Trace and Partial Trace

Partial trace extends from finite matrices to a norm-one map between trace-class spaces. Its rigorous characterization is preservation of all bounded local-observable expectations. The canonical Partial Trace article supplies the subsystem interpretation and index calculations; this page establishes existence, convergence, and the infinite-dimensional limitations of those formulas.

Helpful background. Trace-class operators supplies ideal estimates; trace-norm duality supplies uniqueness and continuity.

The trace-class partial-trace theorem

Let $\mathcal H_A,\mathcal H_B$ be separable Hilbert spaces. There is a unique bounded linear map

$$\operatorname{Tr}_B: \mathcal S_1(\mathcal H_A\otimes\mathcal H_B) \longrightarrow\mathcal S_1(\mathcal H_A)$$

such that, for all bounded $A$,

$$\operatorname{Tr}\bigl(A\operatorname{Tr}_BT\bigr) =\operatorname{Tr}\bigl((A\otimes I_B)T\bigr).$$

It is positive, trace preserving, completely positive, and contractive in trace norm. Hence it sends density operators to density operators. Complete positivity follows by applying the same positive construction with an additional untouched tensor factor; it is stronger than positivity alone.

The identity $I_B$ is bounded even when $\mathcal H_B$ is infinite-dimensional. It need not be trace class: only the global operator $T$ must lie in the trace ideal in this formula.

Constructing the basis series

Choose an orthonormal basis $(b_j)$ of $\mathcal H_B$ and let $V_j:\mathcal H_A\to\mathcal H_A\otimes\mathcal H_B$ be the isometry $V_jx=x\otimes b_j$. For $T\geq0$ trace class, define

$$R_N=\sum_{j=1}^{N}V_j^*TV_j.$$

Every term is positive trace class. Using a product orthonormal basis and nonnegative sums,

$$\sum_j\operatorname{Tr}(V_j^*TV_j)=\operatorname{Tr}T.$$

Therefore $R_N$ is Cauchy in trace norm: the norm of any positive tail is its trace. Its limit $R$ is positive and has trace $\operatorname{Tr}T$.

For arbitrary trace-class $T$, decompose its real and imaginary parts into differences of positive trace-class operators. The same construction extends linearly. In fact the series $\sum_jV_j^*TV_j$ is absolutely convergent in trace norm, as follows by bounding its terms using those four positive parts.

To verify the characterizing identity, pair the finite sums with bounded $A$. The sum on the global space is $(A\otimes P_N)T$, where $P_N$ projects onto the first $N$ vectors $b_j$. Since $A\otimes P_N$ converges strongly and stays uniformly bounded, its trace pairing with $T$ converges. Thus $R$ satisfies the defining formula.

That formula determines $R$ uniquely by trace duality, so the construction is independent of the basis chosen in $\mathcal H_B$. It is the trace-class convergence, rather than a purely symbolic contraction of indices, that justifies the infinite sum.

Contractivity and local indistinguishability

Trace duality and $\|A\otimes I_B\|=\|A\|$ give

$$\begin{aligned} \|\operatorname{Tr}_BT\|_1 &=\sup_{\|A\|\leq1} |\operatorname{Tr}((A\otimes I_B)T)|\\ &\leq\|T\|_1. \end{aligned}$$

Thus tracing out a subsystem cannot increase the trace distance of two states. Any local measurement is also a global measurement with effect $E_A\otimes I_B$, so this inequality matches operational intuition. It does not say that partial trace preserves all distinctions: orthogonal global states can have identical reduced states.

For a trace-class product $T=A_1\otimes B_1$, the result is $A_1\operatorname{Tr}B_1$. By contrast, if $\mathcal H_B$ is infinite-dimensional, $A_1\otimes I_B$ is generally not trace class and has no finite partial trace. One cannot assign $\operatorname{Tr}_B I_{AB}$ a finite operator value by analogy with finite dimension.

A countable entangled expansion

Let $(a_n)$ and $(b_n)$ be orthonormal families and $p_n\geq0$ with $\sum_np_n=1$. The vector

$$|\Psi\rangle=\sum_n\sqrt{p_n}\,|a_n\rangle\otimes|b_n\rangle$$

converges in Hilbert-space norm. Applying the construction to its rank-one projector gives

$$\operatorname{Tr}_B|\Psi\rangle\langle\Psi| =\sum_np_n|a_n\rangle\langle a_n|$$

in trace norm. This is an infinite-rank reduced state even though the global projector has rank one. Choosing a slowly decreasing summable $p_n$ can produce infinite reduced entropy; partial trace does not ensure entropy finiteness.

For an unbounded local observable $H_A\geq0$, apply the bounded identity to spectral truncations and take a monotone limit. The global and reduced energy expectations agree as extended nonnegative numbers, possibly infinity. For a signed observable, require absolute integrability before claiming a finite equality.

Exercises

1. Let $\rho_{AB}$ and $\sigma_{AB}$ have trace distance at most $\epsilon$. Bound the probability difference for an arbitrary local effect $E_A$.

Solution

The global effect $E_A\otimes I_B$ lies between zero and identity. Its probability difference is at most $\epsilon$. Equivalently, partial-trace contractivity bounds the reduced trace distance by $\epsilon$. This connects the theorem to local measurement statistics.

2. Let $A\geq0$ be a nonzero trace-class operator and let $B$ have infinite dimension. Explain why the basis partial sums for $A\otimes I_B$ diverge.

Solution

Every contraction $V_j^*(A\otimes I_B)V_j$ equals $A$, so the first $N$ terms sum to $NA$. Their trace norms equal $N\operatorname{Tr}A\to\infty$. The trace-class hypothesis of the theorem fails.

References

• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019, Chapters 3 and 11 — tensor products, reduced states, and infinite systems. Publisher.
• Simon, Barry, Trace Ideals and Their Applications, 2nd ed., Mathematical Surveys and Monographs 120, American Mathematical Society, 2005 — trace ideal estimates and convergence used in the construction. Publisher.