Asymptotic adaptive bipartite entanglement distillation protocol

Bart De Moor; Erik Hostens; Jeroen Dehaene

REVIEW

Asymptotic adaptive bipartite entanglement distillation protocol

Erik Hostens, Jeroen Dehaene, Bart De Moor

Quantum Physics · Submitted 24 Feb 2006 · 23 references

Pith's verdict: A genuinely new entropy-reduction principle and a concrete protocol, but the quoted yields depend on an under-proved claim about optimality of the final breeding stage.

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CitationBart De Moor, Erik Hostens, Jeroen Dehaene. "Asymptotic adaptive bipartite entanglement distillation protocol." arXiv:quant-ph/0602205 (2006). https://pith.science/paper/X4C4E7TE

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Editor's note

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Dear X,

The short version: this paper has a genuinely new idea—replacing ebit measurements by bilateral Pauli measurements on a few copies, when the outcomes are equiprobable, cuts the entropy of the global state by more than one bit—and it builds a concrete adaptive protocol around that idea that plausibly beats breeding for Werner states. The qualitative claim is believable. The quantitative yields, though, rest on a final-stage optimality assertion that is under-justified.

What’s actually new: the entropy-reduction principle in Sec. IV is correct and clean; the update/decoupling rules (3)–(4) are consistent; the recursive way of forcing equiprobable outcomes in Sec. V.B is a clever construction; and the favored-ordering variant in Sec. V.D is a reasonable refinement. The paper also explains why the earlier Vollbrecht–Verstraete improvement worked, which is a nice contribution. The numerical results for Werner states are concrete and show convergence with q.

The soft spots are in the yield analysis. Eq. (12) subtracts the entropy (9) as the cost of completing the distillation, on the grounds that breeding is optimal for the remaining correlated bracket states [n0,n1]. The only argument given is a short symmetry remark: the paper says all further parity checks must be built from 01 and 10, but from the commutation condition with the already-known 11 parities one would actually think the allowed checks have 00 and 11 blocks. So the statement doesn’t obviously follow, and the paper doesn’t rule out a BPM that merges pairs inside a bracket and yields extra entropy reduction at no extra ebit cost. Rains’ optimality result, cited for this step, covers independent rank-two pairs, not the correlated multi-pair states here. The bracket-state entropy formula S([n0,n1]) in Eq. (10) is also presented without derivation. These are addressable gaps, but they are load-bearing: if the final-stage cost is not tight, the quoted yields are off.

One smaller point: the abstract claims to outperform all existing asymptotic schemes, but the plots only compare to breeding; there is no direct numerical comparison to Vollbrecht–Verstraete.

None of this undercuts the main idea; the protocol is new and the principle is valuable. But the paper needs a tighter proof (or a more explicit conjecture) about the final stage before the yields can be taken at face value. I’d send it to the referee: the work is significant enough and the gaps are fixable.

Best,

Referee report

Significance

If the construction and yield formula are correct, the paper gives a concrete, analytically motivated improvement over previous asymptotic distillation schemes and tightens the lower bound on the distillable entanglement of Bell-diagonal states. The key identity in Eq. (2) is correct, and the paper's binary/Clifford formalism makes the protocol transparent and reproducible in principle. The decoupling rules and the recursive treatment of equiprobable parity checks are elegant and constitute a useful conceptual step beyond the ad hoc constructions of Ref. [13]. However, the final-stage treatment of correlated bracket states is asserted rather than proved, and the numerical comparisons do not include the closest prior adaptive protocol, so the headline claim of outperforming all existing schemes is not yet fully supported.

Major objections (3)

