File:Unambiguous quantum state discrimination with a priori probs - animation.gif
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Size of this preview: 600 × 600 pixels. Other resolutions: 240 × 240 pixels | 480 × 480 pixels | 972 × 972 pixels.
Original file (972 × 972 pixels, file size: 1.83 MB, MIME type: image/gif, looped, 101 frames, 13 s)
Captions
Captions
Unambiguous quantum state discrimination for two pure states with different a priori probabilities
Summary[edit]
| Description |
English: Dependence on a priori probabilities of the form of measurement operators for unambiguous quantum state discrimination for the case of two pure states. When the two a priori probabilities are identical, , the (non-normalized) eigenvectors of measurement operators are situated symmetrically around the (orange) initial states for . If can occur with a very high probability , the POVM measurement reduces to a projective one. Analogously also for . Čeština: Závislost tvaru měřicích operátorů na apriorních pravděpodobnostech pro případ jednoznačného rozlišení dvou čistých kvantových stavů. Pokud jsou obě apriorní pravděpodobnosti totožné, , jsou (nenormalizované) vlastní vektory měřicích operátorů rozmístěny souměrně kolem (oranžových) počátečních stavů pro . Může-li se vyskytnout s velmi velkou pravděpodobností , zredukuje se POVM měření na měření projektivní. Analogicky pak i pro . |
| Date | |
| Source | Own work |
| Author | JozumBjada |
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Source code[edit]
This animation was created using Wolfram language 12.0.0 for Microsoft Windows (64-bit) (April 6, 2019). The source code follows.
(* ::Package:: *)
(* ::Title:: *)
(*Unambiguous quantum state discrimination for varying a priori probabilities*)
(* ::Input::Initialization:: *)
vec[char_,idx_,ang_,italic_:True,len_:1,off_:.15]:=If[len==0,{},{Arrow[{{0,0},len AngleVector[ang ]}],Text[Style[ToString[Ket[Subscript[char,idx]],TraditionalForm],If[italic,Italic,Plain],30,FontFamily->"Times",Background->White],(1+off)AngleVector[ang],{0,0},AngleVector[ang]]}]
(* ::Input::Initialization:: *)
f[x_]:=(1+Clip[1.07Sign[Sin[2\[Pi] x]]Abs[Sin[2\[Pi] x]]^3,{-1,1}])/2
(* ::Input::Initialization:: *)
usdInternal[ang1i_,ang2i_,prob_]:=Module[{ang1=ang1i,ang2=ang2i,p1=prob,p2=1-prob,psi1,psi2,q,r,e1,e2,eQ,ang1e,ang2e,angQe,prefac1,prefac2,prefacQ,disk},
If[prob<1/2,
{p1,p2}={p2,p1};
{ang1,ang2}={ang2,ang1};
];
{psi1,psi2}=AngleVector/@{ang1,ang2};
{q,r}={Abs[Cos[ang2i-ang1i]],Sqrt[p2/p1]};
If[q<=r,
(*generic case*)
e1=1/(1-q^2) Sqrt[1-q r](psi1-q psi2);
e2=1/(1-q^2) Sqrt[1-q/r](psi2-q psi1);
eQ=1/(1-q^2) Sqrt[q/r]((r-q)psi1+(1-q r) psi2);
{ang1e,ang2e,angQe}=ArcTan[#[[1]],#[[2]]]&/@{e1,e2,eQ};
{prefac1,prefac2,prefacQ}=Norm/@{e1,e2,eQ};
,
(*degenerate case*)
e1=1/Sqrt[1-q^2] (psi1-q psi2);
e2={0,0};
eQ=psi2;
{ang1e,ang2e,angQe}={ArcTan[e1[[1]],e1[[2]]],0,ang2};
{prefac1,prefac2,prefacQ}={1,0,1};
];
disk=If[q<=r,{},Disk[{0,0},.1,{ang1e,angQe}]];
If[prob<1/2,
{ang1e,ang2e}={ang2e,ang1e};
{prefac1,prefac2}={prefac2,prefac1};
];
{ang1e,ang2e,angQe,prefac1,prefac2,prefacQ,disk}
]
(* ::Input::Initialization:: *)
grUSDapriori[ang1i_,ang2i_,prob_]:=Module[{prefac1,prefac2,prefacQ,e1,e2,eQ,ang1e,ang2e,angQe,disk},
{ang1e,ang2e,angQe,prefac1,prefac2,prefacQ,disk}=usdInternal[ang1i,ang2i,prob];
Graphics[{
Circle[{0,0},1],
{Thickness[0.003],Dashing[0.02],Line[{{{-1,0},{1,0}},{{0,-1},{0,1}}}]},
{Thickness[0.005],Arrowheads[0.05],
{
{Opacity[Sqrt[prob],Orange],vec["\[Psi]",1,ang1i,False]},
{Opacity[Sqrt[1-prob],Orange],vec["\[Psi]",2,ang2i,False]}
},
{Blue,disk,
vec["e","1",ang1e,True,prefac1,prefac1-1+0.15],
vec["e","2",ang2e,True,prefac2,prefac2-1+0.15],
vec["e","?",angQe,True,prefacQ,Min[prefacQ-1+0.2,0.15]]
},
{
Text[Style[Row[{Style[Subscript["p",1],Italic],"\[VeryThinSpace]=\[VeryThinSpace]",ToString[NumberForm[Chop@N@prob,{2,2}]]}],30,FontFamily->"Times"],ImageScaled[{.83,.55}]],
Text[Style[Row[{Style[Subscript["p",2],Italic],"\[VeryThinSpace]=\[VeryThinSpace]",ToString[NumberForm[Chop@N@(1-prob),{2,2}]]}],30,FontFamily->"Times"],ImageScaled[{.7,.85}]]
},
{Black,PointSize[0.02],Point[{0,0}]}
}
},ImageSize->700,PlotRange->{.7+{-1.25,1.25},{-1.25,1.25}},Background->None]
];
(* ::Input::Initialization:: *)
animationUSDapriori[t_]:=grUSDapriori[0.1,0.8,f[t]]
(* ::Input::Initialization:: *)
exportAnimation[fun_,name_,resolution_:100]:=Module[{seq},
SetDirectory[NotebookDirectory[]];
seq=Table[fun[t],{t,0,1,.01}];
Export[name<>".gif",seq,AnimationRepetitions->Infinity,"DisplayDurations"->0.13(*0.05*)(*0.01*),"TransparentColor"->Automatic,ImageResolution->resolution]
]
(* ::Input:: *)
(*(*Plot[f[x],{x,0,1}]*)*)
(* ::Input:: *)
(*(*Manipulate[grUSDapriori[0.1,0.8,p],{p,0,1},SaveDefinitions->True]*)*)
(* ::Input:: *)
(*(*Manipulate[animationUSDapriori[t],{t,0,1},SaveDefinitions->True]*)*)
(* ::Input:: *)
(*exportAnimation[animationUSDapriori,"animUSDapriori"]*)
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:05, 10 December 2021 | | 972 × 972 (1.83 MB) | JozumBjada (talk | contribs) | Cross-wiki upload from cs.wikipedia.org |
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