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Refuting spectral compatibility of quantum marginals

Felix Huber, Nikolai Wyderka

TL;DR

This work introduces a symmetry-reduced semidefinite programming hierarchy that completely certifies spectral incompatibility among overlapping quantum marginals, with certificates that are valid in all local dimensions when the copy number does not exceed the local dimension ($k\le d$). The approach leverages permutation symmetry, Schur-Weyl duality, and a quantum de Finetti-type argument to guarantee completeness and provide practical scalability (up to modest system sizes). It yields dimension-free incompatibility witnesses, along with concrete numerical results for tripartite and higher-partite systems, and derives entropy- and purity-type inequalities from the dual. Beyond the spectral marginal problem, the framework applies to sums of Hermitian matrices, local unitary invariants, and equivariant state polynomials, suggesting broad applicability to invariant theory and quantum information tasks.

Abstract

The spectral variant of the quantum marginal problem asks: Given prescribed spectra for a set of overlapping quantum marginals, does there exist a compatible joint state? The main idea of this work is a symmetry-reduced semidefinite programming hierarchy that detects when no such joint state exists. The hierarchy is complete, in the sense that it detects every incompatible set of spectra. The refutations it provides are dimension-free, certifying incompatibility in all local dimensions. The hierarchy also applies to the sums of Hermitian matrices problem, the compatibility of local unitary invariants, for certifying vanishing Kronecker coefficients, and to optimize over equivariant state polynomials.

Refuting spectral compatibility of quantum marginals

TL;DR

This work introduces a symmetry-reduced semidefinite programming hierarchy that completely certifies spectral incompatibility among overlapping quantum marginals, with certificates that are valid in all local dimensions when the copy number does not exceed the local dimension (). The approach leverages permutation symmetry, Schur-Weyl duality, and a quantum de Finetti-type argument to guarantee completeness and provide practical scalability (up to modest system sizes). It yields dimension-free incompatibility witnesses, along with concrete numerical results for tripartite and higher-partite systems, and derives entropy- and purity-type inequalities from the dual. Beyond the spectral marginal problem, the framework applies to sums of Hermitian matrices, local unitary invariants, and equivariant state polynomials, suggesting broad applicability to invariant theory and quantum information tasks.

Abstract

The spectral variant of the quantum marginal problem asks: Given prescribed spectra for a set of overlapping quantum marginals, does there exist a compatible joint state? The main idea of this work is a symmetry-reduced semidefinite programming hierarchy that detects when no such joint state exists. The hierarchy is complete, in the sense that it detects every incompatible set of spectra. The refutations it provides are dimension-free, certifying incompatibility in all local dimensions. The hierarchy also applies to the sums of Hermitian matrices problem, the compatibility of local unitary invariants, for certifying vanishing Kronecker coefficients, and to optimize over equivariant state polynomials.
Paper Structure (31 sections, 9 theorems, 64 equations, 1 figure, 2 tables)

This paper contains 31 sections, 9 theorems, 64 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Further applications
  4. Notation
  5. Quantum systems
  6. Symmetric group
  7. Representations
  8. Spectra are polynomial in $\varrho$
  9. Spectrum from $\varrho^{\otimes k}$
  10. Permuting subsystems of copies
  11. Compatibility conditions
  12. Symmetric extension relaxation
  13. Invariance
  14. SDP refutation
  15. Primal and dual programs
  16. ...and 16 more sections

Key Result

Theorem A

Let $\mathcal{A}$ be a collection of subsets of $\{1,\dots,n\}$ with associated marginal spectra $\{ \mu_A \, |\, A \in \mathcal{A} \}$. The spectra are compatible with a joint quantum state on $(\mathds{C}^d)^{\otimes n}$ if and only if every level in the hierarchy eq:hierarchy_dual_symred is feasi

Figures (1)

  • Figure 1: Regions of spectral incompatibility. Consider prescribed eigenvalues $\lambda_{AB}, \lambda_{AC}, \lambda_{BC}$ of rank-$2$ two-body marginals of three-partite states. We plot the regions of certified incompatibility for values in the interval $[0, \frac{1}{2}]$, as the problem is symmetric under the exchange of $\lambda_{ij} \leftrightarrow 1-\lambda_{ij}$. The infeasible regions are below (for $\lambda_{AC} < \lambda_{AB}$) and to the left (for $\lambda_{AC} > \lambda_{AB}$) of the lines. Shown are the boundaries of infeasibility for $k=2$ (dashed lines), $k=4$ (dotted lines), and $k=4$ with factorizing permutations (solid lines), with the height of the Young tableaux $d$ equal to the number of copies $k$. Due Theorem \ref{['thm:dimfree']}, the infeasibility regions are valid for tripartite states of arbitrary local dimensions.

Theorems & Definitions (19)

  • Theorem A
  • Theorem B
  • Proposition 1
  • proof
  • Proposition 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 6
  • proof
  • ...and 9 more