Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels

Benjamin Schumacher; Charles H. Bennett; Gilles Brassard; John A. Smolin; Sandu Popescu; William K. Wootters

REVIEW

Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels

Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, William K. Wootters

Quantum Physics · Submitted 20 Nov 1995 · 13 references

Pith's verdict: The founding paper of entanglement distillation: the constructive lower bound is solid, and the only real soft spot is an upper bound borrowed from an unpublished companion preprint.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 8 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDquant-ph/9511027 (version 2)

Pith IDQFZAXENH, Pith's permanent record of this paper and its review

CitationBenjamin Schumacher, Charles H. Bennett, Gilles Brassard, John A. Smolin, Sandu Popescu, William K. Wootters. "Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels." arXiv:quant-ph/9511027 (1995). https://pith.science/paper/QFZAXENH

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

Significance

This is a foundational contribution: it establishes that noisy entanglement is a usable resource, not merely a degraded one, and it does so with a fully explicit, checkable construction rather than an existence argument. The recurrence of Eq. (7) follows directly from the BXOR table of Eq. (6) and the z-basis measurement, and I verified it independently: writing the denominator as (8F^2 - 4F + 5)/9, one finds F' - F = (2F-1)(1-F)(4F-1)/9 divided by that denominator, which is strictly positive exactly on (1/2, 1) and vanishes at F = 1/2 and F = 1. The threshold F > 1/2 is therefore a sharp, falsifiable prediction, and the comparison with the CHSH threshold (2+3√2)/8 ≈ 0.7803 correctly identifies a window of Werner states that are distillable yet Bell-local. The paper is also candid about what it does not establish: it states explicitly that the optimal D(M) is unknown even for Werner states, that no M is known for which D(M) and E(M) provably differ, and that the asymptotic yield of the variable-blocksize variant is not computed. That candour is a strength and should be preserved.

Major objections (2)

[Eqs. (9)-(10) and the sentence 'In fact [11] these are the least entangled ensembles realizing W_F'] The upper-bound half of the abstract's claim rests on a step that is asserted rather than derived here. The paper exhibits the eight-state ensemble of Eq. (10), notes that each member has entropy of entanglement equal to the right side of Eq. (9a), and then states that these are the least entangled ensemble realizing W_F, citing reference [11], which is listed as an unpublished preprint with overlapping authorship. The lower bound D(M) > 0 is unaffected, since it is constructive, but the abstract's phrase 'upper and lower bounds' inherits this dependence. Either supply a short argument for the optimality of the ensemble (for instance via the invariance of W_F under bilateral rotations and the resulting structure of any ensemble realizing it), or state the dependence explicitly in the text and soften the abstract so that the upper bound is presented as following from [11].
[Text following Eq. (7), and §B1] The headline claim D(M) > 0 requires a positive yield, not merely arbitrarily high purity at vanishing yield, and the text's only explicit route to a positive yield is the variable-blocksize sketch with k(F) ≈ 1/sqrt(1-F), which is given only 'to lowest order in 1-F' and whose asymptotic yield the authors decline to compute. A simpler and fully rigorous route is available and should be stated: for any fixed Fin > 1/2, iterate Eq. (7) a finite number of times until F exceeds 0.8107, at each step succeeding with probability at least 5/9 and consuming two pairs to produce one, and then apply the breeding protocol, whose yield 1 - S(W) is positive in that range. This makes D(M) > 0 follow from the recurrence plus breeding alone, with the variable-blocksize construction demoted to a yield-improvement remark. One sentence making this explicit would close the gap between the stated claim and the argument as presented.

