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We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs.
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We can write H ( r ) = max { P ( r , 0 ) , sup 0 < s < 1 P ( r , s ) } H(r)=\max\left\{P(r,0),\sup_{0<s<1}P(r,s)\right\} , where P ( r , 0 ) = − ln C 0 = 0 P(r,0)=-\ln C_{0}=0 can be neglected and sup 0 < s < 1 P ( r , s ) = sup 0 < s < 1 − r s − ln F 1 − s = { ln 1 F for r ≥ ln 1 F , + ∞ for r < ln 1 F . \sup_{0<s<1}P(r,s)=\sup_{0<s<1}\frac{-r~s-\ln F}{1-s}=\left\{\begin{array}[c]{c}\ln\frac{1}{F}\text{~~for~}r\geq\ln\frac{1}{F},\\ \\ +\infty\text{~~for~}r<\ln\frac{1}{F}.\end{array}\right
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More generally, for any two vectorial operators 𝐚 \mathbf{a} and 𝐛 \mathbf{b} , we can express their commutation relations in the compact form [ 𝐚 , 𝐛 T ] := 𝐚𝐛 T − ( 𝐛𝐚 T ) T [\mathbf{a},\mathbf{b}^{T}]:=\mathbf{ab}^{T}-(\mathbf{ba}^{T})^{T}
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Extension to asymmetric ST states is only technical
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For brevity we do not consider the other case where the ST state is the null hypothesis and the thermal state is the alternative hypothesis. This case is included in the our final analysis for generic ST states
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