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arxiv: 2604.24423 · v1 · submitted 2026-04-27 · 🪐 quant-ph

Dualistic operational characterization of device-dependent correlation sets via convex analysis in the (2,m,2) Bell scenario

Ryosuke Nogami , Jaeha Lee This is my paper

Pith reviewed 2026-05-08 04:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Bell scenariocorrelation setsentanglement witnessesbeyond-quantum statesconvex analysisdevice-dependent measurementsWerner statestwo-qubit systems
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The pith

Support and gauge functions derived via convex analysis characterize the separable, quantum, and beyond-quantum correlation sets generated by fixed dichotomic measurements in the (2,m,2) scenario, with optimal detection thresholds reached a

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit formulae for the support and gauge functions of correlation sets generated by fixed dichotomic measurements on two-qubit systems across separable, quantum, and maximal tensor-product state spaces. These functions serve as optimal witnesses for entanglement and beyond-quantum states while quantifying robustness against depolarizing noise. The analysis reveals that the fundamental limits of these characterizations depend solely on the smaller number of linearly independent measurement directions available to each party. When both parties have three or more such directions, the criteria attain the optimal PPT threshold for Werner states at a noise parameter of 2/3 and achieve equivalent separation for extremal beyond-quantum states.

Core claim

For fixed local dichotomic measurements in the traceless case, the correlation sets admit simple explicit support and gauge functions that furnish optimal entanglement witnesses and noise-robustness quantifiers. Convex-hull representations show that extremal quantum correlations are realized by maximally entangled states. The separation between the three sets and their detection thresholds are governed solely by the minimum number of linearly independent directions on each side, attaining the PPT bound p_crit = 2/3 for Werner states and equivalent performance for beyond-quantum detection precisely when both parties have at least three such directions.

What carries the argument

Support and gauge functions of the correlation sets for separable, quantum, and maximal tensor-product states, derived from convex analysis of device-dependent correlations in the (2,m,2) Bell scenario with fixed dichotomic measurements.

If this is right

  • The support functions supply optimal witnesses that detect entanglement up to the PPT bound when each party has three linearly independent directions.
  • The gauge functions quantify the exact amount of depolarizing noise tolerated before correlations fall inside the separable or quantum set.
  • Extremal quantum correlations in the set are realized exclusively by maximally entangled states under the convex-hull representation.
  • Nontrivial separation between quantum and beyond-quantum correlation sets occurs only when both parties possess at least three independent measurement directions.
  • The convex structures clarify which physical states achieve the boundary of each correlation set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit dependence on the number of independent directions could guide the choice of measurement settings in experiments to reach full detection power with minimal resources.
  • Relaxing the traceless assumption to general dichotomic measurements would likely require numerical optimization but might extend the dual characterizations to a broader class of experimental configurations.
  • The same convex-analytic approach might generalize to other Bell scenarios or to systems with more parties, yielding analogous dual operational descriptions.

Load-bearing premise

The simple explicit formulae and optimal thresholds hold under the assumption of traceless local dichotomic measurements whose directions are linearly independent up to the relevant dimension.

What would settle it

A calculation for any specific set of three linearly independent traceless dichotomic measurements showing that the support-function witness fails to detect a Werner state exactly at visibility parameter 2/3 would falsify the claimed optimality of the threshold.

Figures

Figures reproduced from arXiv: 2604.24423 by Jaeha Lee, Ryosuke Nogami.

Figure 1
Figure 1. Figure 1: FIG. 1. Duality structure of the support and gauge functions. The upper level represents the view at source ↗
read the original abstract

