8 research outputs found
The Minimum Distance Problem for Two-Way Entanglement Purification
Entanglement purification takes a number of noisy EPR pairs and processes
them to produce a smaller number of more reliable pairs. If this is done with
only a forward classical side channel, the procedure is equivalent to using a
quantum error-correcting code (QECC). We instead investigate entanglement
purification protocols with two-way classical side channels (2-EPPs) for finite
block sizes. In particular, we consider the analog of the minimum distance
problem for QECCs, and show that 2-EPPs can exceed the quantum Hamming bound
and the quantum Singleton bound. We also show that 2-EPPs can achieve the rate
k/n = 1 - (t/n) \log_2 3 - h(t/n) - O(1/n) (asymptotically reaching the quantum
Hamming bound), where the EPP produces at least k good pairs out of n total
pairs with up to t arbitrary errors, and h(x) = -x \log_2 x - (1-x) \log_2
(1-x) is the usual binary entropy. In contrast, the best known lower bound on
the rate of QECCs is the quantum Gilbert-Varshamov bound k/n \geq 1 - (2t/n)
\log_2 3 - h(2t/n). Indeed, in some regimes, the known upper bound on the
asymptotic rate of good QECCs is strictly below our lower bound on the
achievable rate of 2-EPPs.Comment: 10 pages, LaTeX. v2: New title, minor corrections and clarifications,
some new references. v3: One more small correction. v4: More small
clarifications, final version to appear in IEEE Trans. Info. Theor
Communicating over adversarial quantum channels using quantum list codes
We study quantum communication in the presence of adversarial noise. In this
setting, communicating with perfect fidelity requires using a quantum code of
bounded minimum distance, for which the best known rates are given by the
quantum Gilbert-Varshamov (QGV) bound. By asking only for arbitrarily high
fidelity and allowing the sender and reciever to use a secret key with length
logarithmic in the number of qubits sent, we achieve a dramatic improvement
over the QGV rates. In fact, we find protocols that achieve arbitrarily high
fidelity at noise levels for which perfect fidelity is impossible. To achieve
such communication rates, we introduce fully quantum list codes, which may be
of independent interest.Comment: 6 pages. Discussion expanded and more details provided in proofs. Far
less unclear than previous versio
Improvement of stabilizer based entanglement distillation protocols by encoding operators
This paper presents a method for enumerating all encoding operators in the
Clifford group for a given stabilizer. Furthermore, we classify encoding
operators into the equivalence classes such that EDPs (Entanglement
Distillation Protocol) constructed from encoding operators in the same
equivalence class have the same performance. By this classification, for a
given parameter, the number of candidates for good EDPs is significantly
reduced. As a result, we find the best EDP among EDPs constructed from [[4,2]]
stabilizer codes. This EDP has a better performance than previously known EDPs
over wide range of fidelity.Comment: 22 pages, 2 figures, In version 2, we enumerate all encoding
operators in the Clifford group, and fix the wrong classification of encoding
operators in version
An Adaptive Entanglement Distillation Scheme Using Quantum Low Density Parity Check Codes
Quantum low density parity check (QLDPC) codes are useful primitives for
quantum information processing because they can be encoded and decoded
efficiently. Besides, the error correcting capability of a few QLDPC codes
exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical
performance analysis of an adaptive entanglement distillation scheme using
QLDPC codes. In particular, we find that the expected yield of our adaptive
distillation scheme to combat depolarization errors exceed that of Leung and
Shor whenever the error probability is less than about 0.07 or greater than
about 0.28. This finding illustrates the effectiveness of using QLDPC codes in
entanglement distillation.Comment: 12 pages, 6 figure
Entanglement Distillation; A Discourse on Bound Entanglement in Quantum Information Theory
PhD thesis (University of York). The thesis covers in a unified way the
material presented in quant-ph/0403073, quant-ph/0502040, quant-ph/0504160,
quant-ph/0510035, quant-ph/0512012 and quant-ph/0603283. It includes two large
review chapters on entanglement and distillation.Comment: 192 page