I have a system that generates a random 7 qubit state and I need a method to always find the orthogonal state.

I'm currently using python and qutip for this, representing this 7 qubit state by a 128-dimensional vector.


Basically, you divide the entire $7$-qubit Hilbert space into two subspaces: the one spanned by your state (let's call the state $|\psi\rangle$), let's call that subspace $W$, and it's orthogonal complement $V = W^{\perp}$. You want any vector from $V$, because this will by definition be orthogonal.

We know that $P_{V}$ + $P_{W} = I$, with $P_{V}$ and $P_{W}$ being the projectors upon the $V$ and $W$ subspaces. Since $P_{W} = |\psi\rangle\langle\psi|$, we can easily calculate $P_{V}$:

$$ P_{V} = I - |\psi\rangle\langle\psi|. $$ We can let this projection matrix act on virtually any state to obtain a state orthogonal on $|\psi\rangle$, lets use the $0$ vector $|00....0\rangle$. An orthogonal state $|\psi^{\perp}\rangle$ to $|\psi\rangle$ is thus:

$$ |\psi^{\perp}\rangle = P_{V}|00...0\rangle = (I - |\psi\rangle\langle\psi|)|00...0\rangle = |00...0\rangle - \langle\psi|00...0\rangle|\psi\rangle $$

which is more or less the Gram-Schmidt process.

Note that you can use (almost) any state instead of $|00...0\rangle$; the only state that you cannot use is $|\psi\rangle$ itself.

In python, this becomes something like:

from numpy import zeros_like, inner

zeros_vect = zeros_like(psi_orig)

psi_orth = zeros_vect - inner(zeros_vect,psi_orig).conj()*psi_orig


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