Partial Trace: First Encounter

The partial trace removes one subsystem’s indices from a joint density operator, leaving the state that predicts all measurements on the remaining subsystem. We work through the basic rule here; partial trace gives the basis-independence proof, block-matrix methods, and algorithms.

Helpful background. Reduced states explains why this operation is useful; bra–ket notation makes the calculation short.

Tracing One Factor of a Dyad

Tracing out $B$ means taking the ordinary trace only on its factor. Thus

$$\operatorname{Tr}_B\!\left( |a\rangle\langle a'|\otimes|b\rangle\langle b'| \right) =\langle b'|b\rangle\,|a\rangle\langle a'|.$$

The order in the overlap matters for complex vectors. Apply this rule linearly to all terms in $\rho_{AB}$. If the input is a density operator, the result is positive and has unit trace; no extra normalization factor is needed.

Bell-State Calculation

Expand the projector onto $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$. Its first diagonal term reduces as

$$\operatorname{Tr}_B(|00\rangle\langle00|) =|0\rangle\langle0|\langle0|0\rangle =|0\rangle\langle0|.$$

One cross term gives

$$\operatorname{Tr}_B(|00\rangle\langle11|) =|0\rangle\langle1|\langle1|0\rangle=0.$$

The other two terms give $|1\rangle\langle1|$ and zero. Retaining the common factor $1/2$,

$$\rho_A=\tfrac12(|0\rangle\langle0|+|1\rangle\langle1|)=I/2.$$

The cross terms vanish because the two $B$ states are orthogonal. Partial trace does not delete every off-diagonal term indiscriminately.

Quick Check

For the product state $|+\rangle_A|0\rangle_B$, what is $\rho_A$? Does its $|0\rangle\langle1|$ coherence survive?

Solution

The joint density operator is $|+\rangle\langle+|\otimes|0\rangle\langle0|$. Its partial trace is $|+\rangle\langle+|$, because $\operatorname{Tr}|0\rangle\langle0|=1$. The $A$ coherence survives: both terms carry the same $B$ ket, whose overlap is one, unlike the Bell example.

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.