Coarse Graining Irreversibility

Coarse-graining discards distinctions that a chosen description does not resolve. It can make a reversible quantum dynamics appear irreversible because phase information and correlations move into unobserved variables. An entropy increase then depends on the description, initial conditions, and dynamical assumptions; it is not a contradiction of unitary evolution.

Required background. Density operators: Use density operators to calculate probabilities and distinguish coherent from incoherent alternatives.

Fine-grained entropy is constant under unitaries

For $\rho(t)=U(t)\rho(0)U^\dagger(t)$, the eigenvalues are unchanged. Hence

$$S(\rho)=-\operatorname{Tr}(\rho\log\rho)$$

is constant for a closed system. Subsystem entropy can nevertheless change because tracing over an environment changes the object being described.

Start with two qubits in $|+\rangle|0\rangle$ and apply a controlled-NOT. The global state becomes $(|00\rangle+|11\rangle)/\sqrt2$, still pure with entropy zero. Each qubit now has entropy $\log2$. The missing local purity resides in correlations, not in a loss of global information.

Dephasing as a coarse description

For a finite-dimensional projective record algebra, $\Delta(\rho)=\sum_aP_a\rho P_a$ removes inaccessible cross-sector terms. This pinching map cannot decrease von Neumann entropy. It describes the loss of access to particular coherences, not the exact closed-system evolution. Infinite-dimensional extensions require care with convergence and possibly infinite entropies.

At a single time, $S(\Delta(\rho))\ge S(\rho)$ does not imply that $S(\Delta(\rho(t)))$ increases monotonically with time under arbitrary unitary evolution. Recurrences and recoherence can reverse the apparent trend.

Likewise, subsystem entropy need not be monotonic: a qubit can first entangle with an environment and later disentangle. Markovian approximations, bath conditions, or thermodynamic limits are extra ingredients in monotonicity claims.

Why reversals become impractical

If a record spreads into many environmental degrees of freedom, reversing it requires coherent control of their correlations. Ignoring a few degrees of freedom can prevent recovery. This explains practical irreversibility without asserting that the microscopic equations cease to be reversible.

The argument also depends on initial conditions. A specially prepared environment containing the right correlations can restore coherence. Ordinary laboratory preparations usually lack those finely arranged correlations, but that asymmetry should be acknowledged rather than derived from reversible dynamics alone.

Records and the thermodynamic arrow

A durable record requires a physical medium, metastability, and a process that correlates the medium with an event. Amplification often dissipates energy. Erasing a memory while returning it to a standard state has thermodynamic costs under specified conditions, whereas merely possessing a correlation is not equivalent to continual heat production.

Classical irreversibility, information loss from a reduced model, and a fundamental nonunitary collapse are therefore different notions. Macroscopic records applies the distinction to measurement.

Exercises

Apply the controlled-NOT a second time to the two-qubit example. What happens to global and local entropies?

Solution

Controlled-NOT is its own inverse, so the state returns to $|+\rangle|0\rangle$. Global entropy remains zero throughout, while both local entropies return from $\log2$ to zero. This exact recurrence disproves an unconditional claim that tracing out an environment always produces monotonic entropy growth.

References

• H. D. Zeh, The Physical Basis of the Direction of Time, 5th edition, Springer (2007).
• W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics 75, 715–775 (2003), doi:10.1103/RevModPhys.75.715.