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topic:qm:density-matrix
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| topic:qm:density-matrix [2023/02/15 18:48] – samuel | topic:qm:density-matrix [2023/02/15 18:56] (current) – samuel | ||
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| ===== Postulates in Density operator form ===== | ===== Postulates in Density operator form ===== | ||
| + | //This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang: | ||
| **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. | **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. | ||
| Line 15: | Line 16: | ||
| the system is $\sum_i{p_i \rho_i}$. | 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' | ||
| + | times $t_1$ and $t_2$, $$\rho' | ||
| + | |||
| + | **3.** | ||
| + | |||
| + | **4.** | ||
| + | |||
| + | ===== Properties ===== | ||
| + | * $\tr(\rho^2) <= 1$ | ||
| + | * $\rho$ is a pure state if and only if $\tr(\rho^2) == 1$. | ||
| ===== Linking state vector and density matrix ===== | ===== Linking state vector and density matrix ===== | ||
| For multiple pure states it is defined as $$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$. | For multiple pure states it is defined as $$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$. | ||
topic/qm/density-matrix.1676486939.txt.gz · Last modified: by samuel
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