Nt1310 Unit 4 Lab Report

1146 Words3 Pages

For the most part, the probability matrix for $P^2$ is the same as the probability matrix for $B^2$; however, there is one important distinction to be made. Which is that while $B^2[5,0] = \frac{1}{3}$ in the quantum simulation $P^2[5,0] = 0$. On a mathematical basis, this is trivially written as
$$\frac{1}{\sqrt{2}}\bigg(\frac{-1+i}{\sqrt{6}}\bigg) + \frac{1}{\sqrt{2}}\bigg(\frac{1-i}{\sqrt{6}}\bigg) = \frac{-1+i}{\sqrt{12}} + \frac{1-i}{\sqrt{12}} = 0$$
This may seem troubling at first, but it must be remembered that photons are subject to particle interference and thus, on the shared target of the windows, the probability drops to 0. Furthermore, one may try to argue that if the experiment was carried out using only one photon, then the …show more content…

The quantum system could also represent the energy level of the photon polarization direction; however, for the purpose of defining the bracket notation consider just the position of the particle.\footnote{This section focuses on the ``ket'' of the bracket notation. The matching ``bra'' $\bra{x}$ denotes the conjugate transpose of $\ket{x}$. This choice is arbitrary, but is the convention which is widely used when talking about quantum computing \cite{non}}\par
The state $$\ket{\psi} = [0,1,0,0,...,0]^T$$ is to represent that the particle is found at position 1. Similarly the state $$\ket{\psi'} = [0,...,1,...,0]^T$$ is said to be found at the position $j$ where the particle can be found. These states, where the particle can be certainly found are known as pure states. A superposition of the general form $$\ket{\phi} = [c_0, c_1,...,c_{j},...,c_{n-1}]^T$$ can be added to another superposition state simply by adding elements individually. Adding the initial to $$\ket{\phi'} = [c_0', c_1',...,c_{j}',...,c'_{n-1}]^T$$ yields
$$\ket{\phi} + \ket{\phi'} = [c_0+c_0', c_1+c_1', ..., c_{j}+c'_{j},..., c_{n-1} + c'_{n-1}]^T$$ This process of adding the complex vector spaces are valid and yield accurate …show more content…

Furthermore, distinct eigenvectors which have distinct eigenvalues of a hermitian matrix are orthogonal. It follows that the set of eigenvectors form a basis for the entire complex vector space which represents the quantum of interest. \cite{intro}. Taking $A\ket{\phi} = m\ket{\phi}$, it becomes obvious that, as stated before, the only part of the state that matters is the direction rather than the length. This means that $m\ket{\phi} = \ket{\phi}$ A critical assumption which can be made following this statement is that if the current state of the quantum system is based on the eigenvector basis, than the system will not

More about Nt1310 Unit 4 Lab Report

Open Document