[Paper Review] Quantum marginal problem and representations of the symmetric group

Alexander A. Klyachko|ArXiv.org|Sep 17, 2004

Advanced Algebra and GeometryMathematics31 references115 citations

TL;DR

This paper provides a complete solution to the quantum marginal problem for bipartite quantum systems by deriving linear inequalities on the spectra of density matrices, using geometric and representation-theoretic methods. It establishes a connection between the quantum marginal problem and the representation theory of the symmetric group, enabling explicit computation of marginal constraints and answering questions about maximal eigenvalues and ranks of states with given margins.

ABSTRACT

We discuss existence of mixed state of multicomponent system with given spectrum and given reduced density matrices. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.

Motivation & Objective

• To solve the quantum marginal problem—determining whether given spectra for a composite quantum system and its subsystems can be realized by a consistent mixed state.
• To derive explicit linear inequalities constraining the spectra of reduced density matrices in bipartite systems.
• To establish a deep connection between the quantum marginal problem and the representation theory of the symmetric group.
• To provide computable criteria for properties like maximal eigenvalue and rank of a state given its marginal spectra.
• To generate and tabulate marginal inequalities for small systems, including up to four qubits, using a combinatorial framework.

Proposed method

• Leverages the Berenstein-Sjamaar theorem applied to the subgroup $\mathrm{SU}(\mathcal{H}_A) \times \mathrm{SU}(\mathcal{H}_B) \subset \mathrm{SU}(\mathcal{H}_A \otimes \mathcal{H}_B)$ to derive spectral constraints.
• Uses filtrations, cubicles, and extremal edges in flag varieties to generate marginal inequalities via cohomological techniques.
• Applies representation theory of the symmetric group to reframe the marginal problem as a decomposition of tensor products of irreducible representations.
• Introduces a systematic method for generating marginal inequalities for arrays of qubits under a standard conjecture (Theorem 4.2.3).
• Employs combinatorial tools such as Schubert polynomials and Kronecker coefficients to analyze spectral compatibility.
• Provides explicit tables of marginal inequalities for systems of rank up to 4, including two-, three-, and four-qubit systems.

Experimental results

Research questions

• RQ1 What linear inequalities on the spectra of reduced density matrices $\rho_A$, $\rho_B$, and $\rho_{AB}$ are necessary and sufficient for the existence of a consistent mixed state $\rho_{AB}$?
• RQ2 How can the quantum marginal problem be reformulated using the representation theory of the symmetric group?
• RQ3 What constraints does the spectrum of a composite quantum state impose on the maximal eigenvalue and rank of the state?
• RQ4 What is the structure of marginal inequalities for small systems such as arrays of qubits or two qutrits?
• RQ5 How are the quantum marginal problem and the Hermitian spectral problem related through representation-theoretic duality?

Key findings

• The paper provides a complete set of linear inequalities for the quantum marginal problem in systems of rank $\leq 4$, with explicit tables of marginal inequalities for two-, three-, and four-qubit systems.
• For arrays of qubits, under a standard conjecture, marginal inequalities can be generated systematically via Theorem 4.2.3, reducing the problem to combinatorial data.
• The maximal eigenvalue of a state with given marginal spectra is bounded by explicit linear combinations of the eigenvalues of the marginals, as shown in Theorem 6.3.1.
• The rank of a state with prescribed marginal spectra is constrained by inequalities derived from the representation-theoretic framework, as formalized in Theorem 6.4.1.
• The connection between the quantum marginal problem and Kronecker coefficients of the symmetric group is established, with applications to the stable support of these coefficients (Section 7).
• The paper demonstrates that Bell-type inequalities in quantum mechanics arise as special cases of the general marginal constraints derived from cohomology and flag variety geometry.