PaperlayerView exact revision on arXiv

Source-linked AI summary

Entanglement-of-formation potentials for feedback-assisted classical communication and randomness distillation

Mark M. Wilde

arXiv:2610.04831v1quant-phcs.ITmath.IT

TL;DR

Feedback-assisted coding and randomness distillation require accounting for classical information alongside quantum correlations retained for later use. The paper develops entanglement-of-formation potentials for one-way bounds and shows that a two-round protocol can exceed the exact one-way isotropic-state rate.

  • Problem

    In feedback-assisted coding, early transmissions can create entanglement that makes later inputs correlated with Bob’s memory, while randomness distillation needs accounting that reflects communication costs and retained correlations.

    Source: §1.1 · p. 4, §9 · p. 35

  • Method

    The paper proves an exact amortization bound for message information plus remaining entanglement, then develops a state analogue yielding a finite-blocklength one-way randomness converse.

    Source: §1.2.1 · p. 5, §4.3 · p. 20, §14.1 · p. 61

  • Results

    A two-round weak-measurement protocol strictly improves on the exact one-way isotropic-state rate throughout the nontrivial noise interval, alongside new channel-capacity and one-way randomness bounds.

    Source: §14.1 · pp. 60–61

  • Takeaways & Limitations

    Gentle first-round decoding preserves quantum correlations used for reverse-direction extraction, so the one-way state potential is not a general two-way converse.

    Source: §14.1 · p. 61

    The proven operational role of the state potential is one-way; accounting for successive two-way rounds and determining whether a proposed hierarchy reaches the true rate remain open.

    Source: §14.2 · p. 61, §14.1 · p. 61

Abstract

from arXiv · show

We develop entanglement-of-formation potentials for feedback-assisted classical communication and one-way randomness distillation. For a finite-dimensional quantum channel without initial shared entanglement, an exact amortization theorem bounds the increase of message information plus entanglement remaining across the communicating laboratories. This proves a classical-feedback capacity bound using a mixed-convex-roof expression previously studied by Winter and Yang in the context of potential capacities. We determine the capacity of flagged mixtures of noiseless and entanglement-breaking channels, improve a published depolarizing-channel converse, and evaluate the new bound exactly for the amplitude damping channel. A state analogue gives a finite-blocklength upper bound on the net shared randomness obtainable with one-way classical communication. Amplitude-damping Choi states provide exact benchmarks, while a full-rank generalized amplitude-damping Choi state demonstrates a useful improvement where we do not evaluate the operational rate exactly. We also distinguish one-way from two-way randomness distillation: a two-round weak-measurement protocol strictly exceeds the exact one-way rate of every nontrivial qubit isotropic state. The protocol preserves quantum side information during its first decoding and extracts additional randomness in the reverse direction. This shows why the one-way state potential is not a general two-way converse. Together, these results establish a common entanglement-based approach to sharper communication and randomness bounds, while revealing how retained quantum correlations enable two-way distillation to surpass one-way limits.

1 Introduction

The introduction frames feedback-assisted communication and randomness distillation as resource-accounting problems: classical information can be extracted while quantum correlations remain available for later use. It presents an entanglement-of-formation potential for feedback capacity and a one-way state bound, then distinguishes those bounds from two-way randomness protocols that exploit retained quantum correlations.

1.1 Background and motivation

  • A feedback-assisted transmission can create entanglement between the laboratories, so bounding each use by its message-information increase alone can fail, as dense coding yields zero then two message bits over two noiseless qubit transmissions.The introduction motivates tracking both extracted information and entanglement available for later use.Source: §1.1 · p. 4

1.2.1 Classical-feedback-assisted classical communication

  • The channel converse tracks message information plus entanglement of formation across the laboratories; its mixed-state amortization proof bounds each channel use by UF and yields a weak feedback-capacity converse.The potential is nonincreasing under local operations and classical feedback; the converse is not an exact finite-error formula or a strong converse.Source: §1.2.1 · p. 5
  • The channel applications determine the capacity of a flagged noiseless/entanglement-breaking mixture, improve a published depolarizing-channel converse, and evaluate the amplitude-damping channel bound exactly.The depolarizing comparison also considers the unassisted capacity and other converse bounds.Source: §1.2.1 · p. 5

