Eigenvalue Distributions of Reduced Density Matrices

Abstract

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

51 pages, 7 figures

References in corpus (17)

• Black holes as mirrors: quantum information in random subsystems
• Aspects of generic entanglement
• Randomizing quantum states: Constructions and applications
• N-representability is QMA-complete
• Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information
• The Pauli principle revisited
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• Hastings' additivity counterexample via Dvoretzky's theorem
• Quantum state transformations and the Schubert calculus
• Gaussian quantum marginal problem
• Connecting N-representability to Weyl's problem: The one particle density matrix for N = 3 and R = 6
• Even Partitions in Plethysms
• Phase transitions for random states and a semi-circle law for the partial transpose
• Unifying typical entanglement and coin tossing: on randomization in probabilistic theories
• Computing Multiplicities of Lie Group Representations
• The absolute positive partial transpose property for random induced states
• A quantum information-theoretic proof of the relation between Horn's problem and the Littlewood-Richardson coefficients

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• Asymptotic properties of entanglement polytopes for large number of qubits
• The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing
• Computing partial traces and reduced density matrices
• Dirac Delta Function of Matrix Argument
• Plethysm and lattice point counting
• Membership in moment polytopes is in NP and coNP
• Quantum marginals from pure doubly excited states
• Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem
• Recoupling coefficients and quantum entropies
• Spectral density of mixtures of random density matrices for qubits
• Calculating eigenvalues of many-body systems from partition functions
• Spectral statistics for the difference of two Wishart matrices
• Uncertainty regions of observables and state-independent uncertainty relations
• Derivative principles for invariant ensembles
• On the joint distribution of the marginals of multipartite random quantum states
• Duistermaat-Heckman measure and the mixture of quantum states
• A variant of Horn's problem and derivative principle
• Projections of Orbital Measures and Quantum Marginal Problems
• Energy spectrum of interacting gas: cluster expansion method
• Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state
• Volume of the set of locally diagonalizable bipartite states
• A variation principle for ground spaces
• Probability density functions of quantum mechanical observable uncertainties
• Refuting spectral compatibility of quantum marginals
• Refining ensemble $N$-representability of one-body density matrices from partial information
• One application of Duistermaat-Heckman measure in quantum information theory
• Detecting quantum many-body states with imperfect measuring devices