[Sec. V.C, Eqs. (9)–(12)] The claim that 'breeding is optimal for all pairs' is not established for the correlated bracket states [n0,n1]. Rains' optimality result [18] applies to independent rank-two Bell-diagonal states; the bracket states arising after BPMs are correlated multi-pair ensembles. The symmetry argument that remaining parity checks are built from 01 and 10 shows only that a particular class of BPMs yields no extra entropy reduction; it does not prove that the Shannon entropy (9) is the minimal ebit cost for finishing distillation on these correlated states, nor does it prove that standard breeding achieves exactly this cost. Since (9) is subtracted in the yield formula (12), the numerical yields in Figs. 2–5 rest on this assertion. Please provide a proof or reference covering the correlated case, or explicitly state that (12) is an achievable lower bound rather than the optimal yield, and adjust the paper's claims accordingly.
[Sec. V.C, Eq. (6); Sec. V.D, Eq. (13)] The frequency update rules in Eq. (6) are stated for all 0(2m) blocks, but after previous BPMs a 0(2m) block can appear inside a bracket state [n0,n1] whose internal distribution is correlated. The transition probabilities in Eq. (6), such as the probability p0(m)^2/(p0(m)^2+p1(m)^2) of splitting into 0(m)0(m), are derived for an independent block conditioned only on its parity; it is not shown that the same probabilities hold when the block is part of a bracket. The conditional fractions eta(0(2m)|A) in Eq. (13) account for the frequency of BPMs but not for possible changes in the split probabilities. Without a proof of this point, the frequencies f(A) entering Eqs. (8)–(12) are not rigorously justified.
[Abstract and Sec. VI] The abstract and introduction claim that the protocol 'outperforms all existing asymptotic schemes,' but Sec. VI compares numerically only with breeding with and without recurrence. No quantitative comparison with the Vollbrecht-Verstraete protocol [13] is presented, even though [13] is the closest prior adaptive asymptotic protocol and is explicitly discussed in the introduction. To substantiate the headline claim, the authors should include a direct numerical comparison with [13], or qualify the claim accordingly.

Minor objections (6)

[Sec. III] The text contains a typo: 'Basicly' should be 'Basically'.
[Sec. V.C] The notation [n0,n1] is used before it is defined; define it at first use, and clarify that it denotes a bracket state up to permutation of the pairs inside the brackets.
[Sec. VI] The truncation of the recursion in Eq. (5)/(13) after 10 steps is described, but no error estimate is given; please state that the terms decay exponentially and bound the truncation error, or provide an alternative justification that the plotted yields are converged.
[References] Reference [3] appears to duplicate reference [1]; if a different source was intended, please correct it.
[Sec. V.D, Eq. (13)] The definitions of L(A) and U(A) involve floor and ceiling of negative logarithms; the limiting cases L(A)=0 and U(A)=1 should be discussed explicitly so that Eq. (13) is well defined in all regimes encountered in the numerical calculations.
[Sec. VI] The relative-difference plot in Fig. 5 should state the baseline explicitly in the caption; currently the caption says 'the relative difference of the yields' without specifying that the denominator is the yield of breeding preceded by the optimal number of recurrence iterations.

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The paper

The paper claims

A new adaptive protocol for asymptotic bipartite entanglement distillation outperforms all existing asymptotic schemes by replacing ebit measurements with bilateral Pauli measurements on finite copies whenever the two outcomes are…

The reading

Entanglement distillation turns many noisy shared pairs into pure Bell pairs, and the asymptotic yield counts how many pure pairs are obtained per input pair. This paper constructs an adaptive protocol that beats every previously known asymptotic scheme by replacing each ebit-consuming parity measurement with a bilateral Pauli measurement on a finite number of copies whenever the two outcomes are equally likely. Because such a measurement identifies two distinct bitstrings with one another, the global entropy drops by more than one bit per measurement rather than exactly one bit. The yield is worked out numerically for Werner states, shifting the distillable-fidelity threshold from about 0.8107 to about 0.7424 and improving further when recurrence steps are prepended. The paper thus tightens the lower bound on the entanglement of distillation for Bell-diagonal states.

What carries the argument

The load-bearing object is the bilateral Pauli measurement (BPM): a parity check $r^T s$ performed directly on the copies by local Clifford operations, a bilateral $\sigma_z$ measurement on one pair, and inverse Clifford operations. A BPM not only reads one parity bit but also identifies the two bitstrings $g$ and $g + P a$, so the entropy reduction is one bit plus the positive collision term $(x+y)H(x/(x+y), y/(x+y))$, where $H$ is the binary entropy function. The protocol's second mechanism is a recursive procedure that forces the two outcomes of a BPM to be equiprobable, together with decoupling rules that express later correlated states as bracket combinations $[n_0, n_1]$.

Extended reading notes

Core claim

The central discovery is that a bilateral Pauli measurement does more than extract the parity bit it is designed to read: it also merges the two bitstrings $g$ and $g + P a$ into a single representative, adding a strictly positive entropy-reduction term whenever the colliding probabilities are comparable. The authors show how to force the outcomes of such measurements to be equiprobable, using the fact that blocks with parity $1_{2m}$ split evenly, and a recursive pairing trick for the unbalanced $0_{2m}$ blocks. The protocol proceeds in stages from $m = 2^q$ down to $m = 2$, with decoupling rules that keep the post-measurement correlations trackable, and the final yield is the fraction of surviving pairs minus the ebits spent before and after the BPM stage. For Werner states the yield converges with $q$ and lies above breeding wherever breeding gives a nonzero yield.

