# Is it possible to formulate quantum mechanics in the equilibrium state?

The standard formulation of quantum theory takes measurement as "part of the postulates" (see for example this post).

It is known that measurement is always associated with an increase in entropy (see for example this post).

Putting these two statements together we logically arrive at the following conclusions:

1. In the state of equilibrium (where entropy has reached a maximal value so cannot increase further) measurement is not possible

This seems reasonable: clearly any description of measurement apparatus is describing a non-equilibrium condition

1. In the state of equilibrium (where measurement is not possible) the standard formulation of quantum mechanics loses meaning

This is more disturbing. Have I missed something here (apart from the obvious "the measurement problem is an unresolved problem") or is there a known way to formulate quantum mechanics in the equilibrium state (which seems on the face of it to be a perfectly reasonable physical state worthy of study)?

If everything in the universe were in equilibrium, measurement would not be our only problem. Nothing useful could happen at all, regardless of the measurement issue. (This is still an interesting scenario, though, and I'll revisit it below.)

If only part of the universe is in equilibrium, then interesting things can still happen, including measurement. The statement that a given subsystem is in "equilibrium" doesn't mean that it is devoid of dynamics; it only means that it's time-evolution is carrying it through some sequence of states that, at any given time, might as well be randomly selected from the set of states that have the given macroscopic constraints (such as total volume and total energy). It is still quite capable of becoming entangled with other systems, even if it remains "in equilibrium" in the sense of having an essentially thermal reduced state. These assertions are presumably consistent with the results reported in [1], [2], and [3].

For example, suppose that we have a room $$R$$ full of air in equilibrium. The room has walls to confine the air. Suppose that a measurement occurs in a laboratory $$L$$ that is located next to the room $$R$$, on the other side of one of the walls. By definition of "measurement," the rest of the system (outside the thing being measured) must be influenced in some practically irreversible way that depends on the measurement outcome. As usual, this can be expressed as $$\big(|a\rangle+|b\rangle\big)\otimes|X\rangle\rightarrow |a\rangle\otimes|X_a\rangle + |b\rangle\otimes|X_b\rangle \tag{1}$$ where $$a$$ and $$b$$ represent two (not necessarily normalized) measurement-eigenstates of the thing being measured, and $$X_a$$ and $$X_b$$ represent two states of the rest of the laboratory that differ from each other in some hopelessly complicated way on the microscopic scale, so that their orthogonality cannot be undone by any feasible future operation. (Confession: the integrity of this whole sentence hinges on the word "feasible", which I don't really know how to define.) Once this has happened in $$L$$, the molecular-scale differences between $$X_a$$ and $$X_b$$ will probably lead them to become entangled with the molecular-scale state of the wall, which in turn will probably lead them to become entangled with the molecular-scale state of the air in the adjacent room $$R$$. And so on forever. We don't have to wait forever, though; as soon as the effects have become practically irreversible, we might as well project onto one of the "outcomes" ($$a$$ or $$b$$), with the relative frequencies dictated by Born's still-unexplained rule.

I think we can make an even stronger statement. Even if we treat the walls of $$R$$ as idealized boundary conditions, so that the air in $$R$$ is mathematically incapable of becoming entangled with anything outside of $$R$$, I think we can say that the locations of individual air molecules in $$R$$ are constantly being measured by virtue of their interactions with each other, simply because two different locations of a single air molecule would quickly lead to hopelessly complex differences between the resulting states of the air overall, which is essentially the defining feature of measurement — albeit a non-deliberate measurement in this case. (Maybe we could even extend this reasoning to the whole universe, but that raises a host of other issues that I'm even less qualified to address.)

To relate this to entropy, we need to be careful about the definition of entropy. For example, if entropy is defined to be the number of mutually-orthogonal microstates compatible with the given macroscopic constraints, then the picture just described is compatible with the non-increase of entropy. Maximum entropy doesn't mean no dynamics; it only means that none of the states through which the system is evolving are "special" in any macroscopically-quantifiable way. (Related: Explain the second principle of thermodynamics without the notion of entropy.) If entropy is defined to be the von Neumann entropy associated with the reduced state of a subsystem, then essentially the same comment applies: maximum entropy in this case means that the subsystem is maximally entangled with the rest of the system. The total system may still be described using a pure state and may still have interesting dynamics involving the subsystem in question. This again is presumably consistent with the results reported in [1], [2], and [3].

It's easy to weave hand-waving arguments, like I did here, but this is always risky because hand-waving arguments often miss some very interesting details. It's too bad that the calculations are so difficult.

References:

[1] Linden et al (2008), "Quantum mechanical evolution towards thermal equilibrium," http://arxiv.org/abs/0812.2385

[2] Popescu et al (2005), "The foundations of statistical mechanics from entanglement: Individual states vs. averages," http://arxiv.org/abs/quant-ph/0511225

[3] Goldstein et al (2005), "Canonical Typicality," http://arxiv.org/abs/cond-mat/0511091