1,006 research outputs found
Duality in Entanglement-Assisted Quantum Error Correction
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is
defined from the orthogonal group of a simplified stabilizer group. From the
Poisson summation formula, this duality leads to the MacWilliams identities and
linear programming bounds for EAQEC codes. We establish a table of upper and
lower bounds on the minimum distance of any maximal-entanglement EAQEC code
with length up to 15 channel qubits.Comment: This paper is a compact version of arXiv:1010.550
Entanglement-assisted quantum error-correcting codes over arbitrary finite fields
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.Supported by the Spanish Ministry of Economy/FEDER: grants MTM2015-65764-C3-1-P, MTM2015-65764-C3-2-P, MTM2015-69138-REDT and RYC-2016-20208 (AEI/FSE/UE), the University Jaume I: grant UJI-B2018-10, Spanish Junta de CyL: grant VA166G18, and JSPS Grant No. 17K06419
Weak Decoupling Duality and Quantum Identification
If a quantum system is subject to noise, it is possible to perform quantum
error correction reversing the action of the noise if and only if no
information about the system's quantum state leaks to the environment. In this
article, we develop an analogous duality in the case that the environment
approximately forgets the identity of the quantum state, a weaker condition
satisfied by epsilon-randomizing maps and approximate unitary designs.
Specifically, we show that the environment approximately forgets quantum states
if and only if the original channel approximately preserves pairwise fidelities
of pure inputs, an observation we call weak decoupling duality. Using this
tool, we then go on to study the task of using the output of a channel to
simulate restricted classes of measurements on a space of input states. The
case of simulating measurements that test whether the input state is an
arbitrary pure state is known as equality testing or quantum identification. An
immediate consequence of weak decoupling duality is that the ability to perform
quantum identification cannot be cloned. We furthermore establish that the
optimal amortized rate at which quantum states can be identified through a
noisy quantum channel is equal to the entanglement-assisted classical capacity
of the channel, despite the fact that the task is quantum, not classical, and
entanglement-assistance is not allowed. In particular, this rate is strictly
positive for every non-constant quantum channel, including classical channels.Comment: 14 pages; v2 has some remarks added and inaccuracies corrected; v3
has new title, improved presentation and additional references; v4 is the
final, accepted version (to appear in IEEE IT), title changed once more and
numerous improvements made - the main one being that we can now show that
nontrivial amortization is necessary in erasure channel
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
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