Load-bearing premise

The yield formula assumes that after the BPM stage, the best way to finish is the standard breeding routine, applied to every remaining pair whose state has only two possible Bell components, including the correlated groups written as $[n_0,n_1]$, even though the cited optimality proof covers only independent pairs.

Editorial extensions

If this is right

The protocol gives a strictly better lower bound on the entanglement of distillation than breeding for every Bell-diagonal state in the regime where breeding already works.
For Werner states the zero-yield fidelity threshold drops from about 0.8107 to about 0.7424 without recurrence, and recurrence extends distillation to lower fidelities.
Any finite equiprobable parity check found during an asymptotic protocol should be performed by BPM rather than by an ebit measurement; this is the general mechanism behind the improvement.
The yield increases with the block-size parameter $q$ and converges, so a modest $q$ such as 6 captures most of the asymptotic gain, with extra gain from favouring BPMs on small blocks.

Reading between the lines

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

Beyond the paper: the same recipe—force equiprobable checks, then apply a BPM—can be tested on Bell-diagonal states beyond Werner, where the gain may be larger or smaller depending on how balanced the colliding probabilities are.
Beyond the paper: the decoupling rules provide a normal form for post-BPM correlations, and that normal form could serve as a search space for better block orderings than the provisional one used here.
Beyond the paper: if an improved residual purification for the correlated bracket states $[n_0, n_1]$ exists, the reported yields would rise for all states, not just Werner states.
Beyond the paper: the recursive forcing procedure suggests a constructive way to generate adaptive stabilizer-style protocols without exhaustive search.

Original abstract

We present a new asymptotic bipartite entanglement distillation protocol that outperforms all existing asymptotic schemes. This protocol is based on the breeding protocol with the incorporation of two-way classical communication. Like breeding, the protocol starts with an infinite number of copies of a Bell-diagonal mixed state. Breeding can be carried out as successive stages of partial information extraction, yielding the same result: one bit of information is gained at the cost (measurement) of one pure Bell state pair (ebit). The basic principle of our protocol is at every stage to replace measurements on ebits by measurements on a finite number of copies, whenever there are two equiprobable outcomes. In that case, the entropy of the global state is reduced by more than one bit. Therefore, every such replacement results in an improvement of the protocol. We explain how our protocol is organized as to have as many replacements as possible. The yield is then calculated for Werner states.

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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, 6,696 input tokens.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.40 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.96 · 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.

We present a new asymptotic bipartite entanglement distillation protocol that outperforms all existing asymptotic schemes. This protocol is based on the breeding protocol with the incorporation of two-way classical communication. Like breeding, the protocol starts with an infinite number of copies of a Bell-diagonal mixed state. Breeding can be carried out as successive stages of partial information extraction, yielding the same result: one bit of information is gained at the cost (measurement) of one pure Bell state pair (ebit). The basic principle of our protocol is at every stage to replace measurements on ebits by measurements on a finite number of copies, whenever there are two equiprobable outcomes. In that case, the entropy of the global state is reduced by more than one bit. Therefore, every such replacement results in an improvement of the protocol. We explain how our protocol is organized as to have as many replacements as possible. The yield is then calculated for Werner states.

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	We present a new asymptotic bipartite entanglement distillation protocol that outperforms all existing asymptotic schemes.	contribution 0.98	0.91	0.89	asserted 1.00
s1	This protocol is based on the breeding protocol with the incorporation of two-way classical communication.	method 0.91	0.53	0.04	asserted 1.00
s2	Like breeding, the protocol starts with an infinite number of copies of a Bell-diagonal mixed state.	method 0.69	0.17	0.03	asserted 1.00
s3	Breeding can be carried out as successive stages of partial information extraction, yielding the same result: one bit of information is gained at the cost (measurement) of one pure Bell state pair (ebit).	background 0.98	0.08	0.03	asserted 1.00
s4	The basic principle of our protocol is at every stage to replace measurements on ebits by measurements on a finite number of copies, whenever there are two equiprobable outcomes.	method 0.83	0.73	0.07	asserted 0.97
s5	In that case, the entropy of the global state is reduced by more than one bit.	result 0.50	0.62	0.08	asserted 1.00
s6	Therefore, every such replacement results in an improvement of the protocol.	interpretation 0.80	0.69	0.20	asserted 1.00
s7	We explain how our protocol is organized as to have as many replacements as possible.	method 0.91	0.86	0.04	asserted 0.91
s8	The yield is then calculated for Werner states.	result 0.55	0.87	0.06	asserted 0.84

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.74.

