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By Dominik Janzing

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By initializing an additional system A′ to |2 and 38 CHAPTER 2. FUNDAMENTAL PROBLEMS applying U once we decode and negate the information on B simultaneously : 20 → 21 and 21 → 12. These operations allow obviously universal computation since every boolean function can be computed from circuits which consist only of NOR gates. We summarize : Theorem 3 (Computing with Heat Engines on 3-level Systems) Given two reservoirs of 3-level systems with temperature TA and TB and energy gap EA and EB , respectively, such that TB TA < 2 EA EB and EA 2> > 1, EB then the ability to implement the optimal heat engine on any chosen pair consisting of one system of type A and one of type B implies the ability to implement universal classical computation on the 3-level systems.

1 ) . , only if one measures the magnetization “the system decides” whether it is a magnet in positive or in negative z-direction. The reason that superpositions like the cat state above are surprising is that some properties of a physical system have such an immediate effect on the surrounding of the system that it does not require a measurement instrument to investigate them. This is, for instance, the case for the mean magnetization of a many-spin system which determines the magnetic field around the system.

2. REVISITING THERMODYNAMICS In the limit of infinite temperature, g is the uniform distribution on the set of energy eigenstates and if g and g˜ have equal dimension the condition Ag = g˜ is simply the condition that A is a double stochastic matrix. Then (p, g) ≥ (˜ p, g˜) if and only if p majorizes p˜. We say that the vector p majorizes p˜ if l l j=1 pj ≥ p˜j j=1 ∀j ≤ n whenever p1 , . . , pn and p˜1 , . . , p˜n are the entries of p and p˜, respectively, in non-decreasing order. Note that the density matrix γ corresponding to a vector g with uniform distribution is the maximally mixed state which commutes with every other matrix.

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