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Finite-blocklength classical communication over the quantum erasure channel with and without classical feedback

Mark M. Wilde

arXiv:2610.00440v1quant-phcs.ITmath.IT

TL;DR

The paper asks for the finite-blocklength optimum for classical messages over a quantum erasure channel with classical feedback, and whether the corresponding no-feedback converse is operationally attainable. It proves a dimension-based bound and matches it with a classical retransmission protocol, while identifying no-feedback gaps and asymptotic comparisons.

  • Problem

    At finite blocklength, it is unclear whether the dimension bound min{1,d^k/M} remains valid with adaptive feedback and quantum memories, or whether one protocol attains it across erasure patterns.

    Source: §1.1 · p. 3, §1.2 · p. 3

  • Method

    The paper proves a dimension-based converse and matches it with a fixed-blocklength protocol that retransmits base-d message digits until acknowledged.

    Source: §1.1 · p. 3, §1.2 · p. 4

  • Results

    Averaging min{1,d^K/M} over the binomial number K of unerased transmissions gives the optimal feedback-assisted success probability for every integer M.

    Source: §1.1 · p. 3, §1.2 · pp. 3–4

  • Takeaways & Limitations

    Classical no-feedback codes retain the bounded-remainder normal approximation and average-success strong-converse exponent, while the feedback-assisted error exponent is exact below capacity.

    Source: §9.1 · p. 41

    The optimal unassisted quantum success probability remains unknown even for four messages and four qubit-channel uses; the tetrahedral code is only a lower bound.

    Source: §9.2 · p. 42

Abstract

from arXiv · show

We determine the optimal success probability for transmitting a fixed number of classical messages through a finite number of uses of the quantum erasure channel, assisted by noiseless classical feedback and without initial shared entanglement. For an input dimension $d$, an erasure probability $p$, a blocklength $n$, and $M$ equiprobable messages, the optimal success probability is equal to $E[\min\{1,d^K/M\}]$, where $K$ is binomial with parameters $n$ and $1-p$. The converse allows arbitrary adaptive quantum encoders, quantum memories, and receiver instruments. Its main ingredient is an elementary dimension bound for noiseless quantum communication with classical feedback, proved by fixing the classical controls without conditioning the sender's state on the observed transcript. A classical protocol that transmits the base-$d$ digits of an integer representing the message, repeating each digit until the receiver acknowledges its reception or the prescribed blocklength is reached, attains the bound for every integer $M$. We also establish a relative-majorization property of erasure-channel outputs and exactly evaluate a hypothesis-testing converse, recovering the same numerical bound without feedback. That converse need not be achievable without feedback: four uses of the qubit erasure channel and four messages give a strict gap. A product-state code employing tetrahedral qubit states nevertheless outperforms every classical binary erasure code with these parameters. We give an explicit message-size formula and show that the bounded-remainder normal approximation and the average-success strong-converse exponent are unchanged without feedback. We also determine the feedback-assisted error exponent below capacity, prove that it agrees with the no-feedback exponent above a critical rate, and give no-feedback bounds at lower rates.

1 Introduction

The paper asks whether a finite-blocklength dimension bound survives arbitrary classical feedback and whether one code attains it for every erasure pattern. It answers these questions, develops no-feedback converse and asymptotic results, and situates them alongside prior work.

