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Quantum Advantage in Multiple Access Wiretap Channels with Entangled Transmitters

Hassan ZivariFard, Xiaodong Wang

arXiv:2609.06808v1cs.ITmath.ITquant-ph

TL;DR

The paper asks whether shared entanglement benefits secure communication over a classical multiple-access wiretap channel. It derives rate bounds and soft-covering results for entangled transmitters, and gives an example where their secrecy capacity exceeds that with classical common randomness.

  • Problem

    The paper examines whether shared entanglement can provide an advantage in secure communication over a classical MAC-WTC.

    Source: §I · p. 1

  • Method

    The authors derive achievable and converse bounds, a regularized secrecy-capacity expression, and strong soft-covering lemmas for MACs with entangled transmitters.

    Source: §I · p. 2, §III · p. 3, §V · p. 10

  • Results

    Shared entanglement strictly enlarges the secrecy-capacity region compared with classical common randomness, as illustrated by a MAC-WTC based on the magic square game.

    Source: 3) Main Result · pp. 7–8, §V · p. 10

  • Takeaways & Limitations

    The results establish a benefit of shared entanglement for secure multi-user communication in the studied MAC-WTC setting.

    Source: §V · p. 10

Abstract

from arXiv · show

We investigate secure communication over a classical multiple-access wiretap channel (MAC-WTC), specifically exploring the benefit of shared entanglement between the transmitters. Under the strict semantic security criterion, we derive an achievable rate region and a regularized expression for the secrecy capacity. We further establish a single-letter upper bound on the secure capacity of the MAC-WTC with entangled transmitters. Our results demonstrate that entanglement strictly enlarges the secrecy capacity of MAC-WTC compared to sharing only classical correlated randomness, a finding we illustrate using a pseudo-telepathy game example. Finally, we establish new strong soft-covering lemmas for the output statistics of multipleaccess channels (MACs) with entangled transmitters. Our results generalize existing results for non-entangled systems.

I. INTRODUCTION

This paper asks whether shared entanglement can benefit secure communication over a classical multiple-access wiretap channel. It develops rate results and an explicit example to examine that question.

I. INTRODUCTION

  • The motivating question is whether shared entanglement between transmitters can offer an advantage when the channel and other system components remain fully classical.The introduction notes that encoder cooperation can boost achievable transmission rates.Source: §I · p. 1
  • Prior work demonstrated through examples that entanglement shared between transmitters can strictly increase the achievable sum rate of a classical MAC; the pseudo-telepathy-game examples are in.Source: §I · p. 1
  • Earlier MAC-WTC studies addressed weak secrecy for the Gaussian model, strong secrecy for the discrete memoryless model, and semantic security, with soft-covering lemmas supporting achievability in and a strong soft-covering lemma in [37].The cited discrete-memoryless MAC-WTC study is; studies MAC-WTC under semantic security.Source: §I · p. 1
  • Shared entanglement can strictly enlarge the secrecy capacity region compared with transmitters sharing only classical common randomness.The paper also presents inner and outer bounds for the general MAC-WTC.Source: §I · pp. 1–2
  • The paper establishes a regularized secrecy-capacity expression and two strong soft-covering lemmas for classical MACs with entangled transmitters.The lemmas yield doubly exponentially decreasing variational distance and enable semantic security through the union bound.Source: §I · p. 2
  • The scheme combines random coding, coded time-sharing, wiretap coding, and channel resolvability, with new strong soft-covering lemmas addressing semantic security.The scheme must use shared entanglement for reliable communication, secrecy, and channel resolvability.Source: §I · p. 2

II. PROBLEM STATEMENT FOR MAC-WTC 

The problem is secure transmission of two confidential messages over a classical MAC-WTC when the transmitters may share entanglement. The paper compares this setting with transmitters sharing classical common randomness.

II. PROBLEM STATEMENT FOR MAC-WTC

  • The MAC-WTC has finite input alphabets for two senders, finite receiver and eavesdropper output alphabets, and a channel transition law.The channel is classical; the entanglement is a resource shared by the transmitters.Source: §II · p. 2
  • A code uses a shared bipartite entangled state, local randomness, encoding POVMs, and a decoder that estimates the message pair from the receiver's channel output.The encoders apply their measurements to their respective shares of the entangled state to produce channel inputs.Source: §II · p. 2
  • The secrecy capacity region is the closure of achievable message-rate pairs, with reliability assessed for uniformly distributed messages and security required independently of message distribution.The uniform-message assumption concerns average error probability and does not restrict the distribution-independent security criterion.Source: §II · p. 3
  • The paper states that the entangled-transmitter secrecy capacity region strictly improves on the region with classical private common randomness.The CR setup includes shared randomness kept secret from both the legitimate receiver and the eavesdropper.Source: §II · p. 3

III. MAIN RESULTS FOR MAC-WTC WITH ENTANGLED TRANSMITTERS

The paper presents rate bounds and a secrecy-capacity characterization for the MAC-WTC with entangled transmitters, then gives an example of a strict advantage over classical common randomness.