1.2.2 Randomness distillation assisted by local operations and classical communication

  • A state analogue gives a finite-blocklength upper bound on net randomness with one-way communication; amplitude-damping Choi states offer benchmarks, while a generalized amplitude-damping Choi state demonstrates an improved bound without an exact operational rate.The bound is explicitly for one-way LOCC-assisted distillable randomness, not unrestricted LOCC-assisted randomness distillation.Source: §1.2.2 · p. 6
  • A two-round weak-measurement protocol strictly exceeds the exact one-way rate for nontrivial qubit isotropic states by preserving quantum side information during decoding and extracting randomness in reverse; this does not establish a general two-way converse.For p = 1/2 and s = 4/5, the protocol achieves 0.2114185019… bits per copy.Source: §1.2.2 · pp. 6–7

1.3 Relation to prior work

  • The channel quantity is a mixed-convex-roof expression previously studied by Winter and Yang [WY16], while the operational contribution is its exact amortization proof for arbitrary adaptive feedback protocols.The two-round construction establishes a specific weak-measurement specialization and strict isotropic-state separation, not a new general idea of retaining quantum outputs after an instrument.Source: §1.3 · pp. 7–8

1.4 Organization

  • The paper proceeds from communication models and channel bounds through channel applications, then develops one-way state bounds, damping examples, isotropic results, and the two-round separation.The appendices include numerical certificates and supporting proofs.Source: §1.4 · p. 8

2 Notation, communication model, and channel bound

The section sets the finite-dimensional communication model and defines the information, entanglement, and channel quantities used to bound feedback-assisted communication. The channel bound optimizes output Holevo information together with entanglement of formation over possibly mixed input ensembles.

2 Notation, communication model, and channel bound

  • The model uses finite-dimensional systems, no initial shared entanglement or free forward classical communication, and exactly n channel uses; feedback-assisted capacity is defined in bits per use.A message-independent initial random seed is permitted, and average-error and maximum-error capacity have the same value in this feedback model.Source: §2 · p. 9, §2.3 · pp. 11–12

2.1 States and information quantities

  • The notation and information quantities specify states, entropies, mutual information, and channel Holevo information used in the analysis.The regularized channel Holevo information equals unassisted classical capacity; no general additivity assumption is made.Source: §2.1 · pp. 9–10

2.2 Entanglement of formation

  • Entanglement of formation is a convex-roof measure of shared entanglement, and conditioning on classical labels means an ordinary average over conditional states.With a readable classical flag, the conditional entanglement of formation equals the average over flagged states, since Bob can distinguish the labels.Source: §2.2 · pp. 10–11

2.3 The feedback-assisted task

  • The task defines finite-error and asymptotic feedback-assisted classical rates for protocols using the channel, with the message uniform unless conditioning is specified.A message-independent initial seed allows conversion of an average-error code to one with each message’s error equal to the average error.Source: §2.3 · pp. 11–12

2.4 The quantity UF

  • The channel bound UF maximizes output Holevo information plus the average reference-output entanglement of formation over ensembles that may contain mixed states.Mixed ensembles cannot generally be replaced by pure ones because the channel input may be entangled with Alice’s retained memory; an optimizing ensemble needs at most d_A^2 members.Source: §2.4 · p. 12

3 A potential that is monotone under classical feedback

The section defines a potential combining message information with entanglement remaining across the laboratories. It is initially zero and cannot increase under local operations or receiver-to-sender classical feedback, but a separate channel-use theorem is needed to bound its increase during transmission.

3.1 Message information and retained entanglement

  • In the two-use noiseless example, each transmission raises the potential by one bit, transferring its contribution from retained entanglement to message information.The first use creates a shared Bell state without revealing the message; after the second, Bob distinguishes four encoded Bell states and the shared entanglement is consumed.Source: §3.1 · p. 13, Figure 2 · p. 14
  • The potential is initially zero, dominates the message information available to Bob, and cannot increase under local channels or receiver-to-sender classical feedback.The entanglement term concerns entanglement in the actual laboratories, rather than mixing correlated with an inaccessible environment.Source: §3.1 · pp. 13–14

3.2 What remains to be proved

  • Free-operation monotonicity alone does not bound a forward channel use; an additional one-step theorem must handle inputs correlated with Bob’s memory and mixed conditional states.The potential uses actual-laboratory entanglement of formation for mixed states, so the pointwise comparison with the earlier entropy-based potential is not by itself an operational converse.Source: §3.1 · p. 13, §3.2 · p. 15

4 Exact amortization and the feedback converse

The section proves that the channel quantity UF bounds the increase of the laboratory potential for an arbitrary channel use, then applies this amortization result to feedback protocols. This yields a finite-blocklength converse and a classical-feedback capacity upper bound.

