Concentrating Partial Entanglement by Local Operations

Benjamin Schumacher; Charles H. Bennett; Herbert J. Bernstein; Sandu Popescu

REVIEW

Concentrating Partial Entanglement by Local Operations

Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, Benjamin Schumacher

Quantum Physics · Submitted 21 Nov 1995 · 12 references

Pith's verdict: The founding entanglement-concentration paper: the central claim is correct and has held for thirty years, with only minor exposition gaps in the standardization lemma and a couple of stray factors.

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arXiv IDquant-ph/9511030 (version 1)

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CitationBenjamin Schumacher, Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu. "Concentrating Partial Entanglement by Local Operations." arXiv:quant-ph/9511030 (1995). https://pith.science/paper/JEDVRFFA

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This is the paper that made entanglement a fungible resource. Worth your time, and worth citing. The core result: n copies of a bipartite pure state with Schmidt coefficients c_i can be concentrated by LOCC into nE − O(log n) singlets, where E is the von Neumann entropy of either reduced state. The Schmidt-projection argument is clean — measure the power k of sinθ, collapse onto a maximally entangled state in a C(n,k)-dimensional subspace, then standardize. The entropy accounting in eq. 8 is right: the binomial outcome entropy grows only logarithmically, so the yield is asymptotically tight. The converse dilution construction, teleporting a locally prepared purifying system, is correct and neat. The no-signaling argument that local operations cannot raise expected entanglement is the right way to say it, and the observation that parties can gamble entanglement rather than only spend it is a genuinely useful distinction. Two things here outlived the paper: the definitions of entanglement of formation and distillable entanglement, and the explicit statement that mixed-state reversibility is open. That framing set the agenda for years. Soft spots, all minor. The standardization lemma in Sec. III is stated for 'any distribution of step sizes' and that is false — steps of 1/2 on the circle never leave {0, 1/2}. The binomial step distributions actually used do work (for n ≥ 3 some C(n,k) is not a power of 2, so the walk is recurrent on the circle), but the paper gives one line and no proof. The stress-test note is right about this, and it is a proof gap in the exposition, not in the result. The stray factor of 2 in the subspace dimensions (2·2^ℓ and 2·(D_m − 2^ℓ)) is unexplained and looks like a slip. The abstract's claim that any mixed state can be built from singlets is only constructed for pure states in Sec. V; the mixed case follows by convex decomposition but is not spelled out. And the Werner numbers (0.1176 ebits, 0.0001258 yield) come from ref. [11], listed as submitted — peripheral to the argument, but worth flagging. None of this touches the main claim, which is correct. Send it out; the report will be short.

Referee report

Significance

This is a foundational paper: it introduces the ebit as a unit, defines entanglement of formation and distillable entanglement, establishes asymptotic reversibility of pure-state entanglement under local operations and classical communication, and separates entanglement concentration from quantum data compression. The central derivation is parameter-free and the asymptotic yield nE − O(log n) is a sharp, falsifiable prediction; the Schmidt-projection and Procrustean constructions are explicit and reproducible. The paper is also candid about what it does not settle, explicitly flagging that no simple mixed-state entanglement measure is known and that reversibility for mixed states (whether distillable entanglement equals entanglement of formation for the Werner state) is open. Those are strengths, not weaknesses.

Major objections (1)

[§III, Eq. (9)] The standardization argument rests on the sentence 'It is elementary to show that for any distribution of step sizes, and any positive ε, such a walk visits the interval [0, log₂(1+ε)] with probability 1.' This is false as stated: a walk with steps of 1/2 on the circle never leaves {0, 1/2}. The lemma is needed only for the specific step distributions mod(log₂ C(n,k), 1). For n = 2 every step is 0 mod 1, so D_m is always an exact power of two and no standardization loss occurs; for n ≥ 3 at least one binomial coefficient is not a power of two (C(3,1) = 3), so log₂ 3 is irrational and the walk is Harris recurrent on the circle, hitting every open set almost surely. The conclusion survives, but the stated justification does not, and the claim that the standardization stage approaches unit efficiency depends on it. I recommend replacing the one-line assertion with the restricted lemma and a short justification, and noting the n = 2 case separately.

