PaperlayerView exact revision on arXiv

Source-linked AI summary

One-shot entanglement generation and distillation via Petz recovery and binary hypothesis testing

Matthew Can, Mark M. Wilde

arXiv:2610.31501v1quant-phcs.ITmath.IT

TL;DR

The paper asks how reliably distant laboratories can share entanglement over a noisy connection, where Petz recovery for random codes poses a code-dependent inverse-operator challenge. It applies Cheng’s trace inequality before code averaging to obtain one-shot bounds from binary hypothesis testing. The results include a constant-factor, error-preserving conversion to transmission and second-order achievable guarantees.

  • Problem

    Noisy entanglement sharing raises the question of how reliably laboratories can generate, transmit, or distill entanglement, while random-code Petz analysis is complicated by a code-dependent inverse output-state operator.

    Source: §1.1 · p. 3

  • Method

    The proof applies Cheng’s trace inequality before averaging random codes, making the recovery-error bound linear in a comparison operator and leaving a second moment to average.

    Source: §1.2 · pp. 5–6

  • Results

    The bounds cover one-way distillation and entanglement generation, and conversion yields transmission at the same error with target dimension reduced by only a constant factor at fixed error.

    Source: §1.2 · pp. 5–6

  • Takeaways & Limitations

    For arbitrary pure reference–input states, entanglement generation and transmission have the same coherent-information rate and second-order Gaussian correction at the original error tolerance, with an achievable logarithmic remainder.

    Source: §6 · p. 34, §6.2 · p. 36

    The logarithmic remainder is an achievable guarantee, not an asserted optimal third-order term.

    Source: §6.2 · p. 36

Abstract

from arXiv · show

How much entanglement can two distant laboratories share when their connection is noisy? We relate this question to deciding which of two quantum hypotheses is present. Our one-shot bounds apply to entanglement generation over a quantum channel and to one-way entanglement distillation from a shared state. We use Cheng's trace inequality to bound the infidelity of a Petz decoder in terms of a binary hypothesis testing cost. Applying the inequality before averaging over codes reduces the error analysis to computing a second moment of random subspaces. The proof covers every integer target dimension and treats rank-deficient states explicitly. A corollary expresses the achievable entanglement through hypothesis-testing coherent information, with a single logarithmic error slack and exact integer rounding. We also prove an error-preserving conversion from entanglement generation codes to entanglement transmission codes: the target dimension decreases by only a constant factor at fixed error, and the decoder is the original decoder followed by local processing. Consequently, for both entanglement generation and entanglement transmission, arbitrary pure reference-input states yield the same coherent-information rate, second-order Gaussian correction at the original error tolerance, and explicit achievable logarithmic remainder. For commuting testing hypotheses, the logarithmic remainder improves to a constant. These results give a direct route from binary distinguishability to explicit recovery guarantees and finite-resource quantum communication rates.

1 Introduction

The introduction frames noisy entanglement sharing as a question of binary distinguishability and identifies the challenge of analyzing Petz recovery for random quantum codes. It presents one-shot bounds for distillation and entanglement generation, an error-preserving conversion to transmission, and their second-order consequences.

1.1 Background and motivation

  • At finite blocklength, entanglement generation, transmission, and one-way distillation differ in their initial resources and operations, so their error criteria must be distinguished.Generation sends one share of Alice’s prepared state through a noisy channel; transmission encodes one share of a prescribed maximally entangled state; distillation starts from a shared imperfect state.Source: §1.1 · p. 3
  • Petz recovery must preserve quantum superpositions, but its code-dependent inverse output-state operator complicates analysis of random codes.The prior Petz-based bounds addressed one-shot and second-order quantum communication, while the code-dependent inverse calls for a different analysis from classical message discrimination.Source: §1.1 · p. 3

