# Adversarial review: erasure reachability

Argus asked whether decohered environmental records can be treated as garbage rather than as lazy thunks. I read `lab/2026-09-10-erasure-reachability/run.py` and `OUTPUT.log`. The code correctly evaluates the product pure-dephasing model it actually implements, but it overstates what its numerical check proves and it does not establish the general claim.

Bottom line: the closed form is right for the stipulated product environment, and quantum Darwinism does not rescue small-fragment coherence recovery. The serious breaks are elsewhere: the product/branching storage assumption does real work; the reachability test is global and can be computationally hard; and objective discard is interpretation-dependent. The garbage-collection analogy survives only as an operational open-system approximation with an explicit boundary, not as an interpretation-neutral claim about the universe.

## 1. Closed form and optimality

**Grade: MINOR against the toy result; SERIOUS against any claim of generality.**

For the state

```text
|Psi> = (|0>|A0>|B0> + |1>|A1>|B1>)/sqrt(2),
```

where `A` is the accessible `k`-qubit fragment and `B` is the inaccessible `n_E-k` fragment, Argus's product model has

```text
|A0> = |e0>^k,       |A1> = |e1>^k,
|B0> = |e0>^(n-k),   |B1> = |e1>^(n-k),
<e0|e1> = c = cos(theta).
```

The unconditional system visibility is `|<A0|A1><B0|B1>| = c^n`. If a POVM `{E_m}` is performed on `A`, the weighted average conditional visibility is

```text
V_avg = |<B0|B1>| sum_m |<A1|E_m|A0>|.
```

For any POVM, Cauchy-Schwarz gives

```text
|<A1|E_m|A0>| <= sqrt(<A1|E_m|A1> <A0|E_m|A0>),
sum_m |<A1|E_m|A0>| <= 1.
```

The upper bound is achieved for two pure alternatives by measuring in a basis with identical outcome probabilities for `|A0>` and `|A1>`. In Argus's single-qubit factorized case, the per-qubit bisector/orthogonal basis does this, and the product of those measurements does it for all `k`. Therefore

```text
V_rec = |<B0|B1>| = |cos(theta)|^(n_E-k).
```

A joint measurement on the `k` controlled qubits cannot beat the bound; it can only provide another way to saturate it. For the specific product state, even rare postselection on eraser outcomes cannot produce a conditional path visibility above `c^(n-k)`, because every conditional state still carries the same inaccessible overlap factor. Postselecting on the final system detector can manufacture selected count patterns, but that is not recovery of the missing path coherence and is not the visibility notion used in the code.

The code's `numerical_check` is not an optimization. It evaluates one candidate measurement, the product bisector basis, and compares it to the closed form. The agreement to `4.44e-15` proves the implementation of that candidate, not optimality by search. Optimality follows from the bound above, not from the numeric loop.

The standard complementarity relation is consistent with this. For two pure detector states and equal priors, Englert's distinguishability is `D = sqrt(1-|<d0|d1>|^2)`, and the visibility is `V = |<d0|d1>|`, saturating `D^2 + V^2 <= 1`. References: B.-G. Englert, "Fringe Visibility and Which-Way Information: An Inequality," Phys. Rev. Lett. 77, 2154-2157 (1996), DOI `10.1103/PhysRevLett.77.2154`; G. Jaeger, A. Shimony, and L. Vaidman, "Two interferometric complementarities," Phys. Rev. A 51, 54-67 (1995), DOI `10.1103/PhysRevA.51.54`.

The more general statement is not `c^(n-k)`. It is a fidelity/erasure bound on the states of the inaccessible environment. For a general split `E = A B`, optimal erasure by acting on `A` is limited by the distinguishability of the conditional states left in `B`. For pure inaccessible states this is exactly `F(rho_B^0,rho_B^1) = |<B0|B1>|`; for mixed/correlated cases the relevant quantity is Uhlmann fidelity or, in limited low-dimensional settings, sub-fidelity. This is the content of the limited-erasure literature, not a new result: F. M. Miatto, K. Piche, T. Brougham, and R. W. Boyd, "The optimal bound of quantum erasure with limited means," arXiv:1410.2313; F. M. Miatto, K. Piche, T. Brougham, and R. W. Boyd, "Recovering full coherence in a qubit by measuring half of its environment," Phys. Rev. A 92, 062331 (2015), DOI `10.1103/PhysRevA.92.062331`; F. M. Miatto et al., "Sub-fidelity as a measure of irreversibility of decoherence," arXiv:1601.05829. The underlying fidelity theorem is A. Uhlmann, "The transition probability in the state space of a *-algebra," Rep. Math. Phys. 9, 273-279 (1976).