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

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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

Figures

**Figure 1.** FIG. 1: the binary entropy function

**Figure 2.** FIG. 2: the yields of the protocol of Sec. V C (solid lines), fo

**Figure 3.** FIG. 3: the yields of the protocol of Sec. V D, where BPM

**Figure 4.** FIG. 4: the yield of our best protocol (solid line) and breed

**Figure 5.** FIG. 5: the relative difference of the yields.

Figures reproduced from arXiv:quant-ph/0602205 by the authors.

References

Works cited (23)

[13]

M. Plenio and S. Virmani, An introduction to entangle- ment measures, quant-ph/0504163

View on Pith Pith review arXiv
[18]

E. Rains, Entanglement puriﬁcation via separable super- operators, quant-ph/9707002

View on Pith Pith review arXiv
[1]

by local Cliﬀord operations on the tensor product and an appended ebit |Bs⟩⊗| B00⟩, followed by the local measurement of the ebit; 3

View on Pith
[2]

by directly performing local measurements on |Bs⟩. We explain the two ways in more detail, and call them appended ebit measurement (AEM) and bilateral Pauli measurement (BPM) respectively. By means of local Cliﬀord operations (1), we ﬁrst transform|Bs⟩⊗| B00⟩ into|Bs⟩⊗| B0 rT s⟩. The sym- plectic matrix C that corresponds to this action is C =   0 ...

View on Pith
[3]

perform local Cliﬀord operations (1) that corre- spond to a symplectic C of which the last row is rT : such a C can always be found, for every r⁄= 0

View on Pith
[4]

then, perform a bilateral σz measurement on the last qubit pair

View on Pith
[5]

ﬁnally, apply the inverse of the local Cliﬀord oper- ations of the ﬁrst step. Note that the result is no longer a tensor product of Bell states, as the last of the qubit pairs is measured in the second step, leaving it in a separable state. Since an AEM leaves the state |Bs⟩ unaﬀected, we only need to worry about the situation after a BPM. The only irreve...

View on Pith
[6]

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 84, 2014 (2000)

View on Pith 2000
[8]

A. Ekert, Phys. Rev. Lett. 67, 661 (1991)

View on Pith 1991
[9]

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

View on Pith 1993
[10]

C. Bennett, D. DiVincenzo, J. Smolin, and W. Wootters, Phys. Rev. A 54, 3824 (1996)

View on Pith 1996
[11]

C. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, and W. Wootters, Phys. Rev. Lett. 76, 722 (1996). 11

View on Pith 1996
[12]

A. Ambainis and D. Gottesman, IEEE Trans. Info. The- ory 52, issue 2, 748 (2006)

View on Pith 2006
[14]

P. Shor and J. Smolin, Quantum error-corrrecting codes need not completely reveal the error syndrome , quant- ph/9604006

View on Pith arXiv
[15]

E. Maneva and J. Smolin, Improved two-party and multi- party puriﬁcation protocols , quant-ph/0003099

View on Pith Pith review arXiv
[16]

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996)

View on Pith 1996
[17]

J. Dehaene, M. Van den Nest, B. De Moor, and F. Ver- straete, Phys. Rev. A 67, 022310 (2003)

View on Pith 2003
[19]

K. Vollbrecht and F. Verstraete, Interpolation of re- currence and hashing entanglement distillation protocols , quant-ph/0404111

View on Pith Pith review arXiv
[20]

F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A 64, 010101 (2001)

View on Pith 2001
[21]

J. Dehaene and B. De Moor, Phys. Rev. A 68, 042318 (2003)

View on Pith 2003
[22]

E. Hostens, J. Dehaene, and B. De Moor, The equiv- alence of two approaches to the design of entanglement distillation protocols, quant-ph/0406017

View on Pith Pith review arXiv
[23]

T. Cover and J. Thomas, Elements of Information The- ory (John Wiley & Sons, Inc., 1991)

View on Pith 1991
[101]

Combining this with for instance PB c = 1000 for 1 (a), we get to know for every 4 bits in which of the following sets they are: S1 = 0 (a)∩ 0(b) ={0000, 0101, 1010, 1111}, S2 = 0 (a)∩ 1(b) ={0001, 0100, 1011, 1110}, S3 = 1 (a)∩ 0(c) ={0010, 0011, 0110, 0111}, S4 = 1 (a)∩ 1(c) ={1000, 1001, 1100, 1101}. It can be veriﬁed that the total number of AEM neede...

