ORIGINAL ARTICLE: Shubhayan Sarkar (2023), Distrustful quantum steering, Phys. Rev. A 108, L040401.
DOI: https://doi.org/10.1103/PhysRevA.108.L040401
Article Abstract: Quantum steering is an asymmetric form of quantum nonlocality where one can trust the measurements of one of the parties. In this work, inspired by practical considerations, we investigate the scenario in which one cannot fully trust their measurement devices, just up to some level of precision. We first find the effect of such an imprecision on standard device-dependent quantum tomography. We then utilize this result to compute the variation in the local bound of any general steering inequality depending on the amount of trust one puts in one of the party’s measurement devices. This is particularly important as we show that even a small distrust on Alice might cause the parties to observe steerability even if the quantum state is unsteerable. Furthermore, this effect becomes more relevant when observing higher-dimensional quantum steering.
Specify the dimension
d=2;
Specify the error
ϵ=0.1;
Basis
Z={{1,0},{0,-1}};X={{0,1},{1,0}};Y={{0,-1},{1,0}};
Noisy basis
NZ=z1Z+z2X+z3Y;​​NX=x1Z+x2X+x3Y;​​NY=y1Z+y2X+y3Y;

Tomography

Specify the state in the Pauli basis
pz=1/4;px=1/8;py=0;​​rho=1/2IdentityMatrix[2]+pzZ+pxX+pyY(*checkyourstate!*)

3
4
,
1
8
,
1
8
,
1
4

qz=1/dTr[rho.NZ];​​qx=1/dTr[rho.NX];​​qy=1/dTr[rho.NY];
Maximising the distance between the states
NMaximize[{d(qz-pz)^2+d(qx-px)^2+d(qy-py)^2,
2
z1
+
2
z2
+
2
z3
1,
2
x1
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2
x2
+
2
x3
1,
2
y1
+
2
y2
+
2
y3
1,4(1-z1)≤ϵ,4(1-x2)≤ϵ,4(1-y3)≤ϵ},{z1,z2,z3,x1,x2,x3,y1,y2,y3}]//Chop
{0.0169162,{z10.975,z2-0.22221,z31.03946×
-8
10
,x1-0.222205,x20.975,x34.32034×
-8
10
,y10.198746,y20.099373,y30.975}}

Correction to LHS bound

Specify the F_ (0 | 0) in the Pauli basis
fooo=1/2;fooz=1/2;foox=0;fooy=0;​​F00=foooIdentityMatrix[2]+foozZ+fooxX+fooyY(*checkF_(0|0)!*)
{{1,0},{0,0}}
Specify the F_ (1 | 0) in the Pauli basis
f1oo=1/2;f1oz=-1/2;f1ox=0;f1oy=0;​​F10=f1ooIdentityMatrix[2]+f1ozZ+f1oxX+f1oyY(*checkF_(1|0)!*)
{{0,0},{0,1}}
Specify the F_ (0 | 1) in the Pauli basis
fo1o=1/2;fo1z=0;fo1x=1/2;fo1y=0;​​F01=fo1oIdentityMatrix[2]+fo1zZ+fo1xX+fo1yY(*checkF_(0|1)!*)
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1
2
,
1
2
,
1
2
,
1
2

Specify the F_ (1 | 1) in the Pauli basis
f11o=1/2;f11z=0;f11x=-1/2;f11y=0;​​F11=f11oIdentityMatrix[2]+f11zZ+f11xX+f11yY(*checkF_(1|0)!*)

1
2
,-
1
2
,-
1
2
,
1
2

Specify the assemblage σ_ (0 | 0) attaining the LHS bound in Pauli basis
sooo=1/2;sooz=Sqrt[2]/4;soox=Sqrt[2]/4;sooy=0;​​σ00=soooIdentityMatrix[2]+soozZ+sooxX+sooyY(*checkσ_(0|0)!*)​​q000=1/dTr[σ00];​​qooz=1/dTr[σ00.NZ];​​qoox=1/dTr[σ00.NX];​​qooy=1/dTr[σ00.NY];​​σ00corr=q000IdentityMatrix[2]+qoozZ+qooxX+qooyY;

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2
,
1
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1
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
Specify the assemblage σ_ (0 | 0) attaining the LHS bound in Pauli basis
s1oo=0;s1oz=0;s1ox=0;s1oy=0;​​σ10=s1ooIdentityMatrix[2]+s1ozZ+s1oxX+s1oyY(*checkσ_(1|0)!*)​​q100=1/dTr[σ10];​​q1oz=1/dTr[σ10.NZ];​​q1ox=1/dTr[σ10.NX];​​q1oy=1/dTr[σ10.NY];​​σ10corr=q100IdentityMatrix[2]+q1ozZ+q1oxX+q1oyY;
{{0,0},{0,0}}
Specify the assemblage σ_ (0 | 0) attaining the LHS bound in Pauli basis
so1o=1/2;so1z=Sqrt[2]/4;so1x=Sqrt[2]/4;so1y=0;​​σ01=so1oIdentityMatrix[2]+so1zZ+so1xX+so1yY(*checkσ_(0|1)!*)​​q010=1/dTr[σ01];​​qo1z=1/dTr[σ01.NZ];​​qo1x=1/dTr[σ01.NX];​​qo1y=1/dTr[σ01.NY];​​σ01corr=q010IdentityMatrix[2]+qo1zZ+qo1xX+qo1yY;

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,
1
2
2
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1
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-
1
2
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
Specify the assemblage σ_ (0 | 0) attaining the LHS bound in Pauli basis
s11o=0;s11z=0;s11x=0;s11y=0;​​σ11=s11oIdentityMatrix[2]+s11zZ+s11xX+s11yY(*checkσ_(1|0)!*)​​q110=1/dTr[σ11];​​q11z=1/dTr[σ11.NZ];​​q11x=1/dTr[σ11.NX];​​q11y=1/dTr[σ11.NY];​​σ11corr=q110IdentityMatrix[2]+q11zZ+q11xX+q11yY;
{{0,0},{0,0}}
Checking the ideal LHS bound
Tr[F00.σ00]+Tr[F10.σ10]+Tr[F01.σ01]+Tr[F11.σ11]//FullSimplify
1+
1
2
Evaluating the corrected LHS bound
NMaximize[{Tr[F00.σ00corr]+Tr[F10.σ10corr]+Tr[F01.σ01corr]+Tr[F11.σ11corr],
2
z1
+
2
z2
+
2
z3
1,
2
x1
+
2
x2
+
2
x3
1,
2
y1
+
2
y2
+
2
y3
1,4(1-z1)≤ϵ,4(1-x2)≤ϵ,4(1-y3)≤ϵ},{z1,z2,z3,x1,x2,x3,y1,y2,y3}]//Chop
{1.84655,{z10.975,z20.222205,z31.17193×
-8
10
,x10.222205,x20.975,x3-2.09538×
-9
10
,y10.026264,y20.109938,y30.993591}}

CITE THIS NOTEBOOK

Distrustful quantum steering​
by Shubhayan Sarkar​
Wolfram Community, STAFF PICKS, November 13, 2023
​https://community.wolfram.com/groups/-/m/t/3064608