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Quantum estimation, channel orders, and private capacity

Christoph Hirche

arXiv:2610.03706v1quant-phcs.ITmath.IT

TL;DR

Whether zero private capacity requires antidegradability, and how zero-capacity channels can activate together, remain central questions because private capacity may require control over arbitrarily many channel uses. This paper connects channel-order comparisons to binary estimation, then establishes comparison criteria and applies them to channel families with zero capacity or activation. Complete comparisons rule out activation with antidegradable helpers, while the Pauli and coherent-ladder constructions extend half-erasure activation to 1/2 ≤ p < 1.

  • Problem

    Determining zero private capacity can require checking every blocklength, while recent constructions show zero-capacity channels may be non-antidegradable or useful together.

    Source: §1 · pp. 2–3

  • Method

    The paper uses binary-estimation MMSE and measured-χ2 comparisons to derive channel-order implications, signed-lift criteria, and a reliability–secrecy bound.

    Source: §1 · pp. 3–4, §6 · p. 29

  • Results

    Complete comparisons prevent activation with antidegradable helpers, whereas Pauli and coherent-ladder constructions activate with a qubit erasure channel throughout 1/2 ≤ p < 1.

    Source: §1 · p. 4, §6 · p. 29

  • Takeaways & Limitations

    The examples distinguish reference-stable complete comparison, which rules out antidegradable-helper activation, from regularized comparisons that can coexist with activation.

    Source: §6 · p. 29

    Whether complete unsmoothed measured-χ2 ordering implies a complete signed lift remains open in general; the examples do not settle the remaining case.

    Source: §6 · p. 29

Abstract

from arXiv · show

Recent examples have shown that zero private capacity need not imply antidegradability and that two channels with zero private capacity can nevertheless transmit private information together. Existing entropic channel orders compare what a receiver and its environment can learn and thereby bound capacities. We connect these orders to the recent constructions through binary estimation, whose minimum mean-square error is governed by measured $\chi^2$ divergence. A new integral representation shows how measured $\chi^2$ on a qubit extension recovers quantum $\chi^2$. It lets us pass from complete measured-$\chi^2$ ordering to complete quantum-$\chi^2$, relative-entropy, and less-noisy ordering. Zhu and Wang used a signed lift to show that their qutrit channel has zero private capacity. We show that Hermitian-smoothed measured-$\chi^2$ ordering characterizes when such lifts exist, even with a quantum reference. Environmental dominance in binary estimation at every blocklength yields a finite-code reliability--secrecy bound. These results explain why complete comparison prevents activation with antidegradable helpers whereas regularized comparison alone does not. Building on that qutrit example, we establish this stability for a range of noisy Werner--Holevo channels, determine sharp private-capacity and antidegradability thresholds, and compute exact complementary capacities in a nondegradable range. By contrast, we extend Pauli half-erasure activation to all $1/2\le p<1$ and establish the same range for a new four-level family. Reference-assisted measured-$\chi^2$ witnesses show why these activating constructions lack complete comparison.

1 Introduction

Private capacity asks how fast a channel can transmit classical information reliably while keeping it inaccessible to the environment; deciding whether it vanishes can require control of arbitrarily many uses. This introduction connects recent zero-capacity and activation examples to channel orders through binary estimation and develops criteria for comparing receiver and environment information.

