Extremal quantum correlations for N parties with two dichotomic observables per site

Ll. Masanes

REVIEW

Extremal quantum correlations for N parties with two dichotomic observables per site

Ll. Masanes

Quantum Physics · Submitted 13 Dec 2005 · 9 references

Pith's verdict: A clean, correct structural theorem for the (N,2,2) quantum set; the only real gaps are expositional, and the M=K=2 restriction is acknowledged.

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CitationLl. Masanes. "Extremal quantum correlations for N parties with two dichotomic observables per site." arXiv:quant-ph/0512100 (2005). https://pith.science/paper/E3B6BS2H

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Editor's note

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This one is worth your time if you care about the geometry of the quantum correlation set. Masanes proves that for N parties with two dichotomic observables each, every extreme point of the quantum set is achievable by measuring an N-qubit pure state with projective observables. That answers Problem 26.A of Krueger-Werner for M=K=2, and it turns membership testing into a finite-dimensional search you can hand to the Groetschel-Lovasz-Schrijver machinery. The SLOCC corollary is a nice bonus: any state violating a Bell inequality can be filtered into an N-qubit state that violates it at least as much. That is a genuinely useful structural fact, not decoration. The proof strategy is the right one: Lemma 1 kills POVMs and mixed states by convexity, Lemma 2 forces the projector ranges into general position for non-factorizable extremes, Lemma 3 pins the local dimension to 2r, and Theorem 4 slices each local space into two-dimensional blocks E_k that carry one vector from each of the four ranges. The construction is correct. I checked the E_k orthogonality myself: it follows in two lines from G_a = A(1|1)A(a|2)A(1|1) and G_0+G_1 = A(1|1), so the simultaneous eigenbasis exists and cross terms vanish. The stress-test note is right that this is not load-bearing. Where I would push back on the reader's report is only in emphasis: the orthogonality is asserted in the text, not derived, and a referee should ask for the two lines. Same for the induction over factorizable extreme points, which is implicit, and the convexity step in Corollary 5, which is one sentence where three would help. These are minor. The bigger honest limitation is the one the author states in the conclusions: the techniques do not extend to larger M or K, so the general Problem 26.A stays open. That is acknowledged, not hidden, and it does not undercut the M=K=2 result. Citation pattern is clean: Werner-Wolf, Tsirelson, Khal-fin, the standard references, nothing self-serving. No data, no fitting, no circularity. Who should read it: anyone working on Bell inequality geometry, quantum set membership, or SLOCC filtering of Bell violation. It deserves a serious referee, and I would expect a smooth review with a request for the orthogonality derivation and a slightly fuller Corollary 5.

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Minor objections (8)

[§II.C, Theorem 4 proof] The assertion that the subspaces E_k^n are mutually orthogonal and span H_n is load-bearing for the convex decomposition (14), but it is stated without proof. The orthogonality follows from ⟨v_k^a|v_l^b⟩ = δ_{ab}⟨v_k|A(a|2)|v_l⟩ = δ_{ab}λ_l^a⟨v_k|v_l⟩ = 0 for k≠l, using G_a|v_l⟩ = λ_l^a|v_l⟩ and A(1|1)|v_k⟩ = |v_k⟩. The spanning follows from dim E_k = 2 and Σ_k dim E_k = dim H_n by Lemma 3. Please add this derivation explicitly, since without it the mixture (14) is not justified as a convex combination.
[§II.C, Theorem 4 proof] The purity of ρ[k] is not explicitly stated. By Lemma 1, for an extreme point P we can take ρ pure; then ρ[k] is pure. Clarify this at the start of the proof, since the theorem statement claims N-qubit pure states.
[§II.B, Lemma 2 proof] The induction that removes vectors is implicit. Clarify that the procedure terminates because the local Hilbert space dimension decreases at each step, and that for an extreme non-factorizable P, the factorizable term in (9) must vanish (π = 0).
[§III, Corollary 5 proof] The convexity step is terse. Spell out that V(ρ) = Σ π[k] V(ρ[k]), so if V(ρ) < 0 then min_k V(ρ[k]) ≤ V(ρ) < 0, giving an equal or larger violation.
[§II.C title and §III] There are several typos: 'Qubtis' should be 'Qubits' in the §II.C title; 'where looking for' should be 'were looking for' in the Corollary 5 proof; 'rise the question' should be 'raise the question' in §I; 'constrains' should be 'constraints' in §I; 'in order decide' should be 'in order to decide' in the abstract; and 'space-like s eparated' should be 'space-like separated' in the abstract.
[§II.C, Theorem 4 proof] The notation E_k is used both for the subspace and for the projector onto it. Consider using a distinct symbol for the projector, e.g., P_k, to avoid confusion in (12) and (13).
[§II.B, Remark after Lemma 2] The remark says that the two observables A(a|1) and A(a|2) 'do not commute for each possible direction'. This phrasing is vague; consider rephrasing in terms of the non-overlap of the ranges, which is what the lemma actually establishes.
[§I] The claim that the result 'answers the question risen in Problem 26.A of [8] for the case M=K=2' could be clarified: the theorem is about extreme points, and the convex hull argument gives the full set. A sentence making this explicit would help the reader.

