The simple reason turbine engines are more efficient at high altitudes is because as the air becomes less dense it requires less fuel as the ratio has to remain constant. This holds true up to isothermal temperature which is usually at 40,000 feet. The turbine compresses the colder air with less fuel required and you have a much lower fuel burn.
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This isn't true. Less dense air at higher altitudes means less oxygen to burn with the fuel. Less burn means less thrust. In the limit of no oxygen, there is no burn, and no thrust at all.