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Let me tell you what I understand, at least what I think I understand, and then ask my question.

  1. I understand that superposition describes that combinations of solutions are also a solution of it. This can help understanding the collapsing of a wave function, and determining the states of multiple waves with only one "read" of the wave state.
  2. This means, I also understand entanglement, and what happens when particle systems become detangled.
  3. I understand a qubit has two possible configurations. The probability of up and down Pup + Pdown = 1. Another example would be below, square both and the probability is 1

$$ |\Psi | = \frac{3}{5}i|\uparrow + \frac{4}{5}|\downarrow $$

It falls apart for me after this. Taken from wikipedia,

"The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition $\psi$ turns into a superposition of A′ and B′ with the same coefficients as A and B. For example, if we have the following

$$|\uparrow \rangle \rightarrow |\downarrow \rangle $$ if state up is true, then down is true? What? $$|\downarrow \rangle \rightarrow \frac{3i}{5} | \uparrow\rangle + \frac{4}{5}|\downarrow\rangle$$

The same with the equation above, I must be missing a core concept between step three and how we start arriving at multiple subequent states. I'm seeing the equation above as if the state is down then it is both 3i/5 up and 4/5 down?

$$c_{1}|\uparrow \rangle + c_{2}|\downarrow \rangle \rightarrow c_{1} \left ( |\downarrow\rangle \right ) + c_{2}\left ( \frac{3i}{5} | \uparrow\rangle + \frac{4}{5}|\downarrow\rangle \right )$$

Helping me understand the disconnect should help me reach the final understanding for the equation above.

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What Wikipedia means by $|\uparrow\rangle \rightarrow |\downarrow \rangle$ means is that the state evolves from $|0 \rangle$ to $|1\rangle$. This means that, because quantum dynamics is unitary, there is some unitary transformation $U$ such that $U |0\rangle = |1\rangle$.

Unfortunately, Wikipedia's example is wrong. Unitary transformations have to preserve inner products. and the inner product of $|\uparrow\rangle$ and $|\downarrow \rangle$ is 0, but the inner product of $|\downarrow \rangle$ and $\frac{3i}{5} | \uparrow \rangle + \frac{4}{5}| \downarrow \rangle$ is $\frac{4}{5}$.

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    $\begingroup$ Yes, the rightward arrow should be considered a violation of intergalactic law in physics, as its use is almost always vague. Indeed, if the arrow indicates "evolves to" then we ought to define a propagator $U$ and write $U \lvert \text{initial state} \rangle = \lvert \text{final state} \rangle$ and do away with the damned arrow. $\endgroup$ – DanielSank Aug 7 '16 at 1:50
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    $\begingroup$ There is no unitary operator can achieve the operation in your example. You can check it. To understand the classical QM system, just regard a pure state as a vector V in a vector space and an unitary operation as an unitary matrix U, the evolution is just the left multiplication of the matrix U with the vector V. $\endgroup$ – XXDD Aug 7 '16 at 4:34
  • $\begingroup$ @X.Dong Okay. Then I am confused on the last equation still. if c1 and c2 undergo evolution of states, why is c1 $|\downarrow\rangle $ but c2 contains the states from the previous cycle? I guess my disconnect here is the understanding between A and A, B and B $\endgroup$ – Chris Clark Aug 7 '16 at 15:05
  • $\begingroup$ Let's take a simple example, the Hadamard gate H that works as $A=0\rightarrow A'=0+1$ and $B=1\rightarrow B'=0-1$. Then H is given by an unitary matrix $H=[1 ,1; 1,-1]/\sqrt{2}$, $A=[1, 0], B=[0, 1]$. Then it's easy to find out the output of H working on any composition of A and B following the linear algebra idea. $\endgroup$ – XXDD Aug 7 '16 at 16:39

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