Mixed-state entanglement and quantum error correction.

@article{Bennett1996MixedstateEA,
  title={Mixed-state entanglement and quantum error correction.},
  author={Bennett and DiVincenzo and Smolin and Wootters},
  journal={Physical review. A, Atomic, molecular, and optical physics},
  year={1996},
  volume={54 5},
  pages={
          3824-3851
        }
}
  • Bennett, DiVincenzo, Wootters
  • Published 23 April 1996
  • Computer Science
  • Physical review. A, Atomic, molecular, and optical physics
Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical… 

Quantum entanglement capacity with classical feedback

TLDR
This work presents a general scheme to convert any quantum error-correcting codes into adaptive protocols for this newly defined quantity of the quantum depolarizing channel, and illustrates with the repetition code and Shor code.

Two-Way Entanglement Purification for Finite Block Size

TLDR
This work considers the analog of the minimum distance problem for QECCs, and shows that 2-EPPs can exceed the quantum Hamming bound and the quantum Singleton bound, and achieves the rate k/n = 1− 3 log2 t/n− h(t/n)−O(1/ n) (asymptotically reaching the quantumHamming bound).

Quantum error correction assisted by two-way noisy communication

TLDR
An error-correcting protocol assisted by two-way noisy communication that is more easily realisable: all quantum communication is subjected to general noise and all entanglement is created locally without additional resources consumed.

Quantum error-correction codes and absolutely maximally entangled states

TLDR
In this network, it is shown how corrections arise to the Ryu-Takayanagi formula in the case of entangled input state, and that the bound on the entanglement entropy of the boundary state is saturated for absolutely maximally entangled input states.

On entanglement distillation and quantum error correction for unknown states and channels

We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any

Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates

In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error distinguishability of arbitrary ensembles of mixed quantum states; (2)

Multiparty quantum protocols for assisted entanglement distillation

TLDR
This work extends the notion of entanglement of assistance to arbitrary states and gives a protocol for extracting pure Entanglement, a new protocol for the task of multiparty state merging, which does not require time-sharing for distributed compression of two senders.

A remark on quantum key distribution with two way communication: the classical complexity in decoding the CSS code can be removed

TLDR
This protocol shows that, this type of 2-EPP protocol can be reduced to an equivalent prepare and measure protocol, and the huge complexity of decoding the classical CSS code is totally removed.

Quantum Versus Classical Advantages in Secret Key Distillation (and Their Links to Quantum Entanglement)

We consider the extraction of shared secret key from correlations that are generated by either a classical or quantum source. In the classical setting, two honest parties (Alice and Bob) use public

Extracting quantum entanglement (general entanglement purification protocols)

TLDR
It is proved that, on average, Alice and Bob cannot increase the fidelity of the input state significantly, and there exist protocols that may fail with a small probability, and otherwise will output states arbitrarily close to EPR pairs with very high probability.
...

References

SHOWING 1-10 OF 79 REFERENCES

Purification of noisy entanglement and faithful teleportation via noisy channels.

TLDR
Upper and lower bounds on the yield of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$ are given, showing $D(M)>0$ if $\bra{Psi-}M\ket-}>\half$.

Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels.

TLDR
The concept of quantum privacy amplification and a cryptographic scheme incorporating it which is provably secure over a noisy channel is introduced and implemented using technology that is currently being developed.

Perfect Quantum Error Correcting Code.

TLDR
A quantum error correction code which protects a qubit of information against general one qubit errors and encode the original state by distributing quantum information over five qubits, the minimal number required for this task.

Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.

An unknown quantum state \ensuremath{\Vert}\ensuremath{\varphi}〉 can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen

Multiple-particle interference and quantum error correction

  • A. Steane
  • Physics
    Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1996
The concept of multiple-particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum

Good quantum error-correcting codes exist.

  • CalderbankShor
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1996
TLDR
The techniques investigated in this paper can be extended so as to reduce the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions.

Concentrating partial entanglement by local operations.

TLDR
Any pure or mixed entangled state of two systems can be produced by two classically communicating separated observers, drawing on a supply of singlets as their sole source of entanglement.

Quantum error correction in the presence of spontaneous emission.

TLDR
The code proposed in this paper has the ability to correct a special kind of error, due to the conditional time evolution, to all orders, an interesting feature, as one would be interested to correct those errors which frequently occur to higher order than rare errors.

Quantum Error-Correcting Codes Need Not Completely Reveal the Error Syndrome

TLDR
A code which does not find the complete error syndrome and can be used for reliable transmission of quantum information through channels which add more than one bit of entropy per transmitted bit.

Error Correcting Codes in Quantum Theory.

  • Steane
  • Physics
    Physical review letters
  • 1996
TLDR
It is shown that a pair of states which are, in a certain sense, “macroscopically different,” can form a superposition in which the interference phase between the two parts is measurable, providing a highly stabilized “Schrodinger cat” state.
...