We analyze device-dependent correlation sets generated by fixed local dichotomic measurements for two-qubit systems in the $(2,m,2)$ Bell scenario. We consider three fundamental state spaces for the composite system: the separable state space, the standard quantum state space, and the maximal tensor-product state space, which contains beyond-quantum states compatible with local quantum measurements. We formulate the corresponding correlation sets for general fixed dichotomic measurements and, in the traceless case, derive particularly simple explicit formulae for their support and gauge functions. These functions furnish dual operational characterizations of the three correlation sets: the support functions give optimal witnesses for entanglement and beyond-quantum states, whereas the gauge functions quantify the robustness of these detections against depolarizing noise. We further derive convex-hull representations that elucidate the extremal structures of the correlation sets and the physical states realizing them, showing in particular that extremal quantum correlations are realized by maximally entangled states. The fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions available to Alice and Bob. When both parties have three linearly independent measurement directions, our entanglement criterion detects Werner states up to the optimal PPT threshold $p_{\mathrm{crit}}=2/3$. For beyond-quantum-state detection, a nontrivial separation from the quantum set occurs only under the same measurement condition; in that case, the same optimal noise threshold is attained for an extremal state in the maximal tensor-product state space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript analyzes device-dependent correlation sets in the (2,m,2) Bell scenario for two-qubit systems via convex analysis applied to three state spaces: separable, quantum, and maximal tensor-product. It formulates the correlation sets for general fixed dichotomic measurements and derives explicit support and gauge functions in the traceless case. These functions yield dual operational characterizations (witnesses via support functions; noise robustness via gauge functions), along with convex-hull representations of extremal structures. The central results state that the fundamental limits are governed solely by the smaller number of linearly independent measurement directions, with Werner-state entanglement detection reaching the optimal PPT threshold p_crit=2/3 when both parties have three such directions, and a nontrivial quantum vs. maximal-tensor-product separation occurring only under the same condition with the same threshold for an extremal beyond-quantum state.

Significance. If the derivations hold, the work supplies a clean convex-geometric framework for device-dependent correlations that unifies entanglement witnesses, robustness measures, and extremal-state characterizations across classical, quantum, and post-quantum regimes. The explicit formulae, convex-hull representations, and identification of linear-independence as the sole governing parameter constitute concrete, falsifiable contributions that could be directly useful for entanglement detection protocols and studies of beyond-quantum correlations.

major comments (1)
  1. [Abstract] Abstract: the claim that 'the fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions' and that nontrivial separation occurs only under the three-direction condition is load-bearing for the optimality of p_crit=2/3. The manuscript itself states that simple explicit formulae exist only in the traceless case and that the general case relies on the dimension-limiting assumption; confirmation is required that no additional extremal points arise once the traceless restriction is lifted or when directions are linearly dependent in non-obvious ways, as this directly controls whether the reported thresholds remain optimal.
minor comments (1)
  1. The abstract and introduction would benefit from a short paragraph explicitly contrasting the derived support/gauge functions with prior convex-analytic treatments of Bell correlations (e.g., those based on the CHSH or other fixed scenarios) to clarify the incremental advance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to explicitly confirm the scope of our claims regarding the governing role of linearly independent measurement directions. We address this point below and propose targeted revisions to enhance clarity while preserving the core results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'the fundamental limits of these dual operational tasks are governed solely by the smaller of the numbers of linearly independent measurement directions' and that nontrivial separation occurs only under the three-direction condition is load-bearing for the optimality of p_crit=2/3. The manuscript itself states that simple explicit formulae exist only in the traceless case and that the general case relies on the dimension-limiting assumption; confirmation is required that no additional extremal points arise once the traceless restriction is lifted or when directions are linearly dependent in non-obvious ways, as this directly controls whether the reported thresholds remain optimal.

    Authors: We agree that explicit confirmation is warranted. The traceless restriction is chosen solely to derive closed-form support and gauge functions; it does not restrict the physical applicability. Any general two-qubit state admits a Bloch-vector decomposition in which the trace component contributes only a uniform shift to the local marginals. Because the measurements are dichotomic (with outcomes summing to the identity), this shift factors out of the correlation probabilities, rendering the correlation sets affinely equivalent to their traceless counterparts. Consequently, the extremal points, convex-hull structure, and the parameter that governs the sets—the smaller of the ranks of the two local measurement spans—remain unchanged when the traceless restriction is lifted. Linear dependence among directions is already accounted for by using the dimension of the span rather than the nominal number m; no additional extremal points appear in non-obvious dependence configurations. We will revise the abstract to state the assumption explicitly, add a short paragraph after Eq. (3) explaining the affine equivalence, and include a remark confirming that the reported p_crit = 2/3 thresholds therefore remain optimal in the general case. revision: partial

