Extractable quantum work from a twomode Gaussian state in a noisy channel
Author
Alexei Zubarev
Title
Extractable quantum work from a twomode Gaussian state in a noisy channel
Description
Program for estimation of extracted work and Szilard engine efficiency.
Category
Academic Articles & Supplements
Keywords
Quantum Information, Quantum Thermodynamics, Szilard Engine
URL
http://www.notebookarchive.org/2022-01-c07ri99/
DOI
https://notebookarchive.org/2022-01-c07ri99
Date Added
2022-01-26
Date Last Modified
2022-01-26
File Size
3.21 megabytes
Supplements
Rights
CC BY 4.0
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This file contains supplementary data for Marina Cuzminschi, Alexei Zubarev and Aurelian Isar, “Extractable quantum work from a two-mode Gaussian state in a noisy channel,” Scientific Reports, 11(1) 2021 24286. https://doi.org/10.1038/s41598-021-03752-4.
Extractable Quantum Work from a Two-Mode Gaussian State In a Noisy Channel (Resonant Case)
Extractable Quantum Work from a Two-Mode Gaussian State In a Noisy Channel (Resonant Case)
Alexei Zubarev
k=1;m=1;(*WesetBoltzmannconstantandmasstoone.*)λ=0.1;(*λ-dissipationparameter*)
a[r_,n1_,n2_]=n1+n2+Cosh[2r];b[r_,n1_,n2_]=n1+n2+Cosh[2r];c[r_,n1_,n2_]=(n1+n2+1)Sinh[2r];σ0[r_,n1_,n2_]=
;(*Herewedefinedthecovariancematrixforinitialtimeintermsofsqueezingparameterr,averagenumbersofthermalphotonsofthefirstandsecondmodecorrespondingly(and).Theinitialstateisasqueezedthermalstate.*)nth[T_]=Coth-1;(*istheaveragenumberofthermalphotonsoftheresourcestate.Weconsideraresonantcasefirst(Thefrequenciesofthemodesare==1).*)
2
Cosh[r]
2
Sinh[r]
1
2
2
Sinh[r]
2
Cosh[r]
1
2
1
2
a[r,n1,n2] | 0 | c[r,n1,n2] | 0 |
0 | a[r,n1,n2] | 0 | -c[r,n1,n2] |
c[r,n1,n2] | 0 | b[r,n1,n2] | 0 |
0 | -c[r,n1,n2] | 0 | b[r,n1,n2] |
n
1
n
2
1
2
1
2T
n
th
ω
1
ω
2
Nn[T_,Rr_]=nth[T](+)+;(*Rristhesqueezingparameterofthenoisychannel.*)MRe[T_,Rr_,ϕ_]=-(2nth[T]+1)Cosh[Rr]Sinh[Rr]Cos[ϕ];MIm[T_,Rr_,ϕ_]=-(2nth[T]+1)Cosh[Rr]Sinh[Rr]Sin[ϕ];(*NoisychannelisdescribedintermsofitsqueezingparameterRr,temperatureTandthephaseϕ.*)
2
Cosh[Rr]
2
Sinh[Rr]
2
Sinh[Rr]
σ∞[T_,Rr_,ϕ_]=
;(*Herewedefinetheasymptoticcovariancematrixfortimet=∞*)σa∞[T_,Rr_,ϕ_]=
;σb∞[T_,Rr_,ϕ_]=
;σc∞[T_,Rr_,ϕ_]=
;(*Herearegiventhetheasymptoticcovariancematricesformode1((∞)),thesecondmode((∞)),andcorrelationsbetweenthem((∞)).*)
1 2 | MIm[T,Rr,ϕ] | 0 | 0 |
MIm[T,Rr,ϕ] | 1 2 | 0 | 0 |
