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In[]:=

CompoundExpression[

]

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Thu 21 Dec 2023 11:00:12

From “linear-estimation.nb: Get Choi representation ...” and linear-estimation-scratch-nov: Choi

Connecting Choi matrix of T to contraction properties of T

Util


Main example

In[]:=

Clear[EE,covStep,choi];X={{1,0},{1,1}};(*X=X/Norm[X,"Frobenius"];*)b=1;d=Length[First@X];ii=IdentityMatrix[d];batches=Subsets[X,{b}];(*Unfoldedoperatorview*)EX2=Mean[#.#&/@batches];EX4=Mean[(#.#)⊗(#.#)&/@batches];Tflat=ii⊗ii-ii⊗EX2-EX2⊗ii+EX4;Tflat≐Mean[(ii-#.#)⊗(ii-#.#)&/@batches];(*Choimatrixgivesoperatorinnatural

2

n

basis*)EE[i_,j_]:=Array[Boole[#1i&&#2j]&,{d,d}];covStep[cov_]:=Mean[(ii-#.#).cov.(ii-#.#)&/@batches];choi=ArrayFlatten@Table[covStep@EE[i,j],{i,1,d},{j,1,d}];unvecMap=Map[unvec[#]&,Partition[Tflat,d],{2}];fl1[l_]:=Flatten[l,{1}];vec2[mat_]:=Join@@Transpose@mat;coords=fl1@Outer[List,Range[d],Range[d]];{"T rearrangement=choi: ",ArrayFlatten[unvecMap]≐choi}{"T vectorized arrangement: ",Tflat≐(vec2[covStep[EE@@#]]&/@vec2@coords)}SeedRandom[1];A=RandomReal[{-1,1},{d,d}];A=A.A;{"Choi construction: ",choi≐choiFromTflat[Tflat]}kraus1=unvec/@symsqrtFactors[choi];kraus=getKraus@Tflat;kraus≐kraus1;osrApply[A_]:=Total[#.A.#&/@kraus];{"Kraus=covStep: ",osrApply[A]≐covStep[A]}t[stuff_]:=StringForm["``",stuff];TableForm[{{"Ai",t[MatrixForm[(ii-#.#)]&/@batches]},{"Kraus",t[MatrixForm/@getKraus[Tflat]]},{"Choi",choi//MatrixForm},{"Choi evals",t@Eigenvalues[choi]},{"T",Tflat//MatrixForm},{"T evals",t@Eigenvalues[Tflat]}}]

Linearly dependent example

Adding 0 observation makes Choi and T matrix have identical eigenvalues up to sign

Out[]//TableForm=

Ai



0	0
0	1

,

0	-1
-1	0

,

1	0
0	0



Kraus



0	1 3
1 3	0

,

0	0
0	1 3

,

1 3	0
0	0



Choi

1 3	0	0	0
0	1 3	1 3	0
0	1 3	1 3	0
0	0	0	1 3

Choi evals



2

3

,

1

3

,

1

3

,0

T

1 3	0	0	1 3
0	0	1 3	0
0	1 3	0	0
1 3	0	0	1 3

T evals



2

3

,-

1

3

,

1

3

,0

T vs M eigenvalues

M eigenvalues close to 1 suggest preservation of quantum states
T eigenvalues can be negative.

Eigenvalues of M close to 1 suggest Fidelity is preserved (effective rank not dropping?)

Out[]=

	1st
	2nd
	1