1.2 Summary of results

  • Fixing feedback controls without conditioning the sender’s state on the transcript lets a dimension bound apply despite adaptive encoders, quantum memories, and receiver instruments.The converse fixes the erasure pattern first, applies the noiseless-feedback bound to surviving transmissions, then averages over patterns.Source: §1.2 · p. 3, §1.1 · p. 3
  • Averaging the fixed-pattern bound min{1,d^k/M} over the binomial number K of surviving transmissions gives the optimal feedback-assisted success probability, attained for every integer M.The result concerns arbitrary message sizes and does not require initial shared entanglement.Source: §1.2 · pp. 3–4, §1.1 · p. 3
  • The repeat-until-received protocol sends base-d digits, repeats each after erasure, and uniformly guesses among compatible messages after the fixed blocklength.Sending least significant digits first makes k successful receptions reveal the message label modulo d^k.Source: §1.2 · pp. 3–4
  • The operational formula also yields a finite-sum maximum message size and extends patternwise to correlated erasures sampled independently of the message.The message-size expression is for a prescribed error tolerance.Source: §1.2 · p. 4
  • Without feedback, relative majorization and an optimized hypothesis-testing converse recover the same numerical bound, but that converse does not establish achievability.The paper notes that the numerical no-feedback converse was already given by Rosati.Source: §1.2 · p. 4, §1.3 · p. 5
  • The paper also treats bounded-remainder normal approximations and strong-converse exponents, while leaving the exact no-feedback reliability function below a critical rate undetermined.Its organization places the channel model first, then the feedback bound and erasure formula, no-feedback converses and separations, and asymptotic results.Source: §1.2 · p. 4, §1.4 · p. 6

1.3 Relation to prior work

  • The paper builds on classical erasure converses and retransmission schemes [PPV11, Theorem 7], while its converse additionally handles adaptive quantum encoders, memories, and protocol-generated entanglement.The authors do not claim the retransmission principle itself as new [DDL+16].Source: §1.3 · p. 5

2 Notation and communication model

This section defines the finite-dimensional channel setting and the communication resources being compared. Erasure flags partition outputs into orthogonal sectors, while the code models fix the forward-use schedule and distinguish feedback-assisted from no-feedback communication.

2 Notation and communication model

  • The paper works with finite-dimensional quantum systems, binary logarithms, and standard channel, instrument, and POVM notation.A POVM is a positive-operator-valued measure whose elements sum to the identity.Source: §2 · p. 6

2.1 The erasure channel and its sectors

  • Erasure flags identify orthogonal sectors indexed by surviving positions; each sector contains the input marginal on its surviving systems, including for inputs entangled across uses.The receiver can identify the sector without disturbing its within-sector state.Source: §2.1 · pp. 6–7

2.2 Codes, feedback, and error

  • Both models use exactly n forward transmissions without initial shared entanglement; feedback permits later inputs to depend on receiver outcomes, but there is no free forward classical channel or backward quantum channel.The schedule is message-independent, and dummy systems may fill remaining uses after decoding becomes possible.Source: §2.2 · pp. 7–8
  • No-feedback codes may use inputs entangled across channel uses, and their average success is optimized over input states and decoding POVMs; maximum error is the largest individual-message error.The section also defines largest feasible message size at a prescribed error tolerance.Source: §2.2 · pp. 7–8

3 A dimension bound for noiseless communication with classical feedback

The section proves a dimension-based success bound for noiseless quantum communication with classical feedback, despite adaptive encoders and receiver instruments. Its key move is to prescribe feedback controls without conditioning on their occurrence, while keeping the transcript probabilities in one receiver POVM.

3.3 An arbitrary number of transmissions

  • For M equiprobable messages, success is at most min{1, D/M} when the transmitted systems have total dimension D.The lemma permits arbitrary local quantum memories and classical feedback, but assumes no initial shared entanglement.Source: §3.1 · p. 8, §3.3 · pp. 12–13

3.1 The elementary bound and the difficulty caused by feedback

  • Prescribing each feedback transcript as classical controls keeps sender states normalized without conditioning on outcomes that could change the message distribution.The receiver's complete instruments form one POVM whose outcomes retain the transcript probabilities.Source: §3.1 · p. 8, §3.2 · p. 10, §3.3 · p. 12

3.2 Three transmissions: the complete construction

  • For three transmissions, commuting operations on disjoint systems yields the actual joint probabilities from transcript-specific states and one complete POVM.This argument does not require product states or postselection on a transcript.Source: §3.2 · pp. 9–10

3.3 An arbitrary number of transmissions

  • The argument extends to any number of transmissions because fixed-control local operations commute even on entangled inputs, while the prescribed transcript probabilities remain intact.The bound excludes initially shared entanglement and backward quantum communication, which can invalidate it.Source: §3.3 · pp. 11–13

4 Exact feedback-assisted performance of the erasure channel

The paper bounds erasure-channel success by conditioning on the message-independent survivor pattern and applying the noiseless dimension bound to surviving systems. A repeat-until-received digit protocol attains the resulting finite-blocklength bound, and the exact error relation yields a maximum-message formula.

