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Classical Commitment over Quantum Channels with Limited Entanglement Assistance

Remi A. Chou

arXiv:2608.28500v1quant-phcs.IT

TL;DR

The paper asks how much classical string commitment is possible over quantum channels when preshared entanglement is limited, including whether interaction helps. It analyzes a Heisenberg–Weyl channel class using reductions between quantum and classical commitment models, obtaining an exact noninteractive capacity and interactive bounds, with a complete interactive characterization for quantum erasure channels.

  • Problem

    The paper studies information-theoretically secure classical string commitment over quantum channels with limited preshared entanglement, including arbitrary quantum-input strategies and interactive communication.

  • Method

    For a channel class based on random Heisenberg–Weyl operations, the paper reduces achievability to an induced classical channel and proves quantum security through post-processing and strategy reduction.

  • Results

    The noninteractive capacity is min{H(Z|F), log2 d + E}; interactively, the rate is at most log2 d + E, and quantum erasure capacity is unchanged by interaction.

  • Takeaways & Limitations

    Interaction does not increase capacity when H(Z|F) ≥ log2 d + E, and it never increases capacity for the quantum erasure channel.

Abstract

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We study classical string commitment over quantum channels with limited preshared entanglement. For noninteractive protocols, we determine the commitment capacity of a class of channels with input dimension $d$ that, at each use, sample a pair of classical random variables $(F,Z)$, apply one of the $d^2$ Heisenberg--Weyl operators indexed by $Z$ to the input, and deliver the transformed quantum system together with $F$ to the receiver. If $E$ is the available entanglement rate in bits per channel use, then the capacity is $\min\{H(Z|F),\log_2d+E\}$. This class of channels encompasses quantum erasure and depolarizing channels, as well as families of Pauli channels. Additionally, for interactive protocols, we show that the commitment rate cannot exceed $\log_2d+E$ bits per channel use, so that when $H(Z|F)\geq\log_2d+E$, interactive communication does not increase the capacity. As a consequence, for interactive protocols, we determine the capacity of the quantum erasure channel.

I. INTRODUCTION

The paper studies information-theoretically secure classical string commitment over quantum channels with limited preshared entanglement, covering noninteractive and interactive protocols. It characterizes noninteractive capacity for a Heisenberg–Weyl channel class and establishes interactive upper bounds, including the quantum erasure channel.

  • Model: The protocol uses independent quantum-channel uses, authenticated noiseless classical communication, and preshared entanglement limited by an entanglement rate E.Security must hold against dishonest parties with arbitrary quantum memories and operations.
  • Channel class: The channel samples classical variables (F, Z), applies a Heisenberg–Weyl operator indexed by Z to a d-dimensional input, and gives the transformed system and F to the receiver.The channel class includes quantum erasure, depolarizing, dephasing, and bit-flip channels.
  • Noninteractive capacity: The noninteractive commitment capacity is min{H(Z|F), log2 d + E}.Achievability reduces the quantum channel to an induced classical channel and uses orthogonal entangled encoding states, while the converse bounds the environment contribution by H(Z|F) and the input-plus-entanglement contribution by log2 d + E.
  • Security: Security against arbitrary quantum adversaries is established by generating Bob’s quantum output from the induced classical output and reducing quantum strategies to equivalent classical dishonest strategies.Classical commitment results alone do not directly establish security for the quantum-input model.
  • Interactive protocols: For interactive protocols, the commitment rate is at most log2 d + E, and this matches the noninteractive capacity whenever H(Z|F) ≥ log2 d + E.For the quantum erasure channel, interaction does not increase capacity at any entanglement rate.
  • Relation to prior work: The quantum-input setting permits dishonest senders to use arbitrary states, including states entangled across channel uses and with private quantum memory.Therefore, prior proofs for classical-input or cq-channel models do not cover the arbitrary quantum-input strategies considered here.

II. NOTATION

The paper fixes finite-field and Heisenberg–Weyl notation, then specifies noninteractive commitment protocols over quantum channels with preshared entanglement and classical reveal communication.

