data = {{0, 2}, {0, 4}, {3 Pi/8, 0}, {3 Pi/8, -2 Sqrt[2]}};
operator1 = {{1 + I, 1 - I}, {1 - I, 1 + I}};
operator2 = {{0, 1}, {1, 0}};
basis = {{1, 0}, {0, 1}};
initialState1 = {1 + I, 1 - I};
That's such a creative next phase! You could define an experiment based on the complex network graph and the QuantumToMultiwaySystem tool. I suppose it's just a matter of defining what types of interactions and transitions between states that we make, quickly, such that we can experimentally measure these things that show up on the graph like quantum phenomena, entanglement properties, specific state transitions, you know how it goes.
BarChart[
data[[All, 2]],
BarSpacing -> 0.3,
ChartLabels -> Placed[data[[All, 1]], Below],
ChartStyle -> "Pastel",
ChartLegends -> Placed[{
"{0, 2}",
"{0, 4}",
"{3 Pi/8,0}",
"{3 Pi/8,-2 Sqrt[2]}"
},
Right],
LabelingFunction -> Above,
AxesLabel -> {"Key", "Value"},
PlotLabel -> "Histogram of Data",
ImageSize -> Large]
It's sort of hard to imagine that this is data in the complex plane. if only we had more access to the current technology before things get more experimental. We've got all these high-energy systems, like let's say we want to design these histograms. It's sort of like designing rockets, those hand-crafted leagues of graphs that could represent anything from a quantum state transition graph or a causal graph, from a multiway system simulation to the visualization of the evolution of quantum states which just goes to show how different quantum states are related to each other. We start out with the binary dimensionality of the data, the Law of the Excluded Middle and then we get this graph and fluff it up a bit.
ListPlot[
Transpose[data],
PlotMarkers -> {"\[FilledCircle]", "\[FilledSquare]"},
PlotLegends -> Placed[{"X-value", "Y-value"}, Above],
AxesLabel -> {"Index", "Value"},
PlotLabel -> "List Plot of Transpose[data]",
GridLines -> Automatic,
Method -> "Frame" -> True,
Background -> Transparent,
PlotRangePadding -> 0.2,
ImageSize -> Large,
GridLinesStyle -> Directive[GrayLevel[0.7], Dashed],
PlotStyle -> {Directive[Orange, PointSize[0.03]],
Directive[Blue, PointSize[0.03]]}
]
The only way I would describe it is as the concept of superposition where a particle can be in multiple states at once until it's measured. It's sort of like posting emojis and doing facial recognition, and going from something like a placeholder to emojis that can link us to different types or groups of states, something we could graph where the direction of the arrows shows the direction of state evolution or time progression...so we can go from emojis to the visualization of quantum phenomena that are not yet experimentally verifiable but for which we have the side-by-side hypotheses based on these simulations in a thousand different avenues that allow us to advance our understanding of quantum mechanics and quantum computing; 500 generations and wow, my whale is talking to me! That's why these plots exist, there's no parallel word to describe what happens in Mathematica. All the founders of these quantum experiments, they use Mathematica whereas the low-level folks they all just use low-level programming languages.
Graphics3D[{
Sphere[{0, 0, 0}, 1],
Text[Style["0", 12], {0, 0, 1}],
Text[Style["1", 12], {0, 0, -1}],
Text[Style["+", 12], {1, 0, 0}],
Text[Style["\[Minus]", 12], {-1, 0, 0}],
Text[Style["+i", 12], {0, 1, 0}],
Text[Style["\[Minus]i", 12], {0, -1, 0}]},
Boxed -> True, Lighting -> "ThreePoint", ImageSize -> 300]
Now, I don't think I was quite able to generate these plots but here's the ball representing the three-point space in which we operate. So what we're going to get is the natural shape of the ball without any data added to it, it's a real live representation of how we can simulate quantum worlds which really just means the fabric of the cosmos, all those acres of land that can allow us to illustrate extensions to our research, like you said, where we can get Sigma5 level type inequality experiments such as Wigner's Friends experiments using the QuantumToMultiwaySystem resource function, so we've got some related function in the Resource Function Repository the only question is which one is it?