1 Introduction

  • The qutrit channel Λ1/2 cannot activate with any antidegradable helper because its complete anti-less-noisy comparison tensorizes.The comparison rules out positive product private capacity with such a helper.Source: §1 · p. 3
  • At every blocklength, if the environment’s binary-estimation MMSE is no greater than the receiver’s for every encoding, the comparison yields a quantitative reliability–secrecy bound and suffices for zero private capacity.A binary decoder with error η has mean-square error 4η; the environment can then form an estimator whose error also vanishes as decoding becomes reliable.Source: §1 · p. 3
  • Complete measured-χ2 ordering implies complete quantum-χ2 ordering, which is equivalent to complete relative-entropy and less-noisy ordering.The implication uses a coherent-qubit realization; the ensuing equivalences connect the result to established entropic comparisons.Source: §1 · p. 4
  • Fixed-record Hermitian smoothing characterizes when a channel admits a complete variance-dominating signed lift through another channel, including with a quantum reference.The criterion minimizes measured χ2 over trace-one Hermitian operators that preserve a fixed classical–quantum measurement record.Source: §1 · p. 4
  • Complete comparisons can tensorize through a common implied order, whereas a regularized comparison alone need not remain valid when paired with another channel.The introduction contrasts the qutrit result with N4, whose regularized relation follows from its zero private capacity but whose complete relation would force zero product capacity with a half-erasure helper.Source: §1 · pp. 3–4
  • Receiver dominance over the environment at every blocklength implies zero private capacity for the complement and equality of private and quantum capacities with their single-use versions.The stated implication also gives P(N)=P^(1)(N)=Q(N)=Q^(1)(N) when the complement is less noisy than N.Source: §1 · p. 4
  • The Pauli channel F extends half-erasure activation to every 1/2 ≤ p < 1, while the coherent-ladder family activates throughout that interval for each finite s ≥ 3.At s = 4, the ladder recovers Zhu and Wang’s channel, for which activation throughout the interval was already proved [12, Theorem 1].Source: §1 · pp. 4–5
  • For 1/2 ≤ x ≤ 1, the Werner–Holevo complements are nondegradable, and the complete comparison makes both complementary capacities equal to one-use coherent information.The maximally mixed qutrit input is optimal by concavity and covariance.Source: §1 · p. 5

2 Quantum estimation

The section develops binary mean-square estimation as an operational form of measured χ², then restricts estimator scores to those determined by a fixed reference measurement record. The restricted comparison is characterized by minimizing over Hermitian completions, which need not be positive.

2.1 Binary estimation

  • For an equiprobable binary state pair, the receiver minimizes mean-square error over POVMs and real-valued estimates; the optimum is expressed through measured χ².The optimal spectral measurement has at most dim B outcomes.Source: §2.1 · pp. 6–7

2.2 Hermitian smoothing

  • Hermitian smoothing restricts the score to observables determined by a fixed classical–quantum record, and its value minimizes measured χ² over trace-one Hermitian completions of that record.The record fixes the measurement outcomes and unnormalized conditional states; for a qubit reference, the Y component remains unconstrained.Source: §2.2 · pp. 8–9
  • Locally, the smoothed task minimizes SLD information over perturbations invisible to the record channel; the corresponding quadratic form is SLD Fisher information.The restricted value for a fixed state pair can depend on the chosen reference basis, while the complete channel order is basis independent.Source: §2.2 · pp. 8–9
  • The completion may be nonpositive, and the restriction applies to the score observable rather than requiring every spectral projector to lie in the record subspace.For a Bell-state example, the optimal completion has an eigenvalue of −1/4.Source: §2.2 · pp. 8–9

3 Quantum channel orders

The section connects complete channel comparisons and signed lifts to quantum χ² and entropic orders, then represents quantum χ² through measured χ² on a coherent-qubit extension. It also shows that receiver dominance at every blocklength removes capacity regularizations.

3.1 Definitions and hierarchy

  • Complete measured-χ² ordering implies quantum-χ² ordering, which is equivalent to complete relative-entropy and less-noisy ordering with arbitrary finite references.The quantum-χ² implication follows by realizing the comparison on a coherent-qubit extension and integrating.Source: §3.1 · p. 11
  • Complete signed lifts persist under probabilistic mixtures, input isometries, and output unitaries.The lift can be block diagonal on an environment that flags the mixture branch.Source: §3.1 · p. 11
  • Hermitian-smoothed measured-χ² ordering characterizes when a complete variance-dominating signed lift exists, including with a quantum reference.The joint tests produce one lift preserving means and the second-moment bound; separate state-dependent estimators do not suffice.Source: §3.1 · pp. 10–11

3.2 Quantum χ² divergence

  • Quantum χ² is an integral of measured χ² divergences on an enlarged system with a coherent qubit; dephasing that qubit removes the off-diagonal signal perturbation.The representation also applies to arbitrary Hermitian directions.Source: §3.2 · pp. 13–14

