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  Conditional Time Evolution increases entropy?

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Question
Does the below calculation conclusively show the idea of conditional time evolution (if state  measured is x I do y else I do z ) increases the Von Neumann entropy? Has this already been established  in the literature?


Calculation
Let's say I have an isolated system containing a measuring apparatus, an experimenter and a Hamiltonian system:

Let the Hamiltonian be:

Hsys=H=(E0+μEΔΔE0μE)

where μ is the dipole moment , E is the electric field and E0 is the ground state energy and Δ is the tunnelling element.

Let the energy eigenstates be represented by:

H|=(E0(μE)2+Δ2)|


H|+=(E0+(μE)2+Δ2)|+

Now, the experimenter play the following game, if he measures the energy and finds the energy in the lower of the 2 energy states he will double the electric field else he will half the electric field. Hence,

H(±)=H+λ±I(μE)

where I is the identity matrix and λ+=1/2 and λ=1

Let the energy eigenstates of H(±) be represented by:

H()|0=(E0(1+λ)2(μE)2+Δ2)|0

H()|1=(E0+(1+λ)2(μE)2+Δ2)|1

H(+)|˜0=(E0(1+λ+)2(μE)2+Δ2)|˜0


H(+)|˜1=(E0+(1+λ+)2(μE)2+Δ2)|˜1

Let use the density matrix formalism for a pure state:

ρ=12(||+|+|+|+|+|++|)

Now lets say I measure the state and find it in |. Then using the sudden approximation:

ρ()=12(|00|(||0|2)+|10|(||0|||1|)+|01|(||0|||1|)+|11|(|1|2))

Similarly:


ρ(+)=12(|˜0˜0|(||˜0|2)+|˜1˜0|(||˜0|||˜1|)+|˜0˜1|(||˜0|||˜1|)+|˜1˜1|(|˜1|2))

 

Hence, the new density matrix is given by:

ρ=ρ(+)(||+++|+|221/2)+ρ()(||++||221/2)=p+ρ(+)+pρ()

 

Now, let write down the explicit Hamiltonian of the isolated system:

Hiso=HIexperimenter + apparatus+IHexperimenter + apparatus+Hint

Since, we suddenly change the electric field H is time dependent and since this energy is supplied by the apparatus and experimenter (since he will expend energy to "calculate" how much to change the energy field by) along with the interaction between the systems. Note: Hiso is time independent (as it is an isolated system)

ρiso=ρ(t)ρR(t)

Where R is the rest of the isolated system excluding the Hamiltonian system. Since the Hiso is time independent so is ρiso. Comparing before (t) and after (t+) the field is turned on:

S(ρiso)=S(ρρR(t))=S(ρρR(t+))

Where S is the Von Neumann entropy of both sides:

S(ρρR(t+))=S(ρρR(t))

Hence,

S(ρ)+S(ρR(t+))=S(ρ)+S(ρR(t))

Substituting ρ+ and using ΔSR=S(ρR(t+))S(ρR(t))

S(p+ρ(+)+pρ())+ΔSR=S(ρ)0

Using an entropy inequality :

p+S(ρ(+))+pS(ρ())+ΔSRp+lnp+plnp0

Hence,

p+S(ρ(+))+pS(ρ())+ΔSRp+lnp++plnp

Since ρ(±) are pure states:

ΔSRp+lnp++plnp

asked Oct 13, 2020 in General Physics by Asaint (90 points) [ revision history ]
edited Oct 13, 2020 by Asaint

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