Due to the probabilistic nature of measurement in quantum mechanics, one invevitably needs to introduce the concept of an ensemble in order to make a well-defined statement about the outcome of a measurement, namely the expectation is introduced which is defined as the average result of measurement of an ensemble of identically prepared systems.(adsbygoogle = window.adsbygoogle || []).push({});

Density matices find utility when the ensemble under consideration consists of more than one quantum-mechanical state. In this case one must distinguish between the coherent superposition within each state and the incoherent superposition of different states which make up the ensemble. Thus one introduces the density operator [itex]\hat{\rho} = \sum_i w_i| \psi_i\rangle \langle \psi |[/itex]. The ensemble expectation is then obtained independently of basis by taking the trace of [itex]\hat{A}\hat{\rho}[/itex].

I am wondering how the density matrix formalism manages to avoid any mention of joint hilbert spaces considering that it deals with more than one quantum-mechanical system. Is it necessary to assume that the states which make up the system are non-interacting? If so, then the state of the systemwould be described by some infinite tensor product state [itex]\otimes_{i=1}^\infty |\alpha_i\rangle \in \mathcal{H}^{\otimes\infty}[/itex]. Why is it that with finite numbers of particles it is important to deal with the combined Hilbert space but with infinite ensembles we manage to avoid this issue completely?

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# Question about density matrices

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