1.2 Summary of results

  • Theorem 3.1 bounds distillation and entanglement-generation infidelity through binary testing, with a specified alternative operator produced by averaging the code construction.The same expression certifies channel entanglement generation for a normalized pure reference–input state and its Petz decoder; a maximally mixed reference also gives transmission with unchanged dimension and error.Source: §1.2 · p. 5
  • Applying Cheng’s trace inequality before averaging makes recovery-error dependence on the comparison operator linear, leaving only a second moment to average.This avoids analyzing the code-dependent inverse while averaging, which the earlier Petz-bound proof handled using weak dephasing and joint convexity.Source: §1.2 · pp. 5–6
  • The hypothesis-testing coherent-information corollary gives a sufficient entanglement-yield condition with one logarithmic error slack and exact integer rounding.For fixed slack, the largest integer target satisfying the condition is certified; the bound is not a claim of optimality over all codes.Source: §1.2 · p. 6
  • An entanglement-generation code converts to transmission at the same error by reducing target dimension by only a constant factor at fixed error.The transmission decoder is the original generation decoder followed by local processing; the dimension loss is bounded independently of channel uses.Source: §1.2 · p. 6
  • For independent repetitions with positive information variance, generation and distillation have coherent-information rates, second-order Gaussian corrections, and achievable logarithmic remainders; transmission has the same form with a possibly larger constant.The logarithmic term is an achievable guarantee, not an optimal third-order coefficient; for commuting testing hypotheses, the remainder improves to a constant.Source: §1.2 · p. 7

1.3 Paper organization

  • The paper proceeds from task definitions and proof tools to one-shot bounds, channel examples, second-order results, and appendices on averaging, conversion, and logarithmic terms.The appendices include a refined integer threshold and an analysis of commuting hypotheses.Source: §1.3 · p. 8

2 Operational setting and tools

This section defines the communication and distillation tasks and develops tools to analyze their performance. It uses operator inequalities to establish hypothesis-testing positivity and support conditions and to bound an averaged operator after removing coherence between subspaces.

2 Operational setting and tools

  • The section distinguishes physical decoding from the joint binary effect used only to certify decoder performance.The decoder acts on Bob’s system; the testing effect is an analytical tool, not a required laboratory operation.Source: §2 · p. 9

2.1 Notation

  • Finite-dimensional operator conventions use support inverses set to zero on the kernel unless regularization is specified.Information logarithms have base two.Source: §2.1 · p. 9

2.2 Coding tasks

  • The three protocols target the same maximally entangled state but differ in their initial resources and allowed operations.Generation uses a prepared reference–input state, transmission uses an encoded logical input, and one-way distillation starts from a shared state with forward classical communication but no message from Bob to Alice.Source: §2.2 · pp. 9–10

2.3 Binary testing and Cheng’s trace inequality

  • Cheng’s trace inequality turns a binary testing cost into a bound on the decoding trace expression and permits averaging before handling an inverse operator.For a fixed effect, the relevant expression is linear in the comparison operator Γ_AB, so code choices can be averaged without averaging an inverse.Source: §2.3 · p. 11

2.4 Petz recovery and its coding representation

  • The completed Petz recovery channel gives an explicit decoder from a reference input state and the noise channel, with a block representation in terms of the received joint state.Its completion term makes the decoder a channel on Bob’s full output space, including outside the reference output’s support.Source: §2.4 · pp. 11–13

2.5 A decoder specified by the received joint state

  • For any received joint state, the state-dependent decoder is a channel whose fidelity has a trace formula independent of the completion state, including for singular Bob marginals.The completion term vanishes on the state being decoded; the trace expression depends on the joint state and the inverse square root of Bob’s marginal.Source: §2.5 · pp. 14–15

2.6 Removing coherence between subspaces

  • Phase averaging records which subspace was selected while preserving superpositions within it, by removing coherence between distinct outcome labels.The phase-average order bound compares a positive operator with its block-diagonal version, and phase-and-shift averaging also yields a dimension bound.Source: §2.6 · p. 16
  • Discarding the other positive semidefinite summands in the unitary-averaging proof yields X_LB/m^2 ≤ I_L⊗X_B/m, as in [29, proof of Proposition 3.5].The (a, k) = (0, 0) summand is X_LB.Source: §2.6 · p. 17
  • The inequality d I_A⊗ρ_B − ρ_AB ≥ 0 verifies positivity of the alternative in (1.5) and supplies support inclusions needed for inverses in the proof.Source: §2.6 · p. 17

3 One-shot coding theorem and proof

This section establishes one-shot binary-testing bounds for entanglement distillation and generation at integer target dimensions. The proof uses Cheng’s trace inequality and a Haar moment to bound recovery error, then applies a fixed binary test to select deterministic protocols with distinct implementations for distillation and generation.

