Undergrad Density matrix on a diagonal by blocks Hamiltonian.

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A Hamiltonian that is diagonal by blocks, such as (H1 0; 0 H2), leads to a density matrix that is also block diagonal if the corresponding particles are not entangled. The density matrix reflects the system's state, which is influenced by the entanglement between particles. If H1 and H2 represent different particles, the density matrix maintains this block structure only in the case of a product state. Therefore, entanglement alters the density matrix's form, breaking the block diagonal structure. The relationship between the Hamiltonian and the density matrix is crucial for understanding the system's quantum state.
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If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?
 
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The density matrix describes the state of the system, so its exact form will depend on the state of the system.

If H1 and H2 correspond to different particles, then the density matrix will be block diagonal only if the particles are not entangled (corresponding to a product state).
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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