Circularity Check

0 steps flagged

No circularity: standard convex geometry applied to explicitly defined state spaces

full rationale

The derivation begins from the three explicitly defined state spaces (separable, quantum, maximal tensor-product) and fixed local dichotomic measurements in the (2,m,2) scenario. Support and gauge functions are obtained via standard convex-analytic operations on the resulting correlation sets; explicit formulae appear only after restricting to the traceless case, and the convex-hull representations follow directly from the geometry of those sets. The statement that extremal limits are governed solely by the smaller number of linearly independent directions is a consequence of the dimension of the linear span inside the convex body, not a self-definitional reduction, a fitted parameter renamed as a prediction, or a load-bearing self-citation. No ansatz is smuggled via prior work, and no uniqueness theorem is imported from the authors' own earlier papers to force the result. The reported thresholds (including p_crit=2/3) therefore remain independent outputs of the convex analysis rather than inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard convex-analytic tools and the established definitions of the three state spaces without introducing new free parameters or postulated entities.

axioms (2)
  • standard math Properties of support and gauge functions for convex sets in finite-dimensional vector spaces.
    Invoked to obtain dual characterizations of the correlation sets.
  • domain assumption Definitions of the separable, quantum, and maximal tensor-product state spaces for two-qubit systems.
    These spaces generate the three correlation sets under fixed dichotomic measurements.

pith-pipeline@v0.9.0 · 5571 in / 1388 out tokens · 54063 ms · 2026-05-08T04:03:09.204863+00:00 · methodology

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Reference graph

Works this paper leans on

80 extracted references · 4 canonical work pages

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    (62) Proof

    In particular, any block-positive state with an orthogonal correlation matrix is uniquely determined by T and takes the form ˆρ = 1 4 ˆI4 + 3X i,j=1 Tij ˆσi ⊗ ˆσj . (62) Proof. See Appendix E. 15 VI. DETECTION OF ENT ANGLEMENT A. Entanglement detection via the support function ϕAB sep The support function ϕAB sep provides a direct criterion for entangleme...

  2. [2]

    Sincethecompletelymixedstateproducesthezerocorrelationmatrix, wehave ΦAB(ˆρp) = (1 − p) CΦ+, where CΦ+ := Φ AB(|Φ+⟩⟨Φ+|)

    For two-qubit systems, the PPT criterion [44] is necessary and sufficient for separability [45]:ˆρp is entangled if and only ifp < 2/3. Sincethecompletelymixedstateproducesthezerocorrelationmatrix, wehave ΦAB(ˆρp) = (1 − p) CΦ+, where CΦ+ := Φ AB(|Φ+⟩⟨Φ+|). By the positive homogeneity of the gauge function, γAB sep (1 − p) CΦ+ = (1 − p) γAB sep (CΦ+), (73...

  3. [3]

    (78) Since m = 2, we haver = min{rank A, rank B} = 2, so γAB sep (CΦ+) = 2 and pcrit = 1/2

    The (2, 2, 2) scenario Consider the standard CHSH measurement settings: ˆA1 = ˆσ3, ˆA2 = ˆσ1, ˆB1 = ˆσ3 + ˆσ1√ 2 , ˆB2 = ˆσ3 − ˆσ1√ 2 . (78) Since m = 2, we haver = min{rank A, rank B} = 2, so γAB sep (CΦ+) = 2 and pcrit = 1/2. For comparison, the CHSH inequality [8] provides a device-independent entanglement test. The Tsirel’son bound [10] gives the maxi...

  4. [4]

    (80) 20 Since A and B both have full rank,r = 3

    The (2, 3, 2) scenario Consider the measurement settingsA = I3 (i.e., ˆAi = ˆσi) and B =   1√ 2 0 1√ 2 0 1 0 1√ 2 0 − 1√ 2   . (80) 20 Since A and B both have full rank,r = 3. The maximally entangled state|Φ+⟩ produces the correlation CΦ+ = ATΦ+B⊤ = TΦ+B⊤ with TΦ+ = diag(1 , −1, 1) ∈ SO−(3). By (75), γAB sep (CΦ+) = 3 and pcrit = 2 /3, saturatin...

