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arXiv:2510.08463 · Linear Algebra and its Applications

Approximating quantum states by states of low rank

Nathaniel Johnston, Chi-Kwong Li

Published Jan 1, 2026• 0 citations
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Abstract

Given a positive integer k , it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k . This problem has already been solved when “distance” is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally-mixed state, but this is not true in many other unitary similarity invariant norms.

DOI
10.1016/j.laa.2026.01.024

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Given a positive integer k , it is natural to ask for a formula for the distance between a given density matrix (i.e., mixed quantum state) and the set of density matrices of rank at most k . This problem has already been solved when “distance” is measured in the trace or Frobenius norm. We solve it for all other unitary similarity invariant norms. We also present some consequences of our formula. For example, in the trace and Frobenius norms, the density matrix that is farthest from the set of low-rank density matrices is the maximally-mixed state, but this is not true in many other unitary similarity invariant norms.
Let Sm\mathcal{S}_{m}Sm​ be the set of all m×mm\times mm×m density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix ρ∈Sm\rho\in\mathcal{S}_{m}ρ∈Sm​ based on outcomes of nnn measurements of observables X1,…,Xn∈HmX_{1},\dots,X_{n}\in\mathbb{H}_{m}X1​,…,Xn​∈Hm​ (Hm\mathbb{H}_{m}Hm​ being the space of m×mm\times mm×m Hermitian matrices) for a quantum system identically prepared nnn times in state ρ.\rho.ρ. Outcomes Y1,…,YnY_{1},\dots,Y_{n}Y1​,…,Yn​ of such measurements could be described by a trace regression model in which Eρ(Yj∣Xj)=tr(ρXj),j=1,…,n.\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.Eρ​(Yj​∣Xj​)=tr(ρXj​),j=1,…,n. The design variables X1,…,XnX_{1},\dots,X_{n}X1​,…,Xn​ are often sampled at random from the uniform distribution in an orthonormal basis {E1,…,Em2}\{E_{1},\dots,E_{m^{2}}\}{E1​,…,Em2​} of Hm\mathbb{H}_{m}Hm​ (such as Pauli basis). The goal is to estimate the unknown density matrix ρ\rhoρ based on the data (X1,Y1),…,(Xn,Yn).(X_{1},Y_{1}),\dots,(X_{n},Y_{n}).(X1​,Y1​),…,(Xn​,Yn​). Let Z^:=m2n∑j=1nYjXj\hat{Z}:=\frac{m^{2}}{n}\sum_{j=1}^{n}Y_{j}X_{j}Z^:=nm2​∑j=1n​Yj​Xj​ and let ρˇ\check{\rho}ρˇ​ be the projection of Z^\hat{Z}Z^ onto the convex set Sm\mathcal{S}_{m}Sm​ of density matrices. It is shown that for estimator ρˇ\check{\rho}ρˇ​ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten ppp-norm distances, p∈[1,∞]p\in[1,\infty]p∈[1,∞] and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator ρˇ\check{\rho}ρˇ​ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten ppp-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.
Given a pure state vector | x ⟩ and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.
The trace distance between two quantum states, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\ensuremath{\rho}$ or $\ensuremath{\sigma}$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.
Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in quantum state tomography (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞ for low rank density matrices estimation by Pauli measurements. In our previous paper [14], these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST. Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers [14] and [25], we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten ppp-norms with 1≤p≤21\leq p\leq21≤p≤2 and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten ppp-norms with 1≤p≤+∞1\leq p\leq+\infty1≤p≤+∞. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in [14] and [25]. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.
For a quantum state represented as an n×nn\times nn×n density matrix σ∈Mn\sigma \in M_nσ∈Mn​, let $\cS(\sigma)$ be the compact convex set of quantum states ρ=(ρij)∈Mm⋅n\rho = (\rho_{ij}) \in M_{m\cdot n}ρ=(ρij​)∈Mm⋅n​ with the first partial trace equal to σ\sigmaσ, i.e., $\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma$. It is known that if m≥nm\ge nm≥n then there is a rank one matrix $\rho \in \cS(\sigma)$ satisfying $\tr_1(\rho) = \sigma$. If m<nm < nm<n, there may not be any rank one matrix in $\cS(\sigma)$. In this paper, we determine the ranks of the elements and ranks of the extreme points of the set $\cS$. We also determine $\rho^* \in \cS(\sigma)$ with rank bounded by kkk such that $\|\tr_1(\rho^*) - \sigma\|$ is minimum for a given unitary similarity invariant norm ∥⋅∥\|\cdot\|∥⋅∥. Furthermore, the relation between the eigenvalues of σ\sigmaσ and those of $\rho \in \cS(\sigma)$ is analyzed. Extension of the results and open problems will be mentioned.