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Mutually-orthogonal unitary and orthogonal matrices

Zhiwei Song, Lin Chen, Saiqi Liu

TL;DR

This work introduces the concepts of $n$-OU and $n$-OO matrix sets—mutually orthogonal collections of unitary and real orthogonal matrices under the Hilbert-Schmidt inner product—and develops a decomposition framework based on these sets. It provides a complete classification of order-$3$ $n$-OO sets, derives universal $d$-OU decompositions for order $d$ matrices, and establishes order-$3$ real-matrix criteria for $n$-OO decompositions, along with two weaker decomposition forms. The results yield concrete applications in quantum information, showing that minimum and maximum numbers of unextendible maximally entangled bases in real $3 imes 3$ systems are $3$ and $4$, and that any bipartite pure state can be prepared as a superposition of at most $d$ maximally entangled states. The findings connect matrix-geometry with entanglement theory and pose open questions about the maximum $n$ for $n$-OO sets at general dimensions, and about $d$-OO decompositions for certain $d$ values linked to Hadamard-type matrices.

Abstract

We introduce the concept of n-OU and n-OO matrix sets, a collection of n mutually-orthogonal unitary and real orthogonal matrices under Hilbert-Schmidt inner product. We give a detailed characterization of order-three n-OO matrix sets under orthogonal equivalence. As an application in quantum information theory, we show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively. Further, we propose a new matrix decomposition approach, defining an n-OU (resp. n-OO) decomposition for a matrix as a linear combination of n matrices from an n-OU (resp. n-OO) matrix set. We show that any order-d matrix has a d-OU decomposition. As a contrast, we provide criteria for an order-three real matrix to possess an n-OO decomposition.

Mutually-orthogonal unitary and orthogonal matrices

TL;DR

This work introduces the concepts of -OU and -OO matrix sets—mutually orthogonal collections of unitary and real orthogonal matrices under the Hilbert-Schmidt inner product—and develops a decomposition framework based on these sets. It provides a complete classification of order- -OO sets, derives universal -OU decompositions for order matrices, and establishes order- real-matrix criteria for -OO decompositions, along with two weaker decomposition forms. The results yield concrete applications in quantum information, showing that minimum and maximum numbers of unextendible maximally entangled bases in real systems are and , and that any bipartite pure state can be prepared as a superposition of at most maximally entangled states. The findings connect matrix-geometry with entanglement theory and pose open questions about the maximum for -OO sets at general dimensions, and about -OO decompositions for certain values linked to Hadamard-type matrices.

Abstract

We introduce the concept of n-OU and n-OO matrix sets, a collection of n mutually-orthogonal unitary and real orthogonal matrices under Hilbert-Schmidt inner product. We give a detailed characterization of order-three n-OO matrix sets under orthogonal equivalence. As an application in quantum information theory, we show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively. Further, we propose a new matrix decomposition approach, defining an n-OU (resp. n-OO) decomposition for a matrix as a linear combination of n matrices from an n-OU (resp. n-OO) matrix set. We show that any order-d matrix has a d-OU decomposition. As a contrast, we provide criteria for an order-three real matrix to possess an n-OO decomposition.
Paper Structure (13 sections, 16 theorems, 29 equations)

This paper contains 13 sections, 16 theorems, 29 equations.

Table of Contents

  1. introduction
  2. preliminaries
  3. $n$-OO matrix and $n$-OU matrix sets
  4. order-three $n$-OO matrix sets
  5. $n$-OU and $n$-OO matrix sets of other orders
  6. $n$-OU and $n$-OO matrix decomposition
  7. The $n$-OU decomposition of complex matrices
  8. The $n$-OO decomposition of order-three real matrices
  9. Two weaker forms of order-three $n$-OO decomposition
  10. Applications
  11. Unextendible maximally entangled state bases in $\mathbb{R}^3\otimes \mathbb{R}^3$
  12. Pure bipartite state prepared by maximally entangled states
  13. conclusions

Key Result

Lemma 3

Suppose two $n$-OO matrix sets $\{M_1,\cdots,M_n\}$ and $\{N_1,\cdots,N_n\}$ are orthogonally equivalent, i.e., $UM_iV=(-1)^{a_i}N_{\pi(i)}$ for $i=1,\cdots,n$. Then for any real numbers $k_1,\cdots,k_n$, the two matrices $\sum_{i=1}^n k_i M_i$ and $\sum_{i=1}^n k_i (-1)^{a_i}N_{\pi(i)}$ have the sa

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Definition 4
  • Lemma 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 11 more