4.1 An entanglement relocation inequality

  • The relocation inequality compares physical output entanglement across A′|B′B with the reference-input cut A′B′|B, accounting for the difference by S(B′|B), which may be negative.The comparison is mathematical and does not move Bob’s old memory between laboratories.Source: §4.1 · pp. 15–16, Figure 3 · p. 16

4.2 The one-step theorem

  • The exact amortization theorem bounds the increase of the potential across one channel use by UF(N), even for arbitrary correlated laboratory states and mixed conditional inputs.The supremum of the one-use increment over finite memories and classical–quantum inputs equals UF(N); an ensemble attaining the channel optimization realizes the optimized increment.Source: §4.2 · pp. 16, 18

4.3 The operational converse

  • Theorem 4.3 bounds an n-use feedback code’s message rate using UF(N) and its average error, yielding C←(N) ≤ UF(N).The argument sums channel-use increments while local processing and feedback contribute nonpositive increments.Source: §4.3 · pp. 18, 20
  • The finite-error converse is a weak converse: its factor 1−ε does not establish a strong-converse threshold at UF(N).The resulting capacity upper bound does not assert that the bounded potential increment is achievable as a communication rate.Source: §4.3 · p. 20

5 Earlier convex-roof bounds and structural properties

The section identifies UF with a mixed-convex-roof expression and establishes properties that support channel bounds and regularization. Its feedback role does not make it equal to a channel’s operational capacity or potential Holevo capacity.

5.1 Identification with the Winter–Yang expression

  • UF equals the mixed-convex-roof upper bound on potential Holevo capacity studied by Winter and Yang [WY16], with mixed ensemble states retained in the optimization.The identification preserves the full mixed-ensemble optimization rather than replacing it with a pure-state roof.Source: §5.1 · pp. 21–22
  • This identification gives UF a feedback amortization property, not an identity with feedback capacity or potential Holevo capacity.The strict depolarizing gap illustrates that ordinary Holevo additivity need not make UF equal to channel capacity.Source: §5.1 · p. 22

5.2 Basic bounds and output processing

  • UF is bounded by log2 dA, is convex in the channel, and cannot increase under local output processing.For entanglement-breaking channels it equals the Holevo information; for erasure channels it recovers the established first-order feedback capacity.Source: §5.2 · pp. 22–23

5.3 Subadditivity and a regularized converse

  • UF is subadditive under tensor products, so inf_k UF(N^⊗k)/k gives a regularized upper bound on feedback capacity.The capacity comparison is justified by running independent copies of an entire feedback protocol in parallel, not by partitioning one adaptive protocol.Source: §5.3 · pp. 23–24

6 Flagged-channel simulations

A readable flag lets Bob distinguish an input-independent noiseless or entanglement-breaking component. The theorem evaluates the flagged channel’s feedback capacity while accounting for entanglement retained from noiseless uses.

6.1 Noiseless and entanglement-breaking components

  • The flagged construction chooses between a noiseless channel and an entanglement-breaking channel independently of the input, and Bob can read the flag.When component output spaces agree, discarding the flag implements their ordinary mixture; every feedback code for that mixture can therefore be used with the flagged channel.Source: §6.1 · p. 24, Figure 6 · p. 25
  • Theorem 6.1 gives an exact feedback-capacity expression for the flagged channel, optimized over pure input ensembles whose average state may be mixed.The converse accounts for entanglement created by noiseless uses and still present during later uses.Source: §6.1 · pp. 24–25
  • The flagged-channel formula is an upper bound by the weighted sum of the component capacities, but need not equal that sum because one ensemble must optimize both terms.Simply averaging component capacities misses entanglement that noiseless uses can leave available for later uses.Source: §6.1 · p. 25

7 The qubit depolarizing channel

For qubit depolarizing noise, the paper derives flagged and direct entanglement-of-formation converses. The direct bound analytically improves on the published Υ bound for 0 < p ≤ 2/3, while its gap above unassisted capacity remains strictly positive below the entanglement-breaking threshold.

