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<p>Variance acts as a vital role in quantum metrology and the Heisenberg uncertainty principle. The variance of the observable for the mixed quantum state can be divided into the quantum and classical parts. We unitize the operator monotonic function to describe the quantum uncertainty and classical mixing uncertainty. Based on its good fundamental properties, the quantum uncertainty covers a large family of uncertainty with different operator monotone functions taken into account, and so does classical mixing uncertainty. As a special case, we mainly focus on the classical mixing uncertainty when the operator monotonic function takes <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4825_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(f(x)=\sqrt{x}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msqrt> <mi>x</mi> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, the classical mixing uncertainty equals the difference between variance and Winger–Yanase skew information that quantifies the quantum uncertainty. The lower and upper bounds for the classical mixing uncertainty and quantum uncertainty are obtained.</p>

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The abundance of uncertainty of the observable

  • 期刊论文

  • 发表日期：

    2025年7月14日

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  • Jun Zhang,
  • Kan He

摘要

Variance acts as a vital role in quantum metrology and the Heisenberg uncertainty principle. The variance of the observable for the mixed quantum state can be divided into the quantum and classical parts. We unitize the operator monotonic function to describe the quantum uncertainty and classical mixing uncertainty. Based on its good fundamental properties, the quantum uncertainty covers a large family of uncertainty with different operator monotone functions taken into account, and so does classical mixing uncertainty. As a special case, we mainly focus on the classical mixing uncertainty when the operator monotonic function takes \(f(x)=\sqrt{x}\) f ( x ) = x , the classical mixing uncertainty equals the difference between variance and Winger–Yanase skew information that quantifies the quantum uncertainty. The lower and upper bounds for the classical mixing uncertainty and quantum uncertainty are obtained.