An m-qubit state, inferring some inequality in QIT

Thread starter billtodd
Start date Jun 14, 2024
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Quantum information

Jun 14, 2024

billtodd

Homework Statement: Let ##\sigma## be an ##m##-qubit such that ##Trace(\sigma^2)\le \frac{1+\epsilon^2}{2^m}##, then ##D(\sigma , I/2^m)\le \epsilon##

Relevant Equations: ##D(\sigma,\rho)=1/2 Trace(|\sigma-\rho|)## where ##\rho,\sigma## are quantum states; and there's an inequality between this and the fidelity, i.e. ##D(\sigma,\rho)\le \sqrt{1-F(\sigma,\rho)^2}##, where ##F(\sigma ,\rho)=Trace(\sqrt{\sigma^{1/2}\rho \sigma^{1/2}})##.

I am not sure how to show this inequality, I guess what is the relation between ##Trace(\sigma^2)## and ##Trace(\sigma)##?
Here's my latest attempt, let me know how to proceed?
$$D(\sigma, I/2^m)\le \sqrt{1-F(\sigma,I/2^m)^2}=\sqrt{1-(Tr(\sqrt{\sigma }))^2/2^m}$$

Last edited: Jun 14, 2024

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