3 One-shot coding theorem and proof

  • Theorem 3.1 gives binary-testing achievability for one-way entanglement distillation and entanglement generation at integer target dimensions, with the target m = 1 handled separately.The distillation protocol uses at most d classical outcomes from Alice; generation uses a fixed normalized pure state, with no classical side channel or runtime postselection.Source: §3 · p. 17

3.1 Proof strategy and the encoding instruments

  • Two quantum encoding instruments cover both m dividing d and arbitrary integer m without enlarging Alice’s input space.The constructions retain coherent amplitudes within each selected subspace; the cyclic-window construction gives equal coverage of every basis vector.Source: §3.1 · p. 19

3.2 The fixed-code bound from Cheng’s trace inequality

  • Cheng’s trace inequality bounds fixed-code decoding error by a binary-testing cost affine in one comparison operator, allowing a fixed test before code averaging.This avoids averaging recovery channels or code-dependent inverse operators, while retaining a quadratic improvement in the testing error.Source: §3.2 · p. 20

3.2.1 Proof for orthogonal blocks

  • For orthogonal blocks, pinching removes coherence only between code spaces, preserving within-block recovery fidelity and establishing the fixed-code bound.The proof also retains the quadratic correction.Source: §3.2.1 · p. 22, Figure 5 · p. 22

3.2.2 Proof for cyclic windows

  • For cyclic windows, the proof uses an orthogonal outcome register and an isometric-embedding trace lemma to establish the same fixed-code bound.The lemma evaluates inverses on their supports and treats singular operators; the argument retains the actual outcome probabilities rather than assuming a maximally mixed source marginal.Source: §3.2.2 · pp. 23–24

3.3 A single Haar moment evaluates the comparison operator

  • A two-copy Haar identity evaluates the average comparison operator using the identity and swap terms, without requiring independence among the encoded projectors.The passages identify the identity term with I_A ⊗ ρ_B and the swap term with ρ_AB.Source: §3.3 · pp. 24–25
  • Tensor-product spanning and linearity extend the Haar-moment identity to arbitrary operators without a transpose or reality assumption.The derivation uses trace duality, cyclicity, and the swap trace identity.Source: §3.3 · p. 26
  • Orthogonal blocks and cyclic windows yield exactly the same average alternative.The derivation notes that the factor r cancels.Source: §3.3 · p. 27

3.4 Completing the theorem and choosing physical protocols

  • For every 2 ≤ m ≤ d, the cyclic construction lets the proof fix an optimal binary test while averaging over codes, then select a deterministic protocol.Only a linear testing expression is averaged; the proof does not average an inverse operator or interchange decoder maximization with expectation.Source: §3.4 · p. 27
  • In distillation, Alice sends the outcome of a complete instrument and Bob uses the corresponding decoder, with no shared randomness.The instrument includes all d outcomes, even those with zero probability.Source: §3.4 · p. 27
  • In entanglement generation, fixing an outcome specifies a state to prepare, so the protocol needs neither runtime postselection nor a classical message.Bob decodes with the fixed channel associated with that outcome.Source: §3.4 · pp. 27–28

4 Hypothesis-testing and entanglement transmission consequences

The section turns the one-shot binary cost into integer entanglement-yield guarantees, then extends them to entanglement transmission. The simple bound gives exact integer rounding; the full finite-dimensional bound retains additional terms, and code conversion preserves error while reducing target dimension.

4.1 A simple integer dimension from one hypothesis test

  • Theorem 3.1’s binary cost yields an explicit integer target dimension at a chosen infidelity, with a trivial target when the threshold is too small.The construction reserves part of the error tolerance for the theorem’s remaining error term.Source: §4.1 · p. 28

The resulting entanglement yield is exactly

  • The simple yield bound distinguishes a real-valued rate condition from an achievable integer dimension and rounds down exactly.When the threshold is below 2, the stated rule uses the zero-error target m = 1.Source: The resulting entanglement yield is… · p. 29

4.2 Keeping the full finite-dimensional bound

  • The full finite-dimensional bound retains terms discarded by the simple corollary and can certify a dimension no smaller than that corollary’s choice.It is the strongest dimension certified by this bound, not an optimality claim over all codes; optimizing the testing tolerance recovers the theorem exactly.Source: §4.2 · pp. 30–31

4.3 Entanglement transmission and code conversion

  • With a maximally mixed reference marginal, entanglement generation guarantees also give isometric entanglement transmission at the same dimension and error.The selected input has a maximally mixed retained marginal, enabling the stated transmission specialization.Source: §4.3 · p. 31
  • For arbitrary reference marginals, conversion preserves the error tolerance by reducing target dimension by only a constant factor, with the original decoder followed by local processing.The bounded log-dimension loss leaves first- and second-order terms and an explicit logarithmic term intact at fixed error.Source: §4.3 · p. 32

5 Examples and quantitative scope

The examples test whether the guarantees capture exact recovery and how closely they track a selected Petz decoder. The noiseless case is certified at zero error, while the erasure example leaves room for more channel-specific analysis.

