PaperlayerView exact revision on arXiv

Source-linked AI summary

Efficient verification of bipartite pure states

Zihao Li, Yun-Guang Han, Huangjun Zhu

arXiv:1901.09783v2quant-ph

TL;DR

Efficient verification of bipartite pure states has been limited, while tomography and direct fidelity estimation require inefficiently many measurements. The paper proposes local-projective protocols using one-way or two-way communication. Their test counts are comparable to the best entangling-measurement protocols, and one protocol is optimal for adversarial verification even without locality restrictions.

  • Problem

    Tomography consumes resources by extracting unnecessary information, direct fidelity estimation has suboptimal measurement scaling with infidelity, and efficient verification was known only for restricted states.

    Source: §I · p. 1

  • Method

    The protocols use adaptive local projective measurements with one-way or two-way classical communication, leveraging mutually unbiased bases or complex projective 2-designs.

    Source: §I · p. 1, D. Alternative protocols based on… · p. 3, §V · p. 5

  • Results

    The test counts are comparable to the best entangling-measurement protocols, and an adversarial-scenario protocol is optimal even among protocols allowing entangling measurements.

    Source: §I · p. 1, §V · p. 5

  • Takeaways & Limitations

    The protocols offer a tool for quantum information processing with bipartite states and insight into foundational studies of entanglement.

    Source: §V · p. 5

Abstract

from arXiv · show

We propose practical and efficient protocols for verifying bipartite pure states for any finite dimension, which can also be applied to fidelity estimation. Our protocols are based on adaptive local projective measurements with either one-way or two-way communications, which are very easy to implement in practice. They can extract the key information much more efficiently than known protocols based on tomography or direct fidelity estimation, and their efficiencies are comparable to the best protocols based on entangling measurements. These protocols highlight the significance of mutually unbiased bases (MUBs) and complex projective 2-designs in quantum state verification and fidelity estimation. Moreover, our protocols can be applied to the adversarial scenario, in which states are controlled by a malicious adversary. In this case, surprisingly, one of our protocols based on local measurements is even optimal among protocols without locality restriction.

I. INTRODUCTION

Tomography and direct fidelity estimation are inefficient for verifying bipartite pure states, and efficient protocols were known only for restricted cases. The paper proposes local-measurement protocols and frames verification efficiency through the strategy’s spectral gap.

I. INTRODUCTION

  • The protocols use local projective measurements with one-way or two-way classical communication, and their test counts are comparable to the best entangling-measurement protocols.Their overhead is at most two times even when one party has access to only two measurement settings.Source: §I · p. 1
  • For adversarially prepared states, one local-measurement protocol is optimal for high-precision verification even among protocols allowing entangling measurements.The authors describe it as the first protocol for this scenario with optimal scaling in the number of tests.Source: §I · p. 1

II. PURE STATE VERIFICATION

  • A verification strategy averages passing-test projectors, and its spectral gap determines the maximum average pass probability for states whose target fidelity is at most 1 − ε.The probability of passing N tests is bounded by [1 − ν(Ω)ε]^N.Source: §II · pp. 1–2
  • At high precision, the required test count is inversely proportional to the spectral gap, so maximizing that gap minimizes the number of tests.Without measurement restrictions, the target-state projector achieves gap 1 and N ≈ ln δ^-1/ε tests.Source: §II · p. 2

A. Construction of tests

The paper constructs tests for bipartite pure states by adapting one party’s local projective measurement to the other party’s outcome. A standard-basis test provides a non-adaptive special case.

A. Construction of tests

  • After Alice measures a basis, Bob tests the normalized conditional state associated with her outcome, ensuring that the target state passes.The construction uses Bob’s projector onto the conditional state as the passing outcome.Source: A. Construction of tests · p. 2
  • When the conditional states are mutually orthogonal, Bob’s measurement need not depend on Alice’s outcome; in the standard basis, both parties simply test whether their outcomes match.The standard test passes when they obtain the same outcome j with s_j > 0.Source: A. Construction of tests · p. 2
  • For an entangled target, each local-projective test has rank at least 2, so verifying it requires at least two distinct tests.The paper assumes s_0 < 1 for the entangled case; separable states can instead be verified with the standard test P_0 alone.Source: A. Construction of tests · p. 2

B. Simplest verification protocols

Any bipartite pure state can be verified with two tests built from mutually unbiased bases. With equal test probabilities, this strategy achieves the best spectral gap among strategies using two distinct local-projective tests.

