QuantumPhysics.dvi

16.7.1 Density matrix for a subsystem

LetHab=Ha⊗Hbbe the Hilbert space organized as the tensor product of two subspacesHa
andHb. Letρabbe the density matrix corresponding to a state of the system. Suppose now that
we observe the system with observables that act only on the subsystema. In all generality, such
observables are of the form,

A=Aa⊗Ib (16.48)

The expectation value ofAof this type of operators in the full system reduces in a systematic
manner to the expectation value with respect to the density matrix of the subsystem. To see this,
we compute the expectation values,

〈A〉 = TrHab

(

ρabA

)

= TrHaTrHb

(

ρab(Aa⊗Ib)

)

= TrHa

(

Aaρa

)

(16.49)

where

ρa= TrHb

(

ρab

)

(16.50)

Note that the normalization TrHa(ρa) = 1 as a result of the normalization ofρab. In particular, the
population fraction of a pure state|ψa〉〈ψa|is given by

pa= TrHa

(

ρa|ψa〉〈ψa|

)

=〈ψa|ρa|ψa〉 (16.51)

16.7.2 Example of relations between density matrices of subsystems

It will be helpful to clarify the relations betweenρab,ρa,ρband the tensor productρa⊗ρbwith the
help of an example involving two spin 1/2 systems. We denote the Pauli matrices for these systems
respectively byσiaandσibwherei= 1, 2 ,3. Consider general density matrices for each subsystem,

ρa =

1

2

(Ia+~a·~σa) |~a|≤ 1

ρb =

1

2

(

Ib+~b·~σb

)

|~b|≤ 1 (16.52)

The tensor product is given by

ρa⊗ρb=

1

4

(

Ia⊗Ib+Ia⊗(~b·~σb) + (~a·~σa)⊗Ib+ (~a·~σa)⊗(~b·~σb)

)

(16.53)

Clearly, when we compute the partial traces, we recover the density matrices of the subsystems,

trHa(ρa⊗ρb) = ρb

trHb(ρa⊗ρb) = ρa (16.54)