Result

Measurement can make individual records look selected while the unconditional physical state retains an entire neutral family:

\[ \boxed{ \text{trajectory collapse} \not\Rightarrow \text{unique unconditional fixed point}. } \]

This is the measurement analogue of the projector-only selection no-go. A record supplies a branch label. It does not, by itself, choose one branch for the ensemble.

Exact weak-measurement witness

For a qubit weakly measured in the (sigma_z) basis, take

\[ M_s= \begin{pmatrix} \sqrt{(1+s\epsilon)/2}&0\\ 0&\sqrt{(1-s\epsilon)/2} \end{pmatrix}, \qquad s=\pm1. \]

Conditioned on outcome (s), the state updates by

\[ \rho_s=\frac{M_s\rho M_s^\dagger}{p_s}, \qquad p_s=\operatorname{tr}(M_s\rho M_s^\dagger). \]

The two outcomes update the population in opposite directions. Repeated records therefore drive an individual trajectory toward one of the two poles.

If the record is not conditioned upon, however,

\[ \mathcal M(\rho)=\sum_sM_s\rho M_s^\dagger = \begin{pmatrix} \rho_{00}&\sqrt{1-\epsilon^2}\,\rho_{01}\\ \sqrt{1-\epsilon^2}\,\rho_{10}&\rho_{11} \end{pmatrix}. \]

Coherence contracts, but both populations are unchanged. Every (sigma_z)-diagonal density matrix is a fixed point. Monitoring creates a pointer basis and a record-indexed branch structure, not a unique unconditional selector.

The verifier evaluates this at (epsilon=0.6). Two conditioned states move toward opposite poles, their probability-weighted average exactly restores the initial populations, and the unconditional coherence multiplier is (0.8).

The second datum is visible

Add a calibrated amplitude-damping channel with strength (gamma). Its energy-basis reservoir maps the excited-state population as

\[ p_1'=(1-\gamma)p_1, \]

and repeated application has the unique ground-state fixed point. The uniqueness is real, but it comes from an additional datum: the reservoir's energy orientation and zero-temperature bias. It is not generated by (sigma_z) measurement alone.

This gives a sharp experimental comparison:

ProtocolConditioned recordsUnconditional populationsFixed set
weak measurement onlybranch toward either poleunchangedall diagonal states
measurement plus outcome feedbackcontroller-dependentcan become biasedsupplied by feedback law
measurement plus amplitude dampingbranch record plus decayexcited population contractsunique reservoir-selected ground state

Measurement and detector equivalence

The detector now has two roles that must not be conflated:

  1. compression: finite resolution maps distinct latent states to the same observed record and loses Fisher information;
  2. backaction: the measurement channel changes the state being inferred.

An experiment can suffer both simultaneously. The correct generative graph is

\[ \text{state} \xrightarrow{\text{instrument outcome}} (\text{conditioned state},\text{record}) \xrightarrow{\text{readout compression}} \text{stored data}. \]

The first arrow is a quantum instrument; the second is an ordinary detector/reconstruction channel. Treating both as one black box makes “selection” ambiguous.

Executable experiment

Use a qubit, qutrit or coupled-mode analogue with tunable measurement strength:

  1. prepare states with equal populations but different coherence;
  2. prepare states with several different populations;
  3. retain every weak-measurement outcome;
  4. reconstruct both conditioned maps and the outcome-averaged channel;
  5. repeat with feedback off and on;
  6. add and remove an independently calibrated damping/selector channel;
  7. account separately for system heat and record-memory reset work.

The signature of measurement-only branching is simultaneous trajectory purification and population neutrality in the ensemble. The signature of genuine added selection is contraction of the unconditional population map toward a unique state when the second datum is active.

Current-research contact

Measurement-induced phase transitions extend this elementary distinction to many-body systems: varying measurement rate can change entanglement structure even when local linear observables do not reveal the transition. A 2026 trapped-ion experiment implemented a postselection-free protocol on tree circuits with universal gates, showing that the record-processing architecture is itself part of experimental reach.

This is a nearby research frontier, not evidence that the corpus selector explains those transitions. Its value is operational: it supplies platforms where channel dynamics, trajectory conditioning, decoding complexity and record formation can be varied separately.

Consequence

The selector programme now has two independent no-go results:

In both cases a second physical datum produces uniqueness, and in both cases that datum must be independently identified rather than hidden inside the update rule.

Primary sources