  5. [5]

    As in Remark VI.5, this simplifies toZ⋆ = G−1 A C⋆G−1 B

    = 3. As in Remark VI.5, this simplifies toZ⋆ = G−1 A C⋆G−1 B . 25 D. Noise tolerance: beyond-quantum states We apply the preceding results to the detection of beyond-quantum states under depo- larizing noise, in direct analogy with Sec. VID. Consider the state ˆρmax :=   1/2 0 0 0 0 0 1 /2 0 0 1 /2 0 0 0 0 0 1 /2   , (94) whichhaseigenvalues...

  6. [6]

    Let X = UXΣXV ⊤ X denote the reduced singular value decomposition of X = A, B, so thatG1/2 X = UXΣXU ⊤ X

    Singular values of A⊤ZB and A+C(B⊤)+ Define the Gram matricesGA := AA⊤ and GB := BB ⊤ ∈ M m(R), whose entries (GA)ij = ai · aj = 1 2 Tr[ ˆAi ˆAj], (99) (GB)ij = bi · bj = 1 2 Tr[ ˆBi ˆBj] (100) are manifestly basis-independent. Let X = UXΣXV ⊤ X denote the reduced singular value decomposition of X = A, B, so thatG1/2 X = UXΣXU ⊤ X. Then A⊤ZB = VA (ΣAU ⊤ A...

  7. [7]

    Determinant of A⊤ZB and A+C(B⊤)+ The support function ϕAB qm and the gauge function γAB qm involve the sign ofdet(A⊤ZB) and det(A+C(B⊤)+), respectively. To verify their basis-independence, we use the identity (see Lemma F.1 in Appendix F) det(A⊤ZB) = 1 6 mX i,j,k=1 mX l,m,n=1 T A ijk zil zjm zkn T B lmn, (105) where T A ijk := (ai × aj) · ak, (106) T B ij...

  8. [8]

    The observables of thep-th party (p = 1,

    Multipartite settings Consider the (n, m,2) scenario, in whichn parties each performm dichotomic measure- ments on a local qubit. The observables of thep-th party (p = 1, . . . , n) are ˆA(p) i = a(p) i · ˆσ, i ∈ {1, . . . , m}, (122) where a(p) i ∈ R3 are unit vectors. We collect them into matrices A(p) :=   (a(p) 1 )⊤ ... (a(p) m )⊤   ∈ M m,3(...

  9. [9]

    This makes the higher-dimensional setting a natural arena for accessing richer correlation structures

    Higher-dimensional local Hilbert spaces For a bipartite system with local Hilbert spaceCd, the generalized Bloch vector of a state lies inRd2−1, so the measurement matrices haved2 − 1 columns and r can be as large as d2 − 1. This makes the higher-dimensional setting a natural arena for accessing richer correlation structures. Consider the bipartite(2, m,2...

  10. [10]

    General case

    Proof of Theorem III.1: Support function of CAB sep Proof. General case. For a product state ˆρA ⊗ ˆρB with Bloch vectors rA, rB ∈ R3 (∥rA∥, ∥rB∥ ≤ 1), the correlation matrix has entries cij = Tr[ˆρA ˆAi] Tr[ˆρB ˆBj] = (ai · rA)(bj · rB) = (ArA)i (BrB)j, (A4) so C = ( ArA)(BrB)⊤. Since the extreme points of Ssep are pure product states (∥rA∥ = ∥rB∥ = 1) a...

  11. [11]

    Proof of Theorem III.2: Support function of CAB qm Proof. Since CAB qm is the image of the state space Sqm under the linear map ΦAB : ˆρ 7→ (Tr[ˆρ ( ˆAi ⊗ ˆBj)])i,j, the support function equals the largest eigenvalue: ϕAB qm (Z) = sup ˆρ∈D(C2⊗C2) Tr[ˆρ ˆSAB(Z)] = λmax( ˆSAB(Z)). (A17) We now compute the spectrum ofˆSAB(Z). Step 1: Diagonalization via spec...

  12. [12]

    Proof of Theorem III.3: Support function of CAB max Proof. Since CAB max is the image of Smax under the linear map ˆρ 7→ (Tr[ˆρ ˆAi ⊗ ˆBj])i,j, we compute the support function by maximizing over the extreme points ofSmax: ϕAB max(Z) = sup ˆρ∈ext(Smax) Tr[ˆρ ˆSAB(Z)]. (A26) 41 The extreme points ofSmax consist of pure quantum states and partial transposes ...

