https://wiki.gyger.at/ 2026-07-26T07:23:02+00:00 https://wiki.gyger.at/ https://wiki.gyger.at/_media/wiki:dokuwiki.svg text/html 2023-02-15T18:56:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://wiki.gyger.at/topic:qm:density-matrix?rev=1676487395&do=diff Density Operator (matrix) While a system state in quantum mechanics can be described by its state vector $\ket{\psi}$, it can also be described by a so called density operator $\rho$. We define the density operator to be a positive operator $\rho$ with a trace equal to one ($\tr{\rho} = 1$$\rho$$\rho_i$$p_i$$\sum_i{p_i \rho_i}$$\rho$$t_1$$\rho'$$t_2$$U$$t_1$$t_2$$$\rho' = U \rho U^\dagger$$$\tr(\rho^2) <= 1$$\rho$$\tr(\rho^2) == 1$$$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$ text/html 2023-02-15T18:37:40+00:00 Anonymous (anonymous@undisclosed.example.com) https://wiki.gyger.at/topic:qm:measurements?rev=1676486260&do=diff Quantum measurements From the general postulates by Nielsen and Chuang, 3 talks about measurements. Quantum measurements are described by a collection ${M_m}$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$$\ket{\psi}$$m$$$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$$$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$$p(m)$$1$$$\sum_m{M_m^\dag M_m} = I$$$M$$ M = \sum_m m P_m $$P_m$$M$$m$$m$$m$$$p(m) = \Brak… text/html 2022-12-14T22:19:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://wiki.gyger.at/topic:qm:nok?rev=1671056356&do=diff Nuggets of Knowledge (NoK) Werner State A mixture of (1-a):a of the Bell State $\ket{\Phi^+}$ and a fully mixed state is called the Werner state.\cite{Werner:1989} $$ \begin{aligned} \rho & =\frac{1-a}{2}(\ket{00}+\ket{11})(\bra{00}+\bra{11})+\frac{a}{4} I \\ & =\frac{1}{2}\left(\begin{array}{cccc} 1-a / 2 & 0 & 0 & 1-a \\ 0 & a / 2 & 0 & 0 \\ 0 & 0 & a / 2 & 0 \\ 1-a & 0 & 0 & 1-a / 2 \end{array}\right) . \end{aligned} $$ text/html 2023-02-15T18:26:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://wiki.gyger.at/topic:qm:postulates?rev=1676485599&do=diff Postulates of Quantum Mechanics This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang:2010}. Four postulates are defined to outline the space quantum mechanics exists in. 1., describes the state of a closed system. 2. describes the evolution of such system in time. 3. describes the influence of observations or information gathering from such a system and 4. defines how multiple systems can be combined.$\ket{\psi}$$\ket{\psi'}$$\ket{\psi'} = U \ket{\psi}$$$ i\hbar \partial_t…