Partial Trace

The Partial Trace ${\mathrm{Tr}}_E$ allows us to map a Density Operator ${\rho }_{SE}$ into the Density Operator ${\rho }_S$ of a Reduced system. ${\rho }_S$ is also known as the Reduced trace It has the key defining properties:

$$\{\begin{array}{l}{\mathrm{Tr}}_E:H_S⨂H_E\to H_S\\ ⟨A⟩={\mathrm{Tr}}_S[(S⨂{\mathbb{I}}_E){\rho }_{SE}]=\mathrm{Tr}[A{\rho }_S]\end{array}$$

In the above:

• $H_S$ and $H_E$ are Quantum Hilbert Spaces
• $⨂$ is their Tensor product.
• $A$ is a Quantum observable in the Quantum Hilbert Space $H_S$
• ${\mathbb{I}}_E$ is the Identity operator on $H_E$.
• $\mathrm{Tr}$ is the Trace.
• ${\rho }_{SE}$ is our Density Operator on the full $H_S⨂H_E$ system.
• ${\rho }_S$ is the Density Operator of the Reduced system, obtained via ${\rho }_S={\mathrm{Tr}}_E[{\rho }_{SE}]$

The definition of the partial trace follows from the condition ${\mathrm{Tr}}_S[(A⨂{\mathbb{I}}_E){\rho }_{SE}]=\mathrm{Tr}[A{\rho }_S]1_E\forall A\in H_S$. It is unclear how this property fully defines the trace. Help me ❓

Properties

• $\mathrm{Tr}[A⨂B]=\mathrm{Tr}[A]\mathrm{Tr}[B]$
• ${\mathrm{Tr}}_B[A⨂B]=A\mathrm{Tr}[B]$
• ${\mathrm{Tr}}_A[A⨂B]=\mathrm{Tr}[A]B$