Knowlpedia

Partial trace

Linear map that traces out one tensor factor to produce a reduced operator.

Let HAH_A and HBH_B be finite-dimensional complex Hilbert spaces, and let XX be an operator on the tensor product HA⊗HBH_A\otimes H_B. The partial trace over BB is the unique linear map

Tr⁡B: L(HA⊗HB)→L(HA)\operatorname{Tr}_B:\ \mathcal{L}(H_A\otimes H_B)\to \mathcal{L}(H_A)

characterized by the identity

Tr⁡ ⁣((MA⊗IB) X)=Tr⁡ ⁣(MA Tr⁡B(X))for all operators MA∈L(HA),\operatorname{Tr}\!\big((M_A\otimes I_B)\,X\big)=\operatorname{Tr}\!\big(M_A\,\operatorname{Tr}_B(X)\big) \quad\text{for all operators } M_A\in \mathcal{L}(H_A),

where Tr⁡\operatorname{Tr} is the usual trace (see ).

Equivalent characterizations

Basis formula

If { ∣j⟩ }\{\,|j\rangle\,\} is any orthonormal basis of HBH_B, then

Tr⁡B(X)=∑j(IA⊗⟨j∣) X (IA⊗∣j⟩).\operatorname{Tr}_B(X)=\sum_j (I_A\otimes \langle j|)\,X\,(I_A\otimes |j\rangle).

This expression is independent of the chosen orthonormal basis.

Basic properties

For operators AA on HAH_A and BB on HBH_B,

Tr⁡B(A⊗B)=Tr⁡(B) A.\operatorname{Tr}_B(A\otimes B)=\operatorname{Tr}(B)\,A.

More generally:

  • Tr⁡B\operatorname{Tr}_B is linear.
  • Tr⁡(Tr⁡B(X))=Tr⁡(X)\operatorname{Tr}(\operatorname{Tr}_B(X))=\operatorname{Tr}(X) (trace-preserving).
  • Tr⁡B\operatorname{Tr}_B is positive and completely positive.

Reduced states

If ρAB\rho_{AB} is a on HA⊗HBH_A\otimes H_B, then

ρA:=Tr⁡B(ρAB)\rho_A := \operatorname{Tr}_B(\rho_{AB})

is a density operator on HAH_A, called the reduced state (or marginal) of subsystem AA. Even when ρAB\rho_{AB} is a , the reduced state ρA\rho_A can be .

Example: maximally entangled state

If dim⁡HA=dim⁡HB=d\dim H_A=\dim H_B=d and

∣Φ⟩=1d∑i=1d∣i⟩⊗∣i⟩,ρAB=∣Φ⟩⟨Φ∣,|\Phi\rangle=\frac{1}{\sqrt d}\sum_{i=1}^d |i\rangle\otimes |i\rangle, \quad \rho_{AB}=|\Phi\rangle\langle\Phi|,

then Tr⁡B(ρAB)=IA/d\operatorname{Tr}_B(\rho_{AB})=I_A/d is maximally mixed.