In[]:=
states=First/@Take[ResourceFunction["MultiwaySystem"][{"Xo""oX","oX""Xo"},"ooooooooXXoooooooo",6,"StateWeights","IncludeStateWeights"True],-14]
Out[]=
{oooooXXooooooooooo,oooooXooooooXooooo,oooooooooXXooooooo,ooooooooXooXoooooo,oooooooXooooXooooo,ooooooXooooooXoooo,ooooooooooXXoooooo,oooooooooXooXooooo,ooooooooXooooXoooo,oooooooXooooooXooo,oooooooooooXXooooo,ooooooooooXooXoooo,oooooooooXooooXooo,ooooooooXooooooXoo}
Pre-measurement, strings represent superpositions
Pre-measurement, strings represent superpositions
"oooooooXooooXooooo"
is a superposition of photons at two different positions
[ There may also be another interpretation ]
Post-measurement, what does the string represent?
Post-measurement, what does the string represent?
"oooooooXoooooooooo"
is clear in what it represents. It is an eigenstate in the position basis with the photon at a certain position.
The goal of the measurement is to get a result that is an eigenstate in the position basis.
< position eigenstate | state we got >
Jonathan’s scheme:
We need to map this onto a specific position eigenstate:
"oooooooXooooXooooo"
How to measure a superposition state
How to measure a superposition state
“oooooooXooooXooooo” roughly represents a wavefunction with support at two distinct points.
Now we do a series of measurements
Now we do a series of measurements
The multiway system has generated a superposition of superposition states
Take 1: just treat the X separately [too classical]
Take 1: just treat the X separately [too classical]
{"oooooXXooooooooooo","oooooXooooooXooooo","oooooooooXXooooooo","ooooooooXooXoooooo","oooooooXooooXooooo","ooooooXooooooXoooo","ooooooooooXXoooooo","oooooooooXooXooooo","ooooooooXooooXoooo","oooooooXooooooXooo","oooooooooooXXooooo","ooooooooooXooXoooo","oooooooooXooooXooo","ooooooooXooooooXoo"}
In[]:=
states=ResourceFunction["MultiwaySystem"][{"Xo""oX","oX""Xo"},"ooooooooXXoooooooo",6,"StateWeights","IncludeStateWeights"True]
Out[]=
ooooooooXXoooooooo1,oooooooXoXoooooooo,ooooooooXoXooooooo,ooooooXooXoooooooo,ooooooooXXoooooooo,oooooooXXooooooooo,oooooooXooXooooooo,oooooooooXXooooooo,ooooooooXooXoooooo,oooooXoooXoooooooo,oooooooXoXoooooooo,ooooooXoXooooooooo,ooooooXoooXooooooo,ooooooooXoXooooooo,oooooooXoooXoooooo,oooooooooXoXoooooo,ooooooooXoooXooooo,ooooXooooXoooooooo,ooooooXooXoooooooo,oooooXooXooooooooo,oooooXooooXooooooo,ooooooooXXoooooooo,oooooooXXooooooooo,oooooooXooXooooooo,ooooooXXoooooooooo,ooooooXooooXoooooo,oooooooooXXooooooo,ooooooooXooXoooooo,oooooooXooooXooooo,ooooooooooXXoooooo,oooooooooXooXooooo,ooooooooXooooXoooo,oooXoooooXoooooooo,oooooXoooXoooooooo,ooooXoooXooooooooo,ooooXoooooXooooooo,oooooooXoXoooooooo,ooooooXoXooooooooo,ooooooXoooXooooooo,oooooXoXoooooooooo,oooooXoooooXoooooo,ooooooooXoXooooooo,oooooooXoooXoooooo,ooooooXoooooXooooo,oooooooooXoXoooooo,ooooooooXoooXooooo,oooooooXoooooXoooo,ooooooooooXoXooooo,oooooooooXoooXoooo,ooooooooXoooooXooo,ooXooooooXoooooooo,ooooXooooXoooooooo,oooXooooXooooooooo,oooXooooooXooooooo,ooooooXooXoooooooo,oooooXooXooooooooo,oooooXooooXooooooo,ooooXooXoooooooooo,ooooXooooooXoooooo,ooooooooXXoooooooo,oooooooXXooooooooo,oooooooXooXooooooo,ooooooXXoooooooooo,ooooooXooooXoooooo,oooooXXooooooooooo,oooooXooooooXooooo,oooooooooXXooooooo,ooooooooXooXoooooo,oooooooXooooXooooo,ooooooXooooooXoooo,ooooooooooXXoooooo,oooooooooXooXooooo,ooooooooXooooXoooo,oooooooXooooooXooo,oooooooooooXXooooo,ooooooooooXooXoooo,oooooooooXooooXooo,ooooooooXooooooXoo