3.3 Receiver dominance and single-letter capacities

  • The complement controls both the gain from regularization and the gap between private and quantum capacity.The paper identifies this role through complementarity bounds.Source: §3.3 · p. 14
  • Receiver dominance at every blocklength removes both capacity regularizations and yields single-letter capacity consequences; dominance at one use alone does not suffice.When the complement has zero quantum capacity, the stated bound gives P(N) = Q(N), without single-letterization.Source: §3.3 · p. 15

4 Zero private capacity

The section converts complete receiver-to-environment estimation dominance into a finite-code reliability–secrecy bound, then contrasts transpose simulation with erasure-assisted activation. It distinguishes consequences of complete comparison from those of regularized comparison alone.

4 Zero private capacity

  • For binary codes, pair-counting and Jensen’s inequality give (1 − 2η)^2 ≤ 2δ when η ≤ 1/2, with δ = dE/2; the bound excludes sequences of reliable secret codes for any message size.Source: §4 · p. 16
  • At each positive rate R, zero private capacity implies squared fidelity to an ideal secret key is at most e−nE(R), for some E(R) > 0.The ideal-key error therefore approaches one exponentially fast.Source: §4 · p. 16
  • Zero private capacity also has an exact regularized channel-order characterization through less-noisy, relative-entropy, and quantum-χ2 order equivalences at each tensor power.The equivalences are combined with the private-capacity expression in (1.2).Source: §4 · p. 16

4.1 Transpose antidegradability

  • If a channel’s complement can be simulated by a map followed by output transposition, transpose antidegradability implies zero private capacity but does not establish complete comparison with a quantum reference.The tensorized simulator gives full transposition, whereas an untouched reference leads to partial transposition.Source: §4.1 · p. 16

4.2 Erasure-assisted communication

  • A reference-assisted information gap becomes positive private information after half-erasure assistance, without requiring a preshared secret key.When P(N) = 0, this construction gives two zero-private-capacity factors with positive product capacity.Source: §4.2 · pp. 17–18
  • A zero-private-capacity channel with support-asymmetric outputs can activate with an antidegradable erasure helper: Bob’s information gain is linear in t while Eve’s leakage is O(t^2).The criterion applies when the environmental output for one input has positive weight outside the support of the other; the helper is antidegradable for p ≥ 1/2.Source: §4.2 · pp. 18–20

5 Examples

These examples distinguish reference-complete channel comparison from comparison at every tensor power, illustrating that zero private capacity, antidegradability, and activation with a zero-private-capacity antidegradable helper need not coincide. Across several channel families, they show stability and activation outcomes, including positive private rates with such a helper, while reference-assisted binary estimation witnesses failure of complete comparison despite Eve’s dominance in every unassisted blocklength comparison.

5 Examples

  • The section contrasts reference-complete comparison with comparisons valid at every tensor power, and relates zero-private-capacity non-antidegradable channels to nondegradable complements.For the Werner–Holevo family, the complementary channel’s common private and quantum capacities are evaluated exactly; the separation between degradability and regularized less-noisiness is identified in [3, Section 4].Source: §5 · p. 20

5.1 Noisy Werner–Holevo channels

  • For noisy Werner–Holevo channels Λx, antidegradability holds exactly for x ≥ 4/7; thus, for 1/2 ≤ x < 4/7, they are not antidegradable, and every family member is NPT.Source: §5.1 · p. 21

5.2 Qubit-to-qutrit channels

  • The qubit-to-qutrit family has complete signed lifts and zero private and quantum capacity, while b>0 gives NPT and 3√2a+4b>3 certifies non-antidegradability.The point (a,b)=(1/2,1/4) satisfies these conditions.Source: §5.2 · p. 22

5.3 Pauli channels

  • For every 1/2 ≤p<1, the fixed Pauli channel and an erasure channel have zero private capacity individually but a positive private rate together.The construction extends activation beyond the p=1/2 cases of Uthirakalyani G, Halder, and Elkouss [14]; Zhu and Wang’s all-p result concerns a different four-level channel [12].Source: §5.3 · p. 23

