Here is what you learn in school.

All observables A must be

Hermitian operators with real

eigenvalues ai and orthogonal

eigenfunctions |ai>. Using summation convention on repeated upper and lower indices and

Dirac’s |ket>

A = a^i |ai>

= 0 if i =/= j orthogonality

for a general state

|Psi> = c^i|ai>

c^i are complex numbers

If the general state is normalized

= 1

therefore

cic*i = 1

The Born ENSEMBLE probability for a strong Von-Neuman measurement of real eigenvalue ai is

p(ai) = |ci|^2

In terms of the density matrix rho

p(ai) = Trace{|ai>

The orthogonality

= 0 if i =/= j

is key.

If it is violated, then for example, if we have a single q-bit with eigenvalues “0” and “1"

|Psi> = c0|0> + c1|1>

rho = |Psi><0| 1=""><1| 0=""><1| 1=""><0|

p(0) = Trace{|0><0rho}

= |c0|^2 + |c1|^2|<1|0>|^2 + c0*c1<1|0> + c1*c0<0|1>

assuming = 1, which is true for the over-complete non-orthogonal macro-quantum coherent physically important Glauber states that are eigenstates of non-Hermitian quantum field creation and destruction operators.

The classical world only exists because of “More is different” (P.W. Anderson) emergent Glauber states of both real and virtual bosons in Higgs-Goldstone-Brout-Englert-Anderson ground and vacuum states with spontaneous broken Lie group symmetries of various kinds.

As shown by Antony Valentini this also leads to entanglement signaling.|ai>|>|aj>|rho}

|psi>|aj>|

|>