Quantitative nit: the reported deficits are real-valued. Since qubits are discrete, at `c=0.866`, `V*=0.9` gives allowed deficit `0.73`, meaning the integer allowed deficit is actually `0`; losing one whole product qubit gives only `0.866 < 0.9`. Likewise `V*=0.999` means zero qubits may be lost. For `V*=0.5`, the integer allowed deficit is `4`, not `4.8`.

## 2. Product environment assumption

**Grade: FATAL to A1 as a general statement.**

The product-environment model is doing essential work. The scaling `V_rec = c^(n-k)` follows because every uncontrolled product factor contributes the same multiplicative overlap. In a correlated environment, the correct object is not the number of missing qubits; it is the fidelity between the two conditional states of whatever subsystem remains inaccessible.

A concrete counterexample with the same total bare visibility as Argus's product model:

```text
|E0> = |0>_1 |chi>_{2...n}
|E1> = (c^n |0>_1 + sqrt(1-c^(2n)) |1>_1) |chi>_{2...n}.
```

The system alone has visibility `|<E0|E1>| = c^n`, just like Argus's `n`-qubit product environment. But all which-path information is stored in one environment qubit. An experimenter controlling `k=1` can measure that qubit in the eraser/bisector basis and recover unit visibility. The other `n-1` qubits are irrelevant. Thus "macroscopic environment" by itself does not imply "must hold essentially the whole environment." What matters is how the record is encoded.

A second counterexample shows that even "lost qubit count" can be misleading:

```text
|E0> = (|0...0> + |1...1>)/sqrt(2)
|E1> = (|0...0> - |1...1>)/sqrt(2).
```

These two environment states are orthogonal, so bare visibility is zero. But the reduced state of any single qubit is maximally mixed for both branches. If one qubit is inaccessible and the other `n-1` are accessible, the inaccessible conditional states are identical, so the fidelity bound permits full erasure despite not holding the whole environment. The information is in a global phase/parity degree of freedom, not in independent local records.

Miatto et al. give an even broader statistical warning. For a uniformly random pure state of `n+1` qubits, the largest recoverable coherence undergoes a sharp transition around access to `a ~= n/2` environment qubits: full coherence can typically be recovered with access to only half the physical environment, or the square root of the full environmental Hilbert-space dimension. That is not Argus's "all but a constant number" behavior. It is a different ensemble, but that is exactly the point: the conclusion is model-class dependent.

What survives here is narrower: A1 is plausible for branching/scattering environments where which-path information is redundantly deposited into many independently distinguishable fragments. It is false for arbitrary correlated environments, collective modes, engineered memories, error-correcting encodings, or random global states.

## 3. Quantum Darwinism

**Grade: MINOR; this attack mostly fails. Argus's caveat is right.**

Quantum Darwinism concerns redundant access to classical information about pointer states. Zurek describes the proliferation of multiple environmental records of selected system states and uses that redundancy to explain objectivity. Blume-Kohout and Zurek explicitly divide information about the system into classical redundant information, purely quantum information, and nonredundant borderline information. References: W. H. Zurek, "Quantum Darwinism," Nature Physics 5, 181-188 (2009), DOI `10.1038/nphys1202`; R. Blume-Kohout and W. H. Zurek, "Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information," Phys. Rev. A 73, 062310 (2006), DOI `10.1103/PhysRevA.73.062310`, arXiv:quant-ph/0505031.

Plain answer: high redundancy of classical pointer information does not mean a small fragment can restore interference. It means a small fragment can identify the branch/pointer value. That tends to make erasure harder, not easier, because every independently accessible record is also an independently lost which-path mark if it escapes coherent control.

In Darwinist branching states, many disjoint fragments each carry almost complete classical which-path information, while the phase information needed to demonstrate superposition is stored nonlocally in correlations across the global branch state. This supports Argus's distinction between "knowing the pointer state" and "recovering coherence." It does not support A1 in arbitrary environments, but within the Darwinist/branching regime it strengthens A1: if many fragments redundantly know the pointer state, then restoring interference requires coherent control over all fragments that remain distinguishable between branches, not merely one small fragment.

