# Creating orthogonal quantum states from a set of given (possibly linearly independent) quantum states

I want to understand how to orthogonalize a system of qubits.

Suppose I have $$n$$ sets of quantum states like

$$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $$i=1, \dots, n$$ is number of states in the sets with $$k$$ tensor products.

Now I am interested in orthogonalizing these set of $$n$$ states. Especially I want to use the Gram-Schmidt orthogonalization procedure here.
But I don't understand how to think from here.

Cross-posted on math.SE

• Crossposted on Mathematics. Jun 19 at 17:19
• are you simply asking how the Gram-Schmidt procedure works for a set of vectors, or is there something else? I ask because you tagged this with and algorithm, but I don't really see the connection with quantum algorithms here
– glS
Jun 19 at 18:35
• Also crossposted on physics. Jun 20 at 6:12
• @gIS: I think the IamKnull is asking how to carry out orthogonalization procedure on a quantum computer. Jun 20 at 6:19

Since you have n sets of quantum states, and every set is just a tensor product of the constituent states, each of them can be represented as a vector. For example, take $${|1_{1} \rangle, |2_{1} \rangle, ..., |k_{1} \rangle}$$. If you actually evaluate the tensor product $${|1_{1} \rangle \otimes |2_{1} \rangle \otimes ... \otimes |k_{1} \rangle}$$ you get a vector which has $$m^{k}$$ dimensions ($$m$$ is the dimension of the vector $$|j_{1} \rangle ,\ j\ \in\ [1,k]$$).
If we take an example, let us say $$k = 4$$, and each individual state is a 2-dimensional qubit state. Then, we will have $$|0_{i} \rangle \otimes |1_{i} \rangle \otimes |3_{i} \rangle \otimes |4_{i} \rangle$$ as the state. This particular vector would have $$2^{4}$$ dimensions and would form one of the vectors which represent one of the state set.
Let us denote the final calculated state of the $$i^{th}$$ set as $$|v_{i} \rangle$$. If you may prove that these vectors, $$|v_{i} \rangle$$ are $$n$$ linearly independent vectors, you shall be able to apply the Gram Schmidt orthogonalization procedure to make the orthonormal basis.