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9 votes
Accepted

Worst Bell inequality violation with non-maximally entangled state?

What you're after is Gisin's theorem. He proved that the maximal violation of the CHSH inequality with a fixed state $|\psi\rangle = c_0 |00\rangle + c_1|11\rangle$ is given by $2\sqrt{1+4 |c_0 c_1|^2}...
Mateus Araújo's user avatar
6 votes

How would a Quantum Computer (network) perform loophole-free Bell tests?

When people talk about a loophole free Bell test, what they really mean is that the two loopholes that most concern the majority of people are closed simultaneously: the measurement loophole and the ...
DaftWullie's user avatar
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6 votes

How to prepare an arbitrary two-qubit state?

GiannisKol is correct in an abstract sense -- you simply want to specify any unitary matrix with the first column containing elements $(\alpha,\beta,\gamma,\eta)$. You then complete the other columns ...
DaftWullie's user avatar
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6 votes
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What is an example of an entangled state whose correlations are describable with a local hidden variable model?

It's easy to generate such a model for specific cases. For example, take the maximally entangled state $|\phi^+\rangle = \frac1{\sqrt2} (|00\rangle + |11\rangle)$, and let the observables in the CHSH ...
Mateus Araújo's user avatar
6 votes

Quantinuum corporation says they have a Bell test for quantum randomness. Why does the simulator pass the test?

In order for Bell's theorem to apply, you need to factor your code into two pieces (an Alice piece and a Bob piece) that can run on separate computers with no messages being passed during the part of ...
Craig Gidney's user avatar
  • 37.6k
5 votes
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Difference between Bell's inequality and CHSH

The difference is as follows: the original Bell inequality requires that outcomes from the same setting are always perfectly anti-correlated. It says nothing about the case where they are even ...
DaftWullie's user avatar
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5 votes
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Definition of locality in Bell experiments

I think that I can explain the definition through the following simple example: Suppose that you perform two experiments in the same house in two separate rooms. In the first you measure the ...
David Bar Moshe's user avatar
5 votes

References examining Bell inequality violations at large distances

I think there is a conceptual thing going on here that needs clarifying (I'll leave the experimental links to others). I presume the question is predicated on the idea that, well, measurements are ...
DaftWullie's user avatar
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5 votes
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Projecting $\lvert ++ \rangle$ on Bell Basis

The Hadamard gate is: $$\frac{1}{\sqrt 2} \left(|0\rangle \langle 0 | + |0\rangle\langle 1| + |1\rangle \langle 0| - |1\rangle \langle 1|\right)$$ And since $|+\rangle$ is $\frac{1}{\sqrt 2}\left(|0\...
Mahathi Vempati's user avatar
5 votes
Accepted

Simultaneous eigenstate of commuting observables and their tensor product

Measuring an observable does not mean applying the observable operator to a quantum state but rather measuring the state in the eigenbasis of the operator. A measurement will basically produce an ...
Sanchayan Dutta's user avatar
5 votes
Accepted

Formal proof of Bell's inequality in CHSH form

I want to emphasize that you are wasting your time with this historical stuff. But since you insist I'll answer anyway. First of all, note that the inequality \begin{equation} \left|\int d\lambda p(\...
Mateus Araújo's user avatar
5 votes
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What is the no-signaling set and how can it be related to other types of correlations?

I'll describe the case of two party correlations but this can straightforwardly extended to more parties. Let's give a box to Alice and a box to Bob. Alice and Bob can interact with their boxes by ...
Rammus's user avatar
  • 5,853
4 votes

How is Bell’s Inequality converted to the CHSH inequality?

... and the question changes again. Re Update 4: I'm not sure what you intended the variable $a$ to be in the question (I think part of the problem is that you've got muddled between the chosen ...
DaftWullie's user avatar
  • 58.7k
4 votes

Determining whether $P(ab|xy)$ factorizes in Bell experiments

I think you're doing things a little bit backwards. You probably shouldn't be calculating $P(a|x)$ or $P(b|y)$ in advance, because you're simply trying to ask: Given a set of $\{P(ab|xy)\}$, do ...
DaftWullie's user avatar
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4 votes
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Why only one of the qubits is rotated in the CHSH circuit?

In an actual implementation of the CHSH game, both Alice and Bob must conditionally rotate their qubit. If one of them doesn't do any conditional operations to the qubit, then all the actions to the ...
Craig Gidney's user avatar
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4 votes
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Is it advantageous for a state to have higher Bell CHSH violation?

