关于$p^2$个正交$p×p$酉矩阵族的刻画及2×2酉矩阵相关问题问询

Hey there! Let's break down your question about extending the 2×2 orthogonal unitary matrix result to general p×p cases, and how to characterize such matrix families.

First, let's clarify the orthogonality condition here: $\mathrm{Tr}(U_i U_j^\dagger) = 0$ for $i≠j$ corresponds to orthogonality under the Frobenius (Hilbert-Schmidt) inner product on $\mathbb{C}^{p×p}$, where the inner product is defined as $\langle A,B\rangle = \mathrm{Tr}(A^\dagger B)$. So your condition just means distinct $U_i, U_j$ are orthogonal in this inner product space.

1. Yes, the 2×2 Result Generalizes to p×p #

For any integer $p ≥ 2$, there do exist families of $p^2$ pairwise orthogonal $p×p$ unitary matrices, and they can be mapped to a "standard" set of orthogonal unitary matrices via unitary transformations (i.e., there exist unitary $V,W ∈ \mathbb{C}^{p×p}$ and complex scalars $\alpha_1, ..., \alpha_{p^2}$ such that each $\alpha_i V U_i W^\dagger$ is part of this standard set).

The standard set we use here is the generalized Pauli matrices (Weyl operators), which are the natural extension of the 2×2 Pauli matrices:

• Define the shift matrix $X$: $X|k\rangle = |k+1 \mod p\rangle$ (in the standard basis $|0\rangle, |1\rangle, ..., |p-1\rangle$)
• Define the phase matrix $Z$: $Z|k\rangle = \omega^k |k\rangle$, where $\omega = e^{2\pi i/p}$ is a primitive $p$-th root of unity
• The generalized Pauli matrices are all products $X^a Z^b$ where $a, b ∈ {0, 1, ..., p-1}$

These $p^2$ operators have two key properties:

• Each $X^a Z^b$ is unitary (products of unitary matrices are unitary)
• They're pairwise orthogonal: $\mathrm{Tr}((X^a Z^b)\dagger X^{a'} Z^{b'}) = p\delta_{a,a'}\delta_{b,b'}$. This implies $\mathrm{Tr}((X^a Z^b)(X{a'} Z^{b'})\dagger) = 0$ whenever $(a,b) ≠ (a',b')$, which matches your orthogonality condition (we just need to scale by $\alpha_{a,b} = 1/\sqrt{p}$ if we want normalized traces, but your question only requires the cross-traces to be zero).

Since any family of $p^2$ pairwise orthogonal $p×p$ matrices forms an orthogonal basis for $\mathbb{C}^{p×p}$ (the space has dimension $p^2$), and the generalized Pauli matrices are also an orthogonal basis, the two bases are unitarily equivalent. That's exactly the generalization of your 2×2 result.

2. Characterizing These Matrix Families #

Here are three key ways to describe any family of $p^2$ pairwise orthogonal $p×p$ unitary matrices:

Orthogonal Basis & Unitary Equivalence #

• At their core, these families are orthogonal bases for $\mathbb{C}^{p×p}$ (under the Frobenius inner product) where every basis element is unitary.
• Any two such families are unitarily equivalent: if ${U_i}$ and ${V_i}$ both satisfy your orthogonality condition, there exist unitary matrices $A,B$ and non-zero complex scalars $\alpha_i, \beta_i$ such that $V_i = \alpha_i A U_i B^\dagger$. This is because all such bases can be mapped to the generalized Pauli matrices via unitary transformations, and composing unitary transformations gives another unitary transformation.

Group Representation Perspective #

• The generalized Pauli matrices are irreducible unitary representations of the finite abelian group $\mathbb{Z}_p × \mathbb{Z}_p$. More broadly, any family of orthogonal unitary matrices like this can be viewed as a unitary representation of an abelian group (or a conjugate of such a representation via unitary matrices).
• Each matrix $U_i$ in the family is a unitary matrix, so all its eigenvalues are complex numbers of modulus 1. For orthogonality, the eigenvalue distributions of distinct $U_i$ must be "orthogonal" in the sense that their weighted inner products (via the trace) cancel out to zero.

Normalization & Completeness #

• If we normalize the matrices so that $\mathrm{Tr}(U_i U_i^\dagger) = p$ (i.e., their Frobenius norm is $\sqrt{p}$), then the family becomes an orthonormal basis for $\mathbb{C}^{p×p}$. In this case, the scalars $\alpha_i$ will have modulus $1/\sqrt{p}$, since the generalized Pauli matrices have Frobenius norm $\sqrt{p}$.

Note that while the generalized Pauli matrices are the most straightforward construction, for composite $p$ there are other orthogonal unitary bases, but all of them are unitarily equivalent to the generalized Pauli family.

内容的提问来源于stack exchange，提问作者Henry Yuen