Quantum Nonlocality Pt. 3: The CHSH Inequality

PHYS 500 - Southern Illinois University

May 3, 2018

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 1 / 10

LHV/QC/NS Correlations

In general, a set of correlations {p(a, b|x , y)} are called local hiddenvariable (LHV) correlations if they can be written as

p(a, b|x , y) =∑λ

p(λ)pA(a|x , λ)pB(b|y , λ) ∀a, b, x , y

for some variable λ with distribution p(λ). Here pA(a|x , λ)pB(b|y , λ) is aproduct of local probability distributions.

The correlations are called quantum correlations (QC) if there exists a

bipartite quantum state ρAB as well as quantum operations {A(x)a }a∈A and

{B(y)b }b∈B for all (x , y) ∈ X × Y such that

p(a, b|x , y) = tr [(A(x)a ⊗ B

(y)b )ρAB(A

(x)a ⊗ B

(y)b )†].

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 2 / 10

LHV/QC/NS Correlations

The correlations are called non-signaling (NS) if∑a

p(a, b|x , y) =∑a

p(a, b|x , y ′) b, x , y , y ′∑b

p(a, b|x , y) =∑b

p(a, b|x ′, y) a, y , x , x ′.

What is the relationship between these classes of correlations?

We have already seen that QC ⊂ NS.

What is the relationship between LHV and QC?

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 3 / 10

LHV ⊂ QC

Do all quantum states generate correlations in LHV?

No. This was originally proven by John Bell in the 1960’s. Here we willshow the result by using a simpler argument originally constructed byClauser, Horn, Shimony, and Holt (CHSH) several years later.

The argument works by first deriving a mathematical condition that allLHV correlations must satisfy (called the CHSH Inequality).

This inequality is then shown to be broken by certain entangled quantumstates.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 4 / 10

The CHSH Inequality

The CHSH Inequality the following linear combination of conditionalprobabilities:

p(00|00)− p(01|01)− p(10|10)− p(00|11).

This is called the CHSH expression.

What is the max/min value for the CHSH expression using LHVcorrelations?

Consider any LHV correlations

p(a, b|x , y) =∑λ

p(λ)pA(a|x , λ)pB(b|y , λ).

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 5 / 10

The CHSH Inequality

The CHSH expression takes the form∑λ

p(λ)

(pA(0|0, λ)pB(0|0, λ)− pA(0|0, λ)pB(1|1, λ)

−pA(1|1, λ)pB(0|0, λ)− pA(0|1, λ)pB(0|1, λ)

).

Since pA(1|1, λ) = 1− pA(0|1, λ) and pB(1|1, λ) = 1− pB(0|1, λ), eachterm in the parentheses takes the form

a0b0 − a0(1− b1)− (1− a1)b0 − a1b1

where 0 ≤ ai , bi ≤ 1.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 6 / 10

The CHSH Inequality

To optimize this expression, we first write it as

a0b0−a0(1−b1)− (1−a1)b0−a1b1 = a0b0−a0− (1−a1)b0 +(a0−a1)b1.

Case a0 ≥ a1: Then the expression increases/decreases as b1increases/decreases. Thus

a0b0 − a0(1− b1)− (1− a1)b0 − a1b1 ≤ b0(a0 − 1) + a1(b0 − 1) ≤ 0

a0b0 − a0(1− b1)− (1− a1)b0 − a1b1 ≥ a0(b0 − 1)− (1− a1)b0

≥ (b0 − 1)− (1− a1)b0 ≥ −1.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 7 / 10

The CHSH Inequality

Case a0 ≤ a1: Then the expression a0b0 − a0 − (1− a1)b0 + (a0 − a1)b1decreases/increases as b1 increases/decreases. thus

a0b0 − a0(1− b1)− (1− a1)b0 − a1b1 ≤ a0(b0 − 1)− (1− a1)b0 ≤ 0

a0b0 − a0(1− b1)− (1− a1)b0 − a1b1 ≥ b0(a0 − 1) + a1(b0 − 1)

≥ b0(a0 − 1) + (b0 − 1) ≥ −1.

We have therefore derived the LHV inequalities:

−1 ≤ a0b0 − a0(1− b1)− (1− a1)b0 − a1b1 ≤ 0.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 8 / 10

The CHSH Inequality

Since these inequalities hold for all λ, we have

−1 ≤∑λ

p(λ)

(pA(0|0, λ)pB(0|0, λ)− pA(0|0, λ)pB(1|1, λ)

−pA(1|1, λ)pB(0|0, λ)− pA(0|1, λ)pB(0|1, λ)

)≤ 0.

This is called the CHSH Inequality (probability form).

If p(ab|xy) are LHV correlations then

−1 ≤ p(00|00)− p(01|01)− p(10|10)− p(00|11) ≤ 0.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 9 / 10

Violation of the CHSH Inequality

Let us now show QC that violate the CHSH Inequality.

Consider the Bell state |Φ+〉 =√

1/2(|00〉+ |11〉) and the local

measurements {A(x)0 ,A

(x)1 } and {B(y)

0 ,B(y)1 } for x , y ∈ {0, 1}, where

A(x)a = |α̂(x)

a 〉〈α̂(x)a | and B

(y)b = |β̂(y)b 〉〈β̂

(y)b | are projectors with

|α̂(0)0 〉 = |+〉, |α̂(0)

1 〉 = |−〉

|α̂(1)0 〉 = |0〉, |α̂(1)

1 〉 = |1〉

|β̂(0)0 〉 = cos π8 |0〉+ sin π

8 |1〉, |β̂(0)1 〉 = sin π8 |0〉 − cos π

8 |1〉

|β̂(1)0 〉 = sin π8 |0〉 − cos π

8 |1〉, |β̂(1)1 〉 = cos π8 |0〉 − sin π

8 |1〉

These measurements generate a CHSH value of 1√2− 1

2 > 0.

PHYS 500 - Southern Illinois University Quantum Nonlocality Pt. 3: The CHSH Inequality May 3, 2018 10 / 10