4.1 Conditioning on the erasure pattern

  • The optimal feedback-assisted success probability is E[min{1, d^K/M}], where K counts surviving uses among n independent erasures.For a fixed pattern with k survivors, the digit protocol succeeds with probability min{1, d^k/M}.Source: §4.1 · pp. 13–14, §4.2 · p. 15

4.2 A protocol that attains the bound

  • An initial message-independent feedback seed makes the optimal maximum error equal the optimal average error; without it, individual message errors need not match for non-power-of-d M.The average-error formula does not require the seed.Source: §4.1 · p. 14, §4.2 · pp. 15–16

4.1 Conditioning on the erasure pattern

  • Fixing the erasure pattern leaves |S| noiseless d-dimensional transfers, so the dimension bound applies to arbitrary adaptive protocols.The pattern is independent of the message; erased inputs can be discarded locally while Bob receives a locally prepared flag.Source: §4.1 · p. 14

4.2 A protocol that attains the bound

  • The repeat-until-received protocol attains the bound for every integer M by sending least-significant base-d digits first and repeating each digit after an erasure.After k successful receptions Bob learns the residue modulo d^k, and the protocol always uses exactly n forward transmissions.Source: §4.2 · pp. 14–15

4.3 An explicit maximum-message formula

  • For a prescribed error tolerance, the exact error relation gives a largest feasible real message size, which is rounded down to obtain the largest integer message count.The cutoff formulation uses cumulative survivor probabilities and dimension-weighted sums; the formulas apply to average error and to maximum error with an initial seed.Source: §4.3 · pp. 16–18

5 Relative majorization and an exactly evaluable converse

The section constructs a common channel for erasure-channel outputs that preserves every sector-uniform auxiliary state, then uses that invariant family to evaluate a no-feedback hypothesis-testing converse exactly. The evaluated converse is also the fully optimized converse, but need not be operationally attainable.

5.1 A simultaneous invariant-family construction

  • One channel converts the pure reference output to any erasure-channel output while preserving every sector-uniform auxiliary state, regardless of the auxiliary sector weights.The construction acts separately in each erasure sector, and the same channel works for every choice of weights qS.Source: §5.1 · p. 19, Thus the same channel implements · p. 20

Thus the same channel implements

  • Within each nonempty sector, a random-unitary channel maps the pure reference direction to the target marginal while fixing the normalized identity.The sector channels are combined on Bob’s full surviving-system space; their operations may entangle those systems.Source: Thus the same channel implements · pp. 19–20, Figure 5 · p. 20

5.2 The hypothesis-testing converse

  • The no-feedback hypothesis-testing converse applies to codes under either average-error or maximum-error criteria.Preserving a common auxiliary state matters because arbitrary message-dependent channels could instead correlate the auxiliary output with the message register.Source: §5.2 · pp. 21–22

5.3 An explicit optimizer, without a minimax exchange

  • The optimized auxiliary distribution is truncated and dimension-weighted, and a matching test and auxiliary state yield the exact testing value without exchanging optimizations.The resulting converse recovers the same numerical no-feedback bound as the sectorwise dimension argument.Source: §5.3 · pp. 22–23

5.4 The fully optimized classical–quantum converse

  • The fully optimized classical–quantum converse for the erasure channel has this same exact value, attained in the supremum by the uniform ensemble of computational-basis product states.The quantity remains a converse expression, not an operational capacity.Source: §5.4 · p. 23
  • The exact converse can exceed operational success: the next section establishes a strict gap using four channel uses.Source: §5.4 · p. 23

6 A strict gap without feedback

Without feedback, one codeword must supply compatible reduced states for every possible erasure pattern, so the dimension-counting converse is not always attainable. Four qubit uses and four messages give a strict gap and an exact separation in feasible message sizes.

