Reduced Density Matrices

A reduced density matrix is obtained by summing the unobserved subsystem’s matching bra and ket indices. It predicts measurements on the retained subsystem without selecting a remote outcome. Reduced density operators explains the full structure; here we focus on reading a matrix in a declared product basis.

Helpful background. Partial trace gives the dyad rule; density operators explains matrix entries and positivity.

Contracting the Unobserved Index

With rows and columns ordered as $|i\rangle_A|j\rangle_B$,

$$(\rho_A)_{ii'}=\sum_j(\rho_{AB})_{ij,i'j}.$$

For two qubits in the order $00,01,10,11$, this reads

$$\rho_A=\begin{pmatrix} \rho_{00,00}+\rho_{01,01}&\rho_{00,10}+\rho_{01,11}\\ \rho_{10,00}+\rho_{11,01}&\rho_{10,10}+\rho_{11,11} \end{pmatrix}.$$

The $B$ index is the same on the bra and ket sides of every retained term. Do not sum all entries of a block or average by dividing by the subsystem dimension.

For example, $\rho_{AB}=\operatorname{diag}(0.1,0.2,0.3,0.4)$ gives $\rho_A=\operatorname{diag}(0.3,0.7)$, whereas tracing out $A$ gives $\rho_B=\operatorname{diag}(0.4,0.6)$.

Checking the Local Prediction

For every bounded local observable $M_A$,

$$\operatorname{Tr}(\rho_A M_A) =\operatorname{Tr}[\rho_{AB}(M_A\otimes I_B)].$$

In the example, $\langle Z_A\rangle=0.3-0.7=-0.4$ in either calculation. The result remains positive with unit trace. It is the marginal state, not a state conditioned on a particular $B$ outcome.

Quick Check

For that diagonal example, condition on a computational-basis result $B=0$. Find its probability and the conditional state of $A$, and compare it with $\rho_A$.

Solution

The retained weights are $0.1$ for $A=0$ and $0.3$ for $A=1$, so $p(B=0)=0.4$. Normalizing them gives $\rho_{A\mid B=0}=\operatorname{diag}(1/4,3/4)$, different from the marginal $\operatorname{diag}(0.3,0.7)$. Averaging all conditional states with their outcome probabilities recovers the marginal.

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010, sec. 2.4.3.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018, ch. 2. Author’s manuscript.