# Joint Distributions and Quantum Nonlocal Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background on Probability and Joint Distributions

^{2}= A, B

^{2}= B. Next, a 2×2 table of nonnegative values over the four pairs of zero/one outcomes will define a joint distribution function if all the values are in the interval [0, 1] and if the assignments to the four cells add up to 1. That’s all that is required for the construction of an arbitrary classical joint distribution, Pr(a, b) = Pr(A = a, B = b), and there are generally an infinite number of ways of assigning probabilities to the cells.

Pr(B|A) = tr[DABA]/tr[DA]

Pr(B|A) = tr[DABA]/tr[DA] × (tr[DA])

## 3. Joint Measurements or Joint Distributions?

## 4. Quantum Outcomes and Joint Distributions

- (i)
- both marginal probabilities for A and B, those on the left in Equation (2), agree with the Born trace-probability rule values given on the right in Equation (2);
- (ii)
- both conditional probabilities, A given B, and B given A, those on the left on Equation (4), agree with the conditional probability rule values given on the right at Equation (4).

^{+}, B

^{+}}, {A

^{+}, B

^{−}}, {A

^{−}, B

^{+}}, {A

^{−}, B

^{−}}.

_{B}= BDB/tr[DB]. Hence, if Condition (i) is valid then Condition (ii) is experimentally testable using just marginal outcomes.

## 5. Joint Distributions and Commutativity

_{i}P

_{i}, A − B = ∑ζ

_{i}P

_{i}, CC* = ∑λ

_{i}ζ

_{i}P

_{i}

_{i}, and eigenvalues λ

_{i}ζ

_{i}. Let S(CC*) be the set of all those projectors P

_{i}appearing nontrivially in the decomposition for CC*, so for such i: λ

_{i}ζ

_{i}> 0. In particular, if CC* = G(A − B) ≠ 0 then there exists at least one projector P

_{i}such that tr[P

_{i}CC*] = λ

_{i}ζ

_{i}≠ 0. For a system in state D = P

_{i}it follows from [9; Proof of Theorem 1, Case (a) and Case (b)] that AP

_{i}≠ 0, BP

_{i}≠ 0. Hence tr[AP

_{i}] = tr[(AP

_{i})*(AP

_{i})] ≠ 0 and tr[BP

_{i}] = tr[(BP

_{i})*(BP

_{i})] ≠ 0. Thus under the assumption that A, B have a joint distribution for the system in state D = P

_{i}, the univariate marginals for both A and B are nonvanishing. Therefore, for D = P

_{i}:

_{i}} as in Equation (9). If A and B are quantum consistent for every state of the form P = ∑δ

_{i}P

_{i}with all δ

_{i}> 0, and P

_{i}in S(CC*), then A and B commute.

_{i}ζ

_{i}δ

_{i}> 0. Hence

_{i}P

_{i}(A − B)G] = ∑λ

_{i}δ

_{i}tr[P

_{i}G] > 0

_{i}, and using (11), it follows that tr[P

_{i}G] = 0, for all i. But this contradicts Equation (12), so only G(A − B) = 0 is possible, and the result follows.

## 6. The Leggett-Branciard Nonlocal Model

**a**,

**b**, directions in two separated labs. Using the notation in [7] the model assumes that the probability of the event (α, β) is given by

**a**,

**b**,

**b′**) of spin measurements, where measurements in the

**b**,

**b′**directions are made in one lab and

**a**is made in a separated lab. That is:

## 7. The Leggett-Branciard Model and Joint Distributions

_{A}(a), while the two joint distributions are f

_{A,B}(a, b) and f

_{A,B′}(a, b′), then a valid joint distribution for the triple is given by

_{A,B,B′}(a, b, b′) = f

_{A,B}(a, b)f

_{A,B′}(a, b′)/f

_{A}(a)

_{1}, A

_{2}). In this case a joint distribution over (A

_{1}, A

_{2}, B, B′) is given by an appropriately notated, right hand side of Equation (15). And then integrating A

_{1}with respect to a suitable density returns a joint distribution for (A

_{2}, B, B′).

_{λ}(α, β, β′|

**a**,

**b**,

**b′**) = {P

_{λ}(α, β|

**a**,

**b**)P

_{λ}(α, β′|

**a**,

**b′**)}/{P

_{λ}(α|

**a**)}

_{1}, φ

_{2}, in a decomposition for G = G(B, B′) with eigenvalues λ

_{1}, λ

_{2}. If both are zero then G = 0 so B and B′ must commute. So suppose λ

_{1}≠ 0 and let the system state be φ = φ

_{1}⊗ φ

_{2}. Write $\tilde{A}=A\otimes I$ and $\tilde{B}=B\otimes I$, and $\tilde{G}=G(\tilde{B},\tilde{{B}^{\prime}})=G\otimes I$. It is straightforward to show that if (B, B′) have a joint distribution then so must $(\tilde{B},\tilde{{B}^{\prime}})$. But then $\tilde{G}\phi ={\lambda}_{1}\phi \ne 0$ and this contradicts $tr\left[D\tilde{G}\right]=0$ for state D = |φ〉〈φ|, using Equation (5.4). Hence $\tilde{G}=0$, and also G = 0. Therefore A and B commute. Since the argument applies to any pair of observables under the model at Equation (13) with no-signaling, it follows that all local observables commute.

_{1}, φ

_{2}, and consider the entangled states

## 8. Conclusions

## Disclaimer

## Author Contributions

## Acknowledgements

## References

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**MDPI and ACS Style**

Malley, J.D.; Fletcher, A.
Joint Distributions and Quantum Nonlocal Models. *Axioms* **2014**, *3*, 166-176.
https://doi.org/10.3390/axioms3020166

**AMA Style**

Malley JD, Fletcher A.
Joint Distributions and Quantum Nonlocal Models. *Axioms*. 2014; 3(2):166-176.
https://doi.org/10.3390/axioms3020166

**Chicago/Turabian Style**

Malley, James D., and Anthony Fletcher.
2014. "Joint Distributions and Quantum Nonlocal Models" *Axioms* 3, no. 2: 166-176.
https://doi.org/10.3390/axioms3020166