6.1 Four messages and four qubit transmissions

  • Without feedback, one codeword must produce suitable reduced states for every erasure pattern, although the receiver may choose a different measurement for each pattern.Attaining the bound would require perfect discrimination on every two-qubit survivor set; purity and orthogonality constraints make those requirements incompatible.Source: §6 · p. 24, §6.1 · pp. 24–25
  • For n = 4, d = 2, M = 4, and 0 < p < 1, optimal average no-feedback success succ(4, 4) is strictly less than dimension-counting success s4,2,p(4), even for mixed, entangled four-input codewords and arbitrary collective measurements.Source: §6.1 · p. 24

6.2 An exact separation of message sizes

  • At n = 4, d = 2, p = 1/2, and ε = 11/64, feedback permits four messages, whereas without feedback only three are achievable under either error criterion.Four messages are achievable with feedback at success 53/64; three computational-basis messages suffice without feedback, and the four-message no-feedback optimum remains undetermined.Source: §6.2 · pp. 25–26

7 A product quantum code that outperforms every classical code

The paper compares a tetrahedral product-state quantum code with the exact classical binary-erasure optimum and the feedback-assisted value. The quantum code improves on every classical code for these parameters without entangling channel inputs, while the optimal quantum no-feedback value remains between the classical and feedback-assisted values.

7 A product quantum code that outperforms every classical code

  • A product-state code using nonorthogonal tetrahedral signals outperforms every classical binary erasure code for these parameters, without entangling channel inputs.The tetrahedral carrier addresses whichever of three binary discrimination questions is selected by the surviving classical coordinate.Source: §7 · p. 26, §7.3 · p. 29

7.1 Tetrahedral states and the four-carrier encoding

  • The four-message encoding places x, y, and x ⊕ y on three classical carriers and a tetrahedral qubit state on the fourth.Any two classical carriers identify the message; one classical carrier together with the fourth leaves a binary pure-state discrimination problem.Source: §7.1 · pp. 26–27, Figure 6 · p. 27

7.2 The exact classical optimum for the same parameters

  • The optimal four-message classical binary-erasure code has equal optimal average and best worst-message success, with a two-survivor projection deficit of 24 versus 22.Linearity makes every nonempty compatible-message class the same size for each erasure pattern.Source: §7.2 · pp. 28–29

7.3 The three-way separation

  • The quantum no-feedback optimum is not evaluated, but the paper’s bounds place it strictly between the classical and feedback-assisted values.Figure 7 distinguishes the explicit tetrahedral code’s gap from the unknown optimum over all quantum codes without feedback.Source: §7.3 · p. 29

8 Asymptotic consequences of the exact formula

This section derives asymptotic consequences of the exact feedback formula, including a bounded-remainder normal approximation and a common strong-converse exponent. It determines the feedback-assisted error exponent below capacity, while without feedback the exponent is established only at and above a critical rate; lower-rate bounds leave its optimal value and any feedback advantage unresolved.

8.1 A normal approximation with a bounded remainder

  • Corollary 8.1 states that, for every positive integer blocklength n, the normal approximation has a uniform bounded remainder: a finite constant Ap,d,ε exists independent of n.Source: §8.1 · p. 30

8.2 Classical codes without feedback

  • Classical random coding supplies deterministic no-feedback codes, including a thresholded maximum-error construction that controls low-survivor errors without multiplying them by the message-removal factor.The resulting achievability supports the same normal approximation without shared randomness.Source: §8.2 · p. 33, There also exists a deterministic… · pp. 34–35

There also exists a deterministic classical code with average error at most

  • The same bounded-remainder normal approximation holds without feedback for unrestricted quantum codes under both average- and maximum-error criteria.A d-ary classical achievability argument combines with the quantum converse; the expansion also holds for the classical d-ary erasure channel without feedback.Source: There also exists a deterministic… · p. 35