III. MAIN RESULTS FOR MAC-WTC WITH ENTANGLED TRANSMITTERS

  • The results include an achievable rate region, a regularized secrecy-capacity expression, and a single-letter upper bound for entangled transmitters.An example demonstrates a strict enlargement of the secrecy-capacity region relative to sharing only classical common randomness.Source: §III · p. 3

A. Semantic Security for MAC-WTC with Entangled Transmitters

The section gives achievable bounds, a regularized secrecy-capacity expression, and a single-letter upper bound. It also explains how entangled encoding measurements generate correlations between channel inputs.

A. Semantic Security for MAC-WTC with Entangled Transmitters

  • The achievable region uses mutual-information rate terms for the receiver offset by eavesdropper terms to ensure secure transmission.The coding proof uses coded time-sharing, wiretap coding, and strong soft-covering results; entangled-state POVMs can induce correlations between inputs.Source: A. Semantic Security for MAC-WTC with… · pp. 3–4
  • The achievable region optimizes over encoding POVMs and shared entangled states, unlike non-entangled schemes that generate inputs through channel prefixing conditioned on classical auxiliaries.The shared-state measurements can deliberately induce correlations between the channel inputs.Source: A. Semantic Security for MAC-WTC with… · p. 4
  • The paper provides a regularized expression for the secrecy-capacity region of the MAC-WTC with entangled transmitters.The expression is stated using blocklength-n auxiliary random sequences and corresponding receiver and eavesdropper information terms.Source: A. Semantic Security for MAC-WTC with… · pp. 4–5
  • A general single-letter upper bound on the entangled-transmitter secrecy capacity region is given with finite cardinality bounds on its auxiliary variables.The stated bounds are |U0| ≤ 5 and |Ui| ≤ 5(|X1||X2| + 1) for i ∈ {1, 2}.Source: A. Semantic Security for MAC-WTC with… · p. 5
  • The regularized and achievable-region results allow unbounded entanglement dimensions, including correlations arising as the dimension tends to infinity.The unions range over the closure of finite-dimensional quantum correlations.Source: A. Semantic Security for MAC-WTC with… · p. 5

B. Entanglement Resources with Limited Rate

This section defines entanglement resources whose size is limited by a rate and gives semantically secure achievable rates under that constraint. The result restricts the shared state to pure states with bounded local entropy.

B. Entanglement Resources with Limited Rate

  • Achievability requires both vanishing average error and semantic security, with leakage maximized over message distributions tending to zero.The definition requires these conditions for all sufficiently large block lengths.Source: B. Entanglement Resources with Limited… · p. 5
  • The rate-limited capacity region consists of rate pairs achievable when the transmitters share entanglement at rate θE.The encoders’ local Hilbert-space dimensions grow with the number of channel uses subject to the rate limit.Source: B. Entanglement Resources with Limited… · p. 5
  • The corollary’s semantically secure achievable rates restrict the shared state to pure states whose local entropies are at most θE.The proof also carries over the cardinality bounds established in Lemma 1.Source: B. Entanglement Resources with Limited… · p. 5

C. Example

The example section presents a classical MAC-WTC in which shared entanglement strictly increases the achievable sum rate. It uses the magic-square game’s deterministic strategy as part of the example.

C. Example

  • The example establishes that shared entanglement can strictly increase the achievable sum rate of a classical MAC-WTC.The section introduces the example as a MAC-WTC with this property.Source: C. Example · p. 5
  • The displayed rate constraints compare mutual-information terms for the receiver and eavesdropper, including individual-rate and sum-rate bounds.The constraints include conditional terms involving U0, U1, and U2.Source: C. Example · p. 5
  • Table 1 presents a deterministic strategy for the magic-square game used in the example.The table caption identifies it as the deterministic strategy.Source: Table 1 · p. 6

1) Mermin-Peres Magic Square Game:

The Mermin-Peres magic-square game asks two players to produce row and column assignments that satisfy parity constraints and agree at their intersection. Classical strategies cannot win every round, whereas an entangled quantum strategy wins with certainty.