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The paper

The paper claims

For any number of parties each choosing between two yes-no measurements, every extreme quantum correlation can be produced with qubits alone, using pure states and projective measurements.

The reading

In a Bell experiment, separated parties each pick one of several measurements and record an outcome; the table of outcome probabilities across all choices is a point in a convex set, and the question is which points quantum mechanics can actually produce. The paper asks how large the local quantum systems must be to reach every extreme point of that set in the simplest nontrivial case: $N$ parties, two measurements each, two outcomes each. The author proves the answer is small: two-level systems (qubits) in a pure state, measured with ordinary projective observables, already reach every extreme point. Because extreme points determine the whole convex set, this turns the question 'is this correlation table quantum?' into a finite search over $N$-qubit pure states, which known algorithms can carry out. A companion corollary says that any state violating a Bell inequality can be pushed by stochastic local operations into an $N$-qubit state that violates the same inequality at least as strongly.

What carries the argument

The load-bearing device is a three-step normal form for extreme points. First, any two-outcome positive-operator-valued measurement can be written as a convex combination of projective measurements, so projectors suffice at extreme points. Second, at a non-factorizable extreme point no nonzero local vector may lie in two of the four outcome subspaces $A_n(a|x)$, otherwise the state can be split into a factorizable piece plus a remainder and the point is not extreme. Third, this disjointness forces all four projectors at a site to have equal rank $r$, so the local space has even dimension $2r$, and it lets the local space be carved into $r$ orthogonal two-dimensional blocks, each holding exactly one vector from each of the four outcome subspaces. Those blocks are the qubit pairs on which the final projective measurements act.

Extended reading notes

Core claim

The central claim is Theorem 4: in the scenario with $N$ parties, two dichotomic observables per site, every extreme point of the quantum correlation set is achievable by measuring an $N$-qubit pure state with projective observables. The proof proceeds by purifying the state, replacing each party's two-outcome measurement by a projective one, and then arguing that at a non-factorizable extreme point each party's local Hilbert space splits into mutually orthogonal two-dimensional subspaces, each containing exactly one vector from each of the four measurement-outcome subspaces. Projecting onto any one of these two-dimensional blocks at every site yields an $N$-qubit state measured with qubit projectors, and the original distribution is recovered as a convex mixture of these. Since an extreme point cannot be a nontrivial mixture of distinct points in the set, it must coincide with one of them. The paper also answers, for this case, the question of whether local dimension $K$ suffices to generate all quantum correlations, posed as Problem 26.A in a published list of open problems.

Load-bearing premise

The proof needs the two-dimensional blocks built at each site to be mutually orthogonal and to fill the whole local Hilbert space, since that is what makes equation (14) a genuine convex mixture with probabilities summing to one; the paper asserts this rather than deriving it, and if it failed the extremality argument would not close.

