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The Qubit in de Broglie-Bohm Interpretation
Michel Gondran, Alexandre Gondran
To cite this version:
Michel Gondran, Alexandre Gondran. The Qubit in de Broglie-Bohm Interpretation. Advances in Quantum Information & Simulations, Nov 2014, Lyon, France. Book of Abstracts, 2014. �hal- 01348981�
The Qubit in de Broglie-Bohm Interpretation Michel Gondran
1and Alexandre Gondran
21
Académie Européenne Interdisciplinaire des Sciences, Paris, France
2
Ecole Nationale de l'Aviation Civile, Toulouse, France
How to explain spin's quantication ?
I Either by the measurement postulates of quantum theory
I Either by Pauli equation with spatial extension of the spinor Representation of the particle with spin
I Complete spinor with spatial extension
Ψ0(z) = (2πσ02)−14e−
z2 4σ2
0
cos θ20e−iϕ20 sin θ20eiϕ20
I Simplied spinor used in quantum information (qubit)
Ψ0 =
cos θ20e−iϕ20 sin θ20eiϕ20
.
Stern-Gerlach experiment
Ψ0(z) = (2πσ02)−14e−
z2 4σ2
0
cos θ20e−iϕ20 sin θ20eiϕ20
I Pure state : θ0 and ϕ0 xed
I Mixed states : θ0 and ϕ0 randomly drawn Pauli equation
I After the magnetic eld: at t + ∆t
Ψ(z, t + ∆t) ' (2πσ02)−14
cos θ20e−
(z−z∆−ut)2 4σ2
0 eimuz+~ ~ϕ+ sin θ20e−
(z+z∆+ut)2 4σ2
0 ei
−muz+~ϕ−
~
I Decoherence in Stern-Gerlach experiment
ρ(z, t + ∆t) ' (2πσ02)−1212 e−
(z−z∆−ut)2 2σ2
0 + e−
(z+z∆+ut)2 2σ2
0
!
−0.6 −0.3 0 0.3 0.6 mm
0 cm
−0.6 −0.3 0 0.3 0.6 mm
1 cm
−0.6 −0.3 0 0.3 0.6 mm
6 cm
−0.6 −0.3 0 0.3 0.6 mm
11 cm
−0.6 −0.3 0 0.3 0.6 mm
16 cm
−0.6 −0.3 0 0.3 0.6 mm
21 cm
I The decoherence time Spots N + and N − appear :
y = vt >16 cm
⇒ the decoherence time :
tD ' 3σ0 − z∆
u = (3σ0 − z∆)mv
µBB00 ∆l = 3 × 10−4s.
How is the transformation done ? Mixed states
θ0 and ϕ0 randomly drawn =⇒ Quantized mixture
θ0 = π and θ0 = 0 Measure quantization's postulates
orPauli equation with spinor spatial extension
Decoherence in Stern-Gerlach experiment
Quantization postulates' proof for Sz = 2~σz
Ψ(z, t + ∆t) ' (2πσ02)−14
cos θ20e−
(z−z∆−ut)2 4σ2
0 ei
muz+~ϕ+
~
sin θ20e−
(z+z∆+ut)2 4σ2
0 ei
−muz+~ϕ−
~
Experimentally, one measures the particle position z˜
I z˜1 ∈ N +
Ψ(˜z1, t + ∆t) ' (2πσ02)−14 cos θ0
2 e−
(˜z1−z∆−ut)2 4σ2
0 ei
muz˜1+~ϕ+
~
1
0
I z˜2 ∈ N −
Ψ(˜z2, t + ∆t) ' (2πσ02)−14 sin θ0
2 e−
(˜z2+z∆+ut)2 4σ2
0 ei
−muz˜2+~ϕ−
~
0
1
Marginal density matrix of spin variables of a pure state
ρS(z, t) = |ψ+(z, t)|2 ψ+(z, t)ψ−∗ (z, t) ψ−(z, t)ψ+∗ (z, t) |ψ−(z, t)|2
!
When t > tD :
ρS(z, t) ' |ψ+(z, t)|2 0
0 |ψ−(z, t)|2
!
Experimental results : z0 randomly drawn
0 5 10 15 20
−0.4
−0.2 0 0.2 0.4 0.6 0.8
y (cm)
z (mm)
0 5 10 15 20
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
y (cm)
z (mm)
Pure state (left) :θ0 = π/3 and ϕ0 = π/4
Mixed states (right) : θ0 ∈ [0; π] and ϕ0 ∈ [0; 2π] randomly drawn
Conclusion on quantum computer
I Spin-based qubit's existence?
Space and spin variables are not factorizable during treatment
I Chuang NMR results' explanation
Each wave function must be physically splitted because it
takes at least two particles to represent the quantum system
=⇒ Signal decay with a factor 2 for each additional qubit
I Statistical qubit (108 spins) exists but not individual qubit
=⇒ Quantum mechanics is not complete
ENS, Lyon, France, November 2014 [email protected] - [email protected] - http://alexandre.gondran.free.fr
