Must the State |O,ready> Be Orthogonal to |O,reads up> and |O,reads down>?

[deneve]

In the measurement ansatz below, must the state |O,ready> be orthogonal to |O,reads up> and |O,reads down>]?

|+x> |O,ready> =1/(sqrt2)[|+z> + |-z>] |O,ready> (1)

---------→ 1/(sqrt2)[|+z> + |-z>] |O,ready> (2)


--------→ 1/(sqrt2)[|+z>|O,reads up> + |-z>|O,reads down>] (3)


If |O,ready> = [a|O,reads up> + b|O,reads down>] was applied to (2) then it would give an entangled state and if a,b were functions of time then a suitable a and b can't be found that a hamiltonian could reach since

1/(sqrt2)[|+z> + |-z>] |O,ready>
= 1/(sqrt2)[|+z> + |-z>] [a|O,reads up> + b|O,reads down>]

=1/(sqrt2)(a|+z>|O,reads up>+b|+z>|O,reads down>+a|-z>|O,reads up>+b|-z|O,reads down>) which cannot equal (3) for any a,b.

Am I correct here?
Many thanks for any help

 

[azntoon]

.Yes, you are correct. The state |O,ready> must be orthogonal to both |O,reads up> and |O,reads down> in order for the measurement ansatz to be valid.