Minor objections (8)

[Abstract] The abstract contains a duplicated and garbled clause: 'thereby achieving faithful transfrom from one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel.' The phrase should appear once, with 'transfrom' corrected to 'transmission' or removed. The macro 'half' in the final sentence should also be checked for definition in the abstract's own scope.
[§B2] The sentence 'Once found, the Ψ± are converted respectively to Φ± by applying a unilateral σy rotation to each of them' is inconsistent with the operation list given earlier, which states that σy maps Ψ± ↔ Φ∓. The correct mapping is Ψ+ → Φ- and Ψ- → Φ+, which is also what makes the subsequent step B3 (bilateral By, then σx) coherent. The phrase should read 'converted respectively to Φ∓'.
[Eq. (7)] The text states only that the protocol succeeds 'with probability greater than 1/4'. The exact success probability is the denominator of Eq. (7), (8F^2 - 4F + 5)/9, whose minimum over 1/2 < F < 1 is 5/9 at F = 1/2. Stating the exact expression, and noting that it is bounded below by 5/9 on the relevant range, would make the yield accounting of the iteration immediate and would strengthen rather than weaken the result.
[Symmetrization step preceding Eq. (4)] The claim that a random bilateral SU(2) rotation 'transforms the initial general two-spin mixed state M into a rotationally symmetric mixture W_F' with the same F is correct but asserted without derivation. A one-line justification, noting that the twirl projects onto the rotationally invariant subspace and that the singlet component is invariant under bilateral rotations, would make the reduction to Werner form self-contained.
[§B1 and the breeding yield, Eq. (8)] The yield accounting should be made explicit: the n(S(W)+δ) prepurified Φ+ target pairs are consumed by the BXOR tests, so the net gain is n(1 - S(W) - δ) rather than n. The text's 'yield 1 - S(W)' is correct in the limit, but the bookkeeping is left to the reader.
[Reference [13]] The Clauser-Horne-Shimony-Holt paper is Phys. Rev. Lett. 23, 880 (1969), not 1980; volume 23 corresponds to 1969. The year should be corrected.
[Fig. 1] The figure is difficult to read as printed: the x-axis label 'Purity-0.5' does not make clear that the axis is F - 1/2 on a logarithmic scale, and the curve labels D0, DR, DM, E overlap near the left edge. The caption should state the axis conventions explicitly and identify which curve corresponds to which protocol, including that E is the upper bound of Eq. (9).
[Notation, Eqs. (3) and (9)] The symbol E is used both for the entanglement of a pure state, E(Υ) in Eq. (3), and for the entanglement of formation of a mixed state, E(W_F) in Eq. (9). A notational distinction, or an explicit sentence identifying the two uses, would prevent confusion when the paper later argues that D(M) cannot exceed E(M).

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Two distant observers who share imperfectly entangled pairs can, by local operations and ordinary classical messages, distil a smaller number of nearly perfect singlets from them, and then use those singlets to teleport unknown quantum stat

The reading

A noisy channel does not deliver perfect singlets; it delivers pairs that are only partly entangled, and the question this paper takes up is whether such damaged pairs are still worth anything. The answer it argues for is yes, provided the pairs are better than half pure: Alice and Bob can sacrifice some pairs to purify the rest, repeating the step until the survivors are arbitrarily close to perfect singlets, and then spend those singlets to teleport unknown quantum states faithfully from one side to the other. The consequence is that a noisy quantum channel, supplemented only by local laboratory operations and classical chatter, can be made to behave like a noiseless one. The paper also puts a ceiling on how much purity can be extracted, by identifying a quantity — the entanglement of formation — that no local protocol can increase.

What carries the argument

The BXOR test is the engine. Alice and Bob each apply a controlled-NOT from one shared pair into another, then measure the target pair along the $z$ axis; the target survives the measurement only if its two spins come out parallel. Because the four Bell states transform under this operation in a small, tabulated way, the test works as a parity check that reveals whether the tested subset contains an even or odd number of $|M|$-type pairs, without disturbing the sources. Running enough tests on random subsets identifies every bad pair with high probability, at a cost asymptotically equal to the state's von Neumann entropy $S(W)$ — the same counting that makes classical parity-check codes work. The companion device is the random bilateral rotation, which exploits the singlet's invariance to symmetrize an arbitrary mixed pair into Werner form without changing its fidelity, so that a single number $F$ becomes a complete description of the resource.

