Completion channels: exact erasure, preserved algebras and residual memory

Name the register that the channel actually changes

The starting point is a tensor factorization, not a claim that all local information becomes irrelevant. Write the finite standard-form space as L⊗R, with both factors of dimension m. Left selector operators act on L and right-carrier observables on R. The supplied expectation is

It sends a selector projector p_a⊗I_R to I_L/m⊗I_R, while fixing every I_L⊗b. These are different operator families. If the projector to be erased belonged to the retained algebra itself, pointwise preservation would require that same projector to remain unchanged. That is the source's projector-erasure obstruction, and it explains the need to identify commuting roles explicitly. No operation copies an arbitrary unknown carrier state into a disposable classical label. Completed records, v21, §4.1 and Definition 4.2, pp.6–7.

For density matrices the same formula replaces the selector by its maximally mixed state and preserves the reduced carrier density matrix. This identification of Schrödinger and Heisenberg formulas works here because E is self-adjoint for the trace inner product; arbitrary channels need not have identical state and observable formulas.

Diagonalize the channel before interpreting its rate

Every operator decomposes uniquely as X=E(X)+X_⊥, with E(X_⊥)=0. For γ>0 the semigroup has the exact form

On the m⁴-dimensional operator space, the eigenvalue one has multiplicity m², and s has multiplicity m⁴−m². For a qubit selector and carrier these multiplicities are four and twelve. The generator γ(E−id) has eigenvalues zero and −γ on the same subspaces. Thus Φ_t+u=Φ_tΦ_u, its positive-time fixed algebra is I_L⊗M_m, and its induced Hilbert–Schmidt distance from E is exactly s. At t=0 the channel is identity and every operator is fixed. V21, Theorem 4.3, p.7.

The relaxation rate γ is a superoperator rate with inverse-time units. It is not the singlet penalty Δ, an excitation energy, or a continuum mass gap. Calling each of these a “gap” without naming the operator would hide the different calculation. Nor does the number of registers turn six nested inner generative levels and the seventh completion into independent time coordinates.

An explicit Kraus description verifies complete positivity

For m=2, let σ₀=I and σ_1,2,3=X,Y,Z act on the selector. The identity E(A)=¼Σ_j=0³σ_jAσ_j follows by expanding A in the Pauli basis. Consequently the selector part of Φ_t is a random Pauli channel:

Tensor each K with I_R. Their squared weights sum to one, so ΣK†K=I and the map is trace preserving; an operator-sum representation supplies complete positivity. The identity observable is fixed as well. Its generator can equivalently be written (γ/4)Σ_j=1³(σ_jAσ_j−A), which attenuates each nonidentity Pauli operator at rate γ.

This is a direct qubit expansion of the finite channel in v21, Definition 4.2 and Theorem 4.3, p.7. It supplies a checkable implementation rather than selecting the Pauli probabilities from temporal ontology. In the reference explorer, the same mathematical depolarization formula is deliberately applied to a different register, physical reference A. That changed assignment changes which carrier algebra is preserved.

A linear inverse is not a physically reversible channel

At every finite time s>0, all superoperator eigenvalues are nonzero. The inverse exists as a linear transformation:

Apply the qubit inverse to |0⟩⟨0|. It produces diag[(1+s⁻¹)/2,(1−s⁻¹)/2]. The second eigenvalue is negative whenever s<1. Therefore the inverse cannot process every density matrix as a positive, let alone completely positive, channel. It can algebraically recover an exact known output inside the image, but that does not provide a physical recovery operation for arbitrary inputs. Noise and estimation errors are also amplified by s⁻¹.

For orthogonal selector preparations, the trace distance between the two outputs is s. At γt=ln2 it is one half; at finite γt it has not become zero. Exact E eliminates their difference, while every unitary conjugation preserves its Hilbert–Schmidt norm. Normalizing that difference gives the source's lower bound ‖α−E‖_2→2≥1 for every same-algebra reversible conjugation α. V21, Theorem 4.6 and its finite-time qualification, pp.7–8.

The obstruction concerns the declared closed algebra. It permits an enlarged reversible evolution followed by the loss of access to an environment.

Locate the erased information in a reversible dilation

Prepare an auxiliary register E_env in I/m and swap it with L. Tracing out E_env gives precisely I_L/m⊗Tr_Lρ_LR. The original selector information has moved into the environment, including any correlations that previously involved L. The carrier's reduced state survives; correlations between protected right carriers at distinct records also survive when the swaps act only on selectors.