8.3 The common strong-converse exponent

  • For rates above capacity, feedback success decays with exponent min_t [D_bin(t||q) + (R − t log2 d)+], and the same average-success exponent holds without feedback.The no-feedback equality applies to unrestricted quantum codes and classical d-ary erasure codes; the exponent changes form at R = t* log2 d.Source: §8.3 · pp. 36–37
  • The no-feedback strong-converse result concerns average success and does not establish equality of finite-blocklength average and worst-message success probabilities.Equality of limiting exponents likewise does not imply equality of finite-blocklength performance.Source: §8.3 · pp. 36–37

8.4 Error exponents below capacity

  • Below capacity, the relevant asymptotic quantity is the decay of error probability, and the constant-factor success comparison does not imply equality of feedback and no-feedback error exponents.The supplied passage sets the range 0 < R < (1 − p) log2 d but gives no exponent result in this part.Source: §8.4 · p. 37
  • Corollary 8.6 gives the exact feedback-assisted error exponent below capacity, and the same exponent holds for maximum error under the initial-feedback-seed convention.The proof uses the exact error formula and establishes the limit by matching exponential upper and lower estimates.Source: §8.4 · pp. 37–38
  • For d = 2 and p = 1/2, the feedback-assisted error exponent is Dbin(R∥1/2) for 0 < R < 1/2.Figure 10 compares this limiting exponent with finite-blocklength feedback error exponents; the limit is taken at fixed positive rate.Source: §8.4 · pp. 37–38, Figure 10 · p. 39
  • The below-capacity limiting exponent as R decreases to zero is one, but R = 0 itself has zero error for a single-message code.Figure 10 assigns no exponent at R = 0; the distinction reflects the theorem's fixed-positive-rate limit and message-size rounding.Source: §8.4 · pp. 38–39
  • For Rcrit ≤ R < q log2 d, the no-feedback error exponent exists and equals the feedback-assisted exponent.The bounds also apply to maximum error without shared randomness and to classical d-ary erasure codes as well as unrestricted quantum codes without feedback.Source: §8.4 · p. 40
  • Below Rcrit, no-feedback random-coding and expurgation bounds are provided, but they do not determine the exact reliability function or prove a feedback advantage.The random-coding exponent is achievable, while the feedback-assisted exponent supplies a converse even for entangled quantum codewords; expurgation gives a further classical achievable bound.Source: §8.4 · pp. 39–40
  • For d = 2 and p = 1/2, Rcrit = 1/3; from that rate to capacity, the classical and quantum no-feedback exponents are determined exactly.At lower positive rates, Figure 11 displays bounds only, and its R = 0 values are limits as R decreases to zero.Source: §8.4 · p. 40, Figure 11 · p. 41

9 Conclusion

The conclusion summarizes exact feedback-assisted finite-blocklength results and contrasts them with unresolved aspects of unassisted quantum coding. It also identifies open questions about low-rate reliability and feedback restrictions.

9.1 Summary of results

  • The converse covers adaptive encoders, arbitrary quantum memories, and receiver instruments through a noiseless-communication dimension bound.Fixing classical controls gives normalized sender states without conditioning on the transcript; the receiver's instrument tree supplies a POVM, without requiring separability during the protocol or a uniform posterior.Source: §9.1 · p. 41
  • The repeat-until-received protocol attains the fixed-pattern dimension bound for every message size, completing the feedback-assisted finite-blocklength result.The conclusion also reports an explicit message-size formula and matching bounded-remainder normal approximation and average-success strong-converse exponent with and without feedback.Source: §9.1 · p. 41
  • The feedback-assisted error exponent is exact at every positive rate below capacity; the no-feedback exponent agrees above a critical rate, with lower bounds below it.The bounded-remainder normal approximation and average-success strong-converse exponent are unchanged without feedback.Source: §9.1 · p. 41
  • Without feedback, the exactly evaluated hypothesis-testing converse is not generally attainable: four messages and four qubit transmissions give a strict gap.For those parameters, a tetrahedral product-state code strictly improves on every classical binary erasure code.Source: §9.1 · p. 41