Minor objections (9)

[§II, after Eq. (7)] The projection is described as splitting the joint system into 'a large space of dimension 2·2^ℓ and a relatively small space of dimension 2·(D_m − 2^ℓ) < ε·2·2^ℓ.' The factor of 2 in both dimensions is unexplained and appears to be a slip: with D_m ∈ [2^ℓ, 2^ℓ(1+ε)] the natural split is a 2^ℓ-dimensional subspace with failure probability (D_m − 2^ℓ)/D_m < ε. Please clarify or correct.
[§I, Eq. (1) versus §II, Eq. (5)] Eq. (1) writes the partly entangled state with a relative minus sign (cos θ|↑↓⟩ − sin θ|↓↑⟩) while Eq. (5) and the figure caption use a plus sign. Phases can be absorbed, so both are legitimate, but the inconsistency should be reconciled or the convention stated once.
[§III, Eq. (8)] The entropy accounting would read more cleanly as an exact identity rather than an asymptotic statement: by the chain rule for the n-fold state, nE = H(k) + Σ_k p_k log₂ C(n,k), where H(k) is the Shannon entropy of the binomial distribution. Stating this identity explicitly makes it immediate that the only entanglement not captured is H(k), which grows as ½ log₂(2πe n p q).
[§V, dilution paragraph] The statement that Schumacher coding reduces singlet consumption 'toward the theoretical minimum' is loose for mixed states: compressing the purification C costs S(ρ_AB) singlets per copy, which is an upper bound on the entanglement of formation, not the minimum itself. Please reword to make clear that this is an achievable upper bound.
[§V, Werner state paragraph] The numerical values 0.1176 ebits and 0.0001258 singlets are quoted from Ref. [11], which is listed as 'submitted to Phys. Rev. Lett. (1995)' and was therefore not available to the reader for verification. Since these numbers are not derived in the manuscript, please flag them as preliminary or supply enough of the construction to make them checkable.
[§IV, two-sided compression paragraph] The claim that the fidelity of the two-sided-compressed state to any maximally entangled state approaches zero with n is supported by the observation that the variance of the Schmidt coefficients increases with n. That is a heuristic; a short quantitative statement (for example, a bound on the fidelity in terms of the coefficient variance) would make the negative result about quantum data compression as a concentration method more convincing.
[§IV, one-sided Schmidt projection paragraph] The phrase 'the entropy of the distribution of the eigenvalues increases absolutely with n' is awkward; presumably 'grows without bound with n' is intended. Please reword.
[§V and §IV, typographical] There are several apparent typographical errors, some of which may be artifacts of the source text: 'engtanglement' in §V, 'conentration' in §IV, and 'Without losss of generality' in §I. A careful copy-edit pass is warranted.
[Figure 1] The figure caption describes the Procrustean curve as 'Λ-shaped' while the text calls it 'inverted-V shaped'; the axis labels in the supplied figure are garbled. Please supply a clean figure with legible axis labels and a single consistent description of the curve shapes.

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The paper

The paper claims

Entanglement can be concentrated: many partly entangled pairs yield nearly as many perfect singlets as their entropy allows.

The reading

The paper asks whether partly entangled pure states are a usable currency for quantum tasks, or whether only perfect singlets count. It shows that two separated observers who share $n$ identical copies of a pure bipartite state can, by local measurements and classical messages, concentrate the entanglement into a number of perfect singlets approaching $nE-O(��log n)$, where $E$ is the entropy of entanglement of one copy. The proposed Schmidt projection measures how many of the $n$ pairs occupy the excited Schmidt branch, leaving a maximally entangled state in a known subspace; a second batching step converts that state into standard singlets. The converse is also argued: singlets plus classical communication can prepare any desired bipartite pure or mixed state, so for pure states entropy of entanglement is the asymptotic exchange rate. The result matters because it turns entanglement into a quantitative resource that can be concentrated, diluted, and spent in teleportation or dense coding.