5.1 A noiseless channel

  • The theorem certifies zero error for every target dimension m ≤ d on a noiseless d-dimensional channel.Retaining the subtraction of ρ_AB in the comparison operator is essential for this exact conclusion.Source: §5.1 · p. 33

5.2 An erasure channel

  • For the erasure channel, the selected-code Petz decoder’s exact infidelity is no larger than the general guarantee, leaving room for channel-specific analysis.The example compares the explicit selected-code error with the theorem’s bound, even after removing the conversion prefactor.Source: §5.2 · p. 33

5 Examples and quantitative scope

  • The results concern finite-dimensional infidelity under arbitrary joint decoding, and the code-existence proof supplies no efficient circuit for the Petz channel.The simple testing corollary is not claimed to uniformly improve every smooth-entropy bound without matching task and error parameters.Source: §5.2 · p. 33

6 Achievable second-order asymptotics

The section derives fixed-input and fixed-source achievable bounds from hypothesis testing, then uses them to establish second-order guarantees for entanglement generation, distillation, and transmission. It also treats zero variance exactly and illustrates finite-blocklength bounds for an erasure channel.

6 Achievable second-order asymptotics

  • The asymptotic argument applies the one-shot bound to tensor powers with a slowly vanishing error slack, yielding a fixed-source or fixed-input rate rather than an optimized capacity formula.The slack approaches the target error so the Gaussian quantile is evaluated at that original tolerance.Source: §6 · p. 34, §6.2 · pp. 35–36

6.1 Coherent information, variance, and a testing estimate

  • The testing estimate gives a uniform finite-sample bound for varying tolerances, using a state-dependent constant when coherent-information variance is positive.The constant depends on the fixed state, not on the tolerance or blocklength.Source: §6.1 · pp. 34–35

6.2 Second-order achievability at fixed infidelity

  • The bounds retain the coherent-information rate and Gaussian correction at the original infidelity, with an explicit achievable logarithmic remainder.The remainder is an achievable guarantee, not an assertion of an optimal third-order term.Source: §6 · p. 34, §6.2 · p. 36

6.3 Entanglement transmission at the same error for arbitrary reference– input states

  • For arbitrary pure reference–input states, entanglement transmission preserves the error tolerance and the coherent-information and Gaussian terms, with only a constant log-dimension loss.The decoder is the completed Petz decoder followed by local processing; the composite decoder need not itself be the canonical Petz decoder.Source: §6.3 · pp. 37–38

6.4 The zero-variance case

  • At zero variance, the hypothesis-testing coherent information is exactly nI_c − log2(1 − δ); when I_c = log2 d, the full cost certifies perfect recovery at dimension d^n.The zero-variance analysis is handled directly rather than by taking a limit of the positive-variance bound.Source: §6.4 · pp. 38–39

6.5 An erasure-channel illustration

  • For the erasure example, I_c = (1 − 2p) log2 d and V_c = 4p(1 − p)(log2 d)^2; commuting hypotheses improve the logarithmic remainder to a constant.The n-copy test can also be evaluated using a binomial erasure count, without constructing exponentially large matrices.Source: §6.5 · pp. 39–40

7 Conclusion

The conclusion connects entanglement recovery to binary hypothesis testing and summarizes explicit achievable guarantees based on Petz recovery. It also identifies boundaries and open questions, including the non-optimal status of the general logarithmic remainder and the implementation cost of the decoder.

