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On the mixed UDA states and additivity

Xinyu Qiu, Lin Chen, Genwei Li, Delin Chu

TL;DR

This work provides a complete framework for characterizing mixed states that are uniquely determined by their local marginals (UDA) across multipartite systems, with a focus on 2-UDA and the additivity of such states. It introduces a systematic method to determine $k$-UDA states, delivering explicit necessary and sufficient conditions for bipartite 1-UDA, tripartite 2-UDA, and several higher-qubit cases, while also detailing when tensor-product states preserve the UDA property. The results have practical impact on quantum state tomography by enabling reconstruction from low-order marginals (LQST) and reducing measurement resources, as well as implications for entanglement certification, quantum communication security, and error correction. A key takeaway is that, although many mixed states are not 2-UDA, the paper identifies precise structural and rank-related conditions under which additivity holds, guiding efficient state certification and tomography in multipartite quantum systems.

Abstract

Mixed states that are uniquely determined among all (UDA) states are vital in efficient quantum tomography. We show the necessary and sufficient conditions by which some multipartite mixed states are UDA by their $k$-partite reduced density matrices. The case for $k=2$ is mostly studied, which requires minimal local information and shows practical benefits. Based on that, we establish a systematic method for determining UDA states and provide a complete characterization of the additivity of UDA bipartite and three-qubit product states. We show the application of mixed UDA states and their characterization from the perspectives of tomography and other tasks.

On the mixed UDA states and additivity

TL;DR

This work provides a complete framework for characterizing mixed states that are uniquely determined by their local marginals (UDA) across multipartite systems, with a focus on 2-UDA and the additivity of such states. It introduces a systematic method to determine -UDA states, delivering explicit necessary and sufficient conditions for bipartite 1-UDA, tripartite 2-UDA, and several higher-qubit cases, while also detailing when tensor-product states preserve the UDA property. The results have practical impact on quantum state tomography by enabling reconstruction from low-order marginals (LQST) and reducing measurement resources, as well as implications for entanglement certification, quantum communication security, and error correction. A key takeaway is that, although many mixed states are not 2-UDA, the paper identifies precise structural and rank-related conditions under which additivity holds, guiding efficient state certification and tomography in multipartite quantum systems.

Abstract

Mixed states that are uniquely determined among all (UDA) states are vital in efficient quantum tomography. We show the necessary and sufficient conditions by which some multipartite mixed states are UDA by their -partite reduced density matrices. The case for is mostly studied, which requires minimal local information and shows practical benefits. Based on that, we establish a systematic method for determining UDA states and provide a complete characterization of the additivity of UDA bipartite and three-qubit product states. We show the application of mixed UDA states and their characterization from the perspectives of tomography and other tasks.
Paper Structure (15 sections, 25 theorems, 57 equations, 1 figure, 2 tables)

This paper contains 15 sections, 25 theorems, 57 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The characterization of multipartite mixed UDA states
  4. Preliminary lemmas
  5. On bipartite mixed UDA states
  6. On tripartite mixed UDA states
  7. On four-qubit mixed UDA states
  8. On $n$-qubit mixed UDA states for $n\geq 5$
  9. Additivity of mixed UDA states
  10. Applications
  11. Conclusion
  12. Proof of preliminary lemmas
  13. Proof of tripartite UDA states
  14. Proof four-qubit UDA states
  15. Proof of $n$-qubit UDA states

Key Result

Lemma 2

If the reduction of the system $(A_1, \ldots, A_m)$ from the global state $\rho_{A_1 \cdots A_m E}$ is a pure state, then the global state is in the form

Figures (1)

  • Figure 1: Process for the determination of $k$-UDA states. Here $\sigma$ is any state $k$-compatible with $\rho$. The first to third step are shown in the cyan, blue and yellow boxes, respectively.

Theorems & Definitions (27)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Proposition 8
  • Proposition 9
  • Example 10
  • ...and 17 more