Information conservation can refer to an invertible state evolution, an entropy preserved by a specified dynamics, or the recoverability of a message. These claims are related in particular models, but they are not interchangeable.
An invertible map preserves distinctions
On a finite set of complete classical states, a reversible deterministic update is a permutation. Distinct inputs remain distinct, and every final state has a unique predecessor. If a probability distribution is carried along by that permutation, its Shannon entropy is unchanged because the probabilities have only been reassigned to state labels.
A restricted observer can still lose information. Let the complete state be two bits, (x,m), and use the reversible map (x,m) → (x,m ⊕ x). Beginning with a fair x and m = 0 gives final states 00 and 11 with equal probability. Joint entropy stays one bit. The memory's entropy rises from zero to one bit because it becomes correlated with x.
Discarding x leaves a perfectly useful record in this example. But other restricted readouts may erase a distinction entirely. Which information remains accessible depends on which variables the observer can inspect, not merely on the invertibility of the complete update.
Unitary evolution and reduced states
For a closed quantum system evolving unitarily, ρ → UρU† preserves the eigenvalues of the density operator and hence its von Neumann entropy. This statement is conditional on the closed-system unitary description. It does not assert that every subsystem's entropy or every measurement's outcome uncertainty stays fixed.
Take a qubit in the state √p|0⟩ + √(1−p)|1⟩ and a memory in |0⟩. A controlled operation produces the joint state √p|00⟩ + √(1−p)|11⟩. The complete state is pure and has entropy zero. Each subsystem has reduced density matrix diag(p,1−p), with entropy h₂(p) bits. For p = 1/2, each has entropy one bit.
The reduced entropy reflects entanglement with the other subsystem. It is consistent with the purity of the joint state. Applying the inverse controlled operation, while both systems remain accessible and coherent, restores the initial product state.
Preserved globally does not mean recoverable locally
The states (|00⟩ + |11⟩)/√2 and (|00⟩ − |11⟩)/√2 have identical reduced density matrices on either subsystem. Their relative sign remains a distinction in the complete state. A suitable joint operation and measurement can reveal it; inspecting either local state alone cannot.
This is closely related to coherence and retained alternatives. Ignoring a correlated environment changes the accessible description. A formal inverse of the global evolution may exist even when the observer cannot control the environmental degrees of freedom needed to implement it.
Neither reduced entropy nor a calculation of decoherence, by itself, specifies why one unique outcome occurs in an individual measurement. An additional interpretation or dynamical proposal must state what it adds and what consequences follow.
Several conservation questions
A precise claim should identify the system boundary and the quantity. Is the question whether complete initial states remain distinguishable, whether a particular entropy is constant, whether a message can be decoded, or whether a coarse record can predict a later observation? These questions can have different answers in the same experiment.
Likewise, a many-to-one observed map does not alone prove fundamental destruction of microscopic information. It may represent neglected variables, noise, an irreversible effective description or a genuinely nonunitary law. The alternatives require different physical models and tests.
Thermodynamic entropy production adds the role of reservoirs and a specified reduction. The discussion of computation and erasure explains why its energetic accounting cannot be replaced by a bare count of output symbols.
Reversible inner dynamics and outer records
A dual-time model may propose reversible inner evolution with a reduced outer description. To establish an information claim, it must specify both dynamics and readout, including the observer's access to correlations. The expanded book discussion of records and decoherence separates the conventional reduction from stronger claims about outcomes.
The published real-commutant phase-selection paper concerns compatible structures within an assumed real representation and flow. The dual-time geometry paper studies temporal geometry and reduced dynamics. Neither should be cited as a substitute for specifying the physical system and its information balance in a particular application.
Sources and further reading
- Entanglement across separation: correlations and the limits of local descriptions.
- An event and a record of it: preparation, copying and retention. The finite-state and two-qubit calculations here are worked teaching examples.
- M. Haj Yousef, Real-commutant phase selection from real Hilbert representations, Physics Letters A 594 (2026), 132085.