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arXiv:2605.18888

Spectral geometric mean and trace characterizations

Airat Bikchentaev, Trung Hoa Dinh, Anh Vu Le, Mohammad Sal Moslehian

quant-phmath.FAmath.OASubmitted May 16, 2026• Announced May 19, 2026
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Abstract

We use nearly parallel pure states to characterize positive linear functionals $φ$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $φ(A \natural B) \leq \sqrt{φ(A) φ(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $φ(A \natural B) \leq φ((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.

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We use nearly parallel pure states to characterize positive linear functionals φφφ on Mn\mathbb{M}_nMn​ as positive multiples of the trace if and only if φ(A♮B)≤φ(A)φ(B)φ(A \natural B) \leq \sqrt{φ(A) φ(B)}φ(A♮B)≤φ(A)φ(B)​ for all positive definite matrices AAA and BBB. Here A♮B=(A−1#B)1/2A(A−1#B)1/2A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}A♮B=(A−1#B)1/2A(A−1#B)1/2 represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality φ(A♮B)≤φ((A+B)/2)φ(A \natural B) \leq φ((A+B)/2)φ(A♮B)≤φ((A+B)/2) for all positive definite matrices AAA and BBB. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.
We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. For a matrix A and a function f, the spectral sum is the trace of f(A), equivalently the sum over eigenvalues of A of f applied to each eigenvalue. Typical examples of spectral sums are the von Neumann entropy, the trace of the inverse of A, the log-determinant, and the Schatten p-norm, where the latter does not require the matrix to be PSD. The current best classical randomized algorithms estimating these quantities have a runtime that is at least linearly in the number of nonzero entries of the matrix and quadratic in the estimation error. Assuming access to a block-encoding of a matrix, our algorithms are sub-linear in the matrix size, and depend at most quadratically on other parameters, like the condition number and the approximation error, and thus can compete with most of the randomized and distributed classical algorithms proposed in the literature, and polynomially improve the runtime of other quantum algorithms proposed for the same problems. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory: approximating the number of triangles, the effective resistance, and the number of spanning trees in a graph.
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Let HHH be a positive semi-definite matrix partitioned in β×ββ\times ββ×β Hermitian blocks, H=[As,t]H=[A_{s,t}]H=[As,t​], 1≤s,t,≤β1\le s,t,\le β1≤s,t,≤β. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.
Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.