View on Pith

How the literature uses it

2 arXiv papers cite this in 2 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

Cites it as background 2 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
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] .

Multiparty quantum protocols for assisted entanglement distillation · Introduction · 0.98

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] [13]

M. Plenio and S. Virmani, An introduction to entangle- ment measures, quant-ph/0504163

View on Pith Pith review arXiv

[2] [18]

E. Rains, Entanglement puriﬁcation via separable super- operators, quant-ph/9707002

View on Pith Pith review arXiv

[3] [1]

by local Cliﬀord operations on the tensor product and an appended ebit |Bs⟩⊗| B00⟩, followed by the local measurement of the ebit; 3

View on Pith

[4] [2]

by directly performing local measurements on |Bs⟩. We explain the two ways in more detail, and call them appended ebit measurement (AEM) and bilateral Pauli measurement (BPM) respectively. By means of local Cliﬀord operations (1), we ﬁrst transform|Bs⟩⊗| B00⟩ into|Bs⟩⊗| B0 rT s⟩. The sym- plectic matrix C that corresponds to this action is C =   0 ...

View on Pith

[5] [3]

perform local Cliﬀord operations (1) that corre- spond to a symplectic C of which the last row is rT : such a C can always be found, for every r⁄= 0

View on Pith

[6] [4]

then, perform a bilateral σz measurement on the last qubit pair

View on Pith

[7] [5]

ﬁnally, apply the inverse of the local Cliﬀord oper- ations of the ﬁrst step. Note that the result is no longer a tensor product of Bell states, as the last of the qubit pairs is measured in the second step, leaving it in a separable state. Since an AEM leaves the state |Bs⟩ unaﬀected, we only need to worry about the situation after a BPM. The only irreve...

View on Pith

[8] [6]

M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 84, 2014 (2000)

View on Pith 2000

[9] [8]

A. Ekert, Phys. Rev. Lett. 67, 661 (1991)

View on Pith 1991

[10] [9]

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

View on Pith 1993

[11] [10]

C. Bennett, D. DiVincenzo, J. Smolin, and W. Wootters, Phys. Rev. A 54, 3824 (1996)

View on Pith 1996

[12] [11]

C. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. Smolin, and W. Wootters, Phys. Rev. Lett. 76, 722 (1996). 11

View on Pith 1996

[13] [12]

A. Ambainis and D. Gottesman, IEEE Trans. Info. The- ory 52, issue 2, 748 (2006)

View on Pith 2006

[14] [14]

P. Shor and J. Smolin, Quantum error-corrrecting codes need not completely reveal the error syndrome , quant- ph/9604006

View on Pith arXiv

[15] [15]

E. Maneva and J. Smolin, Improved two-party and multi- party puriﬁcation protocols , quant-ph/0003099

View on Pith Pith review arXiv

[16] [16]

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996)

View on Pith 1996

[17] [17]

J. Dehaene, M. Van den Nest, B. De Moor, and F. Ver- straete, Phys. Rev. A 67, 022310 (2003)

View on Pith 2003

[18] [19]

K. Vollbrecht and F. Verstraete, Interpolation of re- currence and hashing entanglement distillation protocols , quant-ph/0404111

View on Pith Pith review arXiv

[19] [20]

F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A 64, 010101 (2001)

View on Pith 2001

[20] [21]

J. Dehaene and B. De Moor, Phys. Rev. A 68, 042318 (2003)

View on Pith 2003

[21] [22]

E. Hostens, J. Dehaene, and B. De Moor, The equiv- alence of two approaches to the design of entanglement distillation protocols, quant-ph/0406017

View on Pith Pith review arXiv

[22] [23]

T. Cover and J. Thomas, Elements of Information The- ory (John Wiley & Sons, Inc., 1991)

View on Pith 1991

[23] [101]

Combining this with for instance PB c = 1000 for 1 (a), we get to know for every 4 bits in which of the following sets they are: S1 = 0 (a)∩ 0(b) ={0000, 0101, 1010, 1111}, S2 = 0 (a)∩ 1(b) ={0001, 0100, 1011, 1110}, S3 = 1 (a)∩ 0(c) ={0010, 0011, 0110, 0111}, S4 = 1 (a)∩ 1(c) ={1000, 1001, 1100, 1101}. It can be veriﬁed that the total number of AEM neede...

View on Pith