A finite mixing step can be implemented by a classical control that applies this swap with probability 1−s and does nothing with probability s. Discarding the control and environment yields Φ_t. The full controlled operation can be unitary, while the reduced map remains noninvertible at exact erasure. Keeping access to those registers would enlarge the future test family and change what could legitimately be called forgotten.

The manuscript explicitly supplies fresh registers for selector mixing and phase pinching. Their preparation, replenishment and exclusion from later tests are part of the completion mechanism. The finite clock does not itself furnish that reservoir policy. V21, §8, p.17. These observations identify an operational resource requirement; they do not, without a specified reservoir Hamiltonian and thermodynamic protocol, calculate an energy cost per completed record.

Preserving an algebra is stronger than preserving one marginal

Suppose a unital completely positive joint map fixes every operator a⊗I and I⊗b on two retained carriers. For an operator a in either full algebra, it also fixes a†a and aa†. Equality in the completely positive Schwarz inequality then places a in the map's multiplicative domain. Products obey Ψ((a⊗I)(I⊗b))=(a⊗I)(I⊗b). Such products span the joint matrix algebra, so every joint observable is fixed. V21, Proposition 4.7, p.8.

This is a universal operator statement. Observing I/2 as the output marginal of one qubit is only a statement about one state. Depolarizing one half of a Bell pair preserves both I/2 marginals but attenuates X⊗X, Y⊗Y and Z⊗Z correlations. It fails the premise because X⊗I, for example, is multiplied by s rather than fixed.

The distinction resolves an apparent paradox: selector erasure cannot destroy a prepared right–right pairing while preserving both carrier algebras, but actual noise on a carrier can destroy the pairing. The latter process changes the model's retained observables. The shared-reference calculation quantifies that change rather than treating it as a passive frame relabeling.

Fixed points need not exhaust the surviving information

The simple semigroup above converges to E because its only decaying eigenvalue is s. A joint completion instrument can behave differently. In the source it has observable map Φ(A)=U†E_joint(A)U, with a probe pinching D_P(A)=PAP+QAQ. Its fixed probe algebra is {P,U}′. Replacing U by a function of H requires the stated spectral no-alias condition; it is not automatic at arbitrary sampling times. V21, Theorem 8.1, p.17.

For a transparent example, take a three-state probe with P=|0⟩⟨0| and diagonal H. The operator |1⟩⟨2| lies wholly in Q, so pinching leaves it intact. Unitary conjugation multiplies it by e^{iτ(E₁−E₂)/ℏ}. Unless that phase equals one it is not fixed, yet its norm never decays. Iterating the instrument rotates this coherence instead of approaching the fixed-point projection.

The singlet-loop construction likewise retains dark auxiliary sectors, with an explicit commutant calculation. Temporal completion, v20, Proposition 20.2, p.60. An account of completed data therefore needs both fixed observables and any accessible nondecaying modes, together with the allowed observation times.

Turn the channel statement into a reproducible audit

A useful audit first fixes the tensor-factor assignment and state/observable convention. It then checks trace preservation, complete positivity, the claimed fixed algebra, and the action on a spanning operator basis. Testing only the maximally mixed input would miss an entire family of nontrivial depolarizing channels. For the qubit selector, I,X,Y,Z give a complete basis; tensoring them with a carrier basis tests the joint claim.

Next, separate exact identities from convergence tolerances. The operational projection E is exact, whereas a finite-time approximation Φ_t has residual s in the stated Hilbert–Schmidt operator norm. A target residual ε requires γt≥ln(1/ε). At ε=0.01 the product is ln100≈4.60517, not an unspecified “large” number of inner steps. This bound concerns the selected norm and supplied rate.

Finally, include the environment and reference inventory. A theorem about forgetting a selector does not choose the physical carrier, future tests, comparator, or Hamiltonian. Those interfaces remain explicit in v21, §§5 and 8, pp.9–10 and 16–17. A failed channel identity is a concrete objection to that implementation; a successful identity establishes its finite mathematics while leaving those source-selection questions open.

Continue reading

• The standard-form record construction
• Shared references and the Bell example
• Explore two reference protocols
• Completion relative to future tests