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Optimal verification of entangled states with local measurements
Sam Pallister, Noah Linden, Ashley Montanaro
TL;DR
The paper addresses efficient verification of quantum devices producing a target entangled state under local-measurement constraints. It formulates verification as a worst-case hypothesis test and constructs non-adaptive local projective strategies for two-qubit and stabilizer states. For stabilizer states, using all stabilizer generators yields n_opt ≈ Nϵ−1 ln δ−1, while local strategies require at most twice as many measurements as the optimal non-local strategy.
Problem
The paper addresses efficient verification of quantum devices producing a target entangled state under local-measurement constraints.
Source: Appendix A · p. 6Method
It formulates verification as a worst-case hypothesis test and constructs non-adaptive local projective strategies for two-qubit and stabilizer states.
Source: Stabilizer states · p. 4, Appendix A · p. 6, Appendix C · p. 10Results
For stabilizer states, using all stabilizer generators yields n_opt ≈ Nϵ−1 ln δ−1, while local strategies require at most twice as many measurements as the optimal non-local strategy.
Source: Stabilizer states · p. 4Takeaways & Limitations
Efficient verification is available for arbitrary two-qubit states and stabilizer states using local, non-adaptive, non-collective measurements.
Source: Stabilizer states · p. 4, Supplemental Material: Optimal… · p. 6For arbitrary larger pure states, deriving the strategy and correct constants may be involved.
Source: Stabilizer states · p. 5
Abstract
from arXiv · showhide
Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterise the output state by quantum state tomography; or alternatively to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal amongst local verification strategies for two qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, non-adaptive and non-collective measurements only incurs a constant-factor penalty over a strategy without these restrictions.
Stabilizer states.
The stabilizer-state construction provides efficient local verification with a constant-factor penalty relative to the globally optimal strategy, while exposing a trade-off between measurement settings and total measurements.
Stabilizer states.
- N stabilizer generators are necessary because fewer generators leave an orthogonal state that can fool the verifier with certainty.Using only the generators gives n_s.g. opt ≈ Nϵ^-1 ln δ^-1.Source: Stabilizer states · p. 4
- Using all 2^N−1 stabilizers reduces the measurement scaling relative to the generator-only strategy, but requires more measurement settings.The generator-only strategy scales linearly with N, whereas the full-stabilizer strategy remains bounded by a constant-factor multiple of the optimal non-local strategy.Source: Stabilizer states · pp. 4–5
- 2ϵ^-1 ln δ^-1 measurements suffice for a local strategy for any stabilizer state, at most twice the optimal non-local strategy.This bound follows from using the full set of stabilizers with equal weight.Source: Stabilizer states · p. 4
- The stabilizer strategy is not always exactly optimal because the best local strategy can depend on the individual state structure.For |00⟩, direct projection onto |00⟩ is both local and provably optimal.Source: Stabilizer states · pp. 4–5
- The recipe for arbitrary two-qubit pure states may extend to larger pure states, but deriving the strategy and exact constants is expected to be involved.This marks the scope boundary beyond the explicitly treated two-qubit and stabilizer-state cases.Source: Stabilizer states · p. 5
Supplemental Material: Optimal verification of entangled states with local
The supplemental material formalizes the verification framework, simplifies the protocol optimization, derives the two-qubit strategy, develops stabilizer-state protocols, and supplies the hypothesis-testing framework.
Supplemental Material: Optimal verification of entangled states with local
- The appendices cover the formal framework, protocol simplification, two-qubit optimization, stabilizer-state strategies, and hypothesis testing.Appendices A–E respectively address these components.Source: Supplemental Material: Optimal… · p. 6
Appendix A: Quantum state verification
The appendix formulates verification as a worst-case hypothesis test and shows that certainty-preserving local projective strategies are non-adaptive and nearly globally optimal in scaling.