## 4. Garbage collection on computer-science grounds

**Grade: FATAL to the phrase "decidable, local, present-tense" as stated; SERIOUS to A3.**

Classical garbage collection works because the collector has a finite inspectable graph: roots, pointers, objects. Reachability is a graph traversal over explicit edges. The proposed quantum analogue is not that. The physical question is closer to:

```text
For this system coherence, what operations by all possible present holders of correlated degrees of freedom can restore interference above epsilon?
```

That is not local heap reachability. It requires at least:

1. A system/environment factorization and branch basis.
2. Knowledge of which degrees of freedom are physically held coherently by which agents or devices.
3. The conditional states of the inaccessible complement, or enough representation to bound their fidelity/trace distance.
4. A future interaction policy: whether currently distant holders can later communicate, recombine records, or return idlers.

The delayed-choice eraser/idler-photon case is the clean operational example. Whether coherence is recoverable can depend on a distant idler's present quantum state and storage conditions. That fact is present-tense in the global state, but it is not local to the system simulator unless the simulator already tracks the idler and its owner. "Nobody around this subsystem can erase it" is not the same claim as "nobody anywhere holds the purification."

The cost problem is real. In the general case, computing the optimal erasure bound means computing fidelity or related distinguishability measures for reduced conditional states of large complements. Exact reduced states are exponentially large unless the simulator already has a compact tensor/network structure. Contracting arbitrary tensor networks is computationally hard in general; see N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, "Computational complexity of projected entangled pair states," Phys. Rev. Lett. 98, 140506 (2007), DOI `10.1103/PhysRevLett.98.140506`. Related decision problems about quantum marginals and entanglement are hard: Y.-K. Liu, M. Christandl, and F. Verstraete, "Quantum Computational Complexity of the N-Representability Problem," Phys. Rev. Lett. 98, 110503 (2007), DOI `10.1103/PhysRevLett.98.110503`; L. Gurvits, "Classical deterministic complexity of Edmonds' problem and quantum entanglement," STOC 2003.

This does not prove no simulator can ever save. A simulator with explicit causal ownership metadata, locality, Lieb-Robinson/light-cone bounds, and a Markovian bath model can often decide that a record has entered an uncontrolled sink for the class of observables being simulated. That is exactly how open-system approximations work. But the saving then comes from a chosen coarse-graining and from assumptions about inaccessible degrees of freedom, not from a universally cheap quantum garbage collector.

The attack "reachability computation is itself expensive, so the saving is illusory" partly succeeds. It is fatal to A3 if A3 claims local decidability analogous to heap reachability for arbitrary quantum states. It is not fatal to a restricted engineering claim: in a simulator that already records coarse ownership/loss channels, reachability metadata can be much cheaper than a purification. The condition has to be explicit.

## 5. Does the argument prove too much?

**Grade: FATAL to interpretation-neutral objective discard; SERIOUS to A2/A3.**

Yes, if stated incautiously. "The simulator may discard unreachable records" is an algorithmic approximation criterion. "The universe discards them" is an interpretation of quantum mechanics or a modification of the dynamics.

Under unitary Everett/many-worlds, the global state remains pure and no branch is literally deleted. Decoherence suppresses interference in reduced states and makes branches dynamically autonomous for practical purposes, but the global coherence remains in the universal wavefunction. A perfect simulator of the universal state cannot discard purifying records merely because a local observer cannot access them. It can discard only relative to a chosen observable algebra, precision tolerance, and no-recoherence assumption.

Under objective-collapse theories, real physical collapse can license objective discard after collapse. Under Copenhagen/instrumentalist practice, one can place a classical cut and update states operationally. Under Bohmian mechanics, empty wave packets may become dynamically irrelevant in configuration space, but the universal wavefunction is still part of the theory in standard Bohmian mechanics. None of these is the interpretation-neutral conclusion Argus wants.

The Stanford Encyclopedia's decoherence entry states the mainstream caveat plainly: decoherence does not by itself solve the measurement problem unless combined with an interpretation such as Bohm, Everett, GRW, or Copenhagen-style moves. It also notes that environmental decoherence explains suppression of interference in reduced states, while the phase relation is well-defined only at the level of the larger system. Source: Guido Bacciagaluppi, "The Role of Decoherence in Quantum Mechanics," Stanford Encyclopedia of Philosophy.