  13. [13]

    Let K := conv{A rAr⊤ B B⊤ : rA, rB ∈ R3, ∥rA∥ = ∥rB∥ = 1}

    Proof of Theorem V.1: Convex hull characterization of CAB sep Proof. Let K := conv{A rAr⊤ B B⊤ : rA, rB ∈ R3, ∥rA∥ = ∥rB∥ = 1}. The support function of K is ϕK(Z) = sup C∈K Tr[Z ⊤C] = max ∥rA∥=∥rB∥=1 Tr[Z ⊤A rAr⊤ B B⊤] = max ∥rA∥=∥rB∥=1 r⊤ A(A⊤ZB) rB = ∥A⊤ZB ∥∞, (B1) where the second equality uses the fact that the support function of a convex hull equals...

  14. [14]

    Let K := conv{AQB⊤ : Q ∈ SO−(3)}

    Proof of Theorem V.2: Convex hull characterization of CAB qm Proof. Let K := conv{AQB⊤ : Q ∈ SO−(3)}. The support function ofK is ϕK(Z) = max Q∈SO−(3) Tr[Z ⊤(AQB⊤)] = max Q∈SO−(3) Tr[(A⊤ZB)⊤Q]. (B2) Setting X := A⊤ZB with singular values s1 ≥ s2 ≥ s3 ≥ 0, we evaluate the right-hand side. The optimal value of the Procrustes problemmaxR∈SO(3) Tr[X ⊤R] equal...

  15. [15]

    Let K := conv{AQB⊤ : Q ∈ O(3)}

    Proof of Theorem V.3: Convex hull characterization of CAB max Proof. Let K := conv{AQB⊤ : Q ∈ O(3)}. The support function ofK is ϕK(Z) = max Q∈O(3) Tr[Z ⊤(AQB⊤)] = max Q∈O(3) Tr[(A⊤ZB)⊤Q] = ∥A⊤ZB ∥1, (B4) 43 where the last equality is the variational characterization of the trace norm: ∥X∥1 = maxQ∈O(n) Tr[X ⊤Q] (e.g., Theorem 3.4.1 in [52]). By Theorem II...

  16. [16]

    Proof of Theorem IV.1: Gauge function of CAB sep Proof. Step 1: Derivation via duality.By the duality relation (24) and Theorem III.1, the gauge functionγAB sep satisfies γAB sep (C) = sup Z̸=0 Tr[Z ⊤C] ϕAB sep (Z) = sup Z̸=0 Tr[Z ⊤C] ∥A⊤Z B∥∞ . (C12) We denoteW := A+C (B⊤)+ throughout this proof. When ran C ⊂ ran A and ran C ⊤ ⊂ ran B, the substitution ˜...

  17. [17]

    Since XY and Y X share the same nonzero eigenvalues, these coincide with the eigenvalues of the2 × 2 matrix Q := G−1 α C G−1 β C ⊤

    The nonzero squared singular valuess′2 1 , s′2 2 are the nonzero eigenvalues of W ⊤W = B⊤G−1 β C ⊤G−1 α C G−1 β B. Since XY and Y X share the same nonzero eigenvalues, these coincide with the eigenvalues of the2 × 2 matrix Q := G−1 α C G−1 β C ⊤. (C20) Therefore, s′2 1 + s′2 2 = TrQ = Tr[G−1 α C G−1 β C ⊤], s ′2 1 s′2 2 = det Q = (det C)2 sin2α sin2β . (C...

  18. [18]

    Case m ≥ 3

    Proof of Theorem IV.3: Gauge function of CAB qm Proof. Case m ≥ 3. We follow the same strategy as the proofs of Theorems IV.1 and IV.5. By the duality relation (24) and Theorem III.2, the gauge functionγAB qm satisfies γAB qm (C) = sup Z̸=0 Tr[Z ⊤C] ϕAB qm (Z) = sup Z̸=0 Tr[Z ⊤C] ∥A⊤Z B∥− . (C24) We denoteW := A+C (B⊤)+. 47 If ran C ̸⊂ ran A or ran C ⊤ ̸⊂...

