We obtain a lower bound on the maximum number of qubits, Q n , ε ( N ) , which can be transmitted over n uses of a quantum channel N , for a given non-zero error threshold ε. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the Petz recovery map (also known as the transpose channel). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which Q n , ε ( N ) scales as n .

1.
Barnum
,
H.
and
Knill
,
E.
, “
Reversing quantum dynamics with near-optimal quantum and classical fidelity
,”
J. Math. Phys.
43
(
5
),
2097
2106
(
2002
).
2.
Barnum
,
H.
,
Knill
,
E.
, and
Nielsen
,
M. A.
, “
On quantum fidelities and channel capacities
,”
IEEE Trans. Inf. Theory
46
(
4
),
1317
1329
(
2000
).
3.
Beigi
,
S.
and
Gohari
,
A.
, “
Quantum achievability proof via collision relative entropy
,”
IEEE Trans. Inf. Theory
60
(
12
),
7980
7986
(
2014
).
4.
Bennett
,
C. H.
,
Brassard
,
G.
,
Crépeau
,
C.
,
Jozsa
,
R.
,
Peres
,
A.
, and
Wootters
,
W. K.
, “
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels
,”
Phys. Rev. Lett.
70
(
13
),
1895
(
1993
).
5.
Bennett
,
C. H.
,
DiVincenzo
,
D. P.
, and
Smolin
,
J. A.
, “
Capacities of quantum erasure channels
,”
Phys. Rev. Lett.
78
,
3217
3220
(
1997
).
6.
Buscemi
,
F.
and
Datta
,
N.
, “
Distilling entanglement from arbitrary resources
,”
J. Math. Phys.
51
(
10
),
102201
(
2010
).
7.
Buscemi
,
F.
and
Datta
,
N.
, “
The quantum capacity of channels with arbitrarily correlated noise
,”
IEEE Trans. Inf. Theory
56
(
3
),
1447
1460
(
2010
).
8.
Datta
,
N.
, “
Min- and Max-relative entropies and a new entanglement monotone
,”
IEEE Trans. Inf. Theory
55
(
6
),
2816
2826
(
2009
).
9.
Datta
,
N.
and
Hsieh
,
M.-H.
, “
The apex of the family tree of protocols: Optimal rates and resource inequalities
,”
New J. Phys.
13
(
9
),
093042
(
2011
).
10.
Datta
,
N.
and
Leditzky
,
F.
, “
Second-order asymptotics for source coding, dense coding, and pure-state entanglement conversions
,”
IEEE Trans. Inf. Theory
61
(
1
),
582
608
(
2015
).
11.
Devetak
,
I.
, “
The private classical capacity and quantum capacity of a quantum channel
,”
IEEE Trans. Inf. Theory
51
(
1
),
44
55
(
2005
).
12.
Devetak
,
I.
and
Shor
,
P. W.
, “
The capacity of a quantum channel for simultaneous transmission of classical and quantum information
,”
Commun. Math. Phys.
256
(
2
),
287
303
(
2005
).
13.
Dupuis
,
F.
,
Berta
,
M.
,
Wullschleger
,
J.
, and
Renner
,
R.
, “
One-shot decoupling
,”
Commun. Math. Phys.
328
(
1
),
251
284
(
2014
).
14.
Fawzi
,
O.
and
Renner
,
R.
, “
Quantum conditional mutual information and approximate Markov chains
,”
Commun. Math. Phys.
340
(
2
),
575
611
(
2015
).
15.
Hayashi
,
M.
, “
Second-order asymptotics in fixed-length source coding and intrinsic randomness
,”
IEEE Trans. Inf. Theory
54
(
10
),
4619
4637
(
2008
).
16.
Hayashi
,
M.
, “
Information spectrum approach to second-order coding rate in channel coding
,”
IEEE Trans. Inf. Theory
55
(
11
),
4947
4966
(
2009
).
17.
Hayashi
,
M.
and
Tomamichel
,
M.
, “
Correlation detection and an operational interpretation of the Rényi mutual information
,” in
IEEE International Symposium on Information Theory (ISIT)
(
IEEE
,
2015
), pp.
1447
1451
.
18.
Hayden
,
P.
,
Horodecki
,
M.
,
Winter
,
A.
, and
Yard
,
J.
, “
A decoupling approach to the quantum capacity
,”
Open Syst. Inf. Dyn.
15
(
1
),
7
19
(
2008
).
19.
Hayden
,
P.
,
Jozsa
,
R.
,
Petz
,
D.
, and
Winter
,
A.
, “
Structure of states which satisfy strong subadditivity of quantum entropy with equality
,”
Commun. Math. Phys.
246
(
2
),
359
374
(
2004
).
20.
Hayden
,
P.
,
Shor
,
P. W.
, and
Winter
,
A.
, “
Random quantum codes from Gaussian ensembles and an uncertainty relation
,”
Open Syst. Inf. Dyn.
15
(
1
),
71
89
(
2008
).
21.
Horodecki
,
M.
,
Horodecki
,
P.
, and
Horodecki
,
R.
, “
General teleportation channel, singlet fraction, and quasidistillation
,”
Phys. Rev. A
60
(
3
),
1888
(
1999
).
22.
Junge
,
M.
,
Renner
,
R.
,
Sutter
,
D.
,
Wilde
,
M. M.
, and
Winter
,
A.
, “
Universal recovery from a decrease of quantum relative entropy
,” preprint arXiv:1509.07127 [quant-ph] (
2015
).