Extended reading notes

Core claim

The central claim is that entanglement, unlike classical information, can be concentrated out of noise. Formally, if a shared two-spin state $M$ has fidelity $F = |M| > 1/2$ relative to the singlet $|M|$, then the distillable yield $D(M)$ is strictly positive. The argument is constructive. A random bilateral rotation turns any $M$ into a Werner state $W_F$ — a mixture of the singlet with the three triplets in equal proportion — carrying the same $F$, and two Werner pairs of fidelity $F > 1/2$ can be converted, with probability greater than $1/4$, into one Werner pair of fidelity $F' = (F^2 + (1-F)^2/9)/(F^2 + 2F(1-F)/3 + 5(1-F)^2/9)$, which exceeds $F$ on the whole interval. Iterating drives $F$ toward $1$. A second, more efficient method uses already-pure pairs as targets for parity checks on impure ones, locating and correcting errors at a cost of $S(W)$ tests per pair and leaving a yield of $1 - S(W)$, which is positive for Werner states above $F = 0.8107$. Against this, the paper sets an upper bound: $W_F$ can be written as an equal mixture of eight pure states each with entanglement $H_2(1/2 + |M|)$, and since local operations and classical communication cannot increase expected entanglement, no protocol can distill more than that.

Load-bearing premise

The argument assumes that fidelity relative to the singlet is a sufficient description of a shared pair once it has been symmetrized, so that a single threshold at one half governs every mixed state and not merely the rotationally symmetric ones.

Editorial extensions

If this is right

Any noisy quantum channel whose output pairs exceed fidelity $1/2$ can be upgraded into a faithful channel for quantum states, at some nonzero rate, using only local operations and classical communication.
Werner states with $F > 1/2$ must be regarded as nonlocal and useful, even though they violate the Clauser-Horne-Shimony-Holt inequality only above $F \approx 0.78$; Bell violation is therefore not the boundary of exploitable entanglement.
The von Neumann entropy of a shared mixed state becomes the currency of purification: the breeding method yields $1 - S(W)$ pure singlets per impure pair, a positive rate whenever $S(W) < 1$.
Because the entanglement of formation cannot be increased by local operations and classical communication, it caps every conceivable distillation protocol, not just the ones constructed here.
A one-way-communication variant of the error-locating protocol extends the same reasoning from transmission to storage, protecting quantum information against errors while it sits in memory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The threshold at one half reframes the classical-quantum divide as a rate rather than a property: every fidelity above $1/2$ buys some strictly positive asymptotic yield, so the interesting object becomes the shape of $D(F)$ on the interval, which the paper explicitly leaves undetermined.
Since the BXOR test is a parity check and its cost is an entropy, the purification protocol is structurally the same object as an error-correcting code or a privacy-amplification scheme; one expects a general correspondence in which distillation rate is set by how much syndrome information a state carries.
The visible gap in the paper's own figure between the best achieved yield and the entanglement-of-formation bound is a concrete numerical target: closing it, or proving for some state that distillable entanglement falls strictly below entanglement of formation, would settle whether the two measures ever separate.
The recurrence step needs two-way classical communication while the non-breeding variant needs only one-way, which suggests that the direction and amount of classical communication should be counted as a resource in its own right alongside the entanglement consumed.

Original abstract

Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$, showing $D(M)>0$ if $\bra{\Psi^-}M\ket{\Psi^-}>\half$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,158 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.35 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$, showing $D(M)>0$ if $\bra{\Psi^-}M\ket{\Psi^-}>\half$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets).	contribution 0.52	0.52	0.47	asserted 0.70
s1	These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel.	interpretation 0.75	0.37	0.45	asserted 0.94
s2	We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$, showing $D(M)>0$ if $\bra{\Psi^-}M\ket{\Psi^-}>\half$.	result 0.92	0.93	0.41	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.76 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Log-log plot of entanglement distillable from

Figures reproduced from arXiv:quant-ph/9511027 by the authors.