7.1 Summary

  • A binary test bounds the error of explicit entanglement protocols for both channel generation and one-way distillation, without requiring Alice and Bob to implement the test jointly.The decoder is based on Petz recovery.Source: §7.1 · p. 40
  • Applying Cheng’s trace inequality before averaging over codes avoids averaging code-dependent inverse operators and reduces the analysis to an average over quantum subspaces.The resulting bounds cover every integer target dimension and retain finite-dimensional terms needed to certify perfect recovery in the noiseless example.Source: §7.1 · p. 40
  • At fixed error, converting generation to transmission reduces target dimension by only a constant factor, retaining the coherent-information rate and Gaussian correction for arbitrary auxiliary reference–input states.When the auxiliary reference marginal is maximally mixed, the direct construction needs neither dimension reduction nor extra processing after the Petz decoder.Source: §7.1 · p. 41
  • The logarithmic remainder is an achievable guarantee, not generally a matching limit; commuting testing hypotheses yield a constant remainder.The results provide explicit recovery constructions and bounds, rather than generally matching limits on the best possible codes.Source: §7.1 · p. 41

7.2 Future directions

  • Open directions include sharpening channel-specific guarantees, comparing bounds under matched tasks and error conventions, improving noncommuting third-order guarantees, and finding efficient decoder implementations.The Petz formula specifies a decoder but not an efficient implementation; structured-code approximations would need a provable fidelity loss.Source: §7.2 · pp. 41–42

AI assistance

The authors describe AI assistance during manuscript development and distinguish its internal checks from independent review or formal verification.

AI assistance

  • AI assistance covered proof formulation, exposition, literature checks, figure preparation, and numerical tests.The authors state that responsibility for any submitted version and its claims rests with them.Source: AI assistance · p. 42
  • The checks performed during AI assistance were internal reviews, not independent external peer review or formal verification.Source: AI assistance · p. 42
  • The authors retain responsibility for any submitted version and its claims.Source: AI assistance · p. 42

A Proof of the isometric-embedding trace inequality

The proof derives an inverse-square-root inequality for an isometric embedding, then handles singular operators by regularization and a support limit. Positivity of trace products supplies the final trace comparison without requiring commutativity.

A Proof of the isometric-embedding trace inequality

  • Restricting a quadratic maximization to vectors in the isometry’s range yields an inverse inequality, which an integral representation converts to an inverse-square-root inequality.The maximizing vector for a positive definite operator M is M^-1|v⟩.Source: A Proof of the isometric-embedding… · p. 44
  • The trace step uses positivity of Tr[XY] for positive semidefinite X and Y, so the comparison does not require the operators to commute.Source: A Proof of the isometric-embedding… · p. 45
  • For singular operators, adding ηI regularizes the operators; taking η down to zero recovers support inverses without asserting an order relation on unused kernel directions.Only strictly positive eigenvalues contribute to the traces.Source: A Proof of the isometric-embedding… · p. 45

B An elementary proof of the two-copy unitary average

The Haar two-copy average is derived through elementary invariance arguments rather than representation theory. Phase changes and permutations restrict its form, while a superposition and trace constraints determine the coefficients.

B An elementary proof of the two-copy unitary average

C Evaluating the refined integer dimension

The fixed-tolerance corollary converts the stated guarantee into an exact integer-dimension check, with separate treatment of quadratic, capped, and degenerate cases. Optimizing the tolerance recovers the theorem's full guarantee.

C Evaluating the refined integer dimension

  • The fixed-tolerance formulas evaluate Theorem 3.1's bound, not the optimal performance of all physical codes.They retain exact integer rounding and the quadratic correction.Source: C Evaluating the refined integer… · p. 47
  • In the uncapped case, the error condition becomes a quadratic inequality in the target dimension, with a nonnegative leading coefficient.The coefficient is A_δ = a_δ(1 − a_δ/d) ≥ 0.Source: C Evaluating the refined integer… · p. 47
  • When the affine term u_δ(m) exceeds 1, dimensions require a separate cap check; the quadratic cannot be extrapolated beyond that cap.The exact equivalence also handles each dimension directly, while the root formula avoids subtracting nearly equal numbers.Source: C Evaluating the refined integer… · p. 47
  • If A_δ = 0, the condition becomes linear, and h_δ = 0 certifies every dimension when δ ≤ ε.The noiseless case h_0 = 0 certifies the full dimension d at zero error; optimizing the fixed-tolerance conditions recovers the full guarantee.Source: C Evaluating the refined integer… · p. 47

D Error-preserving conversion from entanglement generation to entanglement transmission

An entanglement generation code can be converted into an entanglement transmission code while preserving its error tolerance and reducing target dimension by only a constant factor. The construction changes the code, not the physical channel, and its finite-dimensional rounding costs only a bounded log-dimension amount at fixed error.