23.
König
,
R.
,
Renner
,
R.
, and
Schaffner
,
C.
, “
The operational meaning of min- and max-entropy
,”
IEEE Trans. Inf. Theory
55
(
9
),
4337
4347
(
2009
).
24.
Kostina
,
V.
and
Verdú
,
S.
, “
Nonasymptotic noisy lossy source coding
,” in
IEEE Information Theory Workshop (ITW)
(
IEEE
,
2013
), pp.
1
5
.
25.
Kretschmann
,
D.
and
Werner
,
R. F.
, “
Tema con variazioni: Quantum channel capacity
,”
New J. Phys.
6
(
1
),
26
(
2004
).
26.
Kumagai
,
W.
and
Hayashi
,
M.
, “
Second order asymptotics for random number generation
,” in
IEEE International Symposium on Information Theory Proceedings (ISIT)
(
IEEE
,
2013
), pp.
1506
1510
.
27.
Li
,
K.
, “
Second order asymptotics for quantum hypothesis testing
,”
Ann. Stat.
42
(
1
),
171
189
(
2014
).
28.
Lloyd
,
S.
, “
Capacity of the noisy quantum channel
,”
Phys. Rev. A
55
(
3
),
1613
(
1997
).
29.
Morgan
,
C.
and
Winter
,
A.
, “
“Pretty strong” converse for the quantum capacity of degradable channels
,”
IEEE Trans. Inf. Theory
60
(
1
),
317
333
(
2014
).
30.
Müller-Lennert
,
M.
,
Dupuis
,
F.
,
Szehr
,
O.
,
Fehr
,
S.
, and
Tomamichel
,
M.
, “
On quantum Rényi entropies: A new generalization and some properties
,”
J. Math. Phys.
54
(
12
),
122203
(
2013
).
31.
Ng
,
H. K.
and
Mandayam
,
P.
, “
Simple approach to approximate quantum error correction based on the transpose channel
,”
Phys. Rev. A
81
(
6
),
062342
(
2010
).
32.
Nielsen
,
M. A.
, “
A simple formula for the average gate fidelity of a quantum dynamical operation
,”
Phys. Lett. A
303
(
4
),
249
252
(
2002
).
33.
Ohya
,
M.
and
Petz
,
D.
,
Quantum Entropy and its Use
(
Springer
,
1993
).
34.
Petz
,
D.
, “
Sufficient subalgebras and the relative entropy of states of a von Neumann algebra
,”
Commun. Math. Phys.
105
(
1
),
123
131
(
1986
).
35.
Petz
,
D.
, “
Sufficiency of channels over von Neumann algebras
,”
Q. J. Math.
39
(
1
),
97
108
(
1988
).
36.
Polyanskiy
,
Y.
,
Poor
,
V.
, and
Verdú
,
S.
, “
Channel coding rate in the finite blocklength regime
,”
IEEE Trans. Inf. Theory
56
(
5
),
2307
2359
(
2010
).
37.
Renner
,
R.
, “
Security of quantum key distribution
,” Ph.D. thesis,
ETH Zürich
,
2005
.
38.
Shannon
,
C. E.
, “
A mathematical theory of communication
,”
Bell Syst. Tech. J.
27
,
379
423
(
1948
).
39.
Shor
,
P. W.
, “The quantum channel capacity and coherent information,” in Talk at MSRI Workshop on Quantum Computation, Berkeley, CA, USA, 2002.
40.
Strassen
,
V.
, “
Asymptotische Abschätzungen in Shannons Informationstheorie
,” in
Transactions of the Third Prague Conference on Information Theory
(
Prague: Publishing House of the Czechoslovak Academy of Sciences
,
1962
), pp.
689
723
.
41.
Tomamichel
,
M.
, “
A framework for non-asymptotic quantum information theory
,” Ph.D. thesis,
ETH Zürich
,
2012
.
42.
Tomamichel
,
M.
,
Berta
,
M.
, and
Renes
,
J. M.
, “
Quantum coding with finite resources
,”
Nat. Commun.
7
,
11419
(
2016
).
43.
Tomamichel
,
M.
,
Colbeck
,
R.
, and
Renner
,
R.
, “
A fully quantum asymptotic equipartition property
,”
IEEE Trans. Inf. Theory
55
(
12
),
5840
5847
(
2009
).
44.
Tomamichel
,
M.
,
Colbeck
,
R.
, and
Renner
,
R.
, “
Duality between smooth min- and max-entropies
,”
IEEE Trans. Inf. Theory
56
(
9
),
4674
4681
(
2010
).
45.
Tomamichel
,
M.
and
Hayashi
,
M.
, “
A hierarchy of information quantities for finite block length analysis of quantum tasks
,”
IEEE Trans. Inf. Theory
59
(
11
),
7693
7710
(
2013
).
46.
Tomamichel
,
M.
,
Wilde
,
M. M.
, and
Winter
,
A.
, “
Strong converse rates for quantum communication
,” in
IEEE International Symposium on Information Theory (ISIT)
(
IEEE
,
2015
), pp.
2386
2390
.
47.
Wang
,
L.
and
Renner
,
R.
, “
One-shot classical-quantum capacity and hypothesis testing
,”
Phys. Rev. Lett.
108
(
20
),
200501
(
2012
).
48.
Wilde
,
M. M.
,
Winter
,
A.
, and
Yang
,
D.
, “
Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy
,”
Commun. Math. Phys.
331
(
2
),
593
622
(
2014
).
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