Appendix A: Quantum state verification
- The protocol uses two-outcome measurements that reject immediately on a fail outcome, and seeks the minimum n satisfying the acceptance constraints.All allowed measurements must pass with certainty on |ψ⟩.Source: Appendix A · p. 6
- The verifier distinguishes perfect outputs from states whose fidelity with |ψ⟩ is at most 1−ϵ, with worst-case failure probability δ.The protocol accepts the good case with certainty and limits bad-case acceptance to δ; 1−δ is the statistical power.Source: Appendix A · p. 6
- Allowing measurements that do not always pass on |ψ⟩ worsens asymptotic measurement scaling quadratically in ϵ.More elaborate adaptive, collective, or multi-outcome protocols are not treated explicitly and are not expected to improve markedly over the local strategies.Source: Appendix A · pp. 6–7
- Without measurement restrictions, projecting onto |ψ⟩ is optimal even among adaptive and collective protocols.Replacing any accepting operator by the projector onto |ψ⟩^⊗n preserves perfect acceptance while not increasing acceptance of other states.Source: Appendix A · p. 6
- Optimal protocols of this form can be taken non-adaptive: neither prior outcomes nor earlier measurement choices affect later choices.The optimization reduces to selecting a fixed strategy operator from allowed projective measurements.Source: Appendix A · p. 7
Appendix B: Verification strategy optimisation
The optimization reduces adversarial verification to analyzing strategy operators that accept |ψ⟩ with certainty, with pure adversarial states sufficient and applicability to arbitrary states unresolved.
Appendix B: Verification strategy optimisation
- The adversary can be assumed to use a pure state with fidelity at most 1−ϵ, parameterized by overlap with |ψ⟩ and an orthogonal state.The worst case is represented as |ψ̄_ϵ⟩=√(1−ϵ)|ψ⟩+√ϵ|ψ^⊥⟩.Source: Appendix B · pp. 7–8
- For a strategy operator satisfying Ω|ψ⟩=|ψ⟩, the adversary chooses the boundary fidelity 1−ϵ to minimize the verifier’s distinguishing gap.The resulting optimization depends on the largest orthogonal eigenvalue of Ω.Source: Appendix B · p. 8
- Any strategy with Ω|ψ⟩=|ψ⟩ is future-proof: it remains independent of ϵ and viable when the target fidelity requirement is tightened.If the initial distinguishing gap is positive, it remains positive for any smaller positive ϵ.Source: Appendix B · p. 8
- Constructing and optimizing viable local-projective strategies for arbitrary n-qubit states remains unclear, so the paper focuses on two-qubit and stabilizer states.The unresolved tasks are constructing suitable Ω families and solving the resulting optimization.Source: Appendix B · p. 8
Appendix C: States of two qubits
This appendix derives optimal local verification strategies for arbitrary pure two-qubit states by reducing the problem to canonical cases and exploiting symmetry and locality constraints. It also treats endpoint and maximally entangled cases separately and decomposes the resulting strategies into a small set of locally implementable projective measurements.
Local equivalence
- Any optimal strategy for a locally equivalent two-qubit state is obtained by conjugating the original strategy with the corresponding local unitary.This permits restricting the analysis to |ψ⟩ = sin θ|00⟩ + cos θ|11⟩ without loss of generality.Source: Appendix C · p. 9
Optimality constraints
- Identity measurements cannot improve verification because adding an identity component weakly increases the maximum acceptance probability q for orthogonal states.Thus an optimal strategy need not contain the always-accept measurement.Source: Appendix C · p. 9
Optimal strategy
- For 0 < θ < π/2 and θ ≠ π/4, every optimal local, trusted, projective strategy accepting |ψ⟩ with certainty can be expressed using four measurement settings.Product-state cases are treated separately, while the maximally entangled case θ = π/4 is excluded from this theorem.Source: Appendix C · pp. 9–10
Certainty and locality constraints
- Certainty acceptance excludes product and one-sided measurement terms for entangled states, while the remaining two-sided term is restricted to the positive eigenspace of ZZ except at θ = π/4.These restrictions follow from the impossibility of projecting an entangled state onto a product projector with certainty and from the corresponding single-party acceptance condition.Source: Appendix C · pp. 10–11
Symmetry reduction
- Averaging over continuous rotations, qubit swapping, and complex conjugation preserves or improves performance and reduces the optimization to symmetric strategies characterized by two eigenvalues.The analysis further shows that one post-averaging trace-three term suffices, and locality imposes a lower bound on its admissible eigenvalue.Source: Appendix C · pp. 11, 12, 14
Optimization
- The optimization reduces to case (a), where λ1 and λ2 intersect and q is minimized subject to λ1 = λ2; case (c) is impossible because λ1(1) = 1 and λ2(1) = 0.In case (b), the derivative with respect to P changes sign around the points satisfying λ1(0,P,T) = λ2(0,P,T), forcing the minimum onto the intersection condition.Source: Appendix C · pp. 14–15
