Quantum Tsallis entropy and projective measurement

It is well known that projective measurement will not decrease the von Neumann entropy of a quantum state. In this paper, it is shown that projective measurement will not decrease the quantum Tsallis entropy of a quantum state, either. Using a similar analysis, it can be shown that projective measurement will not decrease the quantum unified (r, s)-entropy in general.

💡 Research Summary

The paper investigates how projective measurements affect generalized quantum entropy measures beyond the well‑known von Neumann entropy. Starting from the definition of the quantum Tsallis entropy, (S_q(\rho)=\frac{1}{1-q}\bigl(\operatorname{Tr}\rho^{,q}-1\bigr)) for a density operator (\rho) and a real parameter (q>0) with (q\neq1), the authors examine its mathematical properties, especially the convexity (for (q\ge1)) or concavity (for (0<q<1)) of the function (f(x)=x^{q}). They then consider a complete set of orthogonal projectors ({P_i}) describing a projective measurement, which transforms the state as (\rho’=\sum_i P_i\rho P_i). Because the projectors satisfy (\sum_i P_i=I) and are mutually orthogonal, the trace is preserved ((\operatorname{Tr}\rho’=\operatorname{Tr}\rho=1)).

Using Jensen’s inequality, the authors show that (\operatorname{Tr}(\rho’)^{,q}\le\operatorname{Tr}\rho^{,q}) when (q\ge1) and the reverse inequality when (0<q<1). Substituting these bounds into the definition of (S_q) yields the universal inequality
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