What carries the argument

Schmidt projection is the central mechanism: Alice measures the total number $k$ of excited Schmidt branches across all $n$ pairs, not which pairs, projecting onto the subspace with common coefficient $��cos^{n-k}��theta��sin^k��theta$; the residual state is maximally entangled in dimension $��binom{n}{k}$. Because the distribution of $k$ has entropy $O(��log n)$, almost all of the original $nE$ entanglement survives. The standardization stage multiplies dimensions $��binom{n}{k_j}$ over batches until the product lies just above a power of two, then projects onto a $2^��ell$-dimensional subspace; local unitary rotations turn that into $��ell$ singlets. The Procrustean filter handles one pair at a time by absorbing a fraction $��tan^2��theta$ of the more probable branch.

Extended reading notes

Core claim

For any bipartite pure state $|��(A,B)��=��_i c_i|��_i��|��_i��$, the entropy of entanglement $E=-��_i c_i^2��log_2 c_i^2$ is asymptotically conserved by local operations and classical communication: $n$ copies can be converted into $nE-O(��log n)$ maximally entangled pairs, and conversely singlets can be diluted into copies of any bipartite state at the same asymptotic rate. The paper establishes this by Schmidt projection, which projects the $n$-copy state onto a subspace of fixed excitation number $k$ and leaves a maximally entangled state of dimension $��binom{n}{k}$, followed by a standardization step that groups such states until their total dimension is close to a power of two. It also gives a single-copy Procrustean method that probabilistically converts one partly entangled pair into one perfect singlet by filtering the larger Schmidt coefficient.

Load-bearing premise

The central claim depends on the unproved standardization lemma that a random walk on the unit interval with steps drawn from the binomial-coefficient distribution will, with probability one, come within a factor $1+��epsilon$ of a power of two, a lemma the paper calls elementary and states for arbitrary step distributions even though that general form fails for commensurate steps.

Original abstract

If two separated observers are supplied with entanglement, in the form of $n$ pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large $n$, the base-2 entropy of entanglement of the initial partly-entangled pure state. Conversely, 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.

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, 3,208 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.21 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.80 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

If two separated observers are supplied with entanglement, in the form of $n$ pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large $n$, the base-2 entropy of entanglement of the initial partly-entangled pure state. Conversely, 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.

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	If two separated observers are supplied with entanglement, in the form of $n$ pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers.	contribution 0.59	0.60	0.59	hedged possibility 0.55
s1	The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large $n$, the base-2 entropy of entanglement of the initial partly-entangled pure state.	result 0.99	0.80	0.40	asserted 0.99
s2	Conversely, 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.	result 0.83	0.44	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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

levels run

P abstract

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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

References

Works cited (12)

[11]

C.H. Bennett, G. Brassard, S. Popescu, J. Smolin and W.K . Wootters, “Puriﬁcation of Noisy Entanglement and Faithful Teleportation via Noisy Channels” submitted to Phys. Rev. Lett. (1995)

View on Pith 1995
[1]

B. Schumacher, “On Quantum Coding,” Phys. Rev. A (to appear 1995)

View on Pith 1995
[2]

R. Jozsa and B. Schumacher, J. Modern Optics 41, 2343-2349 (1994)

View on Pith 1994
[3]

C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992)

View on Pith 1992
[4]

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

View on Pith 1993
[5]

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1993) page 131-133

View on Pith 1993
[6]

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New- York, 1976), pp.74-83

View on Pith 1976
[7]

N. Gisin (private communication 1993) noted that POVMs c an be simply implemented in polarization space by using a polarization-dependent ab sorber

View on Pith 1993
[8]

C. H. Bennett, G. Brassard, R. Jozsa, D. Mayers, A. Peres, B. Schumacher, and W. K. Wootters, J. Mod. Optics 41, 2307-2314, (1994)