References

Works cited (13)

[11]

C.H. Bennett, D. DiVincenzo, J.A. Smolin, and W.K. Wootters “Mixed State Entanglement and Quan- tum Error Correcting Codes”, preprint (1995)

View on Pith 1995
[1]

C.H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993)

View on Pith 1993
[2]

B. Schumacher, Phys. Rev. A 51, 2738 (1995)

View on Pith 1995
[3]

R. Jozsa and B. Schumacher, J. Modern Optics 41, 2343 (1994)

View on Pith 1994
[4]

W.K. Wootters and W.H.Zurek, Nature 299, 802 (1982)

View on Pith 1982
[5]

P. Shor Phys. Rev. A 52, R2493 (1995)

View on Pith 1995
[6]

C.H. Bennett, H. Bernstein, S. Popescu, B. Schumacher, “Concentrating Partial Entanglement by Local Opera- tions”, submitted to Phys. Rev. A (1995)

View on Pith 1995
[7]

R.F. Werner, Phys. Rev. A 40, 4277 (1989)

View on Pith 1989
[8]

D. Deutsch, Proc. Roy. Soc. A 425, 73 (1989)

View on Pith 1989
[9]

C. Macchiavello (private communication 1995) found an improved recurrence using Bx, σy in place of our step A3

View on Pith 1995
[10]

C.H. Bennett, G. Brassard, C. Cr´ epeau, and U.M. Mau- rer, “Generalized Privacy Ampliﬁcation”, to appear Nov. 1995 in IEEE Trans. on Information Theory

View on Pith 1995
[12]

S. Popescu, Phys. Rev. Lett. 72, 797 (1994)

View on Pith 1994
[13]

J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1980). FIG. 1. Log-log plot of entanglement distillable from Werner states of purity F by various methods, versus F−12 . D0 = breeding method alone (eq. 8); DR = breeding pre- ceded by recurrence method of eq. 7; DM = breeding pre- ceded by recurrence of [9]; E = entanglement of fo...

View on Pith 1980

Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The second laws of quantum thermodynamics
quant-ph 2013-05 accept novelty 9.0 of 10

For microscopic systems, a cyclic thermodynamic transition is possible exactly when a continuous family of Renyi free energies all decrease, not just the ordinary free energy.
Sharp Plucker Geometry for Three-Copy Werner Distillation
quant-ph 2026-07 conditional novelty 7.0 of 10

For three-copy Werner states at the critical noise level, every positive-semidefinite or normal rank-two test operator is shown to satisfy q3(C) >= 0, with the remaining nonnormal case reduced to a crossed-Gram criterion.
A Traversable Wormhole Teleportation Protocol in the SYK Model
hep-th 2019-11 conditional novelty 7.0 of 10

A SWAP-based protocol teleports a qubit through a traversable wormhole in the SYK model, with almost perfect fidelity achievable using a multi-fermion encoding in the large q limit.
Optimal fidelity estimation when one state is pure via algorithmic Uhlmann transform
quant-ph 2026-08 accept novelty 6.0 of 10

When at least one of two quantum states is pure, the Uhlmann fidelity can be estimated with Θ(1/ε) queries and Θ(1/ε²) samples without knowing which state is pure, matching the optimal lower bounds.
Quantum Information Flow under String-Diagram Rewriting
quant-ph 2026-08 conditional novelty 6.0 of 10

The authors define Coecke flow lines as branch-independent paths through quantum protocol diagrams that survive every semantics-preserving rewrite down to a bare wire.