  • II. NOTATION: Heisenberg–Weyl operators are products X(a)Z(b), where X(a) and Z(b) are cyclic-shift and phase operators.
  • II. NOTATION: A Pauli qudit channel randomly applies Heisenberg–Weyl operators to the input.
  • II. NOTATION: The protocol commits a classical k-bit string over n independent channel uses, with entanglement rate E measured through preshared Schmidt rank K.
  • II. NOTATION: During the noninteractive commit phase, Alice prepares channel inputs and a retained private register, while Bob receives channel outputs and performs arbitrary operations without sending information.
  • II. NOTATION: Correctness, hiding, and binding govern the protocol; binding uses a measurement fixed before Alice’s reveal strategy and implies pairwise binding with error at most 2γ.
  • II. NOTATION: The commitment capacity is the supremum of achievable commitment rates, with separate notation for finite, zero-entanglement, and unrestricted-entanglement settings.

IV. MAIN RESULTS

The main results determine noninteractive commitment capacities for random Heisenberg–Weyl channels and derive consequences for Pauli, erasure, depolarizing, and dephrasure channels.

  • IV. MAIN RESULTS: The unrestricted-entanglement capacity remains valid under the weaker condition spanFp Sq = F2^d.
  • IV. MAIN RESULTS: For Pauli qudit channels, Ccom(E; Wq) = min{H(q), log2 d + E}.
  • IV. MAIN RESULTS: For independent shift and phase indices, Ccom(E; WqX,qZ) = min{H(qX) + H(qZ), log2 d + E} under the stated Fourier-coefficient conditions.
  • IV. MAIN RESULTS: For qubit Pauli channels satisfying the stated coefficient conditions, Ccom(E; Pq) = min{H(qI, qX, qY, qZ), 1 + E}.
  • IV. MAIN RESULTS: For qubit depolarizing channels, Ccom(E; Dp) = min{H(qp), 1 + E}.

V. ACHIEVABILITY OF THEOREM 1

The achievability proof converts blocks of the quantum channel into an induced classical channel, then applies a classical commitment code. The construction uses a subspace-based encoding with limited entanglement, and establishes security against quantum adversaries.

  • Security: The resulting quantum protocol preserves correctness, hiding, and binding against arbitrary quantum adversaries.The security reduction measures the channel outputs and retained registers, then uses the classical code’s properties to establish the quantum protocol’s guarantees.
  • Classical commitment code: Every rate R < max_PX H(X|F^ℓ,Y) is achievable when the induced channel is nonredundant.This invokes the classical commitment theorem for finite nonredundant channels without commit-phase public communication.
  • Construction: The proof simulates one use of an induced classical channel using ℓ quantum-channel uses and c maximally entangled qudit pairs.The construction encodes a classical input into ℓ channel uses using a subspace Q_L of dimension d^c.
  • Induced classical channel: Bob measures the channel output and retained register in an orthonormal basis, yielding Y = X + π_L(Z^ℓ) while also observing F^ℓ.Thus the quantum channel is reduced to a classical channel whose noise is determined by the projected Heisenberg–Weyl index.

PLW †

This section constructs the encoded states and analyzes the induced classical channel. A subspace L groups Heisenberg–Weyl indices, while measurement converts the channel action into additive classical noise.

  • State construction: The encoding uses an orthonormal basis of entangled states indexed by the quotient alphabet X_L.For c = ℓ, the shared state becomes a tensor product of ℓ maximally entangled qudit pairs; for c = 0, no preshared entanglement is used.
  • Equivocation: The induced channel’s uniform-input equivocation approaches min{H(Z|F), (1 + e) log_2 d}.The argument uses typicality and concentration bounds for the memoryless variables governing Z^ℓ conditioned on F^ℓ.

C. Nonredundancy of the induced channel

The induced channel is nonredundant precisely under a span condition relating the projected noise support to L^⊥. If the condition fails, distinct inputs produce identical output distributions.

  • Characterization: The induced channel W_L,q is nonredundant if and only if span_Fp S_q,L = L^⊥.This is the section’s stated characterization of when no two distinct inputs induce the same output distribution.
  • Failure of nonredundancy: When span_Fp S_q,L ≠ L^⊥, there exists h ∉ L such that translating the input by h + L leaves every output distribution unchanged.The distinct inputs x̄ and x̄ + h + L are therefore indistinguishable through the induced channel.
  • Sufficiency: When the span condition holds, Fourier-character constraints force any translation preserving the output distribution to lie in L.The proof derives a contradiction from a putative redundant representation and concludes h ∈ (L^⊥)^⊥ = L.
  • Special case: For ℓ = c = 1, the condition reduces to span_Fp S_q = F_2^d.In this case L = {0}, so the projected support must span the full relevant vector space.

