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Zero Knowledge Proofs in Quantum Networks

Tuhin Paul, Srijani Das, Manasi Patra, Ramij Rahaman

arXiv:2609.35339v1quant-phcs.CRcs.SImath-phmath.MP

TL;DR

Existing zero-knowledge constructions rely on computational assumptions, and quantum-verifier results leave device independence and multipartite information-theoretic security open. The paper uses quantum-state indistinguishability to build entanglement-certification protocols, including device-independent network protocols. It establishes information-theoretic zero knowledge through simulation of the verifier’s view and reports robustness under noise.

  • Problem

    Existing constructions rely on computational assumptions, while device-independent and multipartite information-theoretic zero-knowledge protocols remain insufficiently addressed.

    Source: §I · p. 1

  • Method

    The paper builds zero-knowledge certification around quantum-state indistinguishability, using device-independent self-testing of GHZ correlations and measurement observables in quantum networks.

    Source: §I · pp. 1–2

  • Results

    The protocols establish device-independent zero knowledge through simulation of the verifier’s view without access to the prover’s secret state, and the guarantees persist under noise.

    Source: §I · p. 2

  • Takeaways & Limitations

    The framework extends zero-knowledge certification to shared entangled states and multiparty quantum-network settings with information-theoretic security.

    Source: §I · p. 1, §VI · p. 17

    The protocols assume verifiers know the candidate ensemble but not the actual state; verification against a known state is beyond scope.

    Source: §V · p. 13

Abstract

from arXiv · show

Zero-knowledge proofs (ZKPs) enable the verification of a statement without revealing any information beyond its validity and constitute a fundamental primitive in cryptography and information theory. However, existing constructions rely on computational assumptions and are predominantly confined to bipartite settings, leaving their information-theoretic realization in bipartite or network scenarios largely unexplored. Here we develop a framework for zero-knowledge verification based on the indistinguishability of quantum states under operational constraints. Exploiting the fundamental limitations imposed by local operations, we show that a verifier is inherently restricted from extracting information about the underlying state while retaining the ability to verify correctness. We construct explicit protocols for multiparty quantum networks that achieve information-theoretic security, ensuring that no subset of collaborating parties can gain knowledge beyond the validity of the statement, independent of their joint computational power.

I. INTRODUCTION

Classical zero-knowledge proofs rely on computational assumptions, and quantum-verifier results do not resolve device independence or multipartite information-theoretic security. The paper develops quantum-state-indistinguishability protocols for zero-knowledge certification, including device-independent protocols for quantum networks.

I. INTRODUCTION

  • Classical zero-knowledge constructions rely on computational assumptions, while quantum algorithms challenge the hardness of some underlying problems.Prior quantum-verifier results preserve zero knowledge for canonical protocols but lack device independence; multipartite information-theoretic extensions remain largely uncharted.Source: §I · p. 1
  • The paper develops a framework based on quantum-state indistinguishability and constructs multiparty protocols with information-theoretic security against collaborating parties.The stated security allows no subset of parties to learn beyond the proven statement's validity, independent of their computational power.Source: §I · p. 1
  • The work asks whether a shared entangled resource can be certified without revealing its identity, and introduces protocols for Bell and GHZ resources.The construction progresses from Bell-state certification to a device-independent protocol for multipartite networks.Source: §I · pp. 1–2
  • A two-basis Bell-state test closes a single-basis correlation loophole, but higher-dimensional realizations can still reproduce the target correlations.The paper therefore identifies both observable incompatibility and effective two-dimensionality as requirements for secure certification.Source: §I · p. 2
  • For quantum networks, self-testing GHZ correlations and measurement observables supports device-independent zero-knowledge, proven by simulating the verifier’s view; the protocols also remain robust under noise.The simulator does not access the prover’s secret state.Source: §I · p. 2

II. ENTANGLEMENT CERTIFICATION BASED ZKP

The section motivates zero-knowledge certification of shared entanglement by the need to establish a resource’s identity, not merely its presence. It frames the task as proving knowledge of that identity while keeping the state concealed.

II. ENTANGLEMENT CERTIFICATION BASED ZKP

  • Entanglement alone does not guarantee a protocol’s success; its utility can depend on the identity of the shared state.The section notes that performance and often security in tasks such as teleportation and quantum key distribution depend on the specific Bell state.Source: §II · p. 2
  • A party who knows a shared Bell state’s identity can transform it into any of the four Bell states using a local unitary operation.Source: §II · p. 2
  • The certification task asks a prover to demonstrate knowledge of a shared entangled state while concealing its identity from the verifier.Source: §II · p. 2

A. Bell-State Certification ZKP

The section develops Bell-state certification protocols that let a prover demonstrate knowledge of a shared state while concealing its identity. It addresses correlation and device-dimension loopholes, then introduces a device-independent protocol and extends the discussion to multipartite GHZ states.

