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Minimum Hilbert-Schmidt distance for Schmidt rank 2 states

Palash Pandya

TL;DR

This work analyzes the minimum Hilbert-Schmidt distance to the set of separable states as an entanglement quantifier for bipartite states with Schmidt rank 2. It derives a closed-form expression for $D^2_{ ext{HS,min}}$ and constructs the Closest Separable State (CSS), leveraging PPT structure and Verstraete's algorithm, with a key proof that the distance is non-increasing under LOCC for pure Schmidt-2 states via Nielsen's majorization theorem. The results are supported by numerical validation (Gilbert's algorithm) and extended to general Schmidt-2 states, including higher local dimensions, with explicit CSS construction and entanglement-witness utility. The findings strengthen the case for the minimum Hilbert-Schmidt distance as a meaningful entanglement quantifier in this class and open directions for generalizing to higher Schmidt rank and mixed states, including connections to PPT-entangled scenarios.

Abstract

The Hilbert-Schmidt distance between two states is proven to be non-contractive under CPTP maps, and therefore is not considered as an entanglement measure. However, that alone does not imply that the minimum Hilbert-Schmidt distance from the set of separable states is not contractive as well. To the contrary, not only do we provide a closed-form expression, we also provide analytical and numerical proof that minimum Hilbert-Schmidt distance for a given bipartite quantum state of Schmidt rank 2 is non-increasing under LOCC. The minimisation is taken to be over the set of separable states. We apply the algorithm by Verstraete et al [Journal of Modern Optics, 49(8), 2002] for the derivation of the analytical expression and Nielsen's theorem for the proof of monotonicity of the distance under LOCC.

Minimum Hilbert-Schmidt distance for Schmidt rank 2 states

TL;DR

This work analyzes the minimum Hilbert-Schmidt distance to the set of separable states as an entanglement quantifier for bipartite states with Schmidt rank 2. It derives a closed-form expression for and constructs the Closest Separable State (CSS), leveraging PPT structure and Verstraete's algorithm, with a key proof that the distance is non-increasing under LOCC for pure Schmidt-2 states via Nielsen's majorization theorem. The results are supported by numerical validation (Gilbert's algorithm) and extended to general Schmidt-2 states, including higher local dimensions, with explicit CSS construction and entanglement-witness utility. The findings strengthen the case for the minimum Hilbert-Schmidt distance as a meaningful entanglement quantifier in this class and open directions for generalizing to higher Schmidt rank and mixed states, including connections to PPT-entangled scenarios.

Abstract

The Hilbert-Schmidt distance between two states is proven to be non-contractive under CPTP maps, and therefore is not considered as an entanglement measure. However, that alone does not imply that the minimum Hilbert-Schmidt distance from the set of separable states is not contractive as well. To the contrary, not only do we provide a closed-form expression, we also provide analytical and numerical proof that minimum Hilbert-Schmidt distance for a given bipartite quantum state of Schmidt rank 2 is non-increasing under LOCC. The minimisation is taken to be over the set of separable states. We apply the algorithm by Verstraete et al [Journal of Modern Optics, 49(8), 2002] for the derivation of the analytical expression and Nielsen's theorem for the proof of monotonicity of the distance under LOCC.
Paper Structure (12 sections, 2 theorems, 41 equations, 3 figures)

This paper contains 12 sections, 2 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Minimum Hilbert-Schmidt Distance
  3. States with Schmidt rank 2
  4. Two qubit pure state
  5. Numerics
  6. Schmidt rank 2 states
  7. Closest Separable state
  8. Monotonicity under Local Operations and Classical Communication
  9. Conclusions and Outlook
  10. Acknowledgements
  11. Commutator of the partially transposed state and closest separable state using the Gilbert's algorithm.
  12. Positivity of the derivative

Key Result

Theorem 1

(Nielsen's Theorem) A bipartite pure state $\ket{\psi}$ can be transformed into another pure state $\ket\phi$ by LOCC if and only if $\lambda_\psi\prec\lambda_\phi$.

Figures (3)

  • Figure 1: Plot of the expression $4e_1e_2-e_3^2$ as a function of $\alpha$. In the positive valued region, the minimum Hilbert-Schmidt distance is simply $4\alpha^2\beta^2/3$ by using $a=b=1/3$.
  • Figure 2: The comparison of minimum Hilbert-Schmidt distance for a pure two qubit state as a function of the Schmidt coefficient $\alpha$, obtained by three methods: 1) analytical expression (solid blue and green), 2) minimization with positive eigenvalue constraint (dashed, red), and 3) Gilbert's algorithm (blue circles). In the inset, the region where the analytical expression switches is shown. For the minimisation and the Gilbert's algorithm, values of $\alpha$ were taken in steps of $0.01$.
  • Figure :

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Theorem 2
  • proof