7.1 Flagged and erasure simulation bounds

  • The flagged simulation chooses the depolarizing entanglement-breaking threshold channel, and covariance lets one input ensemble optimize both terms of its bound.The uniform computational-basis ensemble simultaneously maximizes input entropy and output Holevo information at that threshold.Source: §7.1 · p. 26

7.2 Reduction to one real variable

  • Unitary covariance and Pauli averaging reduce UF(Dp) to a scalar optimization over input Bloch-vector length, with the entanglement term evaluated by Wootters’ formula.The reduction uses the two-qubit concurrence κ√(1−r^2).Source: §7.2 · pp. 26–27
  • For p ≥ 2/3, UF(Dp) equals the exact feedback capacity 1−h2(p/2); for smaller noise, it can exceed the unassisted rate.The equality is in the entanglement-breaking regime.Source: §7.2 · p. 27

7.3 Comparison with the published converse

  • For 0 < p ≤ 2/3, C(Dp) ≤ C←(Dp) ≤ UF(Dp) ≤ Uflag(p, 2) < Υ(Dp), improving the published converse for ordinary capacity.The comparison does not establish an improved strong-converse threshold.Source: §7.3 · p. 28, Figure 7 · p. 29

7.4 Why a small gap is not equality

  • UF(Dp) exceeds the unassisted capacity for every 0 < p < 2/3, although the gap becomes extremely small near the threshold.The gap is about 2.2118783 × 10^-4 bits/use at p=1/2 and about 2.7549 × 10^-15 at p=0.6; positivity is proved analytically.Source: §7.4 · pp. 28–29, Figure 8 · p. 30
  • A constant-weight entanglement penalty cannot yield a universal sharp per-use increment, an obstruction to this potential method rather than evidence of a feedback advantage.For any fixed η > 0, a sufficiently weakly mixed Pauli-orbit ensemble gives a one-use increase above C(Dp).Source: §7.4 · p. 29

8 An exact evaluation for amplitude damping

The section evaluates the feedback potential for amplitude damping exactly and compares its bound with other channel benchmarks. It proves the evaluation by removing input coherences and establishing convexity, while distinguishing the mathematical environment measurement from receiver access.

8.2 Removing coherences and proving convexity

  • Discarding input coherences lowers the relevant penalty, and the resulting scalar penalty is convex in the excited-state population.A fixed environment measurement proves convexity; it is not performed by a receiver in the communication protocol.Source: §8 · p. 30, §8.2 · pp. 31–32

8.1 A determinant representation

  • A determinant parametrization reduces the amplitude-damping penalty to a scalar difference and yields an analytic evaluation of the full zero-temperature UF optimization.The result is related to environment-correlation penalties for generalized amplitude damping [Fan26].Source: §8.1 · pp. 30–31

8.3 The exact channel-bound formula

  • For every 0≤γ≤1, the UF potential is evaluated exactly and upper-bounds ordinary feedback-assisted capacity.Dephasing and convexity remove the ensemble optimization; the endpoint cases are noiseless at γ=0 and a replacer at γ=1.Source: §8.3 · p. 33
  • Tightness of the channel-bound optimization does not establish an achievable classical rate, because its optimizer can create entanglement without sending message information.The entropy in (8.13) also occurs as the Choi entanglement of formation and, in recent work, as the parallel entanglement cost [TZBW26].Source: §8.3 · p. 33

8.4 Benchmarks and numerical comparison

  • For 0<γ<1, the UF bound strictly improves on (8.19); at γ=1/2, UF is 0.6008760367 bits/use versus 0.4717293906 for unassisted capacity and 0.7715533032 for Cβ.The UF curve is an exact evaluation, and its smaller threshold is proved for ordinary feedback-assisted capacity only.Source: §8.4 · p. 34, Figure 11 · p. 35

8.5 Why revealing a damping event is too generous

  • Giving receivers the Kraus-outcome flag makes the channel transmit one bit perfectly for every γ, so discarding that flag yields only a trivial upper bound.The environment measurement used to prove convexity must not be interpreted as free receiver access to the environment.Source: §8.5 · p. 35

9 Randomness distillation and a one-way state potential

This section develops an entanglement-of-formation state potential for one-way source-based shared-randomness distillation, where communication is charged against the yield, and relates its mixed-convex-roof form to a fixed-average Choi optimization that preserves Alice’s unconditional marginal. Damping examples illustrate an exactly solved benchmark and a full-rank case where the bound informs a rate that is not evaluated.