Yes, the CHSH game/scenario is an example of a robust self-test. What this means is that if you obtain a value that is close to the optimal Bell violation, then the state $\hat{\rho}$ and the ...
Condo's user avatar
  • 2,048
3 votes

Projecting $\lvert ++ \rangle$ on Bell Basis

The four Bell states are $$ |\Phi_{\pm}\rangle=(|00\rangle\pm|11\rangle)/\sqrt{2}\qquad |\Psi_{\pm}\rangle=(|01\rangle\pm|10\rangle)/\sqrt{2}. $$ So, let's consider what happens then we try and ...
DaftWullie's user avatar
  • 58.7k
3 votes

How to prepare an arbitrary two-qubit state?

First of all, I'm assuming you mean the state : $\alpha \left|00\right> + \beta \left|01\right> + \gamma\left|10\right> + \eta \left|11\right>$. What you actually want to do is to act with ...
GiannisKol's user avatar
3 votes
Accepted

Find the Bell States of A and B in the following scenario

Simply start by writing out everything $$ |B_{00}\rangle_{13}|B_{00}\rangle_{24}=\frac12\left(|00\rangle_{13}|00\rangle_{24}+|00\rangle|11\rangle+|11\rangle|00\rangle+|11\rangle|11\rangle\right) $$ ...
DaftWullie's user avatar
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3 votes

Are there Bell-like violations that can be observed without collecting statistics?

Yes, there are. I just wrote a paper about it, actually. You need to define carefully what you mean by obtaining a Bell violation or ruling out local hidden variables. You can't demand to have a ...
Mateus Araújo's user avatar
3 votes

Are there Bell-like violations that can be observed without collecting statistics?

I believe you're thinking of the all-versus-nothing proofs based on GHZ states. You start with a state such as $$ |\Psi\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle) $$ and you select at random ...
DaftWullie's user avatar
  • 58.7k
3 votes
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Are there Bell-like violations that can be observed without collecting statistics?

Are there scenarios in which Bell nonlocality can be observed without such averages, that is, in a single-shot scenario? No, you have to collect statistics. Any single result you see could have been ...
Craig Gidney's user avatar
  • 37.6k
3 votes

References examining Bell inequality violations at large distances

The largest scale Bell test done thus far is the "Cosmic Bell Test" of 2017. It ruled out hidden variables within a distance of 600 light years from Earth. The 16 significant Bell test experiments ...
user1271772 No more free time's user avatar
3 votes

In Bell's inequalities, what is the meaning of assuming that the physical properties $P_Q,P_R,P_S,P_T$ have definite values?

This, in a nutshell, is the whole issue of quantum. You say "of course an electron will have either spin up or spin down". It might seem like it should, but this is founded purely on your ...
DaftWullie's user avatar
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3 votes

What are the 2022 Nobel Prize winning papers of Alain Aspect, John Clauser, and Anton Zeilinger, and how do they fit into the timeline of QIS?

Well, a sidebar in Quanta magazine's article had already created a nice timeline on the experiments and events leading up to the 2022 Nobel Prize in physics. Also as identified by @Rammus in a comment,...
2 votes
Accepted

Why is $P(1,2)_{\text{same}} = \frac{1}{4}$ and not $\frac{1}{2}$ in Preskill's Bell experiment?

I do understand that the sum of these three probabilities is greater than one because there are some constraints already involved; like if we uncover all three coins at least two have to be the ...
DaftWullie's user avatar
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2 votes
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What would the CHSH inequality be equal to if the two qubits were separable?

First, is this value correct? Yes, it is. If you expand out the calculation you're doing, this is the same as $$ \sqrt{2}\langle\psi|X\otimes X+Z\otimes Z|\psi\rangle $$ for any two-qubit state $|\...
DaftWullie's user avatar
  • 58.7k
2 votes
Accepted

What are the two measurements per box being done in the Bell Tests in the IBM Quantum Experience "Entanglement and Bell tests" Section

Both $A$ and $A'$ measure the same qubit, while $B$ and $B'$ are applied to the second qubit. Alice chooses (randomly) which of the two measurement settings ($A$ or $A'$) she will choose for each run ...
M. Stern's user avatar
  • 2,432
2 votes
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Why is the CHSH inequality defined with a minus sign?

Imagine you had a general formula $$ C=a_1QS+a_2RS+a_3RT+a_4QT. $$ Algebraically, we know that if $Q$, $S$, $R$ and $T$ are random variables with values $\pm 1$, then each term such as $QS\in\{\pm 1\}$...
DaftWullie's user avatar
  • 58.7k
2 votes
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Problem about entanglement swapping

I guess the way that I'd start (aside from just getting a computer to do it!) is to remember that the Bell states form an orthonormal basis. So, you can ask, for example, about what the $|\Phi^+\...
DaftWullie's user avatar
  • 58.7k

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