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Verifcation of general multi-qudit pure states

Xiao-Dong Zhang, Bin-Bin Cai, Song Lin

Abstract

Verifying prepared quantum states is crucial for hybrid systems whose subsystems may have different local dimensions. We present a generalized stabilizer framework and associated test that apply to general multi-qudit states, including composite-dimensional and hybrid architectures. Using only adaptive local measurements, our method verifies qutrit-qubit states, arbitrary two-qubit pure states, Bell/Bell-like, GHZ/GHZ-like, graph, hypergraph, multigraph, and multihypergraph states, with efficiencies matching or surpassing the best known schemes.

Verifcation of general multi-qudit pure states

Abstract

Verifying prepared quantum states is crucial for hybrid systems whose subsystems may have different local dimensions. We present a generalized stabilizer framework and associated test that apply to general multi-qudit states, including composite-dimensional and hybrid architectures. Using only adaptive local measurements, our method verifies qutrit-qubit states, arbitrary two-qubit pure states, Bell/Bell-like, GHZ/GHZ-like, graph, hypergraph, multigraph, and multihypergraph states, with efficiencies matching or surpassing the best known schemes.

Paper Structure

This paper contains 15 sections, 6 theorems, 87 equations, 1 figure, 1 table.

Table of Contents

  1. Appendix A: Proof of the Proposition on the Existence of Generalized Stabilizers
  2. Appendix B: Proof of Eq. (5) in the main text
  3. Appendix C: Proof of the Theorem in the Main Text
  4. Appendix D: Generalized Stabilizers for Several Representative Quantum States
  5. Appendix E: Verification Examples of Quantum States
  6. E1: Verification of $\left|\psi_1\right\rangle$ and $\left|\psi_2\right\rangle$
  7. E1.1: Verification of $\left|\psi_1\right\rangle$
  8. E1.2: Verification of $\left|\psi_2\right\rangle$
  9. E2: Verification of qubit and qudit Bell/Bell-like states, GHZ/GHZ-like states
  10. E2.1: Verification of qubit GHZ and Bell states
  11. E2.2: Verification of qubit GHZ-like and Bell-like states
  12. E2.3: Verification of qudit GHZ and Bell states
  13. E2.4: Verification of qudit GHZ-like and Bell-like states
  14. E3: Verification of $\left|G_n^d\right\rangle$, $\left|\widetilde{G}_n^d\right\rangle$, $\left|\widehat{G}_n^d\right\rangle$, and $\left|\widehat{\widetilde{G}}_n^d\right\rangle$
  15. Appendix F: Verification of Composite-Dimensional Quantum States

Key Result

Proposition 1

For any pure state $\left| \psi \right\rangle \in \mathcal{H}^{\otimes n}$, there exists a generalized stabilizer set $S_\psi$ that uniquely determines $\left| \psi \right\rangle$.

Figures (1)

  • Figure 1: Adaptive multi-qudit measurement for $n=4$.

Theorems & Definitions (8)

  • Definition : Qubit generalized stabilizer state zhang2021efficient
  • Proposition : Ref. zhang2021efficient
  • Definition : Generalized stabilizer states for prime-dimensional multi-qudit systems
  • Proposition : Proof provided in the Supplemental Material
  • Theorem : Proof provided in the Supplemental Material
  • Proposition
  • Lemma
  • Theorem