arXiv subjects

Lilong Qian

Publications and source records attributed to Lilong Qian.

4 recordsLinked to original sources

arXiv2026

Separable Counterexamples to Complementary Quantum Correlations, and Why Random Search Missed Them

The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair $m\times n$ with $m,n\ge3$, with closed-form excess at least $1/(8m^2n^2)$ nats, and in every qubit--qudit pair $2\times n$ with $n\ge3$ except $n=3,5$; every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound $S\le8/3$ to $S\le3.8265583\ldots$; the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity $X=1-4\operatorname{Var}(c)$ turns the prime-Fourier full-spark barrier into a variance bound. A $128$-bit interval cover then closes that family at $2\times3$ with gap at least $0.012021$ nats, and at $2\times5$ an exact saturator attaining $S=(14+2\sqrt5)/5$ refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at $2\times2$. Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies $7.3$ standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees ($\sim10^{-21}$ of frames), and extrapolating the sample minimum demands $10^{10}$ to $10^{18}$ samples against the $10^7$ ever run.

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arXiv2019

A matrix inequality related to the entanglement distillation problem

The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of $4\times4$ undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix $A\otimes I + I \otimes B$ does not exceed $(3d-4)/d^2$ with $A,B$ traceless $d\times d$ matrices and $||A||_F^2+||B||_F^2=1/d$ when $d=4$. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices $A$ and $B$ are normal. In this paper, we prove that the conjecture holds when one of matrices $A$ and $B$ is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.

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arXiv2018

Separability of Completely Symmetric States in Multipartite System

Symmetry plays an important role in the field of quantum mechanics. In this paper, we consider a subclass of symmetric quantum states in the multipartite system $N^{\otimes d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was conjectured by L. Qian and D. Chu [arXiv:1810.03125 [quant-ph]] that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. In this paper, we proved that this conjecture is true for both bipartite and multipartite cases. And we proved the completely symmetric state $ρ$ is separable if its rank is at most $5$ or $N+1$. For the states of rank $6$ or $N+2$, they are separable if and only if their range contains a product vector. We apply our results to a few widely useful states in quantum information, such as symmetric states, edge states, extreme states, and nonnegative states. We also study the relation of CS states to Hankel and Toeplitz matrices.

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arXiv2018

Separability of Multipartite Quantum States with Strong Positive Partial Transpose

We generalize the definition of strong positive partial transpose (SPPT) to the multipartite system. The tripartite case was first considered by X.-Y. Yu and H. Zhao [ Int. J. Theor. Phys.,54, 292, (2015)]. In this extension, unfortunately, desired properties such as the PPT of SPPT states the separability of super and pure SPPT states are not preserved. In contrast, this paper provides an alternative generalization to multipartite cases with these properties preserved. We also provide sufficient conditions for the separability of SPPT states.

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