B. Simplest verification protocols

  • Two mutually unbiased bases yield two tests for any bipartite pure state, with the second basis chosen unbiased to the standard basis.The Fourier basis is one example, but the result does not depend on that specific choice.Source: B. Simplest verification protocols · p. 2
  • The two tests’ projectors have orthogonal supports after removing the target-state projector, enabling the strategy’s bound to be saturated with equal test probabilities.The strategy uses ΩI = (P0 + P1)/2.Source: B. Simplest verification protocols · p. 2
  • This strategy’s spectral gap is maximal among strategies with two distinct local-projective test projectors, so its test count is minimal within that class.The supplied passage identifies the comparison class but does not provide the numerical gap or test count.Source: B. Simplest verification protocols · p. 2

C. Efficient protocols based on complete sets of MUBs

For prime-power dimensions, a complete set of MUBs supports a verification protocol using d + 1 tests. For other dimensions, embedding into a prime-power-dimensional space preserves the result of Eq. (11).

C. Efficient protocols based on complete sets of MUBs

D. Alternative protocols based on 2-designs

The paper gives an alternative protocol for any bipartite pure state using a weighted set of bases forming a 2-design. It matches the efficiency of the complete-MUB protocol when that protocol is available.

D. Alternative protocols based on 2-designs

E. Protocols using two-way CCs

The section constructs two-way communication strategies by averaging strategies with opposite communication directions. It compares their efficiency with one-way strategies and, for two qubits, with the earlier strategy in Ref..

E. Protocols using two-way CCs

  • Averaging the Alice-to-Bob and Bob-to-Alice strategies yields a two-way strategy that is at least as efficient as the Alice-to-Bob strategy.The passage supports this comparison through an inequality between the strategies’ operator norms.Source: E. Protocols using two-way CCs · p. 3
  • All four proposed two-qubit strategies are more efficient than the strategy introduced in Ref..The figure specifies ε = 0.01 and δ = 0.01 for the comparison; its caption does not provide exact test counts.Source: E. Protocols using two-way CCs · p. 4, Figure 1 · p. 4
  • The one-way strategy Ω→ I requires only two distinct tests.The passage identifies the two-test strategy but does not specify its performance relative to the two-way variant.Source: E. Protocols using two-way CCs · p. 4

IV. ADVERSARIAL SCENARIO

The section adapts verification to adversarially prepared states using homogeneous strategies and local measurements. It gives optimal or near-optimal performance conditions for verification and fidelity estimation.

IV. ADVERSARIAL SCENARIO

  • In the high-precision adversarial setting, the minimum test count occurs at β(Ω) = 1/e, unlike the nonadversarial optimum β(Ω) = 0.For ε, δ ≤ 0.01, the optimal β(Ω) is usually close to 1/e.Source: §IV · p. 4
  • For the one-way homogeneous strategy, the optimal high-precision choice is p = max{e^-1, s_0^2/(1 + s_0^2)}.When s_0^2 ≤ 1/(e − 1), Ω→ V is optimal and β(Ω) = 1/e.Source: §IV · p. 4
  • The passage reports that even in the worst case β(Ω) = 1/2, the number of tests is 2 ln δ−1/(ε ln 2), about 6% more than the optimal strategy.For fidelity estimation, the optimal value of p is s_0^2/(1 + s_0^2).Source: §IV · p. 4
  • When β(Ω) = 1/e, the homogeneous two-way strategy Ω↔ VI reaches the minimum test count for high-precision adversarial verification.The strategy is realized through local projective measurements.Source: §IV · pp. 4–5
  • For the two-way strategy, fidelity estimation is optimized at p = (s_0^2 + s_1^2)/(2 + s_0^2 + s_1^2).The strategy is obtained by modifying Ω↔ IV and replacing its P0 test with a locally realizable operator.Source: §IV · pp. 4–5

V. SUMMARY

The summary presents MUB- and 2-design-based protocols for general bipartite pure-state verification. It emphasizes low test and measurement requirements, improved efficiency with additional resources, and an adversarial protocol that is optimal even against entangling measurements.

V. SUMMARY

  • The simplest protocol uses two local-projective-measurement tests with one-way communication, with efficiency comparable to the best entangling-measurement protocol and at most twofold overhead.Efficiency can improve with more measurement settings or two-way communication.Source: §V · p. 5
  • MUBs and 2-designs underpin the proposed simple and efficient protocols for general bipartite pure states.The summary identifies verification as the application of these protocols.Source: §V · p. 5
  • In the adversarial scenario, one local-measurement protocol is optimal even among protocols that can access entangling measurements.The paper also presents the study as a tool for quantum information processing and a source of insight into entanglement.Source: §V · p. 5

APPENDIX

The appendix derives Eq. (14) using a representation of the test projector and the standard basis’s assigned weight. It then invokes the 2-design property of a complete MUB set to obtain Eq. (9).

APPENDIX

  • The proof begins by expressing the test projector P_l in the form referenced in the derivation.The supplied passage does not fully show the expression.Source: APPENDIX · p. 5
  • The second equality follows from Eq. (12), using the standard basis B0 with weight 1/(d + 1).Source: APPENDIX · p. 5
  • The proof obtains Eq. (9) by similar reasoning, using that a complete set of MUBs is a 2-design.Source: APPENDIX · p. 5
Loading 1901.09783v2…