1) Mermin-Peres Magic Square Game:

  • Players win when their row and column assignments agree at the selected cell, with even row parity and odd column parity.The referee chooses a cell uniformly and reveals its row to one player and its column to the other.Source: 1) Mermin-Peres Magic Square Game · p. 6
  • 8/9 is the maximum winning probability for classical strategies, including randomized strategies, when the referee selects uniformly.A deterministic assignment loses only when the selected cell is (3, 3).Source: 1) Mermin-Peres Magic Square Game · p. 6
  • An entangled strategy wins with probability 1 by measuring the observables associated with the requested row and column.The players place their measurement outcomes in the corresponding cells of their reported assignments.Source: 1) Mermin-Peres Magic Square Game · p. 6, Table 2 · p. 6

2) Channel Model:

The channel model links the magic-square game’s winning condition to whether the receiver or eavesdropper obtains the inputs. The section also compares secrecy capacity with classical-randomness-assisted MAC capacity and gives a no-resource sum-rate bound.

2) Channel Model:

  • When the game is won, the receiver’s channel is noiseless and the eavesdropper’s output is independent of the inputs; when lost, the opposite information pattern applies.The channel construction makes the game’s cooperation relevant to secure communication.Source: 2) Channel Model · p. 6
  • The game samples question pairs and tests whether the players’ answers belong to the specified winning set.The questions are selected uniformly, and each player answers using a strategy determined by their own question.Source: 2) Channel Model · p. 6
  • The secrecy capacity region with shared classical randomness is contained in the corresponding classical-randomness-assisted MAC capacity region because secrecy adds a constraint.The cited passage also states that ordinary MAC capacity is unchanged by sharing classical randomness [26, Remark 2].Source: 2) Channel Model · p. 7
  • R1 + R2 ≤ 3.02 bounds the sum rate of the corresponding MAC when transmitters share no classical or quantum resources.The bound is illustrated by the orange region in Fig. 2.Source: 2) Channel Model · p. 7

3) Main Result:

Theorem 4 characterizes the secrecy capacity region for the magic-square MAC-WTC with entangled transmitters. Its region contains rate pairs outside the classical-CR outer bound, as illustrated by the green region in Fig. 2.

3) Main Result:

  • The entanglement-assisted secrecy capacity region includes rate pairs outside the outer bound for transmitters sharing classical common randomness.Fig. 2 marks these entanglement-achievable, classically unachievable rate pairs in green.Source: 3) Main Result · p. 7, Figure 2 · p. 7
  • Because no classical strategy wins the game with certainty, the classical-CR outer bound is generally loose for this channel.The passage attributes the looseness to information necessarily leaking to the eavesdropper in the classical setting.Source: 3) Main Result · pp. 7–8
  • The construction samples U1 and U2 uniformly from {1, 2, 3}, has each transmitter measure its row or column observables, and sends its query with the resulting triplet.The queries correspond to the referee’s questions in the pseudo-telepathy game.Source: 3) Main Result · p. 7
  • The perfect entangled strategy delivers Y=(U1,U2) while Z is independent, with I(U1,U2;Y)=2 log2(3)=2 × 1.585 and the stated leakage mutual informations equal to zero.The construction uses 2 qubit pairs per channel use.Source: 3) Main Result · pp. 7–8

IV. OUTPUT STATISTICS OF MAC WITH ENTANGLED TRANSMITTERS

This section establishes two strong soft-covering lemmas for MACs with entangled transmitters, covering i.i.d. output targets and a non-i.i.d. target. The lemmas support semantic security and extend soft-covering to settings with entanglement.

IV. OUTPUT STATISTICS OF MAC WITH ENTANGLED TRANSMITTERS

  • Two strong soft-covering lemmas cover i.i.d. output targets when both transmitters use codebooks and non-i.i.d. targets when only one does.In the second case, one transmitter sends an i.i.d. codeword while the other selects from a codebook.Source: §IV · p. 8
  • Strong soft-covering makes the induced-to-target distribution distance decay exponentially with block length, while failure probability is doubly exponentially small.These guarantees hold with high probability over the random code construction.Source: §IV · p. 8
  • Both lemmas require individual and sum-rate mutual-information thresholds for the auxiliary variables and channel output.Under those conditions, the lemmas provide exponential approximation guarantees for sufficiently large block lengths.Source: §IV · p. 9
  • Coded time sharing and random codebooks encode local randomness; POVMs on shared entangled states can correlate the channel inputs.The measurements depend on classical auxiliary variables at each transmitter.Source: §IV · p. 9
  • The achievable rate regions can also be extended to transmitters sharing entanglement resources of limited rate.Source: §IV · p. 10

V. CONCLUSION

The paper derives inner and outer secrecy-capacity bounds for classical MAC-WTCs with entangled transmitters and shows that entanglement can strictly enlarge capacity relative to classical shared randomness. Its proof constructs random codebooks and has each transmitter encode by measuring its share of the entangled state.