5.3.1 Channel and encoding

  • The Pauli channel has P(F)=0 under all tensor-power comparisons, yet its full complement retains error-label coherences and its positive assisted rate rules out complete comparison.The positive product rate also shows F is not antidegradable and its complement is not completely less noisy; complete measured domination and a complete signed lift therefore fail.Source: §5.3.1 · p. 24

5.3.2 Reference-assisted estimation

  • With a qubit reference, dilution reverses the Pauli channel’s estimation comparison, and diluted coding gives positive rate throughout 1/2≤p<1.For the original reference-assisted pair, Bob’s MMSE is 8/9 and Eve’s is 29/33; at p=1/2, the exact rate bound exceeds 6.117 × 10^-4 bits per product use.Source: §5.3.2 · pp. 24–25, Figure 5 · p. 25

5.4.1 Error-detecting subspaces

  • The coherent-ladder construction uses an error-detecting code whose shared Kraus labels preserve interference between transition amplitudes.Separating those transitions into distinct environmental labels would remove the cancellation; the parameter s sets the central-to-outer intensity ratio for one-step transitions.Source: §5.4.1 · p. 26

5.4.2 Channel family and dilation

  • For every finite s ≥3 and erasure probability 1/2 ≤p < 1, the next theorem gives a transpose-antidegradability criterion for this family and a private-rate bound after erasure assistance.Source: §5.4.2 · p. 27
  • Under the joint code projection, only the identity branch survives: F = ww† has zero background probability and signal probability q = 1/(s + 3).The resulting proposition uses cR = cs, cE = s, and κp = κs,p, with a choice of t* below 1/2.Source: §5.4.2 · pp. 27–28
  • The ladder bound proves positive private rate with an antidegradable helper of zero private capacity, excluding complete less-noisy comparison, antidegradability, and complete measured domination.The regularized comparisons still follow from Corollary 4.2.Source: §5.4.2 · p. 28
  • At s=3, p=1/2, and t=1/80, the exact rate bound exceeds 2.475 × 10^-4 bits per product use, improving ln 2/3132 ≃ 2.213 × 10^-4; at s=4 it recovers 3 ln 2/10927 bits [12].Figure 6(a) plots the closed-form rate bound as s varies for three erasure probabilities.Source: §5.4.2 · p. 28, Figure 6 · p. 28
  • An explicit interval in a reference-assisted experiment makes Bob's MMSE smaller than Eve's, despite Eve dominating every unassisted blocklength comparison.The click measurement gives the classical measured-χ2 value, while a preparation-coin argument bounds Eve's value; that bound is sharp for orthogonal prepared states when only the domination constant is specified.Source: §5.4.2 · pp. 28–29

6 Discussion

The discussion places signed lifts, complete measured-χ2 ordering, entropic orders, and binary-estimation bounds in one hierarchy. Its channel examples distinguish complete comparison, which blocks activation by antidegradable helpers, from regularized comparisons that can coexist with activation.

6 Discussion

  • Fixed-record Hermitian smoothing characterizes complete variance-dominating signed lifts, which imply complete measured-χ2, quantum-χ2, relative-entropy, and less-noisy ordering.Binary estimation also yields a quantitative reliability–secrecy bound.Source: §6 · p. 29
  • Noisy Werner–Holevo channels and their qubit-to-qutrit restrictions can have zero private capacity without antidegradability, while complete comparison prevents activation by antidegradable helpers.Source: §6 · p. 29
  • Pauli and coherent-ladder constructions satisfy comparisons at every tensor power through transpose simulation, yet reference-assisted joint measurements yield positive private information.The ladder family also continuously deforms transition amplitudes and its error-detecting code while preserving destructive interference.Source: §6 · p. 29
  • For the Pauli encoding, Bob has more mutual information for the original pair, whereas Eve has smaller MMSE; dilution gives a finite measured-χ2 reversal and extends erasure activation to 1/2 ≤ p < 1.Source: §6 · p. 29
  • Whether complete unsmoothed measured-χ2 ordering implies a complete signed lift remains open in general; the studied qutrit families satisfy both conditions, while activating examples fail both.Source: §6 · p. 29
  • Further open questions include activation beyond the ladder family’s certified range, optimized achievable rates, and other physical noise models with the required interference and support properties.Source: §6 · p. 29

A Estimation and channel-order proofs

The appendix proof section states where the converse and sufficient converse conditions are established and notes that the output-support convention applies throughout.