D. Quantum-to-classical security reduction

The quantum security proof reduces the protocol to a classical commitment code by measuring Bob’s channel outputs and retained systems. Classical correctness, hiding, and binding then transfer to the quantum protocol.

  • Reduction: The quantum protocol implements the classical commitment code over the induced channel W_L,q by having Bob perform the basis measurement from Lemma 2.When both parties are honest, the resulting conditional distribution equals W_L,q, so correctness errors coincide.
  • Rate: For any distribution P_X, every rate R < H(X | F^ℓ, Y) per use of W_L,q is achievable by the corresponding quantum protocol.This transfers the classical commitment rate guarantee to the quantum construction.
  • Hiding: The hiding proof follows from the classical code’s hiding property and monotonicity of trace distance under Bob’s final quantum operation.The quantum states corresponding to different committed strings remain close after Bob’s processing.
  • Hiding: Bob’s commit-phase operations can be represented as a quantum operation applied after the channel outputs are produced.Because Bob sends no messages to Alice during commitment, this representation supports the hiding reduction.
  • Binding: Binding is reduced by fixing the extraction measurement before Alice chooses her reveal strategy, then applying the classical binding proof to the extracted value.The measurement outcome is the value extracted by the classical commitment code.

E. Achievable rates

The construction uses self-orthogonal subspaces and entanglement-assisted induced channels to achieve every rate below min{H(Z|F), log2 d + E}.

  • Achievable rates: Self-orthogonal subspaces Lℓ have dimension ℓ−cℓ and projected noise entropy converging to the capacity expression.The convergence is to min{H(Z|F), log2 d + E}.
  • Achievable rates: Any R < min{H(Z|F), (1 + e) log2 d} = min{H(Z|F), log2 d + E} is achievable for sufficiently large blocklength.The subspaces are chosen so their projected noise entropy exceeds the target rate.
  • Achievable rates: The induced channel W_Lℓ,q satisfies H(X|Fℓ, Y) > ℓR and is nonredundant, enabling rate ℓR per induced use.This follows from the entropy condition and the cited coding results.
  • Achievable rates: Each induced use consumes ℓ physical channel uses and cℓ maximally entangled qudit pairs, so its rate ℓR becomes R per physical use.The entanglement accounting preserves the target rate normalization.
  • Achievable rates: The preshared entanglement rate is log2(d^Ncℓ)/(Nℓ) = (cℓ/ℓ) log2 d ≤ E, proving achievability under the entanglement constraint.Thus every R below the stated minimum is achievable.
  • Achievable rates: For the single-use maximally entangled construction, the induced channel has H(X|F, Y) = H(Z|F) and is nonredundant.This achieves every rate R < H(Z|F) in that regime.

VI. CONVERSE OF THEOREM 1

The converse bounds noninteractive commitment by combining hiding, environmental information, correctness, binding, and the entanglement-limited support dimension.

  • VI. CONVERSE OF THEOREM 1: The channel environment contributes at most nH(Z|F) information about the committed string.This is obtained by representing the inputs and Bob’s initial system jointly before the channel uses and applying Lemma 6.
  • VI. CONVERSE OF THEOREM 1: Correctness, binding, data processing, and Fano’s inequality control the residual uncertainty of the committed string from the receiver’s view and environment.A measurement of the combined systems produces an estimate eS, whose error probability enters the Fano bound.
  • VI. CONVERSE OF THEOREM 1: Hiding bounds the receiver’s view information by βk plus a binary-entropy correction term.The bound uses a state independent of the committed string and a continuity inequality.
  • VI. CONVERSE OF THEOREM 1: The initial entanglement support and n d-dimensional channel inputs give H(V En) ≤ n log2 d + log2 K.The channel isometry preserves the relevant support dimension.
  • VI. CONVERSE OF THEOREM 1: For entanglement rate E, lim sup n→∞ n^-1 log2 K_n ≤ E, yielding lim sup n→∞ k_n/n ≤ log2 d + E.This is the entanglement-dependent converse bound for noninteractive protocols.

