Maximal rank of extremal marginal tracial states

Hiromichi Ohno

Published Sep 1, 2010• 9 citations

Abstract

States on the coupled quantum system M-n(C)circle times M-n(C) whose restrictions to each subsystem are the normalized traces are called marginal tracial states. We investigate extremal marginal tracial states and compute their maximal rank. Diagonal marginal tracial states are also considered.

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Maximal rank of extremal marginal tracial states

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States on the coupled quantum system Mn(C)⊗Mn(C) whose restrictions to each subsystem are the normalized traces are called marginal tracial states. We investigate extremal marginal tracial states and compute their maximal rank. Diagonal marginal tracial states are also considered.

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Maximal rank of extremal marginal tracial states

Hiromichi Ohno

Published Sep 1, 2010• 9 citations

OpenAlex

Abstract

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Maximal rank of extremal marginal tracial states

Hiromichi Ohno

arXiv:0911.3459Journal of Mathematical PhysicsSep 1, 20108 citations

Extremal Marginal States of Maximal Rank in $(d, d+m)$ 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.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:2605.26920May 27, 20260 citations

Extremal marginal tracial states in coupled systems

Geoffrey L. Price, Shôichirô Sakai

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Wanchen Zhang, Zicheng Han, Xiande Zhang

Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem $A$ , the marginal maximal-mixing defect is defined by $D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2,$ and $Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$ is defined as the maximum number of $k$ -body marginals satisfying $D_A\leq\varepsilon$ in an $n$ -qubit pure state. This counting problem differs from approximate $k$ -uniformity, which requires all $k$ -body marginals to obey a common error bound. For pure states on $4m$ qubits, the following local stability inequality is established: $\sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1).$ It follows that, whenever $\varepsilon<1/(2m+1)$ , the hypergraph of $\varepsilon$ -good $2m$ -subsets is $K_{2m+1}^{(2m)}$ -free. Combined with the known exact eight-qubit construction, this yields the stability plateau $Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15.$ For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$ for $0\leq\varepsilon<1/17$ . These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.Absolutely maximally entangled states require every reduction of at most half of the parties to be maximally mixed, a condition that is both rigid and often impossible for qubit systems. Previous work introduced the quantum extremal number, which maximizes the number of exactly maximally mixed half-body marginals, and determined the exact value Qex(8,4)=56. The present work develops a robust extension of this extremal problem. For a subsystem $A$, the marginal maximal-mixing defect is defined by \[ D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2, \] and $Q_{\mathrm{ex},\varepsilon}^{D}(n,k)$ is defined as the maximum number of $k$-body marginals satisfying $D_A\leq\varepsilon$ in an $n$-qubit pure state. This counting problem differs from approximate $k$-uniformity, which requires all $k$-body marginals to obey a common error bound. For pure states on $4m$ qubits, the following local stability inequality is established: \[ \sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1). \] It follows that, whenever $\varepsilon<1/(2m+1)$, the hypergraph of $\varepsilon$-good $2m$-subsets is $K_{2m+1}^{(2m)}$-free. Combined with the known exact eight-qubit construction, this yields the stability plateau \[ Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15. \] For odd systems of $2k+1$ qubits, the exact forbidden hypergraph $H_k$ is used to derive explicit finite-error stability radii. In particular, $Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120$ for $0\leq\varepsilon<1/17$. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.

quant-pharXiv:2608.13907Aug 17, 20260 citations

Maximal lexicographic spectra and ranks for states with fixed uniform margins

Xin Li

We find the spectrum in maximal lexicographic order for quantum states ρ A B ∈ H A ⊗ H B \rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B with margins ρ A = 1 n I n \rho _A=\frac {1}{n}I_n and ρ B = 1 m I m \rho _B=\frac {1}{m}I_m and discuss the construction of ρ A B \rho _{AB} . By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko’s conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.We find the spectrum in maximal lexicographic order for quantum states ρ A B ∈ H A ⊗ H B \rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B with margins ρ A = 1 n I n \rho _A=\frac {1}{n}I_n and ρ B = 1 m I m \rho _B=\frac {1}{m}I_m and discuss the construction of ρ A B \rho _{AB} . By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko’s conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.

arXiv:1801.00986Proceedings of the American Mathematical SocietyJan 7, 20190 citations

Tensor rank and entanglement of pure quantum states

Wojciech Bruzda, Shmuel Friedland, Karol Życzkowski

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$ -fold tensor product of $n$ -dimensional Hilbert space and can be identified with a tensor with $d$ indices, each running from $1$ to $n$ . We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state ${\bf v}$ is called {\sl entangled}, if it {\sl cannot} be written in the product form, ${\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d$ , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$-fold tensor product of $n$-dimensional Hilbert space and can be identified with a tensor with $d$ indices, each running from $1$ to $n$. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state ${\bf v}$ is called {\sl entangled}, if it {\sl cannot} be written in the product form, ${\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d$, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..

arXiv:1912.06854Dec 14, 20197 citations

Extremal Marginal States of Maximal Rank in

(d, d+m)

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.

A

, the marginal maximal-mixing defect is defined by

D_A=2^{|A|}\operatorname{Tr}(ρ_A^2)-1 =2^{|A|}\left\|ρ_A-\frac{I_A}{2^{|A|}}\right\|_2^2,

and

Q_{\mathrm{ex},\varepsilon}^{D}(n,k)

is defined as the maximum number of

k

-body marginals satisfying

D_A\leq\varepsilon

in an

n

-qubit pure state. This counting problem differs from approximate

k

-uniformity, which requires all

k

-body marginals to obey a common error bound. For pure states on

4m

qubits, the following local stability inequality is established:

\sum_{i\in T}D_{T\setminus\{i\}}\geq1 \qquad (|T|=2m+1).

It follows that, whenever

\varepsilon<1/(2m+1)

, the hypergraph of

\varepsilon

-good

2m

-subsets is

K_{2m+1}^{(2m)}

-free. Combined with the known exact eight-qubit construction, this yields the stability plateau

Q_{\mathrm{ex},\varepsilon}^{D}(8,4)=56, \qquad 0\leq\varepsilon<\frac15.

For odd systems of

2k+1

qubits, the exact forbidden hypergraph

H_k

is used to derive explicit finite-error stability radii. In particular,

Q_{\mathrm{ex},\varepsilon}^{D}(9,4)\leq120

for

0\leq\varepsilon<1/17

. These results turn exact quantum Turán obstructions into quantitative robustness statements and identify intervals on which quantum extremal numbers are stable under imperfect marginal mixedness.

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state

\bf v

of a composite system consisting of

d

subsystems with

n

levels each is viewed as a vector in the

d

-fold tensor product of

n

-dimensional Hilbert space and can be identified with a tensor with

d

indices, each running from

1

n

. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state

{\bf v}

is called {\sl entangled}, if it {\sl cannot} be written in the product form,

{\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d

, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..