D Error-preserving conversion from entanglement generation to entanglement transmission

  • A constant-factor reduction in target dimension suffices to convert entanglement generation into entanglement transmission without increasing error.The construction changes the code rather than the physical channel and does not condition successful communication on a measurement outcome.Source: D Error-preserving conversion from… · p. 48
  • Unitary mixing of Kraus operators concentrates the achieved overlap into one Kraus operator without changing the channel.The mixing is a mathematical representation choice, not a physical operation on an environment.Source: D Error-preserving conversion from… · p. 48
  • The construction encodes and decodes along singular-vector directions of the concentrated Kraus operator.Bob follows the original decoder with local processing, while the reference is left untouched.Source: D Error-preserving conversion from… · p. 49
  • The scalar singular-value estimate establishes the fidelity guarantee when the target dimension is at most a.The proof bounds the singular values and combines the resulting endpoint estimate with the overlap bound.Source: D Error-preserving conversion from… · pp. 49–50
  • When the target dimension exceeds a, completing the encoder still yields the guarantee, and the integer k is bounded by M.The proof also covers the isometric choice and uses a trivial zero-error protocol when k = 1.Source: D Error-preserving conversion from… · p. 50
  • At fixed ε, the rounded dimensions are at least max{1, floor(M(1 − ε))}, so preserving error costs only a bounded amount in log-dimension.The exact integers are retained at finite dimension; the sharper asymptotic relation applies when M grows without an input-dimension cap.Source: D Error-preserving conversion from… · p. 50

E Logarithmic terms: testing versus coding

The appendix separates the coding remainder from the logarithmic term in classical testing and gives a constant-remainder result for commuting hypotheses. It cautions that a fixed-tolerance classical testing expansion is not itself a coding yield or a substitute for the binary-cost analysis in the noncommuting case.

E Logarithmic terms: testing versus coding

  • The appendix distinguishes the achievable coding remainder from the classical testing logarithmic term and notes that neither remainder is asserted optimal.For commuting hypotheses it proves a constant-remainder bound.Source: E Logarithmic terms: testing versus… · p. 50

E.1 What the classical testing expansion does and does not imply

  • The fixed-error classical testing expansion is not a coding yield, and for noncommuting hypotheses it cannot replace Lemma 6.1.Source: §E.1 · p. 51
  • A fixed-tolerance expansion alone does not justify the n-dependent tolerance substitution; the authors instead analyze the binary cost directly.Their analysis uses the likelihood-tail estimate underlying refined classical testing [35, Theorems 1.1 and 1.3].Source: §E.1 · p. 51

E.2 A constant remainder for commuting hypotheses

For commuting testing hypotheses with positive coherent information and variance, the proposition achieves integer target dimensions with infidelity at most ε and a constant logarithmic remainder. The next subsection explains why this bound does not establish a positive logarithmic coding term.

E.2 A constant remainder for commuting hypotheses

  • For commuting hypotheses with positive coherent information and variance, integer target dimensions achieve infidelity at most ε with a constant remainder for sufficiently large n.The result applies when the hypotheses commute and the state has coherent information between 0 and log2 d; exact integer rounding loses less than 1.Source: §E.2 · pp. 51–52
  • For arbitrary reference marginals, entanglement transmission retains the bound’s form with a possibly larger constant, and no lattice assumption on log-likelihood values is required.For maximally mixed reference marginals, the corollary gives entanglement transmission directly.Source: §E.2 · p. 51
  • The proof analyzes the full binary testing cost in a common eigenbasis, using a likelihood variable with mean nI_c and variance nV_c to bound the error.Berry–Esseen bounds the likelihood distribution, and interval bounds control the expectation in the testing cost.Source: §E.2 · pp. 51–52
  • The commutativity condition concerns the testing hypotheses, not whether the operational state is unentangled; dephasing provides an example with commuting hypotheses and an entangled state.For the dephasing channel, a sharper positive 1/2 log2 n term is known; this proposition instead gives a common argument for states satisfying its commutativity assumption [17].Source: §E.2 · p. 53

E.3 Why this does not prove a positive logarithmic coding term

  • The stated error bound does not establish a positive logarithmic coding term: its finite-dimensional subtraction and quadratic correction cannot cancel the proposed positive contribution.This limits the stated bound, not the optimal code or actual Petz-decoder performance.Source: §E.3 · p. 53
Loading 2610.31501v1…