Editorial extensions

If this is right

Deciding whether a given $N$-party, two-measurement, two-outcome correlation table is quantum becomes a search over $N$-qubit pure states and projective measurements, which known algorithms can run to any chosen precision.
Every Bell inequality violation achievable by any state, of any local dimension, is matched or beaten by some $N$-qubit state obtained from it by stochastic local operations.
The minimal local dimension needed to generate all quantum correlations in this scenario is settled at two, resolving the $M=K=2$ case of the open question about whether dimension $K$ always suffices.
Because the full probability table is covered, not just the full-correlation functions, the earlier characterisation of extremal full-correlation functions is subsumed.

Reading between the lines

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

If the result holds, then for this scenario the quantum set is the convex hull of a family of qubit-generated points, so numerical outer approximations built from qubit states are not merely lower bounds but exact in the limit of enough extreme points.
The same normal-form strategy looks like it might extend to more than two outcomes per site by replacing two-dimensional blocks with $K$-dimensional ones, but the author explicitly says the techniques do not carry over directly, which marks a concrete next target.
The SLOCC corollary suggests a way to compare Bell violation across states of different local dimension on a common footing: rank the states by the best violation achievable by an $N$-qubit reduction of them.
A natural stress test is whether the block decomposition survives when the state is mixed at an extreme point of the set of distributions rather than of the underlying state space, since extremality of the distribution is what the argument actually uses.

Original abstract

Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring $N$-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an $N$-partite state $\rho$ violates a given Bell inequality, then, $\rho$ can be transformed by stochastic local operations into an $N$-qubit state that violates the same Bell inequality by an equal or larger amount.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.02 · states quantitative result 0.08 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring $N$-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an $N$-partite state $\rho$ violates a given Bell inequality, then, $\rho$ can be transformed by stochastic local operations into an $N$-qubit state that violates the same Bell inequality by an equal or larger amount.

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	Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables.	background 0.98	0.09	0.02	asserted 0.68
s1	Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are space-like separated.	background 0.95	0.10	0.02	asserted 0.92
s2	In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed.	contribution 0.94	0.84	0.04	asserted 0.81
s3	It is shown that all extreme points of this convex set are attainable by measuring $N$-qubit pure-states with projective observables.	result 1.00	0.91	0.66	asserted 1.00
s4	This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not.	interpretation 0.98	0.45	0.12	hedged possibility 0.94
s5	It is also proven that if an $N$-partite state $\rho$ violates a given Bell inequality, then, $\rho$ can be transformed by stochastic local operations into an $N$-qubit state that violates the same Bell inequality by an equal or larger amount.	result 1.00	0.90	0.19	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

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full text
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references and citing sentences
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S sections
T body sentences
C citations
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References

Works cited (9)

[8]

O. Krueger, R. F. Werner; quant-ph/0504166. www.imaph.tu-bs.de/qi/problems/

View on Pith Pith review arXiv
[7]

M. Gr¨ otschel, L. Lovasz, G. Schrijver; Geomet- ric Algorithms and Combinatorical Optimization (Springer-Verlag, Berlin 1988)

View on Pith 1988
[1]

J. S. Bell; Physics 1, 195 (1964)

View on Pith 1964
[2]

L. A. Khalﬁn, B. S. Tsirelson; Found. Phys. 22, 879 (1992)

View on Pith 1992
[3]

R. Werner, M. M. Wolf; (Review article for the QIC journal) quant-ph/0107093

View on Pith Pith review arXiv
[4]

M. A. Nielsen, I. L. Chuang; Quantum computa- tion and quantum information (Cambridge Univer- sity Press, Cambridge, 2000)

View on Pith 2000
[5]

B. S. Tsirel’son; J. Sov. Math. 36, 557 (1987)

View on Pith 1987
[6]

Ll. Masanes; quant-ph/0309137

View on Pith Pith review arXiv
[9]

R. F. Werner, M. M. Wolf; Phys. Rev. A. 64, 032112 (2001)

View on Pith 2001

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How the literature uses it

45 arXiv papers cite this in 61 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 · 46 sentences without a confident label

Builds on it 2 papers

Scaled Tsirelson-Landau-Masanes inequality The second condition is a generalisation of the TLM inequality [ref] .

