitambar
,
D.
Leung
,
L.
Mančinska
,
M.
Ozols
, and
A.
Winter
, “
Everything you always wanted to know about locc (but were afraid to ask)
,”
Commun. Math. Phys.
328
(
1
),
303
326
(
2014
).
7.
T.
Cubitt
,
J.
Chen
, and
A.
Harrow
, “
Superactivation of the asymptotic zero-error classical capacity of a quantum channel
,”
IEEE Trans. Inf. Theory
57
(
12
),
8114
8126
(
2011
).
8.
D. P.
DiVincenzo
,
P. W.
Shor
,
J. A.
Smolin
,
B. M.
Terhal
, and
A. V.
Thapliyal
, “
Evidence for bound entangled states with negative partial transpose
,”
Phys. Rev. A
61
,
062312
(
2000
).
9.
W.
Dür
,
J. I.
Cirac
,
M.
Lewenstein
, and
D.
Bruß
, “
Distillability and partial transposition in bipartite systems
,”
Phys. Rev. A
61
,
062313
(
2000
).
10.
S. N.
Filippov
,
A. A.
Melnikov
, and
M.
Ziman
, “
Dissociation and annihilation of multipartite entanglement structure in dissipative quantum dynamics
,”
Phys. Rev. A
88
(
6
),
062328
(
2013
).
11.
S. N.
Filippov
,
T.
Rybár
, and
M.
Ziman
, “
Local two-qubit entanglement-annihilating channels
,”
Phys. Rev. A
85
(
1
),
012303
(
2012
).
12.
S. N.
Filippov
and
M.
Ziman
, “
Bipartite entanglement-annihilating maps: Necessary and sufficient conditions
,”
Phys. Rev. A
88
(
3
),
032316
(
2013
).
13.
M.
Hayashi
,
Quantum Information – An Introduction
(
Springer
,
Berlin, Heidelberg
,
2006
).
14.
A. S.
Holevo
and
R. F.
Werner
, “
Evaluating capacities of bosonic gaussian channels
,”
Phys. Rev. A
63
(
3
),
032312
(
2001
).
15.
M.
Horodecki
and
P.
Horodecki
, “
Reduction criterion of separability and limits for a class of distillation protocols
,”
Phys. Rev. A
59
,
4206
4216
(
1999
).
16.
M.
Horodecki
,
P.
Horodecki
, and
R.
Horodecki
, “
Separability of mixed states: Necessary and sufficient conditions
,”
Phys. Lett. A
223
(
1
),
1
8
(
1996
).
17.
M.
Horodecki
,
P.
Horodecki
, and
R.
Horodecki
, “
Mixed-state entanglement and distillation: Is there a bound entanglement in nature?
,”
Phys. Rev. Lett.
80
,
5239
5242
(
1998
).
18.
M.
Horodecki
,
P. W.
Shor
, and
M. B.
Ruskai
, “
Entanglement breaking channels
,”
Rev. Math. Phys.
15
(
06
),
629
641
(
2003
).
19.
C.
King
, “
Maximal p-norms of entanglement breaking channels
,”
Quantum Inf. Comput.
3
(
2
),
186
190
(
2003
).
20.
D.
Kretschmann
and
R. F.
Werner
, “
Tema con variazioni: Quantum channel capacity
,”
New J. Phys.
6
(
1
),
26
(
2004
).
21.
S.
Lloyd
, “
Capacity of the noisy quantum channel
,”
Phys. Rev. A
55
,
1613
1622
(
1997
).
22.
L.
Moravčíková
and
M.
Ziman
, “
Entanglement-annihilating and entanglement-breaking channels
,”
J. Phys. A: Math. Theor.
43
(
27
),
275306
(
2010
).
23.
C.
Morgan
and
A.
Winter
, “
Pretty strong converse for the quantum capacity of degradable channels
,”
IEEE Trans. Inf. Theory
60
(
1
),
317
333
(
2014
).
24.
V.
Paulsen
, in
Completely Bounded Maps and Operator Algebras
(
Cambridge University Press
,
2002
), Vol.
78
.
25.
E. M.
Rains
, “
A semidefinite program for distillable entanglement
,”
IEEE Trans. Inf. Theory
47
(
7
),
2921
2933
(
2001
).
26.
A.
Roy
and
A. J.
Scott
, “
Unitary designs and codes
,”
Des. Codes Cryptogr.
53
(
1
),
13
31
(
2009
).
27.
P. W.
Shor
, “
Additivity of the classical capacity of entanglement-breaking quantum channels
,”
J. Math. Phys.
43
(
9
),
4334
4340
(
2002
).
28.
G.
Smith
and
J. A.
Smolin
, “
Detecting incapacity of a quantum channel
,”
Phys. Rev. Lett.
108
,
230507
(
2012
).
29.
E.
Størmer
, “
Tensor powers of 2-positive maps
,”
J. Math. Phys.
51
(
10
),
102203
(
2010
).
30.
M.
Takeoka
,
S.
Guha
, and
M.
Wilde
, “
The squashed entanglement of a quantum channel
,”
IEEE Trans. Inf. Theory
60
(
8
),
4987
4998
(
2014
).
31.
M.
Tomamichel
,
M. M.
Wilde
, and
A.
Winter
, “
Strong converse rates for quantum communication
,” e-print arXiv:1406.2946 (
2014
).
32.
R. F.
Werner
, “
Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model
,”
Phys. Rev. A
40
(
8
),
4277
(
1989
).
You do not currently have access to this content.