Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels

Asymptotically, S(W) BXOR tests per impure pair suffice to find all the errors.

“Combining these results, and recalling that log 2 ( N 1 N 2 ) = nS ( W ) + O ( √ n ), we obtain the desired result, viz. that asymptotically S ( W ) BXOR tests per impure pair suffice to find all the errors.”

By performing a number of BXOR tests on different subsets of the original impure pairs, all the Ψ states can be found and corrected to Φ states.

“By performing a number of BXOR tests, on different subsets of the original impure pairs, all the Ψ states can be found and corrected to Φ states.”

For the choice k(F) ≈ 1/√(1-F), the expected fraction of pairs discarded at each step, to lowest order, is 2/3(1-F)^{1/2}.

“The expected fraction of the pairs discarded at each step, also to lowest order, is 2 3 (1 -F ) 1 2 .”

A random bilateral rotation on each shared pair transforms any initial general two-spin mixed state M into a rotationally symmetric Werner state W_F of the same purity F.

“Because of the singlet's invariance under bilateral rotations, the symmetrized state W F , which we shall call a Werner state [7] of purity F , has the same F as the initial mixed state M from which it was derived.”

• cites [7] R.F. Werner, Phys. Rev. A 40 , 4277 (1989). sourceNot checked yet

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The purity F of a two-spin mixed state relative to a perfect singlet can be computed from the probability P|| of obtaining parallel outcomes if the two spins are measured locally along the same random axis as F = 1 - 3P||/2.

“one finds that F = 1 -3 P ‖ / 2.”

Two separated observers can prepare a smaller number of entangled pairs of arbitrarily high purity from a supply of not-too-impure entangled states by local operations and classical communication.

“Two separated observers, by applying local operations to a supply of not-too-impure entangled states ( e.g. singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ( e.g. nearperfect singlets).”

A similar procedure can find all the Φ- states and correct them to the desired Φ+.

“A similar procedure is then used to find all the Φ -states and correct them to the desired Φ + .”

For the choice k(F) ≈ 1/√(1-F), to lowest order in 1-F, the iteration formula for purity agrees with Eq. (7): F'(F) = 1 - 2/3(1-F).

“(For this choice of k , to lowest order in 1 -F , the iteration formula for purity agrees with Eq. (7): F ′ ( F ) = 1 -2 3 (1 -F ).”

The yield D(M) of pure singlets distillable from mixed states M is positive if the fidelity 〈Ψ-|M|Ψ-〉 > 1/2.

“We give upper and lower bounds on the yield D ( M ) of pure singlets ( | Ψ -〉 ) distillable from mixed states M , showing D ( M ) > 0 if 〈 Ψ -| M | Ψ -〉 > 1 2 .”

The number of BXOR tests per impure pair required to find all the errors, with arbitrarily small chance of failure, approaches the entropy S(W) = -Tr W log2 W in the limit of large n.

“The number of BXOR tests per impure pair required to find all the errors, with arbitrarily small chance of failure, approaches the entropy of the impure pairs, S ( W ) = -Tr W log 2 W , in the limit of large n .”

• AssumptionRests on an assumption: The probability distribution P_X over n-bit strings x has a typical set containing N1 = 2^{H(X)+O(√n)} members, where H(X) is the Shannon entropy of P_X.
• AssumptionRests on an assumption: The conditional distribution P_{Y|X=x} of n-bit strings y has a typical set containing N2 = 2^{H(Y|X=x)+O(√n)} members.

A BXOR test tells Alice and Bob whether there are an even or odd number of Ψ states in the tested subset.

“Thus a BXOR test, like a parity check on classical data, tells Alice and Bob whether there are an even or odd number of Ψ states in the tested subset.”

For F ≤ 1/2, W_F can be expressed as a mixture of unentangled product states ↑↑, ↓↓, ↑↓, and ↓↑.

“while for F ≤ 1 2 , W F can be expressed as mixture of unentangled product states ↑↑ , ↓↓ , ↑↓ , and ↓↑ .”

For Werner states W_F, an upper bound on distillable entanglement is E(W_F) given by the formula in Eq. (9).

“Fig. 1 compares the yields of several purification methods for Werner states W F with an upper bound E ( W F ) given by”

The noisy channel can be used to share entangled pairs for use in teleportation rather than to transmit the source states directly.

“We describe another approach, in which the noisy channel is not used to transmit the source states directly, but rather to share entangled pairs ( e.g. singlets) for use in teleportation.”