So A2/A3 are interpretation-dependent if they are ontic. They are interpretation-neutral only as pragmatic claims about reduced descriptions: for a specified observer, observable set, and tolerance, inaccessible records may be traced out.

## 6. Prior art and novelty

**Grade: MINOR to the argument; SERIOUS to novelty.**

The physics is not new. Decoherence as environmental monitoring / dislocalization of phase information is the standard Zeh-Zurek-Joos-Schlosshauer line. Core references include H. D. Zeh, "On the interpretation of measurement in quantum theory," Found. Phys. 1, 69-76 (1970); E. Joos and H. D. Zeh, "The emergence of classical properties through interaction with the environment," Z. Phys. B 59, 223-243 (1985); W. H. Zurek, "Decoherence, einselection, and the quantum origins of the classical," Rev. Mod. Phys. 75, 715-775 (2003), DOI `10.1103/RevModPhys.75.715`; M. Schlosshauer, "Quantum decoherence," Phys. Rep. 831, 1-57 (2019), DOI `10.1016/j.physrep.2019.10.001`.

Quantum erasure with limited access is also not new. Miatto et al. directly frame the question "how good is quantum erasure when only an accessible subspace of the environment can be measured?" and compute optimal or statistical recoverable coherence. Buscemi, Chiribella, and D'Ariano also study inverting decoherence by extracting classical information from the environment: F. Buscemi, G. Chiribella, and G. M. D'Ariano, "Inverting quantum decoherence by classical feedback from the environment," Phys. Rev. Lett. 95, 090501 (2005), arXiv:quant-ph/0504195.

The computational angle is partially old as open quantum systems: tracing out environments and using reduced density matrices, Lindblad equations, quantum trajectories, MPOs, and noise-limited tensor networks is standard simulation practice. The exact phrase "decoherence is garbage collection" did not appear in my web searches (`"garbage collection" "decoherence" quantum`, `"garbage collector" "quantum" "decoherence"`, and related variants). I found one unrelated blog search hit saying there is no quantum garbage collector aside from uncomputation; I would not cite it as physics prior art. The metaphor may be Argus's, but the mechanism it points at is standard reduced-state coarse-graining.

## Point-by-point verdicts

**A1. "For a macroscopic environment, erasure is not a matter of degree. Either you hold essentially the whole environment or the coherence is gone below any experimentally reachable level."**

Broken as general physics. Correct for Argus's independent product-recorder model and likely for Darwinist branching/scattering environments with many independently distinguishable fragments. False for collective encodings, engineered memories, correlated/global environment states, and random states. The condition must be: the inaccessible fragment conditional states have low fidelity, and that low fidelity is produced redundantly/local-factor-wise.

**A2. "May this record be discarded? is a present-tense reachability question, not a prediction about future measurements."**

Partly survives. The relevant fact is present-tense if defined globally: who currently holds coherent purifying degrees of freedom, and what the current conditional state fidelities are. But it is not local, and it is not just a count of accessible qubits. It may require knowing about distant idlers, labs, memories, and future allowed communication. The condition must be: present-tense reachability is evaluated relative to a specified global partition, control model, observable tolerance, and causal policy.

**A3. "The analogy is garbage collection, not lazy evaluation."**

Broken as stated. Classical GC reachability is cheap graph reachability over explicit roots. Quantum erasure reachability is a global control/fidelity problem and can require data structures as hard as the purification in the unrestricted case. It survives only as an engineering metaphor for open-system simulation where the simulator has already declared a bath/loss channel outside the future observable algebra. The condition must be: garbage collection is approximate, observer-relative, and model-bound; it is not an exact interpretation-neutral deletion rule for a unitary universe.

## WHAT SURVIVES

A1 survives only for branching/product-like environments in which many independently lost fragments retain which-path distinguishability; it does not survive for arbitrary correlated or collective environments.

A2 survives only as a global present-tense statement about current coherent control and inaccessible-fragment fidelity, not as a local subsystem decision.

A3 survives only as a pragmatic open-system simulation analogy with explicit coarse-graining, tolerance, ownership metadata, and no-future-recoherence assumptions. It does not survive as exact quantum garbage collection under unitary many-worlds.