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REVIEW 3 major objections 5 minor 71 references

Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A prepare-and-broadcast witness certifies exactly two bits of joint randomness at its maximal quantum value.

desk verdict Two-bit certificate is genuinely new and mostly proven, but the abstract drops the finite-dimensional/classical-adversary caveats and the CHSH benchmark is not apples-to-apples. read the letter →

arxiv 2608.08329 v1 pith:FN5USF6W submitted 2026-08-08 quant-ph

Tailan S. Sarubi , Moisés Alves , Vinícius F. Alves , Santiago Zamora , Diogo H. G. Duarte , A. de Oliveira Junior , Rafael Chaves This is my paper · ORCID
classification quant-ph
keywords prepare-and-broadcastscenariosemi-device-independentrandomnesscertificationquantumrandomaccesscodesphase-covariantcloningmin-entropyCHSHinequalitydimensionwitnessesnonlocalitytrade-offs
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the prepare-and-broadcast scenario, in which a single quantum system is broadcast to two receivers, and asks what can be certified from the observed correlations when only the communication dimension is trusted. It establishes quantitative trade-offs: a quantum random-access-code advantage cannot be shared by both receivers, and the boundary coincides with the optimal asymmetric phase-covariant cloning trade-off. Its main result is a randomness certificate: at the maximal quantum violation of a specific prepare-and-broadcast witness, the two receivers' joint output is exactly uniform against a classical adversary, so the certified min-entropy is exactly two bits. This exceeds the approximately 1.23 bits obtainable from the CHSH inequality and remains nonzero at higher noise levels than CHSH-based certificates.

What carries the argument

The carrying object is the prepare-and-broadcast witness $W_{\mathrm{PAB}}$ of Eq. (11), a linear combination of two-receiver correlators conditioned on three preparation inputs, whose classical bound is 6 and quantum maximum is $W_Q=8+2\sqrt{2}$. The proof splits the witness into three preparation blocks with individual bounds $4$, $4$, and $2\sqrt{2}$, so maximal violation saturates each block. Saturation becomes algebraic relations among the dilated observables, and a block decomposition of each receiver's pair of reflections, combined with the rank-two constraint imposed by the qubit message, forces the two local anticommutators to vanish on the generation state. The adversary's guessing probability is bounded by a support-function envelope built from independent semidefinite moment relaxations of the adversarial branches, and the witness's 16-element relabeling symmetry transports the two-bit certificate to a four-input orbit.

What would settle it

A prepare-and-broadcast behavior attainable with a qubit message and finite-dimensional tensor-product receivers that attains $W_{\mathrm{PAB}}=8+2\sqrt{2}$ yet has a non-uniform output distribution at $(1,0,0)$, or an adversarial decomposition of such a behavior with guessing probability above $1/4$, would falsify Theorem 2. Concretely, an experimenter could aim to reach the maximal witness value while checking the generation-input statistics for any bias.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2: for the prepare-and-broadcast witness $W_{\mathrm{PAB}}$ of Eq. (11), at the maximal quantum value $W_Q = 8+2\sqrt{2}$, the guessing probability of a classical adversary at the generation input $s^*=(1,0,0)$ is exactly $G=1/4$, so the certified joint min-entropy is exactly $H_{\min}=2$ bits. The proof shows that at maximal violation every nonzero-weight adversarial branch produces the uniform distribution on the output pair at $s^*$, using an exact dilation to projective form, saturation of the three independent preparation blocks, and a rank-two rigidity argument. The two-bit certificate holds on a four-input orbit of the witness's symmetry group, while other inputs certify at most one bit. A separate trade-off result states that the broadcast 2-to-1 quantum random access code scores satisfy $(P_B-1/2)^2+(P_C-1/2)^2=1/8$, the same boundary as optimal asymmetric phase-covariant cloning, so a quantum advantage on one marginal excludes it on the other.

Load-bearing premise

The two-bit certificate assumes the adversary's strategies are qubit prepare-and-broadcast behaviors with finite-dimensional tensor-product receiver spaces and only classical side information; if an eavesdropper instead holds a quantum system correlated with the preparations, or if infinite-dimensional commuting measurements are allowed, the exact two-bit guarantee is not proven.