9.1 One-way and two-way distillable randomness

  • One-way distillation subtracts the full classical-message cost from shared randomness, whereas two-way protocols subtract the costs of all messages; neither task requires secrecy.Outputs must be nearly uniform and nearly identical, and no shared random seed is supplied for free.Source: §9 · p. 35, §9.1 · p. 37

9.2 A state analogue of the channel potential

  • The one-way state potential equals the mixed-convex-roof quantity GE on the purifying system’s joint state with Bob.Instruments on the purifying system generate ensemble decompositions, while retaining a purification of each conditional component makes the mixed-roof minimization attainable.Source: §9.2 · p. 38

9.3 A finite-blocklength one-way converse

  • The finite-blocklength one-way converse bounds net randomness using the state potential, with a weak-converse factor and an error correction independent of the communication alphabet size.For ε ≤ 1/4, the additive correction is h2(√ε) and the factor is 1 − √ε; the bound is specific to one-way protocols.Source: §9.3 · pp. 40, 42

9.4 Choi sources and the fixed average input

  • The state and channel potentials are related by a fixed-average Choi correspondence, and the average-input constraint cannot generally be dropped.The correspondence optimizes over input states τ_x whose probability-weighted average is π.Source: §9.4 · p. 42
  • The construction realizes every admissible ensemble with Alice retaining its conditional purification, proving both directions of the optimization identity and the final inequality.The final inequality follows because (2.26) optimizes over all average inputs.Source: §9.4 · p. 43
  • Alice can steer the transmitted subsystem of a Choi state but cannot change its unconditional marginal using a local instrument.This physical constraint is retained explicitly in the damping examples.Source: §9.4 · p. 43
  • The damping examples contrast an exactly solved benchmark with a full-rank source that provides nontrivial information about an unevaluated regularized rate.Source: §9.4 · p. 43

10 One-way randomness from damping Choi states

Damping Choi states give both an exact benchmark for one-way randomness and a full-rank thermal example where the potential supplies a useful converse without an exact operational rate. The section reduces the thermal source optimization to a scalar convex envelope and certifies bounds that improve on specified earlier state converses.

10.1 Ordinary amplitude damping as an exact benchmark

  • For ordinary amplitude damping, the state potential upper-bounds one-way randomness and is attained by the identity instrument with a constant outcome.The channel bound has a matching state-side lower bound, yielding an exact benchmark for this family.Source: §10.1 · p. 44

10.2 Exact operational rates and their attribution

  • At γ = 1/2, the forward and reverse operational rates are approximately 0.4566992218 and 0.3991239633, while the forward potential is 0.6008760367 bits per copy.The potential is an upper bound, not the operational rate; the receiver-entropy upper bound is looser.Source: §10.2 · p. 45, Figure 13 · p. 45

10.3 A full-rank finite-temperature source

  • The full-rank thermal source has no exact regularized one-way rate established here; its potential yields a certified upper bound smaller than the earlier computable state-Υ restrictions.The section cautions that failure of the cited methods to evaluate the rate does not rule out other exact results.Source: §10 · p. 43, §10.3 · p. 46, §10.6 · p. 49

10.4 Reducing all instruments to a scalar convex envelope

  • For generalized amplitude damping, the fixed-input potential optimization equals the lower convex envelope of a scalar penalty, with at most four mixed input states needed for attainment.No convexity assumption on the penalty itself is required.Source: §10.4 · p. 47

10.5 A certified interval for the thermal example

  • The thermal example’s certified upper and achievable bounds differ by less than 0.020255 bits per source copy, and neither endpoint is claimed to be the exact rate.The affine certificate is checked over the full domain, rather than inferred from sampled instruments.Source: §10.5 · p. 48

