Reduced States

A reduced state gives every prediction available from one subsystem alone. It can be mixed even when the joint state is pure, because local measurements do not access all the joint correlations. Reduced density operators develops the full theory and detailed examples.

Helpful background. Density operators explains mixed states; bipartite systems fixes the subsystem labels.

The State Seen by Local Measurements

For a joint density operator $\rho_{AB}$, the reduced state is

$$\rho_A=\operatorname{Tr}_B\rho_{AB}.$$

It is the unique density operator satisfying

$$\operatorname{Tr}_A(\rho_A M_A) =\operatorname{Tr}_{AB}\!\left[\rho_{AB}(M_A\otimes I_B)\right]$$

for every bounded local observable $M_A$, and therefore for every local measurement effect. This is an exact statement about the specified joint state, not an approximation or an average over spatial positions.

Reduction means that the description no longer retains information needed for some measurements on $B$ or on the pair. It need not represent a physical measurement or erasure of $B$.

Same Local State, Different Joint Predictions

For the Bell state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$, both reduced states are $I/2$. The product state $(I_A/2)\otimes(I_B/2)$ has those same marginals. Yet simultaneous computational-basis measurements give

$$\begin{aligned} p_{\Phi^+}(00)&=p_{\Phi^+}(11)=\tfrac12, & p_{\Phi^+}(01)&=p_{\Phi^+}(10)=0,\\ p_{\mathrm{product}}(ij)&=\tfrac14 &&\text{for all }i,j. \end{aligned}$$

The local states do not reconstruct the joint correlations. To calculate the reduced state itself, use the partial-trace rule.

Quick Check

Does $\rho_A=I/2$ prove that $A$ is entangled with another system?

Solution

No. The Bell preparation and the uncorrelated product preparation above give the same $\rho_A$. Mixedness certifies entanglement across $A|B$ only when the joint state is known to be pure. Without that assumption, it can also arise from ordinary random preparation.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, ch. 2.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 2. Author’s manuscript.