378 research outputs found
Semidefinite programming converse bounds for classical communication over quantum channels
© 2017 IEEE. We study the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes. We first show that both the optimal success probability of a given transmission rate and one-shot-error capacity can be formalized as semidefinite programs (SDPs) when assisted by NS or NS∩PPT codes. Based on this, we derive SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdii for classical channels. Furthermore, we derive an SDP strong converse bound for the classical capacity of a general quantum channel: for any code with a rate exceeding this bound, the optimal success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bound, we derive improved upper bounds to the classical capacity of the amplitude damping channels and also establish the strong converse property for a new class of quantum channels
Semidefinite programming converse bounds for quantum communication
We derive several efficiently computable converse bounds for quantum
communication over quantum channels in both the one-shot and asymptotic regime.
First, we derive one-shot semidefinite programming (SDP) converse bounds on the
amount of quantum information that can be transmitted over a single use of a
quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes,
Nat. Commun. 7, 2016]. As applications, we study quantum communication over
depolarizing channels and amplitude damping channels with finite resources.
Second, we find an SDP strong converse bound for the quantum capacity of an
arbitrary quantum channel, which means the fidelity of any sequence of codes
with a rate exceeding this bound will vanish exponentially fast as the number
of channel uses increases. Furthermore, we prove that the SDP strong converse
bound improves the partial transposition bound introduced by Holevo and Werner.
Third, we prove that this SDP strong converse bound is equal to the so-called
max-Rains information, which is an analog to the Rains information introduced
in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP
strong converse bound is weaker than the Rains information, but it is
efficiently computable for general quantum channels.Comment: 17 pages, extended version of arXiv:1601.06888. v3 is closed to the
published version, IEEE Transactions on Information Theory, 201
A Minimax Converse for Quantum Channel Coding
We prove a one-shot "minimax" converse bound for quantum channel coding
assisted by positive partial transpose channels between sender and receiver.
The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu
[IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding,
and also enjoys the saddle point property enabling the order of optimizations
to be interchanged. Equivalently, the bound can be formulated as a semidefinite
program satisfying strong duality. The convex nature of the bound implies
channel symmetries can substantially simplify the optimization, enabling us to
explicitly compute the finite blocklength behavior for several simple qubit
channels. In particular, we find that finite blocklength converse statements
for the classical erasure channel apply to the assisted quantum erasure
channel, while bounds for the classical binary symmetric channel apply to both
the assisted dephasing and depolarizing channels. This implies that these qubit
channels inherit statements regarding the asymptotic limit of large
blocklength, such as the strong converse or second-order converse rates, from
their classical counterparts. Moreover, for the dephasing channel, the finite
blocklength bounds are as tight as those for the classical binary symmetric
channel, since coding for classical phase errors yields equivalently-performing
unassisted quantum codes.Comment: merged with arXiv:1504.04617 version 1 ; see version
Semidefinite programming strong converse bounds for classical capacity
© 2017 IEEE. We investigate the classical communication over quantum channels when assisted by no-signaling and positive-partial-transpose-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot -error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel
Information-theoretic aspects of the generalized amplitude damping channel
The generalized amplitude damping channel (GADC) is one of the sources of
noise in superconducting-circuit-based quantum computing. It can be viewed as
the qubit analogue of the bosonic thermal channel, and it thus can be used to
model lossy processes in the presence of background noise for low-temperature
systems. In this work, we provide an information-theoretic study of the GADC.
We first determine the parameter range for which the GADC is entanglement
breaking and the range for which it is anti-degradable. We then establish
several upper bounds on its classical, quantum, and private capacities. These
bounds are based on data-processing inequalities and the uniform continuity of
information-theoretic quantities, as well as other techniques. Our upper bounds
on the quantum capacity of the GADC are tighter than the known upper bound
reported recently in [Rosati et al., Nat. Commun. 9, 4339 (2018)] for the
entire parameter range of the GADC, thus reducing the gap between the lower and
upper bounds. We also establish upper bounds on the two-way assisted quantum
and private capacities of the GADC. These bounds are based on the squashed
entanglement, and they are established by constructing particular squashing
channels. We compare these bounds with the max-Rains information bound, the
mutual information bound, and another bound based on approximate covariance.
For all capacities considered, we find that a large variety of techniques are
useful in establishing bounds.Comment: 33 pages, 9 figures; close to the published versio
Approximate Degradable Quantum Channels
Degradable quantum channels are an important class of completely positive
trace-preserving maps. Among other properties, they offer a single-letter
formula for the quantum and the private classical capacity and are
characterized by the fact that a complementary channel can be obtained from the
channel by applying a degrading channel. In this work we introduce the concept
of approximate degradable channels, which satisfy this condition up to some
finite . That is, there exists a degrading channel which upon
composition with the channel is -close in the diamond norm to the
complementary channel. We show that for any fixed channel the smallest such
can be efficiently determined via a semidefinite program.
Moreover, these approximate degradable channels also approximately inherit all
other properties of degradable channels. As an application, we derive improved
upper bounds to the quantum and private classical capacity for certain channels
of interest in quantum communication.Comment: v3: minor changes, published version. v2: 21 pages, 2 figures,
improved bounds on the capacity for approximate degradable channels based on
[arXiv:1507.07775], an author adde
- …