Tensor products of systems
For distinguishable systems with hilbert spaces $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$ the space of the combined system is given by the tensor product $$\mathcal{H} = \mathcal{H}^{(1)} \otimes \mathcal{H}^{(2)}$$
Properties of the composite space are
- Given two orthonormal bases of $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$, ${\ket{\phi_i^{(1)}}}$ and ${\ket{\phi_j^{(2)}}}$ a basis for $\mathcal{H}$ is given by the tensor product ${\ket{\phi_i^{(1)}}\otimes\ket{\phi_j^{(2)}}}$
- A general element of $\mathcal{H}$ can be written as $$\ket{\Phi} = \sum_{ij}\alpha_{ij} \ket{\phi_i^{(1)}}\otimes\ket{\phi_j^{(2)}}$$
- The dimension of $\mathlca{H}$ is the sum of the dimensions of $\mathcal{H}^{(1)}$ and $\mathcal{H}^{(2)}$
- Any operator $A$ acting on $\mathcal{H}$ can be expressed as a linear combination of tensor products of operators acting on the two spaces $$A = \sum_\ælpha A_\alpha^{(1)} \otimes A_\alpha^{(2)}$$
- The operator can act on a composite state by $$A(\ket{\phi_i^{(1)}}\otimes\ket{\phi_j^{(2)}}) = (A^{(1)}\ket{\phi_i^{(1)}})\otimes (A^{(2)}\ket{\phi_j^{(2)}})$$
- Observables $O^{(1)}$ on the space $\mathcal{H}^{(1)}$ take the form $O^{(1)}\otimes I^{(2)}$ and correspondingly for $\mathcal{H}^{(2)}$
- The density matrix of the composite system factorizes $\rho = \rho^{(1)}\otimes\rho^{(2)}$ if the two subsystems are uncorralated
- For uncorralated subsystems expectation values of operator products factorize $$\braket{A_\alpha^{(1)} \otimes A_\alpha^{(2)}} = \braket{A_\alpha^{(1)}}\cdot\braket{A_\alpha^{(2)}},$$ since $\braket{\braket{A_\alpha^{(i)}} = \text{tr}^{(i)}(A_\alpha^{(i)} \rho^{(i)})$ with the trace over $\mathcal{H}^{(i)}$
When considering observables of only one subsystem $O=O^{(1)}\otimes I^{(2)}$ we can use the reduced density matrix
Definition: reduced density matrix
The reduced density matrix $\rho^{(1)}$ of a composite quantum system $\mathcal{H} = \mathcal{H}^{(1)} \otimes \mathcal{H}^{(2)}$ with respect to one subsystem is obtained by tracing over the second system $$\rho^{(1)} = \text{tr}^{(2)}\rho$$ Such a state is also called improper mixture It completely describes the properties of subsystem $\mathcal{H}^{(1)}$ since all observables $O=O^{(1)}\otimes I^{(2)}$ can be calculated by $\braket{O} = \text{tr}^{(1)}(O^{(1)}\rho^{(1)})$ The partial trace over subsystem $(2)$ is the only function which gives the correct statistical properties of the reduced density matrix in terms of the observables of system $(1)$.