Optimal strategy
- Substituting the optimal parameters and reexpressing them in terms of θ yields the optimal strategy, whose orthogonal-state acceptance probability determines its verification performance.The appendix then derives the corresponding measurement complexity for accuracy ϵ and statistical power 1 − δ.Source: Appendix C · p. 15
Verification cost
- (2 + sin θ cos θ)ϵ^-1 ln δ^-1 measurements suffice asymptotically under the optimal test.This follows from ln((1 − ϵ(1 − qopt))^-1) ≈ (2 + sin θ cos θ)ϵ^-1 ln δ^-1.Source: Appendix C · p. 15
Local implementation
- The optimal strategy can be decomposed into three terms involving separable states, yielding a strategy composed of locally implementable projective measurements.The decomposition provides the required local measurement form.Source: Appendix C · p. 15
Special cases
- For θ = 0 or π/2, the target is |00⟩ or |11⟩, so projecting directly onto the state is locally allowed, with p = 1 and q = 0.The appendix separately notes that θ = π/4 is treated in the main text using equally weighted XX, −YY, and ZZ measurements.Source: Appendix C · p. 16
Appendix D: Stabilizer states
The appendix constructs stabilizer-based verification strategies for N-qubit stabilizer states and characterizes their measurement complexity and scope limits. Using the full stabilizer set achieves the same asymptotic infidelity scaling as the two-qubit strategy, while generator-only strategies are substantially weaker.
Scope
- The appendix does not exclude non-stabilizer strategies from achieving a small constant-factor improvement over the proposed stabilizer strategy.This is a remaining scope boundary rather than a failure of the stabilizer construction.Source: Appendix D · p. 16
Optimal stabilizer strategy
- A stabilizer strategy using 2^N−1 linearly independent stabilizers achieves the optimal scaling in verification measurements, with equal weights across the stabilizer projectors.The construction uses the maximal set excluding the identity and assigns weight 1/(2^N−1) to each stabilizer.Source: Appendix D · pp. 16–17
Optimization argument
- The verifier’s optimal operator equalizes all eigenvalues on the subspace orthogonal to the target state, minimizing the adversary’s largest acceptance eigenvalue.The adversary can select the orthogonal eigenstate with the largest eigenvalue, so equalization is optimal.Source: Appendix D · p. 17
Measurement complexity
- The stabilizer strategy requires at most 2ϵ−1 ln δ−1 measurements, independent of N, and is within a factor of two of optimal non-local verification.The comparison is with direct projection onto the target state.Source: Appendix D · p. 18
Reduced strategies
- Using fewer than N stabilizers fails completely because an orthogonal state can remain stabilized by every selected measurement and therefore always pass.The dimension of the subspace stabilized by k<N stabilizers is at least 2^(N−k), guaranteeing an orthogonal passing state.Source: Appendix D · p. 18
- Restricting the protocol to the N stabilizer generators creates a trade-off: fewer accessible settings require much poorer scaling in N than the full stabilizer construction.The generator weights are optimized uniformly, with μj=1/N.Source: Appendix D · pp. 18–19
Appendix E: Concentration inequalities and the relative entropy
This appendix analyzes verification as asymmetric hypothesis testing and uses relative-entropy concentration to determine measurement requirements. It shows that certainty of acceptance for the target state gives a quadratic asymptotic advantage in infidelity over strategies that do not accept it with certainty.
Main proposition
- Any strategy that accepts the target state with certainty and rejects alternatives with nonzero infidelity needs asymptotically fewer measurements than strategies with imperfect target acceptance.The comparison concerns fixed Type II error in the infidelity parameter ϵ.Source: Appendix E · p. 20
Hypothesis-testing framework
- The relevant hypothesis test treats target acceptance and non-target acceptance as Bernoulli outcomes, with its asymptotic Type II error governed by relative entropy.The Chernoff–Stein lemma supplies the asymptotic relation between error probability, sample count, and relative entropy.Source: Appendix E · p. 20
Concentration analysis
- The Chernoff–Stein bound is asymptotically tight, and the resulting measurement count has distinct limiting forms depending on the relative size of the acceptance gap.One limit yields the quadratic dependence on ϵ−1, while certainty acceptance produces the linear dependence.Source: Appendix E · pp. 20–21
Scaling
- In the worst case, general strategies require quadratically many measurements in ϵ−1, whereas certainty-accepting strategies require only linearly many.This follows from the limiting behavior of the relative-entropy expression for the measurement count.Source: Appendix E · p. 21