Optimal Local Testers for Unitarily Invariant Properties of Bipartite Quantum States

المفاهيم الأساسية

Unitary invariance of a property of bipartite pure states on one part implies the existence of an optimal local tester on the other part. This result leads to a canonical local tester for entanglement spectra and reveals the limitations of purifications in mixed state testing.

الملخص

The paper studies the power of local tests for bipartite quantum states. The central result is that for properties of bipartite pure states, unitary invariance on one part implies an optimal (over all global testers) local tester acting only on the other part.
This suggests a canonical local tester for entanglement spectra (i.e., Schmidt coefficients), and reveals that purified samples offer no advantage in property testing of mixed states.
As applications, the paper settles two open questions by providing:

A matching lower bound Ω(1/ε^2) for testing whether a multipartite pure state is product or ε-far, showing the algorithm of Harrow and Montanaro (2010) is optimal.
The first non-trivial lower bound Ω(r/ε) for testing whether the Schmidt rank of a bipartite pure state is at most r or ε-far.
The paper also shows other new sample lower bounds, for example:

A matching lower bound Ω(d/ε^2) for testing whether a d-dimensional bipartite pure state is maximally entangled or ε-far.
Beyond sample complexity, the paper contributes new quantum query lower bounds, such as:

A query lower bound ̃Ω(√(d/Δ)) for the d-dimensional entanglement entropy problem with gap Δ, improving prior results.
Furthermore, the central result can be extended when the tested state is mixed: one-way LOCC is sufficient to realize the optimal tester.

الإحصائيات

None.

اقتباسات

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الرؤى الأساسية المستخلصة من

Local Test for Unitarily Invariant Properties of Bipartite Quantum States

by Kean Chen,Qi... في arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04599.pdf

استفسارات أعمق

What is the converse implication of the duality between unitary invariance and optimal local testability?

The converse implication of the duality between unitary invariance and optimal local testability states that the existence of an optimal one-way LOCC tester implies both the unitary invariance of the property on one part and that the property only consists of pure states on the other part. This means that if there is an optimal one-way LOCC tester for a property of mixed states, it indicates that the property is unitarily invariant on one part and that the property only consists of pure states. This converse implication strengthens the original duality by providing additional conditions that must be met for the optimality of a one-way LOCC tester.

Can the central result be extended to property testing of mixed states in a more general sense, without restricting the property to only consist of pure states?

The central result can potentially be extended to property testing of mixed states in a more general sense, without restricting the property to only consist of pure states. This extension would involve exploring whether there exists an optimal and canonical tester with some locality constraints for testing properties of mixed states that are unitarily invariant on one part. By removing the restriction of the property to only consist of pure states, the extension would aim to provide insights into the optimal testing of a broader range of properties of mixed quantum states, potentially leading to new findings and applications in quantum property testing.

Can the lower bound Ω(r/ε) for testing the Schmidt rank be further improved?

The lower bound Ω(r/ε) for testing the Schmidt rank represents a significant advancement in quantum property testing. However, there is always room for further improvement in quantum algorithms and complexity analysis. To enhance the lower bound for testing the Schmidt rank, researchers could explore more sophisticated techniques, refine the analysis of the problem, or consider alternative approaches to quantum property testing. By delving deeper into the intricacies of the Schmidt rank testing problem, it may be possible to uncover new insights that could lead to a tighter lower bound, surpassing the current Ω(r/ε) threshold.

Can the quantum sample-to-query lifting technique be applied to show more quantum query lower bounds beyond the entanglement entropy problem?

The quantum sample-to-query lifting technique has proven to be a valuable tool in deriving quantum query lower bounds, as demonstrated in the context of the entanglement entropy problem. This technique can indeed be applied to explore and establish additional quantum query lower bounds beyond the entanglement entropy problem. By leveraging the principles and methodologies of quantum sample-to-query lifting, researchers can analyze various quantum property testing scenarios, formulate appropriate reductions, and derive new lower bounds for different quantum query complexity problems. This approach offers a systematic and effective way to advance the understanding of quantum query complexity and uncover novel insights in quantum computing research.