Editorial extensions

If this is right

  • If the central claim is correct, a prepare-and-broadcast experiment that achieves the maximal witness value can be turned into a semi-device-independent random number generator producing exactly two bits of certified joint randomness per round with only a qubit-dimension assumption.
  • The PAB certificate beats CHSH-based certification both in entropy (two bits versus about 1.23 bits) and in noise tolerance, becoming nontrivial at visibility below the CHSH locality threshold.
  • The QRAC trade-off means a broadcast quantum random access code cannot give both receivers a quantum advantage: if one receiver beats the classical 3/4 success probability, the other necessarily drops below it, with the boundary given by optimal asymmetric phase-covariant cloning.
  • PAM and Bell violations can coexist in the same broadcast realization, but only up to a threshold; beyond it the marginal PAM witness prevents a conditioned CHSH violation.
  • Retaining the full observed behavior rather than a single witness value strictly improves the certified randomness, so data-processing choices matter for the certificate.

Reading between the lines

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

  • Because the generation slice at maximal violation is Bell local, the two-bit certificate is driven by the qubit-dimension constraint rather than by nonlocality; a testable extension is to look for other dimension witnesses, outside the broadcast setting, that certify two uniform output bits without any Bell violation.
  • The exact equality with the asymmetric phase-covariant cloning boundary suggests the same broadcast trade-off may hold for other quantum communication tasks whose optimal encodings are equatorial, and that more than two receivers would be governed by multipartite cloning bounds rather than pairwise monogamy alone.
  • The rigidity proof is tied to finite-dimensional tensor-product receivers and classical side information; if the eavesdropper holds a purification of the preparations, the two-bit guarantee is open, so a natural next step is to decide whether a quantum-adversary version of Theorem 2 holds.
  • Experimentally, the witness's generation correlator is absent from $W_{\mathrm{PAB}}$, so the uniform output at the maximal value is enforced indirectly; this suggests noise-robustness tests could be designed around the input orbit $O_B$ rather than the single setting $(1,0,0)$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper develops the prepare-and-broadcast (PAB) scenario, in which a qubit prepared by Alice is broadcast to two receivers, Bob and Charlie. It reports numerical trade-offs between marginal prepare-and-measure witnesses, conditioned Bell violations, and quantum random access code scores, connects the QRAC boundary to asymmetric phase-covariant cloning, and introduces a semi-device-independent randomness certification framework based on PAB witnesses. The central formal result is Theorem 2: for the witness W_PAB of Eq. (11), at its maximal quantum value W_Q = 8 + 2√2, the guessing probability of a classical adversary (Definition 1) at generation input s* = (1,0,0) is exactly 1/4, so the certified joint min-entropy is exactly two bits. The paper also presents SDP-based robustness curves, a numerical CHSH comparison, and a preliminary quantum-side-information analysis in a marginal S3 scenario. The proof of Theorem 2 in Appendix C is detailed and, within the stated assumptions, appears coherent; I found no obvious gap in the algebraic steps.

Significance. If taken together with its stated assumptions, the two-bit rigidity result is a valuable contribution to semi-device-independent randomness certification: it shows that a broadcast scenario with a dimension-bounded message can certify the full two bits of a binary pair at an exact analytic endpoint, with a public GitHub code repository and conservative SDP envelopes for the noisy regime. The trade-off connection to phase-covariant cloning is also interesting and well motivated. However, the advertised significance is currently inflated: Theorem 2 applies only to a classical adversary with finite-dimensional tensor-product receiver spaces, and the quantum-adversary case is explicitly left open in Remark 14. The headline comparison with the CHSH-based limit therefore compares different adversarial and dimension assumptions. These issues are fixable by qualification, but they are load-bearing for the paper's central claim as presented.

