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A note on Schmidt-number witnesses based on symmetric measurements

Xiao-Qian Mu, Hao-Fan Wang, Shao-Ming Fei

TL;DR

This paper addresses detecting the Schmidt number of high-dimensional entangled states by constructing a class of $k$-positive maps from informationally complete $(N,M)$-POVMs, yielding Schmidt-number witnesses of class $(k+1)$. The authors prove $k$-positivity of the map and derive an explicit Schmidt-number witness $W_k$, which extends previous witnesses for symmetric measurements and reduces to known forms in special cases. They further relate the witnesses to the experimentally accessible Fedorov ratio, enabling dual validation of Schmidt-number claims in SPDC-like setups. The work provides both a theoretical framework and practical guidance for experimental verification of high-dimensional entanglement via symmetric measurements.

Abstract

The Schmidt number is an important kind of characterization of quantum entanglement. Quantum states with higher Schmidt numbers demonstrate significant advantages in various quantum information processing tasks. By deriving a class of k-positive linear maps based on symmetric measurements, we present new Schmidt-number witnesses of class (k + 1). By detailed example, we show that our Schmidt number witnesses identify better the Schmidt number of quantum states in high-dimensional systems. Furthermore, we note that the Fedorov ratio, which coincides with the Schmidt number for pure Gaussian states and provides a close approximation in non-Gaussian cases such as spontaneous parametric down-conversion, serves as an experimentally accessible tool for validating the proposed (k +1)-class Schmidt-number witnesses.

A note on Schmidt-number witnesses based on symmetric measurements

TL;DR

This paper addresses detecting the Schmidt number of high-dimensional entangled states by constructing a class of -positive maps from informationally complete -POVMs, yielding Schmidt-number witnesses of class . The authors prove -positivity of the map and derive an explicit Schmidt-number witness , which extends previous witnesses for symmetric measurements and reduces to known forms in special cases. They further relate the witnesses to the experimentally accessible Fedorov ratio, enabling dual validation of Schmidt-number claims in SPDC-like setups. The work provides both a theoretical framework and practical guidance for experimental verification of high-dimensional entanglement via symmetric measurements.

Abstract

The Schmidt number is an important kind of characterization of quantum entanglement. Quantum states with higher Schmidt numbers demonstrate significant advantages in various quantum information processing tasks. By deriving a class of k-positive linear maps based on symmetric measurements, we present new Schmidt-number witnesses of class (k + 1). By detailed example, we show that our Schmidt number witnesses identify better the Schmidt number of quantum states in high-dimensional systems. Furthermore, we note that the Fedorov ratio, which coincides with the Schmidt number for pure Gaussian states and provides a close approximation in non-Gaussian cases such as spontaneous parametric down-conversion, serves as an experimentally accessible tool for validating the proposed (k +1)-class Schmidt-number witnesses.

Paper Structure

This paper contains 3 sections, 2 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $k$-positive map and Schmidt-number witness based on $(N,M)$-POVM
  3. Conclusions and Discussions

Key Result

Theorem 1

The following linear map is $k$-positive for any density operator $X$, where $h_{x}=\frac{1}{kd-1}\sqrt{\frac{M(M-1)}{x(M^2x-d)}}$ and $\mathbb{I}_d$ is the $d\times d$ identity operator.

Figures (1)

  • Figure 1: comparison

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • Example 1