0 | 0 | 1 2 | MIm[T,Rr,ϕ] |
0 | 0 | MIm[T,Rr,ϕ] | 1 2 |
σ∞[T,Rr,ϕ][[1,1]] | σ∞[T,Rr,ϕ][[1,2]] |
σ∞[T,Rr,ϕ][[2,1]] | σ∞[T,Rr,ϕ][[2,2]] |
σ∞[T,Rr,ϕ][[3,3]] | σ∞[T,Rr,ϕ][[3,4]] |
σ∞[T,Rr,ϕ][[4,3]] | σ∞[T,Rr,ϕ][[4,4]] |
σ∞[T,Rr,ϕ][[1,3]] | σ∞[T,Rr,ϕ][[1,4]] |
σ∞[T,Rr,ϕ][[2,3]] | σ∞[T,Rr,ϕ][[2,4]] |
σ
a
σ
b
σ
c
σ[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=Exp[-λt]σ0[r,n1,n2]+(1-Exp[-λt])σ∞[T,Rr,ϕ];(*Thetimeevolutionofthetwo-modestateinanoisychannelispresentedabove.*)
σa[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=
;σb[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=
;σc[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=
;(*Thematrixoftheandmodesaredefined,togetherwiththematrixoftheircorrelations.*)
σ[t,r,n1,n2,T,Rr,ϕ][[1,1]] | σ[t,r,n1,n2,T,Rr,ϕ][[1,2]] |
σ[t,r,n1,n2,T,Rr,ϕ][[2,1]] | σ[t,r,n1,n2,T,Rr,ϕ][[2,2]] |
σ[t,r,n1,n2,T,Rr,ϕ][[3,3]] | σ[t,r,n1,n2,T,Rr,ϕ][[3,4]] |
σ[t,r,n1,n2,T,Rr,ϕ][[4,3]] | σ[t,r,n1,n2,T,Rr,ϕ][[4,4]] |
σ[t,r,n1,n2,T,Rr,ϕ][[1,3]] | σ[t,r,n1,n2,T,Rr,ϕ][[1,4]] |
σ[t,r,n1,n2,T,Rr,ϕ][[2,3]] | σ[t,r,n1,n2,T,Rr,ϕ][[2,4]] |
st
1
nd
2
Ref[θ_]=Cos[θ]IdentityMatrix[2]-Sin[θ]
;(*θisthemeasurementangle.*)γπb[θ_,μ_]=Ref[θ].
.Transpose[Ref[θ]];(*istheseedofmeasurementmatrix.μ=0Homodynedetection,μ=1Heterodynedetection*)
0 | - |
| 0 |
μ 2 | 0 |
0 | 1 2μ |
π
b
γ
σaπb[t_,r_,n1_,n2_,T_,Rr_,ϕ_,θ_,μ_]=σa[t,r,n1,n2,T,Rr,ϕ]-σc[t,r,n1,n2,T,Rr,ϕ].Inverse[σb[t,r,n1,n2,T,Rr,ϕ]+γπb[θ,μ]].Transpose[σc[t,r,n1,n2,T,Rr,ϕ]];(*isthecovariancematrixofthestateofthefirstmodereachedafterthemeasurementofthemodehasbeenperformed.Thestatechangesduetoback-reaction.*)
σ
π
b
a
nd
2
W[t_,r_,n1_,n2_,T_,Rr_,ϕ_,θ_,μ_]=Log;(*Thequentumworkextractedwhenthestateofthemode1reachesthenewequilibriumwiththenoisychannelintermsofRenyientropy.*)Δ[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=4Det[σa[t,r,n1,n2,T,Rr,ϕ]]+4Det[σb[t,r,n1,n2,T,Rr,ϕ]]+8Det[σc[t,r,n1,n2,T,Rr,ϕ]];(*Seralianofthetwo-modestates.*)(*-Δν+Det[σ]=0*)ν1[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=Sqrt;ν2[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=Sqrt;(*Simplecticeigenvaluesofthetwo-modestates.*)S[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=Log2-Log2+Log2-Log2;(*VonNeumannEntropyofatwo-modestate.*)Werasnou[t_,r_,n1_,n2_,T_,Rr_,ϕ_]=TS[t,r,n1,n2,T,Rr,ϕ];Wmnou[r_,n1_,n2_,T_,Rr_,ϕ_]:=FindMaximum[{Werasnou[t,r,n1,n2,T,Rr,ϕ],0<20<100},{t,1}][[1]](*Wetakethemaximalentropyintime.*)ηnou[t_,r_,n1_,n2_,T_,Rr_,ϕ_,θ_,μ_]:=;(*Theentropyisobtained*)