Single-basis certification

  • In the single-basis protocol, an honest prover is accepted with certainty, while an uninformed prover’s success probability falls exponentially with repeated rounds; the verifier’s local state and transcript reveal no Bell-state identity.The uninformed prover succeeds with probability 1/2 in one round.Source: A. Bell-State Certification ZKP · p. 3

Correlation loophole and two-basis repair

  • A separable state can pass the single-basis test, but randomly testing complementary Z and X bases rules out that correlation-only strategy.The single-basis test checks observed correlations without certifying the promised Bell-state structure.Source: A. Bell-State Certification ZKP · pp. 3–4

Generalization

  • The protocols generalize to the complete Bell-state ensemble, and the paper distinguishes demonstrating state knowledge from distinguishing the state by LOCC.The four Bell states cannot be perfectly distinguished by LOCC.Source: A. Bell-State Certification ZKP · p. 4

Higher-dimensional loophole

  • The two-basis test still admits a higher-dimensional separable simulation, so security requires certifying both measurement incompatibility and effective two-dimensionality.The simulation reproduces the target Bell-state correlations in the tested bases.Source: A. Bell-State Certification ZKP · pp. 4–5

Device-independent ZKP

  • The device-independent protocol maps selected Bell states to |ψ−⟩ and uses a CHSH test; honest provers achieve the maximal quantum violation, while uninformed provers cannot exceed the local-realistic bound.The verifier’s reduced state and the CHSH transcript are independent of the original Bell-state identity.Source: A. Bell-State Certification ZKP · pp. 5–6

Multipartite extension

  • The work extends entangled-state certification to a multipartite network, where one prover seeks to convince multiple verifiers about shared GHZ states.The stated target is a randomly selected sequence of shared n-qubit GHZ states.Source: A. Bell-State Certification ZKP · p. 6

B. GHZ certification ZKP

The section develops GHZ-state certification protocols that let a prover demonstrate knowledge of shared-state identities while keeping them concealed. It identifies a loophole in the basic prepare-and-measure test and introduces a device-independent version based on self-testing.

Basic GHZ certification

  • In the basic GHZ protocol, the prover uses the shared-state identity and other verifiers’ X-basis outcomes to predict a withheld verifier’s outcome with certainty.The withheld verifier accepts when the announced prediction matches that outcome; repeating rounds suppresses soundness error exponentially.Source: B. GHZ certification ZKP · pp. 6–7
  • The basic protocol’s X-basis test does not certify its promised GHZ entanglement: a separable state reproduces its perfect correlations and can pass.The authors motivate self-testing from observed nonlocal correlations to certify the underlying state and observables.Source: B. GHZ certification ZKP · p. 7

Device-independent GHZ certification

  • The device-independent protocol samples a state and an operator, has the prover correct and measure the selected state, then checks the parties’ eigenvalue relations.For odd n, the protocol certifies the prover’s knowledge of the shared-state identity.Source: B. GHZ certification ZKP · pp. 7–8
  • The device-independent protocol has perfect completeness: an honest prover attains the maximum Bell value n + 1 and is accepted with probability 1.The verifiers’ eigenvalue relations self-test the selected correlations to the canonical GHZ state.Source: B. GHZ certification ZKP · p. 8
  • Without state knowledge, a local-realistic strategy satisfies at most n of the n + 1 eigenvalue constraints, suppressing soundness error exponentially in k/n.With prior knowledge probability p, single-round acceptance is bounded by 1 − (1 − p)/(n + 1).Source: B. GHZ certification ZKP · p. 8
  • The prover’s correction maps each of four possible GHZ states to one canonical state before measurement, so even collective verifier measurements reveal no state identity beyond claim validity.The same guarantee is asserted for untrusted verifier apparatus because certification uses correlations reaching the algebraic maximum.Source: B. GHZ certification ZKP · p. 8
  • For constant accuracy and failure probability, the device-independent protocol uses O(rn) computation and O[r(n + log N)] classical communication.It consumes r shared GHZ states.Source: B. GHZ certification ZKP · pp. 8–9

III. SIMULATION OF THE VERIFIER’S VIEW

The section frames zero knowledge as equality or indistinguishability between the verifier’s real view and a simulator’s view independent of the prover’s secret. It states an idealized independence assumption and points to robust treatment of coherent attacks.

