Go to fastboot, conect phone to pc and in terminal type:
fastboot oem device-info
Watch for devices unlocked, true or false. I guest you activate developer mode, with enable oem unlock and usb debugging?
Wheu you have permission for unlock, it`s really simple:
http://xiaomitips.com/guide/how-to-fix-couldnt-verify-device-error-during-bootloader-unlock-process/
My phone came with 6.5.19, which is global dev. Unlock with MiUnlock work like a charm. Unlock official work with any China ROM and global devel, but not with global stable. Still, why you need unlock, for EU ROMS? Or for twrp/root?
You can unlock unofficially, but is tricky and somehow risky.
http://xiaomitips.com/guide/unlock-redmi-note-3-bootloader-without-official-permission/
http://forum.xda-developers.com/redmi-note-3/how-to/unlock-bootloader-waiting-permission-t3349886
HIGHLY RECOMMEND you to try this tutorial...
http://forum.xda-developers.com/showthread.php?t=3034811
http://dl-xda.xposed.info/framework/sdk23/arm64/xposed-v86-sdk23-arm64.zip
Copy unninstaler to phone, just in case.
You dont need root for changing versions. But you need to unlock it first. Search "miui unlock" in google, first link its what you need to do first. Anyway, if you a not a geek guy, dont buy phone with chinese version if you need global.