T
2
Det[σa[t,r,n1,n2,T,Rr,ϕ]]
Det[σaπb[t,r,n1,n2,T,Rr,ϕ,θ,μ]]
2
ν
Δ[t,r,n1,n2,T,Rr,ϕ]-Sqrt[-64Det[σ[t,r,n1,n2,T,Rr,ϕ]]]
2
Δ[t,r,n1,n2,T,Rr,ϕ]
2
Δ[t,r,n1,n2,T,Rr,ϕ]+Sqrt[-64Det[σ[t,r,n1,n2,T,Rr,ϕ]]]
2
Δ[t,r,n1,n2,T,Rr,ϕ]
2
ν1[t,r,n1,n2,T,Rr,ϕ]+1
2
ν1[t,r,n1,n2,T,Rr,ϕ]+1
2
ν1[t,r,n1,n2,T,Rr,ϕ]-1
2
Abs[ν1[t,r,n1,n2,T,Rr,ϕ]-1]
2
ν2[t,r,n1,n2,T,Rr,ϕ]+1
2
ν2[t,r,n1,n2,T,Rr,ϕ]+1
2
ν2[t,r,n1,n2,T,Rr,ϕ]-1
2
Abs[ν2[t,r,n1,n2,T,Rr,ϕ]-1]
2
W[t,r,n1,n2,T,Rr,ϕ,θ,μ]
Wmnou[r,n1,n2,T,Rr,ϕ]
In[]:=
PlotNIntegrate[W[0,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],NIntegrate[W[1,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],NIntegrate[W[5,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],{T,0,5},Frame->True,FrameLabel{"T","W"},FrameStyle26,ImageSize1000,PlotStyle{Red,Green,Blue},PlotLegendsPlaced[{Style["t=0",FontSize26],Style["t=1",FontSize26],Style["t=5",FontSize26]},Above]
1
π
1
π
1
π
Out[]=
| |||||||
 |
In[]:=
PlotNIntegrate[W[t,1.8,0,0,0.1,0.2,π/4,π/4,0],{θ,0,π}],NIntegrate[W[t,1.8,0,0,1,0.2,π/4,π/4,0],{θ,0,π}],NIntegrate[W[t,1.8,0,0,2,0.2,π/4,π/4,0],{θ,0,π}],{t,0,5},Frame->True,FrameLabel{"t","W"},FrameStyle36,ImageSize1000,PlotStyle{Red,Green,Blue},PlotLegendsPlaced[{Style["T=0.1",FontSize30],Style["T=1",FontSize30],Style["T=2",FontSize30]},Above]
1
π
1
π
1
π
Out[]=
Plot[{W[0,1.8,0,0,0.1,Rr,π/4,π/4,0],W[0,1.8,0,0,1,Rr,π/2,π/4,0],W[0,1.8,0,0,2,Rr,π/4,π/4,0]},{Rr,0,5},Frame->True,FrameLabel{"R","W"},FrameStyle22,ImageSize1000,PlotStyle{Red,Green,Blue},PlotLegendsPlaced[{Style["t=0",FontSize26],Style["t=1",FontSize26],Style["t=5",FontSize26]},Above]]
Out[]=
(*t,r,n1,n2,T,Rr,ϕ,θ,μ*)
In[]:=
PlotNIntegrate[W[0,r,0,0,0.1,0.2,π/4,θ,0],{θ,0,π}],NIntegrate[W[1,r,0,0,0.1,0.2,π/4,θ,0],{θ,0,π}],NIntegrate[W[2,r,0,0,0.1,0.2,π/4,θ,0],{θ,0,π}],{r,0,1.8},Frame->True,FrameLabel{"r","W"},FrameStyle36,ImageSize1000,PlotStyle{Red,Green,Blue},PlotLegendsPlaced[{Style["t=0",FontSize30],Style["t=1",FontSize30],Style["t=2",FontSize30]},Above]
1
π
1
π
1
π
Out[]=
(*W[t,r,n1,n2,T,Rr,ϕ,θ,μ]*)
Plot3DNIntegrate[W[t,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],{t,0,5},{T,0,5},PlotRangeAll,AxesLabel{"t","T","W"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
In[]:=