10.6 Comparison with previous state converses

  • For ρ⋆, the upper bound 0.45252360 is strictly below both optimized state-Υ restrictions, without establishing an ordering against unrestricted Γ.The comparison concerns the vanishing-error one-way rate, not a two-way or strong-converse bound.Source: §10.6 · p. 49, Table 2 · p. 50

10.7 A nearby state outside the thermal family

  • For the nearby perturbed state, the stated upper curve is a convexity estimate, not an exact evaluation of the state potential.The convexity argument is limited to mixtures with the same reference marginal.Source: §10.7 · pp. 50–51, Figure 15 · p. 51

10 One-way randomness from damping Choi states

  • The thermal examples demonstrate a useful one-way converse even though their operational rates are not exactly solved.The ordinary amplitude-damping family remains the exact benchmark against which potential slack can be measured.Source: §10 · p. 43, §10.7 · p. 51

11 Isotropic states and the state–channel bridge

Qubit isotropic states provide an exact one-way randomness benchmark and identify their state potential with a constrained depolarizing-channel quantity. The potential is strictly above the one-way rate for non-entanglement-breaking noise, but agrees with it in the entanglement-breaking regime.

11.1 Exact one-way distillable randomness

  • The exact one-way randomness rate for every qubit isotropic state is 1 − h2(p/2), in either communication direction.Measuring each share in the computational basis attains the rate, while additivity bounds collective measurements.Source: §11.1 · p. 52

11.2 Equality of the static and dynamical potentials

  • The isotropic-state potential equals the depolarizing-channel potential for every p.The Choi-instrument correspondence identifies the state optimization with the channel objective constrained to average input I/2.Source: §11.2 · pp. 52–53, Figure 16 · p. 53
  • For 0 < p < 2/3 the potential strictly exceeds the exact one-way rate, while for p ≥ 2/3 both are equal.In the latter regime, the depolarizing channel is entanglement breaking.Source: §11.2 · p. 53

12 Two-round randomness distillation from isotropic states

A two-round protocol for qubit isotropic states alternates randomness extraction while preserving quantum side information through the first decoding. It strictly exceeds the exact one-way rate for every nontrivial isotropic state, showing that the one-way potential is not a general two-way converse.

12.1 Bob’s weak measurement and the first message

  • Bob weakly measures first and compresses his outcome using Alice’s quantum system as side information; Alice’s gentle decoder preserves the residual state for the next round.The preserved correlations are the resource used in the second round.Source: §12 · p. 53, §12.1 · p. 54, Figure 17 · p. 55

12.2 Alice’s complementary measurement and the second message

  • After decoding Bob’s outcome, Alice measures in the complementary basis and compresses her new outcome using Bob’s retained system and the shared first outcome as side information.The final shared variable is the pair (X^n, Y^n), with both message lengths subtracted from the gross yield.Source: Figure 17 · p. 55, §12.2 · p. 55

12.3 Achievable rate and strict two-way improvement

  • For every 0 < p < 1, the two-round construction achieves strictly more net randomness than the exact one-way optimum.The strict separation follows analytically from the positive quadratic coefficient in the small-measurement-strength expansion.Source: §12 · p. 53, §12.3 · p. 56
  • At p = 1/2, the fixed measurement strength s = 4/5 gives a numerical improvement over the one-way value 0.188721875541 bits, without requiring numerical optimization.Separately, numerical optimization over the one-parameter protocol family gives approximately 0.21152695 at p = 1/2; it does not evaluate the unrestricted two-way rate.Source: §12.3 · p. 56

12.4 Why the one-way potential cannot be a two-way converse

  • The violation shows that the one-way state potential is not an unrestricted LOCC monotone in the form needed for a two-way converse.Existing Γ bounds and their fidelity-smoothed variants remain valid; this result does not improve those general two-way bounds.Source: §12.4 · p. 57

13 Numerical evaluation and reproducibility

The section distinguishes analytical results from numerical evaluation and consistency checks, including which computations are certificates and which are not. It provides reproducibility details for the depolarizing, damping, and randomness-distillation calculations.