V. CONCLUSION

  • Shared entanglement can strictly enlarge MAC-WTC secrecy capacity compared with transmitters sharing only classical common randomness.Source: §V · p. 10

APPENDIX A PROOF OF THEOREM 1

  • The proof constructs codes that satisfy both reliability and semantic-security constraints when transmitters share n copies of a bipartite state.Source: Appendix A · p. 10

A. Random Codebook Generation

B. Encoding

  • Each transmitter measures its share of the entangled state using codeword-dependent POVMs, then sends the measurement outcome over the channel.Source: B. Encoding · p. 10

C. Decoding and Error Probability Analysis

Under the listed mutual-information rate constraints, the average decoding error vanishes as block length grows. The decoder selects the unique message pair whose codewords are jointly typical with the received sequence.

C. Decoding and Error Probability Analysis

D. Security Analysis

This section analyzes security by establishing semantic-security achievable rate regions for the two-message and single-confidential-message cases, alongside single-letter rate upper bounds. Its proofs use time-sharing, typical- and atypical-output-statistics bounds, and supporting purification, cardinality, and weak-secrecy outer-bound arguments.

D. Security Analysis

  • Under the three randomization-rate inequalities in (28a–c), the probability that a randomly generated code sequence is semantically secure approaches 1 as blocklength grows.For each message pair, the security deviation exceeds exp(−γ1n) with probability at most exp(−exp(γ2n)) for sufficiently large n.Source: D. Security Analysis · pp. 12–13
  • Eliminating R′1 and R′2 from the reliability and security constraints yields the achievable rate region B1 in Theorem 1.Source: D. Security Analysis · p. 13
  • For a single confidential message with R2 = 0, region B2 is achievable when R′1 > I(U1; Z|U0, U2), using Lemma 2 or Lemma 3 according to the second transmitter’s randomization rate.Source: D. Security Analysis · p. 13

APPENDIX B PROOF OF LEMMA 1

  • Both inner and outer regions can be achieved using pure shared states and auxiliary variables with bounded cardinalities.Source: Appendix B · p. 13

A. Inner Bound

2) Cardinality Bounds:

APPENDIX C PROOF OF THEOREM 2

  • The outer bound established under weak secrecy also applies to semantic security, which is strictly stronger.Source: Appendix C · p. 15

APPENDIX D PROOF OF THEOREM 3

  • Theorem 3’s outer-bound proof uses weak secrecy, and the resulting region remains a valid outer bound under semantic security.Source: Appendix D · p. 16
  • Theorem 3 bounds nR1 by I(U1;Y|U0,U2) − I(U1;Z|U0,U2) plus n(δn + ϵn), and analogously bounds nR2 with U1 and U2 swapped.Both bounds include the same n(δn + ϵn) term.Source: Appendix D · p. 17

APPENDIX E PROOF OF LEMMA 2

  • Lemma 6 states a weaker version of Lemma 2 for a discrete memoryless MAC with entangled transmitters, with positive γ1 and γ2 for sufficiently large n.Lemma 2 is then proved using a time-sharing argument.Source: Appendix E · p. 17
  • The proof develops bounds on typical and atypical sets, including a typical-set bound for randomly generated codeword collections.The typical-set bound is stated for finite random variables with a joint probability mass function; the atypical-set bound is introduced separately.Source: Appendix E · pp. 18–19
  • Time-sharing between the two corner points establishes Lemma 2’s region: continuity and monotonicity yield a parameter λ⋆ for which both rates exceed their corresponding interpolated leakage terms.The argument uses R1 + R2 > I(U1, U2; Z|U0) to obtain λ2 < λ⋆ < λ1.Source: Appendix E · p. 21

APPENDIX F PROOF OF LEMMA 3

  • Lemma 3’s region R1 follows under R1 > I(U1; Z|U0), R2 > I(U2; Z|U0), and R1 + R2 > I(U1, U2; Z|U0).The proof obtains these conditions from Lemma 2 and concludes the region R1.Source: Appendix F · pp. 21–22
  • For region R2, the proof combines atypical-set bounds with a typical-term bound from Lemma 9, completing the argument when R1 > I(U1; Z|U0, U2).The analysis covers both R2 ≤ I(U2; Z|U0) and R2 > I(U2; Z|U0).Source: Appendix F · pp. 22–24
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