Plot3DNIntegrate[ηnou[t,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],{t,0.1,5},{T,0.1,5},PlotRangeAll,AxesLabel{"t","T","η"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5},PlotStyleCyan
1
π
In[]:=
Plot3DNIntegrate[ηnew[t,1.8,0,0,T,0.2,π/4,θ,0],{θ,0,π}],{t,0.0001,5},{T,0.0001,5},PlotRangeAll,AxesLabel{"t","T","η"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5},PlotStyleCyan
1
π
Out[]=
(*W[t,r,n1,n2,T,Rr,ϕ,θ,μ]*)
In[]:=
Plot3D[W[t,1.8,0,0,3,0.2,π/4,θ,0],{t,0,5},{θ,0,2π},PlotRangeAll,AxesLabel{"t","θ","W"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5}]
Out[]=
(*W[t,r,n1,n2,T,Rr,ϕ,θ,μ]*)
In[]:=
Plot3DNIntegrate[W[t,1.8,0,0,3,0.2,ϕ,θ,0],{θ,0,π}],{t,0,5},{ϕ,0,6π},PlotRangeAll,AxesLabel{"t","ϕ","W"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
In[]:=
Plot3DNIntegrate[W[t,1.8,0,0,3,0.2,0,θ,μ],{θ,0,π}],{t,0,5},{μ,0,6},PlotRangeAll,AxesLabel{"t","μ","W"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
In[]:=
Plot3DNIntegrate[ηnew[t,1.8,0,0,3,0.2,0,θ,μ],{θ,0,π}],{t,0,5},{μ,0,6},PlotRangeAll,AxesLabel{"t","μ","η"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5},PlotStyleCyan
1
π
Out[]=
(*t,r,n1,n2,T,Rr,ϕ,θ,μ*)
In[]:=
Plot3DNIntegrate[W[t,r,0,0,3,0.2,0,θ,0],{θ,0,π}],{t,0,5},{r,0,1.8},PlotRangeAll,AxesLabel{"t","r","W"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
In[]:=
Plot3DNIntegrate[ηnew[t,r,0,0,3,0.2,0,θ,0],{θ,0,π}],{t,0,5},{r,0,1.8},PlotRangeAll,AxesLabel{"t","r","η"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5},PlotStyleCyan
1
π
Out[]=
In[]:=
Plot3D[W[t,1.8,n1,0,3,0.2,0,π/4,1],{t,0,5},{n1,0,3},PlotRangeAll,AxesLabel{"t","","W"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5}]
n
1
Out[]=
In[]:=
Plot3D[ηnew[t,1.8,n1,0,3,0.2,0,π/4,1],{t,0,5},{n1,0,3},PlotRangeAll,AxesLabel{"t","","η"},AxesStyle36,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5},PlotStyleCyan]
n
1
Out[]=
Plot3DNIntegrate[W[t,1.8,0,n2,3,0.2,0,π/4,1],{t,0,5}],{n2,0,3},PlotRangeAll,AxesLabel{"t","n2","W"},AxesStyle22,PlotPoints75,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
(*W[t,r,n1,n2,T,Rr,ϕ,θ,μ]*)
In[]:=
Plot3DNIntegrate[W[t,1.8,0,0,3,Rr,0,θ,0],{θ,0,π}],{t,0,5},{Rr,0,2},PlotRangeAll,AxesLabel{"t","R","W"},AxesStyle36,ImageSize1000,ViewPoint{4,-10,5}
1
π
Out[]=
Cite this as: Alexei Zubarev, "Extractable quantum work from a twomode Gaussian state in a noisy channel" from the Notebook Archive (2022), https://notebookarchive.org/2022-01-c07ri99
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