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Paper 1
Quantum soundness of testing tensor codes
Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen
- Year
- 2021
- Journal
- arXiv preprint
- DOI
- arXiv:2111.08131
- arXiv
- 2111.08131
A locally testable code is an error-correcting code that admits very efficient probabilistic tests of membership. Tensor codes provide a simple family of combinatorial constructions of locally testable codes that generalize the family of Reed-Muller codes. The natural test for tensor codes, the axis-parallel line vs. point test, plays an essential role in constructions of probabilistically checkable proofs. We analyze the axis-parallel line vs. point test as a two-prover game and show that the test is sound against quantum provers sharing entanglement. Our result implies the quantum-soundness of the low individual degree test, which is an essential component of the MIP* = RE theorem. Our proof also generalizes to the infinite-dimensional commuting-operator model of quantum provers.
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Structured Negativity: A physically realizable measure of entanglement based on structural physical approximation
Anu Kumari, Satyabrata Adhikari
- Year
- 2022
- Journal
- arXiv preprint
- DOI
- arXiv:2209.03909
- arXiv
- 2209.03909
Quantification of entanglement is one of the most important problem in quantum information theory. In this work, we will study this problem by defining a physically realizable measure of entanglement for any arbitrary dimensional bipartite system $ρ$, which we named as structured negativity $(N_S(ρ))$. We have shown that the introduced measure satisfies the properties of a valid entanglement monotone. We also have established an inequality that relate negativity and the structured negativity. For $d\otimes d$ dimensional state, we conjecture from the result obtained in this work that negativity coincide with the structured negativity when the number of negative eigenvalues of the partially transposed matrix is equal to $\frac{d(d-1)}{2}$. Moreover, we proved that the structured negativity not only implementable in the laboratory but also a better measure of entanglement in comparison to negativity. In few cases, we obtain that structure negativity gives better result than the lower bound of the concurrence obtained by Albeverio [Phys. Rev. Lett. \textbf{95}, 040504 (2005)].
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