A Estimation and channel-order proofs

A.1 Complete signed-lift characterization

The proof characterizes complete signed lifts by coupled quadratic tests and a minimax choice of one unbiased observable map. It constructs that map by matching observable means and verifies its reference-assisted variance bound.

A.1 Complete signed-lift characterization

  • Coupled quadratic tests use positive block weights whose off-diagonal blocks couple scores, and complete restricted ordering implies the corresponding cost inequality.Source: §A.1 · p. 31
  • A faithful-state perturbation with a reference realizes the tests, while an optimal score retains only the off-diagonal blocks fixed by the reference.Substituting those blocks yields the quadratic comparison, which extends by continuity as the perturbation vanishes.Source: §A.1 · p. 31
  • Matching means starts with a traceless Hermitian output basis and feasible observables, using ker M ⊆ ker N to obtain im N† ⊆ im M†.Source: §A.1 · p. 31
  • The weighted inequality follows by minimizing the Lagrangian over Hermitian variables; it holds first for positive definite weights and then for semidefinite weights by a limit.Source: §A.1 · p. 32
  • Minimax over trace-one positive block weights selects one feasible Hermitian observable tuple whose cost is bounded for every weight.The proof uses continuity, convexity, coercivity, and existence of a minimizing tuple.Source: §A.1 · pp. 31–32
  • The resulting map preserves the chosen observable means, and the reference-assisted construction converts its second-moment comparison into the required variance bound.Source: §A.1 · p. 32

A.2 Full-output-span converse

Under complete measured ordering and a full-output-span assumption, the converse constructs a unique unbiased map and extends its variance bound to every quantum reference. The appendix also identifies symmetric extension as the certificate used for antidegradability tests.

A.2 Full-output-span converse

  • With complete measured ordering and a full-output-span assumption, kernel inclusion yields a unique Hermiticity-preserving map J satisfying M†J = N† and J(I) = I.The proof subtracts (Tr X)σ for nontraceless X and uses surjectivity to make M† injective.Source: §A.2 · p. 32
  • Optimizing the variational comparison over Hermitian outputs gives a second-moment bound for each target score Y.The displayed supremum equals Tr M(σ)J(Y)^2 and is bounded by Tr N(σ)Y^2.Source: §A.2 · p. 32
  • The same unique unbiased map satisfies the second-moment bound for every reference and input, yielding the required variance bound.The amplified source remains surjective, so its unbiased map is id_R ⊗ J.Source: §A.2 · p. 32

B Symmetric-extension certificates

  • Antidegradability is equivalent to symmetric extendibility of the normalized Choi state, the certificate used for the Werner–Holevo threshold, full-rank mixtures, and a qubit-to-qutrit obstruction.The criteria support Theorem 5.1 and the separation claims in Examples 5.3 and 5.4.Source: §B.1 · p. 33, B Symmetric-extension certificates · p. 33

B.1.1 Invariant qutrit Choi states

For invariant qutrit Choi states, symmetric-extension witnesses yield an antidegradability criterion and a sharp threshold for the stated full-rank mixture. A separate support-based argument establishes a qubit-to-qutrit non-antidegradability obstruction.

B.1.1 Invariant qutrit Choi states

  • For ω = p0Φ + (p1/3)P− + (p2/5)Pst, antidegradability is characterized by the stated symmetric-extension criterion.The invariant witness W has extension operator H(W); its expectation on ω is p0z0 + p1z1 + p2z2.Source: §B.1.1 · p. 33
  • For the mixture Mϵ, antidegradability fails below ϵ = 1/81 and holds at ϵ = 1/81; convexity and the replacer component extend the threshold to larger ϵ.For ϵ > 0 the replacer term makes the Choi state positive definite, and the mixture has a complete signed lift.Source: §B.1.1 · p. 34

B.1.2 Qubit-to-qutrit channels

  • For the qubit-to-qutrit family, violating a necessary extension condition establishes non-antidegradability; for b > 0, the channel is also NPT.The partial transpose has a zero diagonal entry coupled to a basis vector by a nonzero −b/2 entry.Source: §B.1.2 · p. 35

C.1 Pauli-channel calculations

The Pauli-channel calculations reduce receiver and complementary outputs to tractable blocks and establish the information values used in the activation example. A positive private rate with an antidegradable helper also implies non-antidegradability of the channel.