VII. INTERACTIVE PROTOCOLS

The paper extends commitment to interactive protocols with authenticated classical communication, proves a universal upper bound, and determines the quantum erasure channel’s capacity.

  • Interactive commitment protocols: Interactive protocols allow authenticated classical messages in both directions before, between, and after channel uses.Each channel input remains d-dimensional, with arbitrary local operations on current registers.
  • Interactive commitment protocols: Theorem 3 gives an interactive commitment-rate upper bound of log2 d + E for every channel with input dimension d.The bound applies at every entanglement rate E.
  • Interactive commitment protocols: When the noninteractive construction attains log2 d + E, the interactive capacity is determined by the same value.The paper identifies this matching condition for the channels covered by Theorem 1.
  • Interactive commitment protocols: For the channel Nq, Ccom,int(E; Nq) = log2 d + E.This is stated in Corollary 6.
  • Interactive commitment protocols: For qubit Pauli channels with all three coefficients nonzero and 0 ≤ E ≤ 1, Ccom,int(E; Pq) = 1 + E.This follows from Corollaries 3 and 6.
  • Interactive commitment protocols: For the quantum erasure channel, interaction does not increase capacity, and Ccom,int(E; Eϵ,d) = Ccom(E; Eϵ,d) = min{2ϵ log2 d, log2 d + E}.The equality holds for 0 < ϵ < 1 and E ≥ 0.
  • Interactive commitment protocols: The paper leaves open whether interaction can increase capacity in general and identifies finite-blocklength bounds and efficient constructions as open directions.These limitations concern the general interactive setting and construction efficiency.

APPENDIX A

The appendices establish the classical commitment construction and the structural lemmas used to bound supports and verify security in interactive protocols.

  • APPENDIX A: Lemma 7 provides commitment codes for finite nonredundant classical channels at every R < H(X|Y), with message-rate and output-distribution guarantees.The construction uses maps ϕN and sets B(N) for sufficiently large blocklength.
  • APPENDIX A: Alice commits by sampling r uniformly and sending the codeword ϕN(s, r), while Bob decodes using the channel output and a validity test.The reveal procedure sends (s, r), after which Bob sets the decoded string and test flag.
  • APPENDIX A: The construction’s correctness follows from the stated code property, while hiding follows from trace-distance monotonicity and the output-distribution bound.The protocol has no commit-phase public communication.
  • APPENDIX A: Self-orthogonality makes the Heisenberg–Weyl operators commute on the chosen subspace, enabling simultaneous diagonalization and construction of the projector P_L.The image of P_L is the common eigenspace Q_L.
  • APPENDIX A: The projector P_L is orthogonal onto Q_L, and the associated subspace dimension is d^(ℓ−c).The orthogonality follows from the trace properties of the operators.
  • APPENDIX A: In interactive protocols, classical communication does not increase the dimension bound on conditional quantum states.Alice’s messages preserve support inclusion, while Bob’s local maps send the support into a subspace of no larger dimension.
  • APPENDIX A: After i channel uses, conditioned on Bob’s classical record, the relevant state support has dimension at most Kd^i.The induction uses the d-dimensional channel input and the initial Schmidt-rank bound K.
  • APPENDIX A: For each classical record c_i, a subspace K_c_i independent of the committed string contains the conditional state support and has dimension at most Kd^i.This is the statement of Lemma 9 and underlies the interactive converse.

APPENDIX D

The appendix proves the interactive converse by combining the log2 d + E bound with an additional erasure-dependent entropy bound. It tracks the erasure pattern in Bob’s view and bounds the resulting conditional-entropy contribution using the input dimension and erasure probability.

  • log2 d + E is established as a converse bound by Theorem 3, while achievability follows from Corollary 1.
  • The remaining task is to prove the 2ϵ log2 d bound for a fixed interactive protocol.
  • The proof introduces Gi to indicate whether the ith input is erased and includes the final erasure indicator Gn in Bob’s view.
  • Conditioning on Gn yields an average difference of conditional entropies involving W n, Bob’s view V′, and the auxiliary system S.
  • On each erasure pattern, the isometric extension leaves W n supported on a space of dimension d, enabling an absolute conditional-entropy bound proportional to the number of erasures.
  • The final bound uses E[Gi] = ϵ for every channel use, converting the erasure count into the 2ϵ log2 d term.
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