Show 7 more citing papers

Additivity of disjoint interval entanglement in quasiparticle excited states
quant-ph 2026-01 conditional novelty 6.0 of 10

For quasiparticle excited states with large momentum differences, double-interval reflected entropy, mutual information, and logarithmic negativity add: X_{K1∪K2} = X_{K1} + X_{K2}.
Certification of high-dimensional entanglement in continuous-variable systems
quant-ph 2026-10 accept novelty 5.0 of 10

A new fidelity-based Schmidt-number witness for two-mode continuous-variable states is evaluated from the covariance matrix alone and certifies entanglement dimension for Gaussian and non-Gaussian states.
Quantum Utility in Simulating the Real-time Dynamics of the Fermi-Hubbard Model using Superconducting Quantum Computers
quant-ph 2025-09 conditional novelty 5.0 of 10

A 104-qubit IBM quantum computer simulates the 1D Fermi-Hubbard model's staggered-magnetization dynamics with constant-depth Trotter circuits, matching MPS-TDVP up to time 4 but not at later times.
The thread embodiment of holographic quantum entanglement
hep-th 2025-01 conditional novelty 5.0 of 10

Holographic entanglement is represented by geodesic threads whose fluxes equal half conditional mutual information, and kinematic space is treated as the input board of a quantum circuit that reproduces holographic co...
Comments on Entanglement Negativity in Holographic Field Theories
hep-th 2014-06 accept novelty 4.0 of 10

The logarithmic negativity of a thermofield double state equals the free-energy difference at temperatures T and 2T, which yields the negativity of CFT vacua on spherical regions in various dimensions.
Quantum entanglement
quant-ph 2007-02 accept novelty 2.0 of 10

Entanglement is a manipulable, quantifiable resource whose structure (separable, PPT, bound, distillable) is organized by the LOCC paradigm and detected in practice by entanglement witnesses.
Entanglement Certification $-$ From Theory to Experiment
quant-ph 2019-06 unverdicted

Reviews paradigmatic entanglement quantifiers and state-of-the-art detection/certification methods, with emphasis on assumptions about states and measurements.

How later papers used it

384 later papers cite it

Evidence: substantial. 382 of the citing papers share no author with it, and 49 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 556 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

11 later papers by other authors are listed below: 10 used its method or tool and 1 compared its own results with it.

Uses its method or tool 10 of 34 papers with this label

Used the BBPSSW scheme from this paper as the base for entanglement distillation.

Efficient computation of the waiting time and fidelity in quantum repeater chains (2019) · 1.00 · paraphrased: the licence does not allow quoting
Had Alice and Bob undertake the breeding protocol from this paper to guess bit-flip errors in the (2i-1)th pair.

A Study of Channel Estimation and Postprocessing in Quantum Key Distribution Protocols (2009) · 0.99 · paraphrased: the licence does not allow quoting
A plethora of purification protocols exists, but in this work we will focus on one of the most simple non-trivial one, named the IBM (BBPSSW) [ref] recursive protocol, whose recursive expression reads: [equation] This recursive protocol increases the fidelity of an entanglement if and only if the input fidelity is above the value $0.5$ and stops purifying when fidelity reaches a value of $1.0$.

Purification Strategy Optimization for Entanglement Routing in Quantum Networks (2026) · Entanglement and Fidelity · 0.98
Demonstrated the use of an entanglement purification protocol from this paper to correct errors from amplitude damping of microwave photons.

Entanglement purification and protection in a superconducting quantum network (2022) · 0.98 · paraphrased: the licence does not allow quoting
Considered the distillation scheme introduced in this paper, recalled in an appendix.

Entanglement distribution in two-dimensional square grid network (2023) · 0.98 · paraphrased: the licence does not allow quoting
In this work, the entanglement distillation protocol we consider is the BBPSSW protocol introduced in Ref. [ref] .

Reducing classical communication costs in multiplexed quantum repeaters using hardware-aware quasi-local policies (2024) · Entanglement distillation with the BBPSSW protocol · 0.98
Said that the error leaving the qubits in the state |11><11| is efficiently detected by the purification scheme described in this paper.

Long-distance entanglement distribution using individual atoms in optical cavities (2015) · 0.96 · paraphrased: the licence does not allow quoting
With the purification on noisy measurements, Alice and Bob apply the entanglement distillation protocol in Ref. [ref] .