9.2 Directions for future research

  • The exact unassisted quantum success probability remains unknown even for four messages and four qubits; the tetrahedral code is a lower bound, not an optimality proof.The conclusion identifies whether other product-state ensembles or entangled codewords improve the result as open questions.Source: §9.2 · p. 42
  • Below the critical rate, the lower and upper no-feedback reliability bounds need not coincide, and possible gains from nonorthogonal or entangled codewords remain open.These questions concern the asymptotic error exponent, separately from the proved finite-blocklength separation.Source: §9.2 · p. 42
  • Characterizing when equality holds in the sectorwise dimension bound may reveal further exact unassisted formulas or quantitative gaps.The stated equality conditions involve simultaneous distinguishability and purity of many reduced code states.Source: §9.2 · p. 42
  • The attaining feedback protocol needs only acknowledgments after each forward use, plus an initial seed for maximum-error symmetrization; delayed or noisy acknowledgments remain open settings.The conclusion proposes studying which backward-communication restrictions preserve an exact formula.Source: §9.2 · p. 42

Statement on AI-assisted preparation

AI tools were used extensively in mathematical and editorial work, while the author directed the research and remains responsible for the manuscript.

Statement on AI-assisted preparation

  • AI assistance covered mathematical exploration, proof formulation and checking, drafting, revision, bibliography checks, and preparation of technical materials.The listed materials include LaTeX, TikZ, and numerical-checking code.Source: Statement on AI-assisted preparation · p. 42
  • The author directed the research problem and successive revisions, supplying detailed comments on mathematical arguments and presentation.Source: Statement on AI-assisted preparation · p. 42
  • The review passes were internal rather than independent peer review or formal verification; numerical checks supported finite-dimensional examples but did not replace proofs.The author accepts responsibility for the mathematical statements, citations, and presentation.Source: Statement on AI-assisted preparation · p. 42

A Symmetry reduction of the fully optimized hypothesis-testing converse

The appendix reduces the optimized hypothesis-testing converse by symmetry and data processing, restricting auxiliary states to a sector-uniform family. The uniform computational-basis ensemble then attains the resulting upper bound.

A Symmetry reduction of the fully optimized hypothesis-testing converse

  • The reduction uses data processing on the joint classical–quantum system, rather than assuming convexity of the hypothesis-testing divergence.Source: A Symmetry reduction of the fully… · p. 45
  • Pinching into erasure sectors and averaging local phase unitaries preserve the first hypothesis while making the auxiliary state diagonal within each sector.The sector weights remain unchanged.Source: A Symmetry reduction of the fully… · p. 45
  • Averaging over cyclic shifts preserves sector weights and restricts the auxiliary states to ones uniform within each erasure sector.The reduction shows that optimizing over this family gives the same infimum as optimizing over all auxiliary states.Source: A Symmetry reduction of the fully… · p. 45
  • The uniform computational-basis ensemble attains the evaluated upper bound, completing the proof of the proposition.Source: A Symmetry reduction of the fully… · p. 46

B The classical expurgated error bound

The appendix derives a low-rate achievability bound by expurgating a random computational-basis code. Pairwise error is controlled by the chance that differing coordinates are erased, yielding a deterministic code and an exponent bound.

B The classical expurgated error bound

  • Classical expurgation gives a low-rate achievability bound using computational-basis inputs, with no shared randomness required.The argument bounds each retained message's error.Source: B The classical expurgated error bound · p. 46
  • For two input words at Hamming distance h_ij, their surviving projections agree only if all differing positions are erased, an event of probability p^h_ij.Uniform guessing among compatible messages bounds an individual message's error.Source: B The classical expurgated error bound · p. 46
  • Drawing 2M independent uniform words and retaining the M smallest-cost messages produces a deterministic code with bounded retained costs.The argument uses that two independent symbols agree with probability 1/d and are independent across coordinates.Source: B The classical expurgated error bound · p. 46
  • For M_n = ⌈2^nR⌉, the construction yields the lower estimate on the error exponent for every fixed s ≥ 1, and taking the supremum over s proves the bound.No interchange of a limit with a varying choice of s is needed.Source: B The classical expurgated error bound · p. 46
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