For pure states, entanglement can be efficiently concentrated into singlets by local operations and classical communication, with yield m/n approaching E(Υ) as n→∞.

“For pure states, this entanglement can be efficiently concentrated into singlets by the methods of [6], which use local operations and classical communication to transform n input states Υ into m singlets with a yield m/n approaching E (Υ) as n → ∞ .”

• cites [6] C.H. Bennett, H. Bernstein, S. Popescu, B. Schumacher, 'Concentrating Partial Entanglement by Local Operations', submitted to Phys. Rev. A (1995). sourceNot checked yet

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For any two distinct n-bit strings, the probability that they agree on the parities of r independent random subsets of their bits is ≤ 2^{-r}.

“For any two distinct n -bit strings the probability that they agree on the parities of r independent random subsets of their bits is ≤ 2 -r [10].”

• cites [10] C.H. Bennett, G. Brassard, C. Cr´ epeau, and U.M. Maurer, 'Generalized Privacy Amplification', to appear Nov. 1995 in IEEE Trans. on Information Theory .Not checked yet

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Subsequent purification steps can be carried out without regard to any properties of the original mixed state M, or of the noisy channel(s) that may have generated it, except for the purity F.

“Therefore, subsequent steps in the purification can be carried out without regard to any properties of the original mixed state M , or of the noisy channel(s) that may have generated it, except for the purity F .”

• AssumptionRests on an assumption: Two mixed states having the same density matrix are physically indistinguishable.

Given n shared singlets, local actions and classical communication suffice to prepare m arbitrarily good copies of Υ with yield m/n approaching 1/E(Υ) as n→∞.

“Conversely, given n shared singlets, local actions and classical communication suffice to prepare m arbitrarily good copies of Υ with a yield m/n approaching 1 /E (Υ) as n →∞ .”

• cites [6] C.H. Bennett, H. Bernstein, S. Popescu, B. Schumacher, 'Concentrating Partial Entanglement by Local Operations', submitted to Phys. Rev. A (1995). sourceNot checked yet

  Not checked in the free run.

Before entangled pairs can be used to teleport reliably, they must be purified from the mixed entangled states that result from transmission through the noisy channel.

“But before they can be used to teleport reliably, the entangled pairs must be purified-converted to almost-perfectly entangled states from the mixed entangled states that result from transmission through the noisy channel.”

Even without prepurified pairs, using only impure Bell-diagonal states W as input, it is possible to design a sequence of BXORs and local rotations that eliminate approximately half the candidates for x or y at each step, achieving the same asymptotic yield 1 - S(W) as the breeding method.

“Even without the prepurified pairs, using only impure Bell-diagonal states W as input, it is possible [11] to design a sequence of BXORs and local rotations that eliminate approximately half the candidates for x or y at each step, achieving the same asymptotic yield 1 -S ( W ) as the breeding method.”

• cites [11] C.H. Bennett, D. DiVincenzo, J.A. Smolin, and W.K. Wootters 'Mixed State Entanglement and Quantum Error Correcting Codes', preprint (1995).Not checked yet

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The overall result is to simulate a noiseless quantum channel by a noisy one, supplemented by local actions and classical communication.

“The overall result is to simulate a noiseless quantum channel by a noisy one, supplemented by local actions and classical communication.”

An arbitrary unknown qubit can be faithfully transmitted via a pair of maximally entangled qubits previously shared between sender and receiver and a 2-bit classical message.

“Using teleportation, an arbitrary unknown qubit can be faithfully transmitted via a pair of maximally-entangled qubits ( e.g. two spin1 2 particles in a pure singlet state) previously shared between sender and receiver, and a 2-bit classical message from the sender to the receiver.”

• cites [1] C.H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70 , 1895 (1993). sourceNot checked yet

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Because expected entanglement cannot be increased by local actions and classical communication, a mixed state's distillable entanglement D(M) cannot exceed its entanglement of formation E(M).

“Because expected entanglement cannot be increased by local actions and classical communication [11], a mixed state's distillable entanglement D ( M ) cannot exceed its entanglement of formation E ( M ).”

• cites [11] C.H. Bennett, D. DiVincenzo, J.A. Smolin, and W.K. Wootters 'Mixed State Entanglement and Quantum Error Correcting Codes', preprint (1995).Not checked yet

  Not checked in the free run.