major comments (3)
  1. [Abstract; §IV C, Theorem 2; Remarks 8 and 14] The headline claim is narrower than stated. Theorem 2 proves G(BC|Λ,s*;W_Q)=1/4 only for the classical-adversary model of Definition 1, in which every branch has a qubit message and finite-dimensional tensor-product receiver Hilbert spaces, and in which Eve holds no purification of the preparations. The proof is load-bearing on finite-dimensionality: Proposition 7 uses Jordan's lemma and the rank-two transfer on finite-dimensional receiver factors, and the paper itself states in Remark 8 that the tensor form of the commuting algebras becomes an extra assumption in infinite dimensions; Remark 14 states that security against a quantum adversary holding a purification remains open. The abstract nonetheless states 'the maximal quantum violation certifies two bits of joint randomness' without either caveat. Please qualify the abstract, the introduction, and Section V, and state before Theorem 2 that this is a finite-dimensional, classical-side-information result.
  2. [§IV C, Fig. 6, Eqs. (35)–(37)] The comparison with CHSH is not apples-to-apples. The 'approximately 1.23 bits' CHSH limit is obtained with a dimension-unbounded NPA relaxation, as the text itself states, and the PAB certificates use the qubit-bounded L(1,2) relaxation against classical side information. The paper acknowledges this in the paragraph around Eqs. (35)–(37) by calling the two families 'two separately optimized white-noise benchmarks,' but the abstract and introduction present 'exceeding the limit achievable from the CHSH inequality' as an unqualified advantage. This comparison should either be performed under the same adversarial and dimension assumptions, or explicitly labeled in the abstract and conclusions as a comparison between different models. Otherwise the central significance claim overstates the result.
  3. [§III, Eq. (15) and Appendix B] The marginal QRAC trade-off boundary (P_B − 1/2)^2 + (P_C − 1/2)^2 = 1/8 is presented as if it were a derived result: the text says the two bounds 'agree within numerical precision and give Eq. (15)' and later that the cloning boundary 'maps exactly to Eq. (15).' In fact, Eq. (15) is a numerical boundary obtained by matching a nonconvex lower-bound search to an SDP upper bound at the L(1,2) level, as Appendix B makes clear. The cloning interpretation is a heuristic explanation rather than a proof of optimality. Please state explicitly in Section III that Eq. (15) is certified only at the relaxation level used and is not an analytically proven trade-off. This does not affect Theorem 2, but it is part of the paper's first advertised contribution.
minor comments (5)
  1. [§IV C opening paragraph] The text refers to 'the broadcast witness W^{(2)}_{CC}', but the witness of Eq. (11) is called W_PAB throughout the paper. Please either define W^{(2)}_{CC} or correct this notation.
  2. [§II, Eq. (11) and Ref. [29]/[34]] Equation (11) is said to be derived in Ref. [34], which is the companion numerical code repository, while later text says the witness W_PAB was introduced in Ref. [29]. Please clarify the provenance of the witness and cite the original framework for the inequality, with the code repository cited only as code.
  3. [Appendix B and Figures 4–6, 8] The numerical certificates rely on 'agreement within numerical precision' and on conservative SDP envelopes, but no solver tolerance or stopping criterion is reported. Please state the numerical accuracy used for the SDPs and for the reported endpoint values, so that the robustness claims can be reproduced.
  4. [Eq. (C16)] The bound in Eq. (C16) is explicitly an upper estimate and is not attainable because the three signs in the generation distribution cannot be chosen independently. This is fine, but it would help to add one sentence clarifying that the subsequent endpoint conclusion uses only the case where all defect terms vanish, so the looseness of the intermediate bound is irrelevant to Theorem 2.
  5. [Figure 2 caption] The caption distinguishes green filled circles as explicit-strategy lower bounds and red open circles as SDP upper bounds, but the pale-blue 'certified' regions are not described as outer or inner approximations. Please specify in the caption which regions are inner (lower-bound) and which are outer (relaxation) so readers can interpret the unshaded gaps correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-bit certificate is proven by an exact rigidity argument within the paper's explicitly stated model, and the witness's origin in the authors' prior work is not load-bearing.

full rationale

The central derivation chain (Theorem 2, Appendix C) is self-contained. The witness W_PAB is an explicit linear functional (Eq. (11)), and the paper proves the quantum bound W_Q = 8+2√2 internally in Lemma 5, including an explicit attaining strategy; the theorem therefore does not import its endpoint from prior work. At maximal violation, the block-deficit argument forces saturation of each preparation block, Lemma 6 bounds the generation moments by anticommutator defects, and Proposition 7 uses Jordan's lemma and a rank-two transfer to force those defects to vanish, with Lemma 9 extending the result to mixed preparations. Every step is an exact derivation from the stated qubit-message, finite-dimensional tensor-product model, and the numerical SDP certificates in Sec. IV are explicitly conservative upper bounds rather than fitted predictions. The witness is said to be introduced in the authors' prior work [29] and derived in their numerical code [34], but this is a normal citation of an input object: the quantum bound and rigidity are re-proven in this paper, so the self-citation is not load-bearing. The finite-dimensional tensor-product assumption (Remark 8) and the restriction to classical side information (Remark 14) are explicitly acknowledged scope limitations; they make the abstract's unqualified phrasing less general than the theorem, but they do not make the derivation circular. No equation is defined in terms of the quantity it is said to predict, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data: all witnesses and bounds are fixed by the scenario. The results rest on domain assumptions about the communication dimension, the broadcast channel, receiver structure, and the eavesdropper model. No new physical entities are introduced; the broadcast channel and Eve are standard constructs from the authors' prior framework.

