Phase Differential Theory
Phase Differential
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R05

Exclusion dynamics and realisation

Specified exclusion instruments, their implementations and the distinction between exact exclusion and robustness under noise.

Current foundationsPaper overview and reading guideManuscript PDF available

This page is a reading guide written for the website. It is not the manuscript and does not reproduce its proofs. The manuscript PDF is linked below; authors, version and release date will be shown once verified. “Current foundations” does not by itself mean peer reviewed or experimentally confirmed.

Purpose

The question addressed

R05 specifies exclusion instruments that realise a quantum measurement as a last-survivor process, and separates exact exclusion from robustness under noise.

Mathematical starting point

Objects and premises used

  • The quantum instruments of R04 and the last-survivor process of R03.
  • Nonzero mutually orthogonal projectors PjP_j summing to the identity.

Main results

What the paper establishes

  • The complete last-survivor instrument (Theorem R05.1).
  • The obstruction to unrestricted robustness of exact zero exclusion (Theorem R05.4).

Theorem R05.1: The complete last-survivor instrument

Result statement (summary)
For nonzero mutually orthogonal projectors PjP_j with ∑jPj=I\sum_j P_j = I and an active set SS of m≥2m \ge 2 labels, the operators KS,r=(PS−Pr)/m−1K_{S,r} = (P_S - P_r)/\sqrt{m-1} satisfy ∑r∈SKS,r†KS,r=PS\sum_{r \in S} K_{S,r}^{\dagger}K_{S,r} = P_S. Every positive-probability history reaches one survivor after m−1m-1 exclusions, and survivor jj occurs with probability Tr⁡(ρPj)\operatorname{Tr}(\rho P_j).
Assumptions and scope: The projective instrument is supplied. Agreement with its outcome probabilities does not independently select nature’s instrument.
Link to this statementWhy the operators sum to the projector

The full proof is given in the R05 manuscript (PDF).

Theorem R05.4: Obstruction to unrestricted robustness of exact zero exclusion

Result statement (summary)
Arbitrarily small unrestricted full-rank perturbations can remove exact zero effects. Exact exclusion and operationally reliable records therefore require different claims and tests.
Assumptions and scope: Unrestricted full-rank perturbations of the instrument.
Link to this statement

The full proof is given in the R05 manuscript (PDF).

Explanation

Worked example: three labels

With populations pA=12p_A = \tfrac12, pB=13p_B = \tfrac13 and pC=16p_C = \tfrac16, the probability of excluding rr first is (1−pr)/2(1 - p_r)/2. The six exclusion histories have probabilities summing to 1.
Survivor totals are A 12\tfrac12, B 13\tfrac13 and C 16\tfrac16, reproducing the initial populations.

More on the mathematics page: The exclusion construction →

Physical interpretation and scope

What connects this to physics

The construction represents a supplied instrument; it does not independently select nature’s instrument. A small positive population rounded to zero is not a physical realisation, and reliable records need their own claims and tests.

Read, download and cite

Using this paper

View the R05 manuscript (PDF)Download PDF

R05. “Exclusion dynamics and realisation.” Phase Differential Theory current foundations. https://www.phasedifferentialtheory.com/foundations/r05

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Reading order is expository and is not a claim that each premise is derived in the preceding paper.