Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1403.1108 · Electronic Journal of Linear Algebra

Ranks and eigenvalues of states with prescribed reduced states

Chi-Kwong Li, Yiu‐Tung Poon, Xuefeng Wang

Published Jan 1, 2014• 5 citations
arXivHTMLPDF

Abstract

For a quantum state represented as an $n\times n$ density matrix $\sigma \in M_n$, let $\cS(\sigma)$ be the compact convex set of quantum states $\rho = (\rho_{ij}) \in M_{m\cdot n}$ with the first partial trace equal to $\sigma$, i.e., $\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma$. It is known that if $m\ge n$ then there is a rank one matrix $\rho \in \cS(\sigma)$ satisfying $\tr_1(\rho) = \sigma$. If $m < n$, 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 $k$ such that $\|\tr_1(\rho^*) - \sigma\|$ is minimum for a given unitary similarity invariant norm $\|\cdot\|$. Furthermore, the relation between the eigenvalues of $\sigma$ and those of $\rho \in \cS(\sigma)$ is analyzed. Extension of the results and open problems will be mentioned.

DOI
10.13001/1081-3810.2882

Related papers

Approximating quantum states by states of low rank

Nathaniel Johnston, Chi-Kwong Li

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.

arXiv:2510.08463•Linear Algebra and its Applications•Jan 1, 2026•0 citations

n-Representability Problem for Reduced Density Matrices

Hans Kummer

In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8). Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.

Journal of Mathematical Physics•Oct 1, 1967•207 citations

Estimation of low rank density matrices: Bounds in Schatten norms and other distances

Dong Xia, Vladimir I. Koltchinskii

Let $\mathcal{S}_{m}$ be the set of all $m\times m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $\rho\in\mathcal{S}_{m}$ based on outcomes of $n$ measurements of observables $X_{1},\dots,X_{n}\in\mathbb{H}_{m}$ ($\mathbb{H}_{m}$ being the space of $m\times m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $\rho.$ Outcomes $Y_{1},\dots,Y_{n}$ of such measurements could be described by a trace regression model in which $\mathbb{E}_{\rho}(Y_{j}|X_{j})={\rm tr}(\rho X_{j}),j=1,\dots,n.$ 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 $\rho$ 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 $\check{\rho}$ be the projection of $\hat{Z}$ onto the convex set $\mathcal{S}_{m}$ of density matrices. It is shown that for estimator $\check{\rho}$ 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 $\check{\rho}$ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.

arXiv:1604.04600•Electronic Journal of Statistics•Jan 1, 2016•16 citations

Extremal Marginal States of Maximal Rank in $(d, d+m)$

Indu Bala, Swapan Rana

We study the extreme points of the convex set $\mathcal{C}(ρ_1,ρ_2)$ of bipartite quantum states with fixed marginals $ρ_1$ and $ρ_2$. We construct extreme points in $(d,\,d+m)$ dimension, of rank $d+m$, matching the highest possible value, for all $d\geq 3$, $m > \frac{d^2-2d-2}{2}$ (when $d=2$, $m\geq 1$). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of $\mathcal{C}(ρ_1,ρ_2)$, it is sufficient to study the special case $\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)$, where the marginals are diagonal. Additionally, we observe that it is sufficient to consider $d_1\leq d_2$. Thus, our results show that apart from possibly a few finite cases, for each $d_1$, the maximal rank is achieved almost all times.

quant-pharXiv:2605.26920•May 27, 2026•0 citations

On the one-particle reduced density matrices of a pure three-qutrit quantum state

Atsushi Higuchi

We present a necessary and sufficient condition for three qutrit density matrices to be the one-particle reduced density matrices of a pure three-qutrit quantum state. The condition consists of seven classes of inequalities satisfied by the eigenvalues of these matrices. One of them is a generalization of a known inequality for the qubit case. Some of these inequalities are proved algebraically whereas the proof of the others uses the fact that a continuous function of the state must have a minimum. Construction of states satisfying these inequalities relies on a representation of the convex set of the allowed set of eigenvalues in terms of corner points. We also present a result for a more general quantum system concerning the nature of the boundary surface of the set of the allowed set of eigenvalues.