assumptions (4)
  • domain assumption The communicated system is a qubit (d=2) and the broadcasting device is an arbitrary quantum channel; the receivers perform arbitrary local POVMs on finite-dimensional tensor-product Hilbert spaces.
    Defines Q_br and the semi-device-independent model; Theorem 2's rigidity proof (Appendix C, Prop 7 and Remark 8) requires finite-dimensional tensor-product receivers.
  • domain assumption Eve's side information is a classical random variable Lambda that selects a PAB branch p_lambda in Q_br, independent of the inputs (Definition 1, Sec. IV A).
    The two-bit certificate holds against this classical adversary; quantum side information is only handled for the S3 witness in Sec. IV D, and the two-bit claim is explicitly open for quantum adversaries (Remark 14).
  • standard math Every PAB behavior admits a projective isometric dilation with commuting measurements on separate tensor factors (Stinespring and Naimark, Lemma 4).
    Standard dilation theorems for quantum channels and POVMs; used throughout Appendix C.
  • standard math Jordan's lemma holds for pairs of reflections on finite-dimensional Hilbert spaces, giving joint invariant subspaces of dimension at most two (Appendix C, Prop 7).
    Used to analyze the anticommutator defects at the maximal violation.

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Cite this review

Pith. "Pith review of Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario." pith.science (2026). https://pith.science/paper/FN5USF6W

@misc{pith2026260808329,
  author       = {Pith},
  title        = {Pith review of: Randomness Certification and Trade-offs in the Prepare-and-Broadcast Scenario},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FN5USF6W}},
  note         = {Machine review of arXiv:2608.08329}
}
read the original abstract

We investigate the prepare-and-broadcast scenario, a multipartite extension of dimension-constrained prepare-and-measure experiments in which a quantum system is distributed to multiple receivers. We derive fundamental trade-offs between prepare-and-measure witnesses, Bell nonlocality, and quantum random access code performance. We further develop a semi-device-independent randomness certification framework based on prepare-and-broadcast witnesses, showing that the maximal quantum violation certifies two bits of joint randomness, exceeding the limit achievable from the CHSH inequality while remaining robust to noise. Finally, we show that the prepare-and-broadcast scenario naturally accommodates stronger adversarial models in which the eavesdropper retains a quantum system correlated with the measurement device, providing a natural framework for semi-device-independent randomness certification against quantum side information.

Figures

Figures reproduced from arXiv: 2608.08329 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: collects the resulting certificates as functions of the visibility v. Two features stand out. First, for S 3, the fixed￾distribution certificate reaches one full bit at v = 1, the alge￾braic maximum for a binary outcome, in agreement with the unbiased marginal of the o…
Figure 5
Figure 5. Figure 5: shows the resulting certificates for every genera￾tion input directly as functions of the observed witness value WPAB. The certified randomness depends strongly on the input used for generation. For suitable inputs, among them s = (1, 0, 0), the joint certificate grows…
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

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    Trade-offcurves LetG 1 andG 2 be two linear scores of the broadcast behav- ior and let RQ =conv G1[p],G 2[p] :p∈Q br (B1) be the set of attainable pairs, convex because the devices may share classical randomness. Instead of fixing a target value of one coordinate and maximizin...

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    max g∈GBC Fg(τ)−τw # ,(B10) which, combined with the inclusion of the physical set in its moment relaxation, yields the certified chain G(BC|Λ,s;w)≤G q,t(BC|Λ,s;w) ≤inf τ≥0

    Randomness envelopes We now apply the construction to the guessing-probability curves of Sec. IV, stating it for the joint randomness of the broadcast pair, the marginal case following by restriction. Solving the coupled branch program of Eq. (23) pointwise certifies the guess...

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    Exact dilation to the projective isometric form Lemma 4(exact dilation).Every p∈Q br admits a represen- tation p(b,c|x,y,z)=tr Vρ xV† ΠB b|y ΠC c|z⊗1 R ,(C3) with qubit statesρ x, an isometry V:C 2→H B⊗H C⊗H R be- tween finite-dimensional spaces, and projective measurements wh...