13.1 Stable depolarizing maximization

  • The depolarizing optimization uses a small-eigenvalue coordinate, subdivision estimates, and high-precision recomputation, but its global tolerance is not a formal numerical certificate.The quoted tiny gap at p = 0.6 is supported by high-precision evaluation and the analytic proposition, not the coarser tolerance.Source: §13.1 · p. 58

13.2 Amplitude damping and matrix checks

  • Finite matrix and instrument checks found no positive violations, but sampled tests do not replace proofs for arbitrary memories or convex roofs.The reported checks cover restricted finite-dimensional test families.Source: §13.2 · p. 59

13.3 Randomness-distillation checks

  • At p = 1/2, numerical optimization gives 0.2115269457 bits per copy, while the fixed choice s = 4/5 gives 0.2114185019; the all-parameter strict separation is analytic.The optimizer–grid discrepancy is below 3.9 × 10^-10 bits per copy.Source: §13.3 · p. 59

13.4 Damping-source plots and numerical certificates

  • The thermal curve’s numerical minimization checks the displayed plot only; the affine bound is established on the full continuous domain by an interval calculation.The package includes the rational certificate matrices, partition, plotting data, and regeneration scripts.Source: §13.4 · p. 60
  • The thermal certificates use outward interval arithmetic, while the depolarizing global subdivision is an ordinary floating-point estimate rather than a formal certificate.The thermal computations are not verified by a formal proof assistant.Source: §13 · p. 58, §13.4 · p. 60

14 Conclusion

The paper develops entanglement-of-formation bounds for feedback-assisted communication and one-way randomness distillation, then contrasts them with a two-round protocol that improves on one-way isotropic-state rates. It also identifies limits and open directions for extending these results.

14.1 Summary

  • The exact amortization theorem yields a feedback-capacity converse for finite-dimensional channels without initial shared entanglement, improving a depolarizing-channel converse and evaluating amplitude damping in closed form.The same method determines capacity for a flagged mixture of noiseless and entanglement-breaking channels.Source: §14.1 · p. 60
  • A weighted decoder test gives a finite-blocklength converse for net one-way randomness yield without an error term proportional to communication length.The generalized amplitude-damping Choi state has a one-way rate between 0.43226906 and 0.45252360 bits per copy, while the exact rate remains unevaluated.Source: §14.1 · p. 61
  • A two-round weak-measurement protocol strictly improves on the exact one-way isotropic-state rate throughout the nontrivial noise interval.Gentle first-round decoding preserves quantum correlations for reverse-direction extraction, and the protocol exceeds the one-way state potential in part of the parameter range.Source: §14.1 · p. 61

14.2 Directions for further work

  • The channel converse is weak; a stronger potential would need to preserve free-operation monotonicity and a valid channel-step inequality.The regularized bound inf_k k^-1 UF(N ⊗ k) is also identified as a target for future work.Source: §14.2 · p. 61
  • Future randomness bounds should account for successive communication rounds in both directions while subtracting communication cost; the two-round construction suggests a hierarchy whose convergence remains open.The paper also calls for matching finite-blocklength analysis and notes that neither strict operational feedback advantage for depolarizing or amplitude-damping channels nor optimal two-way isotropic randomness rates are established.Source: §14.2 · p. 61

14.1 Summary

  • The thermal-state comparisons do not order the new bound against unrestricted Γ optimization or replace its two-way and strong-converse properties.For full-rank thermal examples, the exact one-way rate is not established, and exact evaluation of the state potential does not prove equality with that rate.Source: §14.2 · p. 61

Statement on AI-assisted preparation

The appendix records the paper's preparation statement and develops technical calculations and certificates. These include concurrence after qubit channels, a symmetry reduction, a global optimizer analysis, and interval bounds for the thermal source potential.

Statement on AI-assisted preparation

  • The author reports extensive AI assistance, followed by internal review and revisions, while noting that these were not independent peer review or formal verification.Numerical tests served as consistency checks, and responsibility for the mathematical statements, citations, and presentation remained with the author.Source: Statement on AI-assisted preparation · p. 62

A Concurrence after a qubit channel

  • The appendix separates the established concurrence identity from the subsequent channel-bound optimization.For amplitude damping, the channel-induced entanglement of formation is expressed as H((1 − γ) det ρ).Source: A Concurrence after a qubit channel · p. 66

B.1 Symmetry reduction of the comparison set

  • Twirling reduces the depolarizing comparison map and its feasible witnesses to a symmetry-compatible form, preserving the constraint and optimized relative entropy.Covariance permits a maximally entangled reference input in this calculation.Source: §B.1 · p. 67

