You are using a browser not supported by the Wolfram Cloud
Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.
I understand and wish to continue anyway »
WOLFRAM NOTEBOOK
On quantum amplitudes, correlations and negativity
Introduction
Introduction
This piece is my short attempt at a deeper understanding of some of the most problematic parts of quantum theory concerning quantum computing . Starting from a particular example of the CHSH game, it goes through a diagrammatic representation of quantum theory, non-classical correlations, the emergence of negative probabilities, a multiway version of the Many Worlds interpretation, the nature of qubits, anti-particles, and the role of quantum resources in beyond quantum theories .
Thepathtoquantumenlightenment@SabineHossenfelder
In the CHSH game, there are two cooperating players, Alice and Bob, and a referee, Charlie. At the start of the game, Charlie chooses bits uniformly at random, and then sends to Alice and to Bob. Alice and Bob must then each respond to Charlie with bits respectively. Now, once Alice and Bob send their responses back to Charlie, Charlie tests if . If this equality holds, then Alice and Bob win, and if not then they lose.
x,y∈{0,1}
x
y
a,b∈{0,1}
a⊻b=x∧y
We can easily construct a circuit representing the setup of this game using the Quantum Framework:
PacletInstall["Wolfram/QuantumFramework"]<<Wolfram`QuantumFramework`
Out[]=
Let’s compute the joint probability distribution for all possible outcomes of the experiment:
In[]:=
measurement=chsh[]
Out[]=
QuantumMeasurement
Out[]=
We can also get the underlying multivariate categorical distribution:
In[]:=
distribution=measurement["MultivariateDistribution"]
Out[]=
CategoricalDistribution
Computing probability of winning now is as straight-forward as taking an expectation of the stated criteria :
a⊻b=x∧y
In[]:=
probabilityOfWinning=Simplify@Expectation[Boole[BitXor[a,b]==BitAnd[x,y]],{a,x,y,b}distribution]
Out[]=
1
4
2
)In[]:=
N[probabilityOfWinning]
Out[]=
0.853553
We get a very well known result of achieving ~85% probability of winning using this quantum strategy.
Ok, but what’s the big deal here?
This game is one of the simplest examples of beyond classical correlations (in the Appendix it is shown that no classical strategy can’t reach more than 75% probability of winning at this game).
This game is one of the simplest examples of beyond classical correlations (in the Appendix it is shown that no classical strategy can’t reach more than 75% probability of winning at this game).
It is widely believed that complex numbers are the cause of all the quantum weird behaviour. But actually most of quantum features can just be explained with the right interpretation and classical probability theory (cf. epistricted complementarity, Spekkens toy model, generalize probability theory etc.), but not the case of violating CHSH inequalities and reaching that 85% winning rate in this game.
Imaginary components do not survive until one gets to the actual probability distribution of any real experimental outcome.
So how far one can get with a theory that only operates on probability distributions? Not far enough! And the missing ingredient to get as far as quantum theory are negative probabilities. To which we’ll get later, but first let’s understand how probability distributions differ from quantum states with their complex amplitudes.
Imaginary components do not survive until one gets to the actual probability distribution of any real experimental outcome.
So how far one can get with a theory that only operates on probability distributions? Not far enough! And the missing ingredient to get as far as quantum theory are negative probabilities. To which we’ll get later, but first let’s understand how probability distributions differ from quantum states with their complex amplitudes.
Classical vs Quantum wires
Classical vs Quantum wires
Multiway picture of entanglement
Multiway picture of entanglement
CHSH
CHSH
Negative outcomes
Negative outcomes
Wigner basis and phase space
Wigner basis and phase space
Beyond quantum correlations
Beyond quantum correlations
Appendix
Appendix
Appendix A: Classical CHSH
Appendix A: Classical CHSH
Appendix B: Computing CHSH correlation factor
Appendix B: Computing CHSH correlation factor
Outtakes
Outtakes