C.1 Pauli-channel calculations

  • A positive private rate with an antidegradable helper already establishes non-antidegradability of the Pauli channel.The calculations use the input letters and dilation of [14].Source: §C.1 · p. 35

C.1.1 Output reduction

  • The receiver states are diagonal, while the complementary states reduce to three-dimensional blocks for the information calculation.Shared eigenvalues 1/6, 1/6, 1/6, 0 cancel in the Holevo information; the remaining average-state eigenvalues are (3 ± 3)/12.Source: §C.1.1 · p. 35

C.1.2 Estimation and support bounds

The appendix computes the measured-χ2 estimation values and a sharp support bound used in the Pauli example. It also introduces the ladder-channel calculations used for Theorem 5.6.

C.1.2 Estimation and support bounds

  • The appendix derives measured-χ2 values for the Pauli states across dilution parameters and uses selected parameters to establish the stated estimation comparisons.The common I2/6 block contributes zero, and tensoring with IC/2 leaves measured χ2 unchanged.Source: §C.1.2 · pp. 35–36
  • At the support boundary, the reported value is 32/45, while the corresponding receiver value is infinite, with the passage attributing that infinity to a zero receiver coordinate.Source: §C.1.2 · p. 36
  • The support calculation establishes that 2 + 3 is the smallest constant in (5.18).The proof uses equal determinants, a rank-two state difference, and generalized eigenvalues.Source: §C.1.2 · p. 36

C.2 Coherent ladder channels

  • The next appendix subsection proves the transpose-simulator and PPT claims, then gives environmental domination bounds for the coherent ladder channels.Source: §C.2 · p. 36

C.2.1 Transpose-antidegrading maps

The proposed map is verified to be completely positive, trace preserving, and to simulate the transpose of the channel’s complement for s ≥ 3. The channel’s Choi partial transpose is positive exactly for s ≥ 3 in the CPTP domain.

PPT condition

  • The partially transposed Choi matrix is positive exactly when s ≥ 3 in the CPTP domain, as its last block has eigenvalues 2s(s + 1) and 2s(s − 3).The other blocks are positive for the stated range.Source: §C.2.1 · p. 36

Construction of the simulator

  • The simulator is built from H_s and W_s, with trace preservation reduced to Tr_B H_s = W_s² and the composition identity checked by substitution.The blocks (H_s)_{ef} are 4 × 4 blocks indexed by environmental labels e and f.Source: §C.2.1 · p. 37

Transpose-antidegrading map

  • For s ≥ 3, D_s is a CPTP transpose-antidegrading map, with trace preservation and the simulation identity verified on all input matrix units.Choi positivity follows from the rank-one sum; linearity extends the identity to every input operator.Source: §C.2.1 · p. 37

C.2.2 Complementary-channel support bounds

The section bounds signal operators by background operators, including coherences between environmental labels. Tracing out Eve’s reference yields a stronger bound with constant s.

Complementary-channel support bounds

  • The proof collects eight background columns and four signal columns into amplitude matrices, then bounds their coefficient matrix to control coherences.The column index is the receiver level traced out in the complement.Source: §C.2.2 · pp. 37–38
  • The weighted operator inequality η_1 ⪯_c s η_0 holds including environmental-label coherences, and both bounds provide the support containment required by Proposition 4.4.The bound follows by congruence with (I_R ⊗ W_s)/√ν_s.Source: §C.2.2 · p. 38
  • When Eve does not receive R, tracing it out gives a stronger bound with constant s, and the reduced unweighted matrices are diagonal.The passage introduces their entries but does not state them here.Source: §C.2.2 · p. 38
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