Can loss of information create an arrow of time?

Forgetting can make reversal inaccessible

When a completed record retains less information than the source history that produced it, an observer may be unable to reconstruct the past. This gives DTT a clear route into questions about irreversibility: inner generation can remain richly structured while an outer record exposes only selected information.

The stronger claim is that this information loss explains a persistent physical arrow of time. That requires more than a many-to-one display. One must specify the dynamics, permitted future tests, initial preparation and resource budget, then prove an appropriate monotonicity or directional statistic. A coarse observation can hide reversibility without making every entropy increase.

The projection-induced irreversibility guide provides finite models for this discussion. The useful aim is to identify exactly which source or completion property produces the observed direction.

A visible reset can be a reversible transfer

Take two classical bits, a visible register x and an environment register e. Prepare e=0 and apply the reversible swap (x,e)↦(e,x). Afterwards the visible register is always zero. If x was initially unknown, the visible operation looks like an irreversible reset. The environment now carries the original x.

With access to both registers, another swap restores the original state. Without access to e, the visible output alone cannot distinguish the two possible inputs. Thus the loss is operationally real for the restricted observer while the complete update remains one-to-one. The direction came partly from a special preparation: the environment started in a known state that could receive information.

The visible Shannon entropy decreases from one bit to zero for an initially unbiased x. Therefore information being discarded from the display does not imply that the display's own entropy increases. A thermodynamic statement must count the other degrees of freedom and specify what is meant by entropy in the chosen physical setting.

A coarse record can become uncertain and then certain again

Let a microscopic state move around a four-state cycle, 0→1→2→3→0. Record only whether the state belongs to A={0,1} or B={2,3}. This display is many-to-one, so it discards the microscopic position within each class.

Prepare an equal mixture of states zero and one. Initially the displayed record is certainly A. After one step the microscopic mixture is one and two, so the display is evenly split between A and B and has one bit of uncertainty. After another step the mixture is two and three, so the display is certainly B and its uncertainty returns to zero.

The underlying update is reversible, the display is coarse, and the displayed entropy rises and falls. Furthermore, the label A does not determine its own next state: microscopic zero stays in A after one step, while microscopic one moves to B. Treating the public labels as an autonomous memoryless process would require extra information or an additional averaging assumption.

This is why the hidden-variable elimination guide connects projection with memory. Discarding variables can create history dependence, rather than an exact irreversible law on the displayed state alone.

A monotone law needs additional structure

There are simple dynamics for which a visible entropy is provably nondecreasing. Consider a classical bit that flips independently with fixed probability ε at each step, with 0≤ε≤1/2. If p is the probability of state one, the next probability is p′=ε+(1−2ε)p.

The distance from the uniform distribution contracts. The binary entropy −p log₂p−(1−p)log₂(1−p) therefore increases or stays constant. This conclusion uses a memoryless symmetric transition rule. A reset channel, a time-dependent preparation or a hidden correlated environment does not necessarily obey the same argument.

For a source theory, the question becomes why the retained dynamics has this kind of contractive or mixing structure. If it emerges only after a separation of time scales and a prepared bath approximation, the monotonicity should be claimed in that regime. The structural-functional article provides another route: identify the functional and the exact hypotheses under which its derivative has a definite sign.

Retained records consume a prepared memory resource

An indefinitely growing archive needs indefinitely growing storage, or a rule for overwriting old records. If old information is copied into another accessible register, the complete evolution may remain reversible. If it is erased while the apparatus is restored for reuse, the thermodynamic account includes the degrees of freedom that absorb the discarded information.

In the standard idealized setting of an unbiased degenerate bit reset cyclically using a thermal bath at temperature T, without useful side information, the reversible-limit heat cost is at least k_B T ln2. This is a conditional physical statement about reset, not a claim that every measurement, gate or recorded bit must dissipate that amount. Bennett's account of Landauer's principle discusses the distinction between reversible information processing and erasure.

A DTT completion model should therefore specify whether the seventh-stage record accumulates, overwrites or transfers information. The six nested inner roles can generate the record, but the storage resource and the subsequent allowed access determine whether its history is recoverable. A supply of fresh blank registers is an initial resource whose origin and consumption should remain visible.

Why a closed finite reversible source cannot have a strict global arrow

For a finite state space with an invertible update T, every trajectory lies on a cycle. Suppose a proposed state functional F satisfies F(Tx)≥F(x) for every state. Going once around a cycle gives a chain of inequalities that returns to the original value. Every inequality on that cycle must therefore be an equality.

A strictly increasing state functional is impossible on such a closed finite reversible system. This short result does not rule out a useful observed arrow over a finite window. It identifies what that arrow must depend on: restricted preparations, inaccessible memory, coarse tests, an open environment, an unbounded resource or an approximation that neglects eventual return.

The distinction also helps assess a proposed memory equation. A reduced equation can be dissipative even when the larger source evolution is reversible, provided the hidden-state preparation and access restrictions are specified. Deriving the reduced law and estimating its validity time are the physical work; the word projection alone does not establish them.

A useful finite task for contributors

Give a reversible source update, a finite prepared memory and an explicit completed-record map. Choose one directional observable, such as a forward-versus-reverse sequence probability or an entropy functional, and state the observation window. Derive the direction from the same update while counting the memory that becomes inaccessible or must be reset.

Then vary the initial preparation and allow a controlled recovery of some hidden memory. The result should predict when the observed arrow persists, weakens or reverses. A theorem with an error bound and finite validity time would establish a concrete mechanism for temporal asymmetry. It would also distinguish the arrow experienced through completed records from an unproved claim that the full source dynamics has no inverse.

Continue reading

• Toy models of projection-induced irreversibility →
• Eliminating hidden variables creates memory →
• A monotone structural functional →
• When a readable record is enough to predict the future →
• Temporal asymmetry: reading the published model →

Read this discussion as part of a path

• Meaning, explanation and distinction →