arXiv:quant-ph/0309186•Sep 25, 2003•12 citations

Matrix Results And Techniques In Quantum Information Science And Related Topics

Diane Christine Pelejo

In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at $2^n\times 2^n$ unitary matrices, which describe operations on a closed $n$-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general $2^n\times 2^n$ unitary matrix to the product of no more than $2^{n-1}(2^n-1)$ single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element $\Phi$ among a given set of quantum channels $\mathcal{S}$ that will produce the optimal value of a scalar function $D(\rho_1,\Phi(\rho_2))$, on two fixed quantum states $\rho_1$ and $\rho_2$. Some of the functions we considered for $D(\cdot,\cdot)$ are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when $\mathcal{S}$ is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers $a_1\geq\ldots\geq a_{2n}\geq 0$, we want to determine if there exist a $2n\times 2n$ density matrix $\rho$ having eigenvalues $a_1,\ldots,a_{2n}$ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on $a_1,\ldots,a_{2n}$ for $n\leq 6$ and state some observations on general values of $n$. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that $\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}$ for given $\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_n$ and $\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_m$, (b) constructing a multipartite state $\rho$ having a prescribed set of reduced states $\rho_1,\ldots, \rho_r$ on $r$ of its subsystems, (c) constructing a multipartite state$\rho$ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix $A$ can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets $K_1$ and $K_2$ of $\IC$ given by $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that $K_1K_2$ is star-shaped when $K_1$ and $K_2$ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set $K_1K_2$ is star-shaped if $K_1$ is a line segment or a circular disk.

•Jan 1, 2016•0 citations
For a quantum state represented as an n×nn\times nn×n density matrix σ∈Mn\sigma \in M_nσ∈Mn​, let \cS(σ)\cS(\sigma)\cS(σ) 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., \tr1(ρ)=ρ11+⋯+ρmm=σ\tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma\tr1​(ρ)=ρ11​+⋯+ρmm​=σ. It is known that if m≥nm\ge nm≥n then there is a rank one matrix ρ∈\cS(σ)\rho \in \cS(\sigma)ρ∈\cS(σ) satisfying \tr1(ρ)=σ\tr_1(\rho) = \sigma\tr1​(ρ)=σ. If m<nm < nm<n, there may not be any rank one matrix in \cS(σ)\cS(\sigma)\cS(σ). In this paper, we determine the ranks of the elements and ranks of the extreme points of the set \cS\cS\cS. We also determine ρ∗∈\cS(σ)\rho^* \in \cS(\sigma)ρ∗∈\cS(σ) with rank bounded by kkk such that ∥\tr1(ρ∗)−σ∥\|\tr_1(\rho^*) - \sigma\|∥\tr1​(ρ∗)−σ∥ is minimum for a given unitary similarity invariant norm ∥⋅∥\|\cdot\|∥⋅∥. Furthermore, the relation between the eigenvalues of σ\sigmaσ and those of ρ∈\cS(σ)\rho \in \cS(\sigma)ρ∈\cS(σ) is analyzed. Extension of the results and open problems will be mentioned.
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.
In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8). Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.
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.
Extremal Marginal States of Maximal Rank in (d,d+m)(d, d+m)(d,d+m)
We study the extreme points of the convex set C(ρ1,ρ2)\mathcal{C}(ρ_1,ρ_2)C(ρ1​,ρ2​) of bipartite quantum states with fixed marginals ρ1ρ_1ρ1​ and ρ2ρ_2ρ2​. We construct extreme points in (d, d+m)(d,\,d+m)(d,d+m) dimension, of rank d+md+md+m, matching the highest possible value, for all d≥3d\geq 3d≥3, m>d2−2d−22m > \frac{d^2-2d-2}{2}m>2d2−2d−2​ (when d=2d=2d=2, m≥1m\geq 1m≥1). This proves the existence of extremal states with relatively large rank and also covers all the known examples. We further show that, in order to analyze the extreme points of C(ρ1,ρ2)\mathcal{C}(ρ_1,ρ_2)C(ρ1​,ρ2​), it is sufficient to study the special case C(D1,D2)\mathcal{C}(\mathcal{D}_1,\mathcal{D}_2)C(D1​,D2​), where the marginals are diagonal. Additionally, we observe that it is sufficient to consider d1≤d2d_1\leq d_2d1​≤d2​. Thus, our results show that apart from possibly a few finite cases, for each d1d_1d1​, the maximal rank is achieved almost all times.
We present a necessary and sufficient condition for three qutrit density matrices to be the one-particle reduced density matrices of a pure three-qutrit quantum state. The condition consists of seven classes of inequalities satisfied by the eigenvalues of these matrices. One of them is a generalization of a known inequality for the qubit case. Some of these inequalities are proved algebraically whereas the proof of the others uses the fact that a continuous function of the state must have a minimum. Construction of states satisfying these inequalities relies on a representation of the convex set of the allowed set of eigenvalues in terms of corner points. We also present a result for a more general quantum system concerning the nature of the boundary surface of the set of the allowed set of eigenvalues.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.