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    Block bounds, attainability, and saturation Lemma 5(bounds, attainability, saturation).In the represen- tation(C3)with receivers of arbitrary finite dimension, the three blocks obey W [0]≤4, W [1]≤4, and W [2]≤2 √ 2, so that WPAB ≤W Q, and the value W Q is attained by an ex- p...

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    Moreover, if the preparations are pure and WPAB =W Q, then B0C0|ϕ0⟩=|ϕ 0⟩,B 1C1|ϕ0⟩=|ϕ 0⟩,(C7) B0C1|ϕ1⟩=|ϕ 1⟩,B 1C0|ϕ1⟩=−|ϕ 1⟩,(C8) {B0,B 1}|ϕ2⟩=0,{C 0,C 1}|ϕ2⟩=0.(C9) Proof.We first establish the three block bounds. EachB yCz is a product of two commuting Hermitian involution...

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    The bound is attained onC 2⊗C 2 byB 0 =C 0 =Z,B 1 = C1 =X(acting on the respective factors) and |ϕ0⟩= 1√ 2 |00⟩+|11⟩ , |ϕ1⟩= 1 2 |00⟩+|01⟩−|10⟩+|11⟩ , |ϕ2⟩= |ϕ0⟩−|ϕ 1⟩ q 2− √ 2

    The three blocks involve three different preparations, so each expectation is bounded separately and WPAB≤8+2 √ 2. The bound is attained onC 2⊗C 2 byB 0 =C 0 =Z,B 1 = C1 =X(acting on the respective factors) and |ϕ0⟩= 1√ 2 |00⟩+|11⟩ , |ϕ1⟩= 1 2 |00⟩+|01⟩−|10⟩+|11⟩ , |ϕ2⟩= |ϕ0⟩−...

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    Then⟨B 0C0⟩[0] +⟨B 1C1⟩[0] =2 with each term in [−1,1] forces both to equal 1; for a Hermi- tian involutionMand a unit vector with⟨M⟩=1, the spectral splitM=M +−M− gives∥M−ϕ∥=0 andM|ϕ⟩=|ϕ⟩, which proves (C7), and the signs in (11) give (C8). For (C9), set u= 1√ 2(C0 +C 1) andv...

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    The next lemma ties all three to two anticommutator defects, so that their simultaneous vanishing forces a uniform generation distribution

    Generation moments and the guessing bound The distribution ats ∗ involves the three moments⟨B 0⟩[1], ⟨C0⟩[1], and⟨B 0C0⟩[1], none of which appears inW PAB. The next lemma ties all three to two anticommutator defects, so that their simultaneous vanishing forces a uniform genera...

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    Insert- ing (C15) into (C6) and maximizing each sign yields (C16)

    The sameRwith X=B 0C0, together with∥B 1C0ϕ1∥=1, gives the third bound; 20 andS=B 0C1 withσ= +1,X=C 0 gives the second. Insert- ing (C15) into (C6) and maximizing each sign yields (C16). The bound (C16) is an upper estimate and is not claimed at- tainable, since the three sign...

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    In particular∆ B = ∆C =0and p(b,c|s ∗)= 1 4

    Exact rigidity at the endpoint Proposition 7(endpoint rigidity).In the representation(C3) with pure preparations, W PAB =W Q implies P bad|ϕ0⟩= Pbad|ϕ1⟩=0, where P bad projects onto the joint Jordan blocks on which the two local anticommutators do not both vanish. In particula...

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    HenceW PAB[p]=W Q forcesWPAB[pλ]=W Q on every branch of nonzero weight

    Proof of Theorem 2 For a classical adversary, the independenceq(λ|x,y,z)= q(λ) makesW PAB affine over the adversarial decomposition, and each branch obeysWPAB[pλ]≤W Q by the block bounds of Lemma 5. HenceW PAB[p]=W Q forcesWPAB[pλ]=W Q on every branch of nonzero weight. Dilati...

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    Symmetry and the four-input orbit The witness (11) is the coefficient tensorγ xyz on the cor- relators⟨B yCz⟩[x], withγ 000 =γ 011 =γ 101 =2,γ 110 =−2, γ200 =γ 201 =γ 211 =−1,γ 210 =1, and all other entries zero; no single-party terms appear. A relabeling of the experiment is ...

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    Decoupling of the channel environment at the endpoint Lemma 12(environment decoupling).In the representation (C3)with pure preparations and WPAB =W Q, the conditional state of the Stinespring environmentH R given the outputs at s∗ is outcome-independent: writingρ (b,c) R for t...

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