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--- topic:qm:density-matrix [2022/12/14 22:08] – samuel
+++ topic:qm:density-matrix [2023/02/15 18:56] (current) – samuel
@@ Line 1: / Line 1: @@
-====== Density Matrix ======
+====== 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 matrix $\rho$.
+While a system state in quantum mechanics can be described by its state vector $\ket{\psi}$,
-This density matrix describing the state of the system is defined as $$ \rho = \sum_i p_i{\ket{psi_i}\bra{psi_i}} $$.
+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 ($\tr{\rho} = 1$).
+===== Postulates in Density operator form =====
+//This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang:2010}.//
+**1.** Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system.
+The system is completely described by its density operator,
+which is a positive operator $\rho$ with trace one,
+acting on the state space of the system.
+If a quantum system is in the state $\rho_i$ with probability $p_i$, then the density operator for
+the system is $\sum_i{p_i \rho_i}$.
+**2.** The evolution of a closed quantum system is described by a unitary transformation.
+That is, the state $\rho$ of the system at time $t_1$ is related to the state
+$\rho'$ of the system at time $t_2$ by a unitary operator $U$ which depends only on the
+times $t_1$ and $t_2$, $$\rho' = U \rho U^\dagger$$.
+**3.**
+**4.**
+===== Properties =====
+  * $\tr(\rho^2) <= 1$
+  * $\rho$ is a pure state if and only if $\tr(\rho^2) == 1$.
+===== Linking state vector and density matrix =====
+For multiple pure states it is defined as $$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$.

Kollektanee - Samuel Gyger

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