Observables on a Composite System

Don’t forget that Operator can sometimes be Observable.

Let F∈L(HA),G∈L(HB)F\in L(H_A), G \in L (H_B) be operators.

Then F⊗G∈L(HA⊗HB)F\otimes G \in L(H_A\otimes H_B) is defined as

(F⊗G)∣v,w⟩≜F∣v⟩⊗G∣w⟩(F\otimes G)\left\lvert v, w \right\rangle \triangleq F\left\lvert v \right\rangle\otimes G\left\lvert w \right\rangle

Upgrading/promoting subsystem operators gets us

F∈L(HA)→F⊗I∈L(HA⊗HB)F\in L(H_A)\rightarrow F\otimes I \in L(H_A\otimes H_B) G∈L(HB)→G⊗I∈L(HA⊗HB)G\in L(H_B)\rightarrow G\otimes I \in L(H_A\otimes H_B)

Where LL is the Set of all linear operators on a Hilbert space. This means that any operator acting on a subsystem HAH_A can be upgraded to an operator that makes it act on a larger dimension HA⊗HBH_A\otimes H_B.

This is because

(F⊗I)(I⊗G)∣v,w⟩=(F⊗I)(∣v⟩⊗G∣w⟩)=F∣v⟩⊗G∣w⟩(F\otimes I)(I \otimes G)\left\lvert v, w \right\rangle =(F\otimes I)(\left\lvert v \right\rangle\otimes G\left\lvert w \right\rangle) = F\left\lvert v \right\rangle\otimes G\left\lvert w \right\rangle

Conversely,

(I⊗G)(F⊗I)(∣v⟩⊗∣w⟩)=F∣v⟩⊗G∣w⟩(I\otimes G)(F\otimes I)(\left\lvert v \right\rangle\otimes \left\lvert w \right\rangle)=F\left\lvert v \right\rangle\otimes G\left\lvert w \right\rangle

Notation wise, let

F(A)≜F⊗IF^{(A)}\triangleq F\otimes I G(B)≜I⊗GG^{(B)}\triangleq I\otimes G

This is form is overloaded, so don’t get confused with notation of states which also can have an superscript (A)(A).

Note

[F(A),G(B)]=[F⊗I,I⊗G]=0\boxed{[F^{(A)}, G^{(B)}]= [F\otimes I, I\otimes G]=0}

For a composite system H=HA⊗HBH = H_A \otimes H_B, an operators acting only on one subsystem extends to the full space by tensoring with the identity on the other.

Subsystem AASubsystem BB
OperatorQ(A)=Q⊗IQ^{(A)} = Q \otimes IR(B)=I⊗RR^{(B)} = I \otimes R
Eigenvaluesqaq_arbr_b
Eigenvectors∣a⟩\left\lvert a \right\rangle∣b⟩\left\lvert b \right\rangle
Index rangea=1,2,…,dAa = 1, 2, \dots, d_Ab=1,2,…,dBb = 1, 2, \dots, d_B

Sum

Q(A)+R(B)=Q⊗I+I⊗RQ^{(A)} + R^{(B)} = Q \otimes I + I \otimes R

  • Eigenvectors: ∣a⟩⊗∣b⟩\left\lvert a \right\rangle \otimes \left\lvert b \right\rangle

  • Eigenvalues: qa+rbq_a + r_b

Product

Q(A)R(B)=Q⊗RQ^{(A)} R^{(B)} = Q \otimes R

  • Eigenvectors: ∣a⟩⊗∣b⟩\left\lvert a \right\rangle \otimes \left\lvert b \right\rangle

  • Eigenvalues: qa rbq_a\, r_b