View on Pith 1994
[9]

R. Jozsa, J. Mod. Optics 41, 2315-2323 (1994)

View on Pith 1994
[10]

R. F. Werner, Phys. Rev. A 40, 4227 (1989), S. Popescu, Phys. Rev. Lett. 72, 797 (1994)

View on Pith 1989
[12]

G.Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A 51, 54-67 (1995). 19 FIGURE CAPTION

View on Pith 1995

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How later papers used it

362 later papers cite it

Evidence: substantial. 362 of the citing papers share no author with it, and 31 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 499 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

4 later papers by other authors are listed below: 1 built on it and 3 used its method or tool.

Builds on it 1 of 4 papers with this label

Generalized this paper's Tiles UPB in $ℂ^3⊗ℂ^3$ in its construction.

Family of bound entangled states on the boundary of Peres set (2018) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 3 of 16 papers with this label

Adopted the so-called Procrustean method of filtering from this paper, following the approach in another work.

Loss-tolerant quantum key distribution with detection efficiency mismatch (2024) · 1.00 · paraphrased: the licence does not allow quoting
Used entanglement and LOCC as related to this paper's approach to form approximate copies of a bosonic Gaussian resource state.

Entanglement cost and quantum channel simulation (2018) · 0.96 · paraphrased: the licence does not allow quoting
We measure the degree of entanglement between the two species by computing the von Neumann entanglement entropy [ref] using the reduced density matrices $ρ_2=Tr_1(ρ)$ or $ρ_1=Tr_2(ρ)$

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How the literature uses it

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Papers whose main result relates to this one's

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States the same result 1 paper

If two parties, Alice and Bob, share some number, n, of partially entangled pairs of qubits, then it is possible for them to concentrate these pairs into some smaller number of maximally entangled states.

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Now let us consider an interpretation of pseudo entropy from the view point of quantum entanglement by extending the well-known argument [ref] for the entanglement entropy.

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Now, the idea of forming $n$ approximate copies $ψ_A^'B^' ^ω$ is then the same as it is in [ref] : Alice prepares the state $ψ_A^'nB^'n^ω$ locally, Alice and Bob require beforehand a maximally entangled state of Schmidt rank no larger than $2^n[ H(B^')_ψ^ω+δ]$, and then they perform quantum teleportation [ref] to teleport the $B^'n$ system to Bob.

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The entanglement entropy $S_EE(r)$, determined by the Von Neumann entropy on the reduced density matrix of one of the subsystems, is defined by [ref] [equation] The required $2×2$ reduced density matrix of the state $|ψ_e(r)⟩$ can be expressed as [equation] where $λ_1(r)$ and $λ_2(r)$ are the two nonzero eigenvalues of $ρ_e(r)$.

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Cites it as background 35 papers

The central result on transformations between bipartite pure states with asymptotically vanishing error is that the optimal rate is the ratio of the (von Neumann) entropies of entanglement of the two states [ref] .

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Entanglement distillation protocols [ref] in quantum information theory allow the extraction of maximally entangled Bell states from multiple copies of weakly entangled or noisy states.

Nonlocality distillation can outperform 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

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, G. Brassard, S. Popescu, J. Smolin and W.K . Wootters, “Puriﬁcation of Noisy Entanglement and Faithful Teleportation via Noisy Channels” submitted to Phys. Rev. Lett. (1995)

View on Pith 1995

[2] [1]

B. Schumacher, “On Quantum Coding,” Phys. Rev. A (to appear 1995)

View on Pith 1995

[3] [2]

R. Jozsa and B. Schumacher, J. Modern Optics 41, 2343-2349 (1994)

View on Pith 1994

[4] [3]

C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992)

View on Pith 1992

[5] [4]

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

View on Pith 1993

[6] [5]

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1993) page 131-133

View on Pith 1993

[7] [6]

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New- York, 1976), pp.74-83

View on Pith 1976

[8] [7]

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