[Paper Review] Estimation of low rank density matrices: bounds in Schatten norms and other distances

Dong Xia, Vladimir Koltchinskii|arXiv (Cornell University)|Apr 15, 2016

Quantum Information and Cryptography24 references3 citations

TL;DR

This paper proposes a minimal distance estimator for low-rank quantum density matrices using trace regression with random measurements. It establishes that the estimator achieves minimax optimal rates up to logarithmic factors in Schatten p-norms, Bures-Hellinger, and quantum relative entropy distances, with efficient computation via eigenvalue projection onto the simplex.

ABSTRACT

Let ${\mathcal S}_m$ be the set of all $m imes m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $ρ\in {\mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,\dots, X_n\in {\mathbb H}_m$ (${\mathbb H}_m$ being the space of $m imes m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $ρ.$ Outcomes $Y_1,\dots, Y_n$ of such measurements could be described by a trace regression model in which ${\mathbb E}_ρ(Y_j|X_j)={ m tr}(ρX_j), j=1,\dots, n.$ The design variables $X_1,\dots, X_n$ are often sampled at random from the uniform distribution in an orthonormal basis $\{E_1,\dots, E_{m^2}\}$ of ${\mathbb H}_m$ (such as Pauli basis). The goal is to estimate the unknown density matrix $ρ$ based on the data $(X_1,Y_1), \dots, (X_n,Y_n).$ Let $$ \hat Z:=\frac{m^2}{n}\sum_{j=1}^n Y_j X_j $$ and let $\check ρ$ be the projection of $\hat Z$ onto the convex set ${\mathcal S}_m$ of density matrices. It is shown that for estimator $\check ρ$ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten $p$-norm distances, $p\in [1,\infty]$ and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator $\check ρ$ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.

Motivation & Objective

• Address the problem of estimating low-rank quantum density matrices from noisy, randomly sampled measurements in quantum state tomography.
• Close the gap between existing upper bounds and minimax lower bounds for estimation error in various quantum distance measures.
• Provide theoretical guarantees for a computationally efficient estimator in high-dimensional quantum systems.
• Extend optimality results beyond Hilbert-Schmidt (Frobenius) norm to include Schatten p-norms, Bures-Hellinger, and quantum relative entropy distances.

Proposed method

• Formulate the quantum state tomography problem as a trace regression model: $ Y_j = \langle \rho, X_j \rangle + \xi_j $, where $ X_j $ are random Hermitian observables and $ Y_j $ are measurement outcomes.
• Construct the minimal distance estimator $ \hat{Z} = \frac{m^2}{n} \sum_{j=1}^n Y_j X_j $, followed by projection $ \check{\rho} = \pi_{\mathcal{S}_m}(\hat{Z}) $ onto the set of density matrices.
• Use eigen-decomposition of $ \hat{Z} $ and project its eigenvalues onto the probability simplex $ \Delta_m $ to compute $ \check{\rho} $, ensuring positivity and unit trace.
• Leverage matrix concentration inequalities and Weyl's perturbation theorem to bound the operator and Schatten norms of estimation error.
• Establish equivalence between the projection of a matrix and the projection of its eigenvalues onto the simplex via unitary invariance.
• Analyze the modified estimator for quantum relative entropy by adjusting the projection step, achieving minimax optimality up to logarithmic factors.

Experimental results

Research questions

• RQ1 Can a computationally simple estimator achieve minimax optimal rates in Schatten p-norms for low-rank density matrix estimation?
• RQ2 To what extent do the minimax lower bounds for low-rank density matrices hold for Schatten p-norms and quantum divergences?
• RQ3 Does the minimal distance estimator $ \check{\rho} $ attain the minimax rate up to logarithmic factors in Bures-Hellinger and quantum relative entropy distances?
• RQ4 How does the computational complexity of the minimal distance estimator compare to iterative methods like matrix LASSO or SVT in high-dimensional settings?
• RQ5 Can the projection of eigenvalues onto the simplex be used to efficiently compute the density matrix estimator while preserving statistical optimality?

Key findings

• The minimal distance estimator $ \check{\rho} = \pi_{\mathcal{S}_m}(\hat{Z}) $ achieves minimax optimal rates up to logarithmic factors in all Schatten $ p $-norms for $ p \in [1, \infty] $.
• The estimator attains the minimax lower bound up to logarithmic factors in the Bures version of the quantum Hellinger distance.
• A slightly modified version of $ \check{\rho} $ achieves minimax optimality in quantum relative entropy (Kullback-Leibler) distance, also up to logarithmic factors.
• The computational complexity of $ \check{\rho} $ is $ O((n + m)m^2) $, dominated by the eigen-decomposition and simplex projection, making it significantly faster than iterative solvers like SVT.
• The projection of the eigenvalues of $ \hat{Z} $ onto the simplex $ \Delta_m $ yields the optimal density matrix estimator, with an explicit solution of complexity $ O(m) $.
• The estimator is robust to random design matrices sampled from an orthonormal basis (e.g., Pauli basis), which is common in quantum tomography experiments.