1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
|
---
title: "GHZ paradox"
firstLetter: "G"
publishDate: 2021-03-29
categories:
- Physics
- Quantum mechanics
- Quantum information
date: 2021-03-29T15:15:41+02:00
draft: false
markup: pandoc
---
# GHZ Paradox
The **Greenberger-Horne-Zeilinger** or **GHZ paradox**
is an alternative proof of [Bell's theorem](/know/concept/bells-theorem/)
that does not use inequalities,
but the three-particle entangled **GHZ state** $\ket{\mathrm{GHZ}}$ instead,
$$\begin{aligned}
\boxed{
\ket{\mathrm{GHZ}}
= \frac{1}{\sqrt{2}} \Big( \ket{000} + \ket{111} \Big)
}
\end{aligned}$$
Where $\ket{0}$ and $\ket{1}$ are qubit states,
for example, the eigenvalues of the Pauli matrix $\hat{\sigma}_z$.
If we now apply certain products of the Pauli matrices $\hat{\sigma}_x$ and $\hat{\sigma}_y$
to the three particles, we find:
$$\begin{aligned}
\hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x \ket{\mathrm{GHZ}}
&= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_x \ket{0}
+ \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_x \ket{1} \Big)
\\
&= \frac{1}{\sqrt{2}} \Big( \ket{1} \otimes \ket{1} \otimes \ket{1} + \ket{0} \otimes \ket{0} \otimes \ket{0} \Big)
= \ket{\mathrm{GHZ}}
\\
\hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y \ket{\mathrm{GHZ}}
&= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_y \ket{0} \otimes \hat{\sigma}_y \ket{0}
+ \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_y \ket{1} \otimes \hat{\sigma}_y \ket{1} \Big)
\\
&= \frac{1}{\sqrt{2}} \Big( \ket{1} \otimes i \ket{1} \otimes i \ket{1} + \ket{0} \otimes i \ket{0} \otimes i \ket{0} \Big)
= - \ket{\mathrm{GHZ}}
\end{aligned}$$
In other words, the GHZ state is a simultaneous eigenstate of these composite operators,
with eigenvalues $+1$ and $-1$, respectively.
Let us introduce two other product operators,
such that we have a set of four observables,
for which $\ket{\mathrm{GHZ}}$ gives these eigenvalues:
$$\begin{aligned}
\hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x
\quad &\implies \quad +1
\\
\hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y
\quad &\implies \quad -1
\\
\hat{\sigma}_y \otimes \hat{\sigma}_x \otimes \hat{\sigma}_y
\quad &\implies \quad -1
\\
\hat{\sigma}_y \otimes \hat{\sigma}_y \otimes \hat{\sigma}_x
\quad &\implies \quad -1
\end{aligned}$$
According to any local hidden variable (LHV) theory,
the measurement outcomes of the operators are predetermined,
and the three particles $A$, $B$ and $C$ can be measured separately,
or in other words, the eigenvalues can be factorized:
$$\begin{aligned}
\hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x
\quad &\implies \quad +1 = m_x^A m_x^B m_x^C
\\
\hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y
\quad &\implies \quad -1 = m_x^A m_y^B m_y^C
\\
\hat{\sigma}_y \otimes \hat{\sigma}_x \otimes \hat{\sigma}_y
\quad &\implies \quad -1 = m_y^A m_x^B m_y^C
\\
\hat{\sigma}_y \otimes \hat{\sigma}_y \otimes \hat{\sigma}_x
\quad &\implies \quad -1 = m_y^A m_y^B m_x^C
\end{aligned}$$
Where $m_x^A = \pm 1$ etc.
Let us now multiply both sides of these four equations together:
$$\begin{aligned}
(+1) (-1) (-1) (-1)
&= (m_x^A m_x^B m_x^C) (m_x^A m_y^B m_y^C) (m_y^A m_x^B m_y^C) (m_y^A m_y^B m_x^C)
\\
-1
&= (m_x^A)^2 (m_x^B)^2 (m_x^C)^2 (m_y^A)^2 (m_y^B)^2 (m_y^C)^2
\end{aligned}$$
This is a contradiction: the left-hand side is $-1$,
but all six factors on the right are $+1$.
This means that we must have made an incorrect assumption along the way.
Our only assumption was that we could factorize the eigenvalues,
so that e.g. particle $A$ could be measured on its own
without an "action-at-a-distance" effect on $B$ or $C$.
However, because that leads us to a contradiction,
we must conclude that action-at-a-distance exists,
and that therefore all LHV-based theories are invalid.
## References
1. N. Brunner,
*Quantum information theory: lecture notes*,
2019, unpublished.
2. J.B. Brask,
*Quantum information: lecture notes*,
2021, unpublished.
|