data = {
{0, 2},
{0, 4},
{(3 \[Pi])/8, 0},
{(3 \[Pi])/8, -2 Sqrt[2]},
{(2 \[Pi])/8, 1},
{(2 \[Pi])/8, -2},
{(4 \[Pi])/8, -1}
};
colors = {Red, Green, Blue, Purple};
plot1 = Plot[{Sin[2 k], 2 Cos[3 k], 3 Cos[4 k], 4 Sin[2 k]},
{k, 0, \[Pi]}, PlotStyle -> colors, Filling -> Axis,
PlotLegends ->
SwatchLegend[
colors, {"Sin[2k]", "2Cos[3k]", "3Cos[4k]", "4Sin[2k]"}],
FillingStyle -> Directive[Opacity[0.1], colors]];
colorsWarm = {Orange, Brown, Yellow, Pink};
plot2 = Plot[{Sin[k], Cos[k], Tan[k], Cot[k]},
{k, 0, \[Pi]}, PlotStyle -> colorsWarm, Filling -> Axis,
PlotLegends ->
SwatchLegend[colorsWarm, {"Sin[k]", "Cos[k]", "Tan[k]", "Cot[k]"}],
FillingStyle -> Directive[Opacity[0.1], colorsWarm]];
colorsCool = {Cyan, Magenta, LightBlue, DarkGreen};
plot3 = Plot[{2 (3 Cos[2 k] - Cos[3 2 k]), 4 Cos[2 k],
3 Cos[2 k] - Cos[3 2 k], 6 Cos[2 k]},
{k, 0, \[Pi]}, PlotStyle -> colorsCool, Filling -> Axis,
PlotLegends ->
SwatchLegend[
colorsCool, {"2(3Cos[2k]-Cos[3*2*k])", "4Cos[2k]",
"3Cos[2k]-Cos[3*2*k]", "6Cos[2k]"}],
FillingStyle -> Directive[Opacity[0.1], colorsCool]];
Show[plot1, plot2, plot3,
ListPlot[data, PlotStyle -> Directive[PointSize[Large], Black]],
Ticks -> {
{0.001 \[Pi], \[Pi]/8, \[Pi]/4, 3 \[Pi]/8, \[Pi]/2, 5 \[Pi]/8,
6 \[Pi]/8, 7 \[Pi]/8, \[Pi]},
{4 Sqrt[2], 2, 4, 2 Sqrt[2], -2 Sqrt[2], -4, -6}},
PlotRange -> All, ImageSize -> Large, AxesStyle -> Thick,
GridLines -> Automatic, GridLinesStyle -> Directive[Gray, Dashed]]
If there's one thing I really enjoyed about your post! I think it's these kinds of graphs, I don't know how many times I read your post and saw how we can describe a more general approach to explore quantum entanglement. Who is we? Well, we have this post about the four phases in the life cycle, of entanglement that are Completion, Propagation, Collapse, or Transfer. Maybe we could challenge the way that quantum mechanics is traditionally taught and put your polarization coincidence measurements between detector pairs into the limelight, the vectorization technique that it provides. And we've got all these string literals not to mention DarkGreen, that's the kind of everlasting color schemes that we might find in the function repository. It's sort of an inversion of the idea that we select everything at once and then pick out the different variables we want, reading different variable names, it's similar to learning Python in one week except when it's not. Because the changes here are more subtle. It's not just the complex patterns that suggest a lifecycle of entanglement, it's not just the collapse of the wave function. It's more like a completion with entanglement concluding in a collapse or transfer phase.
\[Alpha] = 2 \[Pi]/3;
\[Beta] = \[Pi]/4;
a = \[Pi]/8;
b = 0;
styledGrid = Grid[
{{"Polarizer Angle Names", Style["\[Alpha]", Italic],
Style["\[Beta]", Italic], Style["a", Italic], Style["b", Italic],
""},
{"Polarizer Angle Values", \[Alpha], \[Beta], a, b}},
BaseStyle -> {FontSize -> 12},
Frame -> All,
Background -> {None, {LightCyan, None}},
FrameStyle -> Directive[GrayLevel[0.7], Thickness[0.003]],
Alignment -> {Center, Center},
ItemSize -> {{Automatic, Automatic, Automatic, Automatic, Automatic,
2}},
Spacings -> {2, 1},
Dividers -> {{False, False, False, False, False, True}, {True,
True}}]
So what we get is this autonomous scientific endeavor to reconcile quantum mechanics with general relativity and see what these experiments cough up. I thought that the loops might symbolize things like states or conditions that recur, the theme of feedback unto ourselves, which is exactly what we need. How do we need it? I am an idiot. We can have a state lead to itself either directly or through a series of transitions. You could do something like the Force in Star Wars as a field of entanglement and bridge the gap between this phase and that next phase by exploring stuff like quantum decoherence and information storage. You should also do Schrödinger's cat and demonstrate how entangled particles exhibit a connection that transcends space. That would make things both educational and entertaining.
operator1 = {{1 + I, 1 - I}, {1 - I, 1 + I}};
basis = {{1, 0}, {0, 1}};
results = Table[
ResourceFunction["QuantumToMultiwaySystem"][
<|"Operator" -> operator1, "Basis" -> basis|> -> "Random",
{1 + I, 1 - I}, t, "EvolutionCausalGraphStructure"],
{initialConditions, initialConditionsList},
{t, 2, 10}]
And last but not least, it's not like we have some dark mode of matter we're just building upon a common feature of complex systems wherein a quantum system can evolve in multiple ways and lead to different possible outcomes and that's how we get the direction of time progression or state evolution...whether it's photons, atoms, or electrons..and we need some quantum optics equipment, detectors..it's like the mathematical walkthrough of how to tell the story of the state transitions and outcomes that take place by nature which really just means that it depends on how you define your experimental design. I was drawn in by the extraordinary interactions described in the present experimental setup, like the method for initiating state transitions and measuring outcomes as they interact with the theoretical predictions that we make.