1
2
1
2
1
8
1
4
1
8
1
4
1
8
1
8
1
24
1
4
1
12
1
8
1
4
1
8
1
12
1
24
1
96
1
8
1
32
1
24
1
8
1
12
3
16
1
48
1
16
1
12
1
8
1
24
1
48
1
32
1
96
1
320
1
16
1
80
1
64
5
32
5
64
1
8
1
64
1
32
5
32
1
8
1
32
5
64
1
16
1
64
1
64
1
80
1
320
1
1280
3
128
1
256
3
640
27
256
27
640
15
256
9
1280
3
256
5
64
15
256
9
64
3
128
5
64
1
256
1
64
15
256
27
256
15
256
3
256
3
128
27
640
3
128
3
640
1
256
9
1280
1
256
1
1280
In[]:=
sweights=GatherBy[Catenate[Map[Thread[Map[First,StringPosition[First[#],"X"]]Last[#]]&,states]],First]
Out[]=
91,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,101,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,7,7,7,7,7,7,7,7,7,7,7,7,7,7,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,6,6,6,6,6,6,13,13,13,13,13,13,13,13,13,13,5,5,5,5,5,5,14,14,14,14,14,14,4,4,4,15,15,15,3,16
1
2
1
4
1
8
1
8
1
12
1
4
1
24
1
32
1
8
1
12
1
8
1
96
1
80
5
64
5
32
1
16
1
320
1
256
27
640
5
64
15
256
27
256
3
128
1
1280
1
2
1
8
1
4
1
8
1
24
1
4
1
12
1
96
1
8
1
8
1
12
1
32
1
320
1
16
5
32
5
64
1
80
1
1280
3
128
27
256
5
64
15
256
27
640
1
256
1
2
1
8
1
4
1
4
1
8
1
12
3
16
1
48
1
24
5
32
1
64
1
8
1
64
9
1280
15
256
9
64
3
128
15
256
3
640
1
2
1
4
1
8
1
8
1
4
1
24
3
16
1
12
1
48
1
64
1
8
5
32
1
64
3
640
15
256
9
64
15
256
3
128
9
1280
1
8
1
12
1
8
1
8
1
48
1
16
5
64
1
8
1
32
27
256
3
128
5
64
1
256
3
256
1
8
1
8
1
12
1
16
1
8
1
48
1
32
1
8
5
64
3
256
5
64
27
256
3
128
1
256
1
24
1
32
1
24
1
16
1
64
1
32
27
640
15
256
1
256
1
64
1
24
1
24
1
32
1
32
1
16
1
64
1
64
15
256
27
640
1
256
1
96
1
80
1
64
3
128
9
1280
3
256
1
96
1
64
1
80
3
256
3
128
9
1280
1
320
1
256
3
640
1
320
3
640
1
256
1
1280
1
1280
In[]:=
Sort[{#[[1,1]],Total[Values[#]]}&/@sweights]
Out[]=
3,,4,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,
1
1280
3
256
31
384
661
1920
767
768
1681
768
27
8
27
8
1681
768
767
768
661
1920
31
384
3
256
1
1280
In[]:=
ListLinePlot[%]
Out[]=
In[]:=
Module[{states,sweights},states=ResourceFunction["MultiwaySystem"][{"Xo""oX","oX""Xo"},"ooooooooXXoooooooo",8,"StateWeights","IncludeStateWeights"True];sweights=GatherBy[Catenate[Map[Thread[Map[First,StringPosition[First[#],"X"]]Last[#]]&,states]],First];Sort[{#[[1,1]],Total[Values[#]]}&/@sweights]]
Out[]=
1,,2,,3,,4,,5,,6,,7,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,
1
17920
19
17920
17
1792
479
8960
407
1920
121
192
5557
3840
10463
3840
251
64
251
64
10463
3840
5557
3840
121
192
407
1920
479
8960
17
1792
19
17920
1
17920
In[]:=
ListLinePlot[%]
Out[]=
Take 2: the phase of each string is given by the difference of X positions
Take 2: the phase of each string is given by the difference of X positions
Each string can be written in the position basis as an average single X position, together with a phase associated with the distance between Xs
Each pair gives the position basis position, together with the phase associated with this term in the superposition
Fundamental objective: find to which position basis eigenstate each X...X configuration corresponds
Two different phases: phase of photon itself; phase of photon wave function
Two different phases: phase of photon itself; phase of photon wave function
Phase of photon itself is related to position in position basis