# 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 } }

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…

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## References

SHOWING 1-10 OF 79 REFERENCES

### Purification of noisy entanglement and faithful teleportation via noisy channels.

- Computer SciencePhysical review letters
- 1996

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.

- Computer SciencePhysical review letters
- 1996

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.

- Computer Science, PhysicsPhysical review letters
- 1996

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.

- PhysicsPhysical review letters
- 1993

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

- PhysicsProceedings 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.

- PhysicsPhysical review. A, Atomic, molecular, and optical physics
- 1996

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.

- PhysicsPhysical review. A, Atomic, molecular, and optical physics
- 1996

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.

- Computer Science, Physics
- 1997

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

- Computer Science
- 1996

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.

- PhysicsPhysical review letters
- 1996

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.