III. SIMULATION OF THE VERIFIER’S VIEW

  • A simulator constructed without the prover’s secret should reproduce the verifier’s real view, so indistinguishable views establish zero knowledge.The section says it constructs such simulators for two device-independent protocols, with the remaining protocols following from them.Source: §III · p. 9
  • For arbitrary quantum verifiers, the initial auxiliary state is assumed independent of the shared prover–verifier entangled resource in the idealized setting.The passage notes that this factorization need not hold with imperfections or coherent correlations.Source: §III · p. 9
  • The section says coherent attacks and their robust treatment are addressed later.The verifier’s auxiliary system is initially assumed tensor-product with the shared resource in the idealized analysis.Source: §III · p. 9

A. Simulator for Protocol 3

For Protocol 3, the simulator uses a fixed reference secret to reproduce the verifier’s view without learning the actual secret string. The argument obtains perfect zero knowledge by making the selected states and resulting view independent of that string.

A. Simulator for Protocol 3

  • The prover’s local corrections map selected Bell states to the same canonical state, independent of the secret string.The selected subsystem is effectively |ψ−⟩⊗k; the passage states the global phase has no physical consequence.Source: A. Simulator for Protocol 3 · p. 9
  • For any two secret strings, the joint outcome distribution for fixed challenges and settings is identical.The verifier’s conditional state remains secret-independent under its arbitrary CPTP maps.Source: A. Simulator for Protocol 3 · pp. 9–10
  • The simulator chooses a fixed reference string and interacts with the verifier using the prescribed challenge and CHSH-setting interfaces.It internally executes the verifier and generates the corresponding prover responses.Source: A. Simulator for Protocol 3 · p. 10
  • Protocol 3 is perfect zero knowledge against arbitrary malicious quantum-capable verifiers with a classical communication interface.The section derives this by matching the simulated view to the real view for every secret string.Source: A. Simulator for Protocol 3 · p. 10

B. Simulator for Protocol 5

For Protocol 5, the simulator prepares a canonical GHZ reference state and reproduces the verifier’s measurement transcript without using the hidden state identity. The resulting transcript matches the real protocol exactly.

B. Simulator for Protocol 5

  • The prover’s identity-conditioned correction maps each challenged GHZ state to the canonical target, making subsequent measurement statistics invariant under the initial state label.The simulator therefore needs no hidden identity for the challenged state.Source: B. Simulator for Protocol 5 · pp. 10–11
  • In an honest execution, the outcome distribution is independent of the challenged index and hidden state identity while satisfying the GHZ parity constraints.The passage specifies the parity relation for settings i in {0, . . . , n}.Source: B. Simulator for Protocol 5 · p. 11
  • The simulator samples the challenge and setting, measures the canonical reference state, and combines the resulting outcomes to reproduce the ideal distribution.It initializes an independent verifier auxiliary register in the same state and applies the transcript dephasing operation.Source: B. Simulator for Protocol 5 · p. 11
  • The real and simulated transcript states are equal, establishing perfect zero knowledge for the classical transcript.The simulator generates its view without reference to the true state identity.Source: B. Simulator for Protocol 5 · p. 11

IV. SELF-TESTING OF GHZ CORRELATIONS

For odd n, the GHZ eigenvalue relations rule out local-realistic correlations and uniquely identify the target GHZ state. The paper extends this self-testing approach to generalized GHZ-basis states using multiple simultaneous constraints rather than maximal Bell violation.

IV. SELF-TESTING OF GHZ CORRELATIONS

  • Multiple simultaneous eigenvalue constraints, rather than a Bell inequality or maximal-violation condition, characterize the target GHZ correlations and support security analysis against coherent attacks.The paper contrasts this approach with inequality-based self-testing.Source: §IV · p. 11
  • For odd n, the GHZ eigenvalue relations are incompatible with local-realistic correlations and uniquely identify |ϕ−_n⟩.The relations also self-test the target state up to local unitaries.Source: §IV · p. 11
  • The proof combines a local-realistic contradiction with uniqueness of the quantum state satisfying the eigenvalue relations.Multiplying all relations yields opposite products for deterministic local-realistic values and the required eigenvalues; the quantum solution is |ϕ−_n⟩ up to global phase.Source: §IV · pp. 11–12
  • Jordan’s-lemma invariant sectors reduce the untrusted observables to blocks whose saturating states are fixed by the eigenvalue relations.Local isometries extract the ideal GHZ state on ancillary systems, with residual sector degrees of freedom retained separately.Source: §IV · p. 12
  • The GHZ paradox self-tests generalized n-qubit GHZ-basis states for odd n.The results extend beyond the previously described self-testing of |ϕ±_n⟩ at maximal Bell-score saturation.Source: §IV · pp. 11, 13

V. ROBUSTNESS UNDER NOISE

The section studies whether a prover can demonstrate knowledge of a shared state’s identity without revealing the state. Its soundness analysis bounds false certification from the prover’s limited information about GHZ correlations and extends from i.i.d. rounds to coherent attacks.

