Characterizing Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform

• URL: http://arxiv.org/abs/2303.17203v1
• Date: Thu, 30 Mar 2023 07:55:21 GMT
• Title: Characterizing Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform
• Authors: Ying-Hui Yang, Bing-Bing Zhang, Xiao-Li Wang, Shi-Jiao Geng, Pei-Ying Chen
• Abstract summary: We show that for the uncertainty diagram of the DFT matrix which is a transition matrix from basis $mathcal A$ to basis $mathcal B$, there is no hole" We present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation.
• Score: 6.344765041827868
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we investigate the Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform (DFT) in a $d$ dimensional system. The uncertainty diagram of complete incompatibility bases $\mathcal {A},\mathcal {B}$ are characterized by De Bi\`{e}vre [arXiv: 2207.07451]. We show that for the uncertainty diagram of the DFT matrix which is a transition matrix from basis $\mathcal {A}$ to basis $\mathcal {B}$, there is no ``hole" in the region of the $(n_{\mathcal {A}}, n_{\mathcal {B}})$-plane above and on the line $n_{\mathcal {A}}+n_{\mathcal {B}}\geq d+1$, whether the bases $\mathcal {A},\mathcal {B}$ are not complete incompatible bases or not. Then we present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation $n_{\mathcal {A}}(\psi)n_{\mathcal {B}}(\psi)\geq d$, where $n_{\mathcal {A}}(\psi)$ and $n_{\mathcal {B}}(\psi)$ count the number of nonvanishing coefficients in the basis $\mathcal {A}$ and $\mathcal {B}$ representations, respectively. That is, a state $|\psi\rangle$ is KD nonclassical if and only if $n_{\mathcal {A}}(\psi)n_{\mathcal {B}}(\psi)> d$, whenever $d$ is prime or not. That gives a positive answer to the conjecture in [Phys. Rev. Lett. \textbf{127}, 190404 (2021)].

Related papers

• On the Complexity of Pure-State Consistency of Local Density Matrices [0.0]
  We investigate the pure consistency of local density matrices ($mathsfPureCLDM$) and pure $N$-representability ($mathsfPure$-$N$-$mathsfRepresentability$) problems. We prove that both $mathsfPure$-$N$-$mathsfRepresentability$ and $mathsfPureCLDM$ are complete for this new class.
  arXiv  Detail & Related papers  (2024-11-05T13:43:21Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• $\mathcal{P}\mathcal{T}$-symmetric $-g\varphi^4$ theory [0.0]
  Hermitian theory is proposed: $log ZmathcalPmathcalT(g)=textstylefrac12 log Z_rm Herm(-g+rm i 0+rm i 0+$. A new conjectural relation between the Euclidean partition functions $ZmathcalPmathcalT$-symmetric theory and $Z_rm Herm(lambda)$ of the $lambda varphi4$ is presented.
  arXiv  Detail & Related papers  (2022-09-16T12:44:00Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• The planted matching problem: Sharp threshold and infinite-order phase transition [25.41713098167692]
  We study the problem of reconstructing a perfect matching $M*$ hidden in a randomly weighted $ntimes n$ bipartite graph. We show that if $sqrtd B(mathcalP,mathcalQ) ge 1+epsilon$ for an arbitrarily small constant $epsilon>0$, the reconstruction error for any estimator is shown to be bounded away from $0$.
  arXiv  Detail & Related papers  (2021-03-17T00:59:33Z)
• Linear Bandits on Uniformly Convex Sets [88.3673525964507]
  Linear bandit algorithms yield $tildemathcalO(nsqrtT)$ pseudo-regret bounds on compact convex action sets. Two types of structural assumptions lead to better pseudo-regret bounds.
  arXiv  Detail & Related papers  (2021-03-10T07:33:03Z)
• On the computational and statistical complexity of over-parameterized matrix sensing [30.785670369640872]
  We consider solving the low rank matrix sensing problem with Factorized Gradient Descend (FGD) method. By decomposing the factorized matrix $mathbfF$ into separate column spaces, we show that $|mathbfF_t - mathbfF_t - mathbfX*|_F2$ converges to a statistical error.
  arXiv  Detail & Related papers  (2021-01-27T04:23:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.