Extremal points of the quantum set in the CHSH scenario: conjectured analytical solution · Ishizaka's set of conditions for quantum points' extremality · 0.96
The following Lemma proves that the POVMs maximizing $I_d+1$ for any dimension $d$ are such that one of their elements is a projector on a pure state, thus generalizing a result from [ref] .

Robustness of Device Independent Dimension Witnesses · Numerical optimization of $I_d+1$ · 1.00

Uses its method or tool 5 papers

First, by considering all possible pure qubit states and projective measurements [ref] , we maximized the Bell-CHSH expression ([ref]) under the given zero probability constraints to find that the maximum value is zero, i.e., the Bell-CHSH inequality is not violated, and hence there are no quantum nonlocal points on these faces.

Geometry of the quantum set on no-signaling faces · Four-dimensional faces · 0.80
In order to find the optimal value of this Bell inequality, we use the following lemma proved by Masanes in [ref] .

Steering is an essential feature of non-locality in quantum theory · Conclusion and Open Questions · 0.99
To estimate its minimal quantum value, we use the fact that all extreme distributions in the two inputs/two outputs scenario can be obtained by measuring a two-qubit system with the following projectors [ref] : [equation] where $|ψ_1,2⟩ = cos(θ_1,2)|0⟩ + sin(θ_1, 2)|1⟩$.

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Compares with it 2 papers

Complementing the findings of [ref] , which show that any two-setting/two-outcomes Bell inequality can be maximally violated by pure entangled qubits, our result suggests that this is also necessarily the case, at least for the family given by ([ref]).

Robust Self Testing of Unknown Quantum Systems into Any Entangled Two-Qubit States · Self Testing of Partially Entangled Qubits · 0.88
This condition, which, as we said, is weaker than [ref], had previously been obtained in [ref] .

A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations · Analytic conditions for quantum behaviors with two inputs and two outputs · 0.91

Cites it as support 1 paper

More importantly for us, it follows from the work of Masanes [ref] that an extremal point of the (2,2,2) quantum set can be generated by measuring a singlet if and only if it satisfies [ref] (see [ref] for the spelled-out argument).

All the self-testings of the singlet for two binary measurements · Main result · 0.97

Cites it as background 5 papers

In the bipartite scenario $[(2,2),(2,2)]$, it has been proven that qubits are sufficient to produce all convexly extremal correlations [ref] .

Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations · Existing Results · 0.90
A covering of the set of extreme points $∂_e$, which results in the cosine parametrization (b) and its analog in the N220-case, were found in [ref] , see also [ref] .

Quantum Correlations in the Minimal Scenario · Short Review of Previous Work · 0.83
This problem has been tackled through analytical solutions for the smallest setting [ref] , conjectured generalizations [ref] and additional results using semidefinite programming techniques [ref] .

Certifying nonlocal properties of noisy quantum operations · Complete decoherence · 0.87

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] [8]

O. Krueger, R. F. Werner; quant-ph/0504166. www.imaph.tu-bs.de/qi/problems/

View on Pith Pith review arXiv

[2] [7]

M. Gr¨ otschel, L. Lovasz, G. Schrijver; Geomet- ric Algorithms and Combinatorical Optimization (Springer-Verlag, Berlin 1988)

View on Pith 1988

[3] [1]

J. S. Bell; Physics 1, 195 (1964)

View on Pith 1964

[4] [2]

L. A. Khalﬁn, B. S. Tsirelson; Found. Phys. 22, 879 (1992)

View on Pith 1992

[5] [3]

R. Werner, M. M. Wolf; (Review article for the QIC journal) quant-ph/0107093

View on Pith Pith review arXiv

[6] [4]

M. A. Nielsen, I. L. Chuang; Quantum computa- tion and quantum information (Cambridge Univer- sity Press, Cambridge, 2000)

View on Pith 2000

[7] [5]

B. S. Tsirel’son; J. Sov. Math. 36, 557 (1987)

View on Pith 1987

[8] [6]

Ll. Masanes; quant-ph/0309137

View on Pith Pith review arXiv

[9] [9]

R. F. Werner, M. M. Wolf; Phys. Rev. A. 64, 032112 (2001)

View on Pith 2001