V. ROBUSTNESS UNDER NOISE

  • Verification assumes verifiers know only the candidate ensemble, not the actual shared state; verification against a known state is outside the work’s scope.The states may be prepared by the prover or an independent dealer.Source: §V · p. 13
  • A prover’s limited state knowledge constrains its ability to reproduce GHZ-game correlations, linking winning probability to its guessing probability for the verifiers’ joint outcomes.The analysis first establishes a single-round soundness bound under an i.i.d. assumption.Source: §V · p. 13
  • The soundness analysis extends to arbitrary coherent attacks using the entropy accumulation theorem, while allowing an external eavesdropper arbitrary side information and full collaboration with the prover.Source: §V · p. 13

A. GHZ paradox under noise

The section characterizes states that approximately satisfy the GHZ paradox constraints. It uses correlation-tensor and stabilizer structure to show that such states admit a decomposition around the target GHZ state.

A. GHZ paradox under noise

  • An n-qubit density operator is represented by real Pauli correlation-tensor elements, with the identity component normalized to one.Local Pauli conjugation on the first qubit preserves the state’s form while changing specified tensor-component signs.Source: A. GHZ paradox under noise · p. 13
  • The GHZ stabilizer generators commute and have a one-dimensional common +1 eigenspace that specifies the target pure state.The group average gives the corresponding rank-one projector.Source: A. GHZ paradox under noise · p. 13
  • Any state satisfying the GHZ paradox constraints within tolerance ϵ admits the decomposition stated in Eq. (12).The analysis relates the noise parameter to the tolerance through η ≤ ϵ/2 and imposes trace normalization and global positivity.Source: A. GHZ paradox under noise · pp. 13–14

B. ZKP security against coherent attack

The section uses entropy accumulation and independently certified single-round entropy rates to analyze Protocol 5 against coherent quantum attacks. It derives finite-round entropy, soundness, and zero-knowledge bounds without requiring an i.i.d. assumption.

B. ZKP security against coherent attack

  • The EAT lifts single-round entropy bounds to sequential executions, while the single-round entropy rate must be certified independently using device-independent SDP hierarchies.The NPA hierarchy computes the constrained guessing probability used to bound the accumulated entropy rate.Source: B. ZKP security against coherent attack · pp. 14, 15, 17
  • The EAT formulation permits quantum memory between rounds and does not impose an independent-and-identically-distributed assumption on GHZ tests.Its Markov-chain condition is the structural requirement for entropy accumulation with arbitrary inter-round quantum correlations.Source: B. ZKP security against coherent attack · p. 14
  • Passing the GHZ tests certifies at least Nt − O(√N) bits of smooth min-entropy in the transcript, independent of the adversary’s quantum memory.The finite-round rate supports extractor privacy and soundness even when cheating provers share arbitrary entangled states across rounds.Source: B. ZKP security against coherent attack · p. 15
  • If the GHZ Protocol 5 achieves Bell violation within tolerance ϵZK for odd n, the theorem bounds the verifiers’ reduced view’s distinguishing probability.The displayed bound is given as ϵZK/n.Source: B. ZKP security against coherent attack · pp. 15–16
  • The soundness criterion bounds cheating probability exponentially over N rounds, with asymptotic decay rate set by the quantum-classical margin (2 − ϵ)/(n − 1).The bound applies to N consecutive rounds.Source: B. ZKP security against coherent attack · p. 16

VI. CONCLUSION

The paper develops information-theoretically secure zero-knowledge certification for shared entangled states, with explicit bipartite and multipartite protocols. Its device-independent network protocols retain their guarantees under noise and require no computational assumptions.

VI. CONCLUSION

  • The framework certifies shared entangled states with information-theoretic security through explicit bipartite and multipartite protocols using Bell and GHZ resources.The work also establishes robustness against noise in the shared quantum resources.Source: §VI · p. 17
  • A correlation loophole in naive Bell-state certification motivates a two-basis scheme and then a device-independent formulation that excludes higher-dimensional realizations.Source: §VI · p. 17
  • For quantum networks, self-testing of GHZ correlations and measurement structure yields device-independent zero-knowledge protocols, with security proven by simulating the verifier’s view.Completeness and soundness hold independently of computational assumptions.Source: §VI · p. 17
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