Manuscript En02-D@4 · En02-D-v4.tex · theorem · e19ec871

Theorem: Feasibility and fixed probabilities

Sector feasibility · lines 662–692 · proof included or linked

En02-D@4#thm:feasibility

An energy-preserving isometry exists if and only if

d_{E} \leq ν \sum b_{ν, E}

for every $E$ . A specified map is isometric precisely when

ν \sum A_{ν, E}^{†} A_{ν, E} = I_{C_{E}} .

Prescribed probabilities $p_{ν ∣ E}$ , independent of every state in $C_{E}$ , are realizable precisely when they are nonnegative, sum to one, and

b_{ν, E} \geq d_{E} whenever p_{ν ∣ E} > 0.

Their branch effects must equal $p_{ν ∣ E} I_{C_{E}}$ .

Proof. The total-energy output subspace must contain $d_{E}$ orthogonal vectors, and this dimension condition is sufficient. Different total energies are orthogonal. Orthogonal radiation outcomes give the isometry equation.

State-independent expectations imply the scalar effect identity by polarization. A positive scalar identity has rank $d_{E}$ . Conversely, choose an isometry into each sufficiently large branch and multiply it by $p_{ν ∣ E}$ .

Displayed formulas in this passage

unlabeled · line 665
$d_{E} \leq ν \sum b_{ν, E}$
unlabeled · line 670
$ν \sum A_{ν, E}^{†} A_{ν, E} = I_{C_{E}} .$
unlabeled · line 676
$b_{ν, E} \geq d_{E} whenever p_{ν ∣ E} > 0.$

Theorem: Feasibility and fixed probabilities

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