We use nearly parallel pure states to characterize positive linear functionals <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {M}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">M</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> as positive multiples of the trace if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\phi (A \natural B) \le \sqrt{\phi (A) \phi (B)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>♮</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msqrt> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation> for all positive definite matrices A and B. Here <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>♮</mo> <mi>B</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>#</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mi>A</mi> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>#</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\phi (A \natural B) \le \phi ((A+B)/2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>♮</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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.

Spectral geometric mean and trace characterizations

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发表日期：

2026年6月1日

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

摘要

We use nearly parallel pure states to characterize positive linear functionals $\phi $ $ϕ$ on $\mathbb {M}_n$ $M_{n}$ as positive multiples of the trace if and only if $\phi (A \natural B) \le \sqrt{\phi (A) \phi (B)}$ $ϕ (A ♮ B) \leq \sqrt{ϕ (A) ϕ (B)}$ for all positive definite matrices A and B. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ $A ♮ B = {(A^{- 1} # B)}^{1 / 2} A {(A^{- 1} # B)}^{1 / 2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\phi (A \natural B) \le \phi ((A+B)/2)$ $ϕ (A ♮ B) \leq ϕ ((A + B) / 2)$ 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.