Page 72 - GHES-2-1

Global Health Econ Sustain Quantum Data Lake for epidemic analysis

Table 11. Equations and matrices presented in Section 3.4.2 Table 11. (Continued)
PT‑symmetry
0 − i 0 0 
+
†
H PT =   0 mg  ≠ H (XXI) p PTbroken mγ 5 DIRAC =   0 0 i 0   (XXV.III)
 − 0  mg µ 0 0 00 
 
i  0 0  0
a  + ia b + ib 
H PT =  1 2 1 2  ≠ H † (XXII)
 b − ib 2 a − ia 2  000 0 
1
1
 00 i 0 
 ig k  p µ PTbroken mγ 5 DIRAC =  000 − i  (XXV.IV)
H PT =   k 1 ig 2   ≠ H † (XXII.I)  i  00 0  

 ig k  0 m + m 2 0 0 
1
†
H PT =   k − ig   ≠ H (XXII.II) p γ 5WEYL =   0 0 0 0 0 m − 0 m    (XXVI.I)
0
µ
b
Im
Re
a +
H PT =    Re a − Im b − Re a + Im b   (XXII.III)   0 0 0 1 0 2  
b
a +
Re
Im
0 i 0  0
θ   θ    0000 
|ψ cos   |0 + sin   e ϕi |1 (XXIII) p µ PT mγ 5WEYL =  000 i   (XXVI.II)
2  
2  
 
 0000 
2
e
1  i α /   r 
|ψ + =   α =  sin θ 0 i 0 0 

2 cosα e − αi /2     bc   (XXIV.I)  000 0 

p µ PTbroken mγ 5WEYL =  000 − i  (XXVI.III)
2
1  e − αi /   r    000 0  
|ψ − =   α =  sin θ
2
2 cosα   −e α i /     bc   (XXIV.II) [PT, H APT ] = PTH APT +H APT PT=0 (XXVII.I)
PT
[PT, H ] = PTH – H PT=0 (XXVII.II)
PT
PT
 re iθ b 
H PT =   ≠ H † APT  re iθ b 
  c re i −θ   (XXIV.III) H = i    c re − θ  (XXVIII.I)
i 

re iθ  be iϕ   − ig ik
H PT =   ≠ H † H APT =
  ce i −ϕ e r i −θ   (XXIV.IV)   ik ig   (XVIII.II)
 0 − p + m 1 0 0  APT  − ig − ik
0
 0 0 − p + m 0  H =  − ik  (XXVIII.III)
p γ 5DIRAC =   0 0 3 0 2 p + m   (XXV.I)  ig 
µ
 0 1 
p +
 3 m 2 0 0 0 
Hermitian Hamiltonian as presented in Equation XXIV.
0 i 0  0 IV (Table 11), where r, b, c, θ, and φ are real. For PT-qubit
 00 i 0  non-Hermitian Hamiltonian, PT-symmetry is unbroken
2
p µ PT mγ 5DIRAC =  000 i   (XXV.II) if bc > r sin θ. Dogra et al. (2021) demonstrated the
2
  PT-symmetry breaking for a single-qubit (phase transition
i  000  is associated with a loss of state distinguishability).
Naghiloo et al. (2019) described the evolution of qubit in
(Cont’d...) the broken and unbroken regimes; in the PT-symmetric
Volume 2 Issue 1 (2024) 21 https://doi.org/10.36922/ghes.2148