# Density matrix in quantum computation and quantum statistical mechanics

What is the difference between the density matrix for quantum statistical mechanics and density matrix for quantum information theory?

• Why do you assume there is a difference? – By Symmetry Apr 8 '18 at 10:20
• There isn't one - they are the same. – Nathaniel Apr 8 '18 at 14:17

Conceptually they are same. The only difference between them is the number of particles. In quantum information theory, we deal with single particles (Let's call this particle $m$). In this case, the density matrix ($\rho_m$) encodes probabilities, coherence, and decoherence for that single particle. Quantum statistical mechanics deals with many particles (lets say $N$ particles) so the density matrix ($\rho_t$) is taken to be average of the density matrices of each particle, $\rho_t = \frac{1}{N}\sum_{m=1}^N \rho_m$. However, they are same ($\rho_t = \rho_m$) if all the particles are identical, have same probabilities, coherence and we are only looking at the density matrix of the internal states of the particles. Importantly, you will get the same results if you isolate a particle and perform your measurement, and perform your measurement on all particles at once.