Re-express Box 4.3’s pedagogical example of quantum decoherence and entropy increase in the language of the quantum mechanical density operator p̂ (Box 4.2). Use this example to explain the meaning of the various statements made in the next-to-last paragraph of Sec. 4.7.2.

Data from Section 4.7.2.

When one reexamines these issues in quantum mechanical language, one discovers that the entropy increase is caused by the physicist’s discarding the quantum mechanical correlations (the off-diagonal terms in the density matrix of Box 4.2) that get built up through the systems’ interaction with the rest of the universe. This discarding of correlations is accomplished through a trace over the external universe’s basis states (Box 4.2), and if the state of system plus universe was originally pure, this tracing (discarding of correlations) makes it mixed. From this viewpoint, then, it is the physi- discarding quantum correlations (decoherence) causes entropy increase cist’s choice to discard correlations with the external universe that causes the entropy increase and the evolution toward statistical equilibrium. Heuristically, we can say that the entropy does not increase until the physicist actually (or figuratively) chooses to let it increase by ignoring the rest of the universe. For a simple example, see Box 4.3 and Ex. 4.9.

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BOX 4.2. DENSITY OPERATOR AND QUANTUM STATISTICAL MECHANICS T2 Here we describe briefly the connection of our probabilistic distribution function p to the full quantum statistical theory as laid out, for example, in Sethna (2006, Sec. 7.1.1) and Cohen-Tannoudji, Diu, and Laloë (1977, complement E), or in much greater detail in Pathria and Beale (2011, Chap. 5) and Feynman (1972). Consider a single quantum mechanical system that is in a pure state ). One can formulate the theory of such a pure state equally well in terms of v) or the density operator = )(. For example, the expectation value of some observable, described by a Hermitian operator Â, can be expressed equally well as (A) = (A) or as (A) = Trace(@Â).² If our chosen system interacts with the external universe and we have no knowledge of the correlations that the interaction creates between the system and the universe, then the interaction drives the system into a mixed state, which is describable by a density operator ê but not by a ket vector [v). This ê can be regarded as a classical-type average of 1)( over an ensemble of systems, each of which has interacted with the external universe and then has been driven into a pure state ) by a measurement of the universe. Equivalently, ê can be constructed from the pure state of universe plus system by "tracing over the universe's degrees of freedom."