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The answer is: no, it is not true that any $n$ exchangeable state is a linear combination of density matrices of states in the symmetric subspace (that is supported on the symmetric subspace). Actually, there are even pure state counterexamples when $n=2$. Consider the state $$\rho = |\phi\rangle\langle \phi|,$$ where $$|\phi\rangle = \frac{1}{\sqrt{2}}(|... 2 This is due to how the \mathbf{A} matrix was defined; from that same tutorial page we have:$$\tag{1} \mathbf{A} = \sum_{n} c_n A_n  where each $A_n$ is unitary and $c_n$ is complex (and in the original VQLS paper they further impose $\lVert {\mathbf{A}}\rVert<1$ and bounded condition number) but $\mathbf{A}$ is never required to be unitary. Therefore,...

4

The eigenvalues of a matrix are independent of the choice of basis in which we represent it. This remains true for choices of bases that are not orthogonal. Consider then a matrix $A=\sum_k P_k$, where $P_k\equiv\lvert u_k\rangle\!\langle u_k|$, and $\{|u_k\rangle\}$ is a set of normalized (not necessarily orthogonal) vectors. Observe that $A=UU^\dagger$, ...

1

In a similar way to how the global phase difference of a state makes no physical difference, neither does amplitude of a state. We normalise states to have unit magnitude for mathematical convenience in the same way we don't carry around an $e^{i\phi}$ factor for arbitrary $\phi$ with all our states. This is because having unit vectors means we don't need to ...

5

The answer is no. Define X=[[0,1,0],[0,0,1],[1,0,0]] Z=[[1,0,0],[0,w,0],[0,0,w^2]], w^3=1 Then the Pauli group is generated by X and Z and is of order 27. With H being your matrix, you can check that H'XH and H'ZH are not in the group. Calculations like this are easy to do in gap The dim=3 counterpart of the Hadamard gate is the 3 dimensional Fourier ...

3

Personally, I'd jump straight to Mathematica. It took me all of a minute: zero = ({ {1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0} }); one = ({ {0, 0, 0, 0, 0, 0,...

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