  19. [19]

    Subcase r ≤ 2

    Therefore, γAB qm (C) ≥ ∥ W ∥+/1 = ∥W ∥+, and combined with (C25), we concludeγAB qm (C) = ∥A+C (B⊤)+∥+. Subcase r ≤ 2. Since CAB qm = CAB max, the gauge function ofCAB qm coincides with that ofCAB max (Theorem IV.5). Case m = 2. Since CAB qm is a convex compact set containing the origin in its interior, the gauge function uniquely determines the set: CAB...

  20. [20]

    We follow the same strategy as Step 1 of the proof of Theorem IV.1

    Proof of Theorem IV.5: Gauge function of CAB max Proof. We follow the same strategy as Step 1 of the proof of Theorem IV.1. By the duality relation (24) and Theorem III.3, the gauge functionγAB max satisfies γAB max(C) = sup Z̸=0 Tr[Z ⊤C] ϕAB max(Z) = sup Z̸=0 Tr[Z ⊤C] ∥A⊤Z B∥1 . (C34) When ran C ⊂ ran A and ran C ⊤ ⊂ ran B, the substitution ˜Z := A⊤Z B a...

  21. [21]

    Let r := min {rank A, rank B}

    Achievable singular values of A⊤ZB Lemma D.1. Let r := min {rank A, rank B}. For any s1 ≥ · · · ≥ sr ≥ 0, there exists Z ∈ M m(R) such that the singular values ofA⊤ZB are s1, . . . , sr with the remaining singular values equal to zero. Furthermore, the sign ofdet(A⊤ZB) can be chosen arbitrarily. Proof. Since rank A ≥ r, we choose orthonormal vectors u1, ....

  22. [22]

    Proof of Theorem VI.3 Proof. By Theorems III.2 and III.1, the ratio of support functions equals ϕAB qm (Z) ϕAB sep (Z) = s1 + s2 − ϵ s3 s1 , (D4) where s1 ≥ s2 ≥ s3 ≥ 0 are the singular values ofA⊤ZB, and ϵ := sgn(det(A⊤ZB)). When r = 3: Since s1 ≥ s2 ≥ s3 ≥ 0, we have ϕAB qm (Z) ϕAB sep (Z) ≤ s1 + s1 + s1 s1 = 3. (D5) 50 Equality is attained by choosings...

  23. [23]

    Proof of Theorem VII.2 Proof. By Theorems III.2 and III.3, the ratio of support functions equals ϕAB max(Z) ϕAB qm (Z) = s1 + s2 + s3 s1 + s2 − ϵ s3 , (D8) where s1 ≥ s2 ≥ s3 ≥ 0 are the singular values ofA⊤ZB, and ϵ := sgn(det(A⊤ZB)). When r = 3: Since s1 ≥ s2 ≥ s3 ≥ 0, we have ϕAB max(Z) ϕAB qm (Z) ≤ s1 + s2 + s3 s1 + s2 − s3 ≤ s1 + s1 + s1 s1 + s3 − s3...

  24. [24]

    Since C ∈ C AB qm, the convex hull representation (Theorem V.2) givesC = AQB⊤ with Q ∈ Conv(SO−(3))

    Proof of Theorem VI.4 Proof. Since C ∈ C AB qm, the convex hull representation (Theorem V.2) givesC = AQB⊤ with Q ∈ Conv(SO−(3)). In particular, ran C ⊂ ran A and ran C ⊤ ⊂ ran B, so γAB sep (C) is 51 finite. Since γAB sep is a convex function, it attains its maximum over the convex setCAB qm at an extreme pointC = AQB⊤ with Q ∈ SO−(3). For suchC, Theorem...

  25. [25]

    Since C ∈ C AB max, the convex hull representation (Theorem V.3) givesC = AQB⊤ with Q ∈ Conv(O(3))

    Proof of Theorem VII.3 Proof. Since C ∈ C AB max, the convex hull representation (Theorem V.3) givesC = AQB⊤ with Q ∈ Conv(O(3)). In particular, ran C ⊂ ran A and ran C ⊤ ⊂ ran B, so γAB qm (C) is finite. Since γAB qm is a convex function, it attains its maximum over the convex setCAB max at an extreme pointC = AQB⊤ with Q ∈ O(3). Case (i): r = 3. Since r...

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