Protocol for Purifying Noisy Preparation and Measurements of Qubits (2025) · Distilling Entanglement with Noisy SPAM · 0.94
Used the BBPSSW protocol from this paper, which it adapted by bringing the output state back into Werner form.

Efficient optimization of cut-offs in quantum repeater chains (2020) · 0.94 · paraphrased: the licence does not allow quoting
Additionally, Pauli twirling [ref] of the entangling CZ gates is implemented with 500 randomizations, and 100 measurements per randomization.

Observation of hadron scattering in a lattice gauge theory on a quantum computer (2025) · Methods · 0.93

Compares with it 1 of 14 papers with this label

Showed the performance of the asymptotic protocols of this paper and its own protocol on the same states when preceded by an optimal number of iterations of this recurrence protocol.

Heuristic Lookahead Distillation Protocol Search (2026) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

44 arXiv papers cite this in 59 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 22 sentences without a confident label

Uses its method or tool 1 paper

* Then, Alice and Bob undertake the breeding protocol [ref] to guess bit-flip errors in the $(2i-1)$th pair for all $i$.

A Study of Channel Estimation and Postprocessing in Quantum Key Distribution Protocols · Relation to Entanglement Distillation Protocol · 0.99

Cites it as background 28 papers

There is currently no simple formula for computing the two-way distillable entanglement of a bipartite state $ψ^AB$, which has been studied extensively by Bennett et al. and others in [ref] .

Assisted Entanglement Distillation · Introduction · 0.98
In this chapter, we focus on two different information-processing tasks relevant for bipartite quantum interactions, the first being entanglement distillation [ref] and the second secret key agreement [ref] .

Bipartite Quantum Interactions: Entangling and Information Processing Abilities · Entropy of Entanglement · 0.93
The pioneering works of quantum information theory sought to quantify entanglement [ref] , and provide conditions for transformations between entangled states using only Local Operations and Classical Communication (LOCC) in the limit of sharing many copies of the same state.

Entanglement fluctuation theorems · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [11]

C.H. Bennett, D. DiVincenzo, J.A. Smolin, and W.K. Wootters “Mixed State Entanglement and Quan- tum Error Correcting Codes”, preprint (1995)

View on Pith 1995

[2] [1]

C.H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993)

View on Pith 1993

[3] [2]

B. Schumacher, Phys. Rev. A 51, 2738 (1995)

View on Pith 1995

[4] [3]

R. Jozsa and B. Schumacher, J. Modern Optics 41, 2343 (1994)

View on Pith 1994

[5] [4]

W.K. Wootters and W.H.Zurek, Nature 299, 802 (1982)

View on Pith 1982

[6] [5]

P. Shor Phys. Rev. A 52, R2493 (1995)

View on Pith 1995

[7] [6]

C.H. Bennett, H. Bernstein, S. Popescu, B. Schumacher, “Concentrating Partial Entanglement by Local Opera- tions”, submitted to Phys. Rev. A (1995)

View on Pith 1995

[8] [7]

R.F. Werner, Phys. Rev. A 40, 4277 (1989)

View on Pith 1989

[9] [8]

D. Deutsch, Proc. Roy. Soc. A 425, 73 (1989)

View on Pith 1989

[10] [9]

C. Macchiavello (private communication 1995) found an improved recurrence using Bx, σy in place of our step A3

View on Pith 1995

[11] [10]

C.H. Bennett, G. Brassard, C. Cr´ epeau, and U.M. Mau- rer, “Generalized Privacy Ampliﬁcation”, to appear Nov. 1995 in IEEE Trans. on Information Theory

View on Pith 1995

[12] [12]

S. Popescu, Phys. Rev. Lett. 72, 797 (1994)

View on Pith 1994

[13] [13]

J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880 (1980). FIG. 1. Log-log plot of entanglement distillable from Werner states of purity F by various methods, versus F−12 . D0 = breeding method alone (eq. 8); DR = breeding pre- ceded by recurrence method of eq. 7; DM = breeding pre- ceded by recurrence of [9]; E = entanglement of fo...

View on Pith 1980