B.2 Optimizing the commuting divergence

  • The commuting-divergence optimization has an unconstrained regime for p ≤ d/(d + 1) and a boundary optimum when p is larger.The qubit specialization evaluates the limiting Υ expression, not every finite-order Rényi version.Source: §B.2 · p. 68

C Global structure of the depolarizing optimizer

  • The depolarizing scalar objective has a global characterization with at most one interior stationary point, though an optimizer near r = 1 requires careful numerical resolution.The implementation uses δ = (1 − r)/2 and high-precision arithmetic for selected parameters.Source: C Global structure of the depolarizing… · p. 69

D A global certificate for the thermal source potential

  • The thermal-source certificate uses outward interval arithmetic to verify a global inequality rather than relying on a numerical search to exclude better ensembles.The certificate need not locate a stationary point or prove convexity of the minimized penalty.Source: D A global certificate for the thermal… · p. 69, §D.1 · p. 69

D.1 The function to be bounded

  • The certified target is nonnegativity of F* over 0 ≤ t ≤ 1 and 0 ≤ u ≤ t(1 − t).The check establishes a strictly positive margin for a deliberately rounded-down affine function.Source: §D.1 · p. 69

D.2 Two elementary rectangle bounds

  • Two rectangle bounds handle the boundary and differentiable regions, with outward interval evaluation proving positivity across each rectangle.The first bound needs no differentiability at u = 0; the mean-value bound applies away from that boundary.Source: §D.2 · p. 70

D.3 Coverage, margins, and resulting enclosure

The appendices certify the thermal-state potential comparison and prove the formation marginal inequality used in the one-way randomness bound. They detail interval verification, supporting-hyperplane bounds, and pure- and mixed-state steps.

D.3 Coverage, margins, and resulting enclosure

  • 983 tree nodes cover 475 verified nonempty terminal rectangles and 17 discarded empty rectangles; outward intervals certify the operational upper estimate separately from the feasible-ensemble evaluation.The smallest verified residual margin exceeds 8.3 × 10^-9.Source: §D.3 · p. 70

E Lower certificates for the earlier state converses

  • Global supporting-hyperplane certificates bound both earlier state-Υ restrictions without requiring the chosen reference operator to be an optimizer or feasible for the older minimization.The target state is full rank, and certificates may use different σ0 and Q operators.Source: E Lower certificates for the earlier… · p. 71, §E.1 · p. 72

E.1 A relative-entropy supporting hyperplane

  • A relative-entropy supporting hyperplane contracts feasible-constraint inequalities into a lower bound for the state-Υ restrictions.The argument uses positivity and partial-transpose constraints, with full rank of ρA and Tr L ≤ 1 supporting the final step.Source: §E.1 · p. 71

E.2 Explicit certificates for the thermal state

  • The explicit thermal-state certificates use block-structured matrices and interval checks of positivity constraints; their decimal values are enclosures, not claims of exact old-program minimizers.The calculation avoids numerical integration or a matrix-logarithm derivative approximation.Source: §E.2 · p. 72

E.3 Certificates under an additional small perturbation

  • At η = 1/100, the supporting-hyperplane argument retains the stated comparison for a nearby perturbed state without assuming continuity of either optimizer.The perturbed marginal is used when calculating tB.Source: §E.3 · p. 73

F A self-contained proof of the formation marginal inequality

  • The formation marginal inequality is proved from definitions so the finite-blocklength argument does not require it as an additional assumption.The proof measures only E2 and relates its classical outcome information and remaining average entropy to the two-pair formation terms.Source: F A self-contained proof of the… · p. 73

F.1 The pure-state inequality

  • For a pure four-partite state, rank-one POVMs on E2 suffice, and the measurement leaves the unconditional state of the unmeasured systems unchanged.Refining a POVM cannot decrease outcome–B2 mutual information.Source: §F.1 · pp. 73–74

F.2 Passing to an arbitrary mixed state

  • Averaging the pure-state inequality over decompositions yields the mixed-state bound without additivity or strong superadditivity assumptions; the second marginal contributes G_E2, not E_F(B2; E2).The resulting iteration gives the finite-copy upper bound.Source: §F.2 · p. 75
Loading 2610.04831v1…