Decoherence Effects in Qubits Projektpraktikum Peter Wriesnik - - PowerPoint PPT Presentation

▶
decoherence effects in qubits
SMART_READER_LITE
LIVE PREVIEW

Decoherence Effects in Qubits Projektpraktikum Peter Wriesnik - - PowerPoint PPT Presentation

Feb 05, 2023 157 likes •1.29k views

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence Effects in Qubits Projektpraktikum Peter Wriesnik Institute of Theoretical and Computation Physics Graz University of Technology November 28,

slide-1
SLIDE 1

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence Effects in Qubits

Projektpraktikum Peter Wriesnik

Institute of Theoretical and Computation Physics Graz University of Technology

November 28, 2012

slide-2
SLIDE 2

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

slide-3
SLIDE 3

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits

slide-4
SLIDE 4

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

slide-5
SLIDE 5

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

slide-6
SLIDE 6

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix

slide-7
SLIDE 7

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

slide-8
SLIDE 8

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

slide-9
SLIDE 9

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation

slide-10
SLIDE 10

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

slide-11
SLIDE 11

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

slide-12
SLIDE 12

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels

slide-13
SLIDE 13

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Outline

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-14
SLIDE 14

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-15
SLIDE 15

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing

slide-16
SLIDE 16

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing → has to have 2 distinct states

slide-17
SLIDE 17

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing → has to have 2 distinct states Any 2-level-system could be used. Examples: Photon polarization

slide-18
SLIDE 18

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing → has to have 2 distinct states Any 2-level-system could be used. Examples: Photon polarization Energy levels in molecules / atoms

slide-19
SLIDE 19

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing → has to have 2 distinct states Any 2-level-system could be used. Examples: Photon polarization Energy levels in molecules / atoms Spin of an electron

slide-20
SLIDE 20

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Qubits in Quantum Computing

Qubits: analogon to bits in classical computing → has to have 2 distinct states Any 2-level-system could be used. Examples: Photon polarization Energy levels in molecules / atoms Spin of an electron Logical operations (NOT, OR, ...) have to be performed on the qubit

slide-21
SLIDE 21

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Mathematical Description of Qubits

Qubits are elements of a 2-dimensional Hilbert-space H2:

slide-22
SLIDE 22

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Mathematical Description of Qubits

Qubits are elements of a 2-dimensional Hilbert-space H2: |ψ = a(t) |↑ + b(t) |↓

slide-23
SLIDE 23

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Mathematical Description of Qubits

Qubits are elements of a 2-dimensional Hilbert-space H2: |ψ = a(t) |↑ + b(t) |↓ → coefficients a(t), b(t) hold dynamics

slide-24
SLIDE 24

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Mathematical Description of Qubits

Qubits are elements of a 2-dimensional Hilbert-space H2: |ψ = a(t) |↑ + b(t) |↓ → coefficients a(t), b(t) hold dynamics → could be denoted as

  • a(t), b(t)

T

slide-25
SLIDE 25

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

2-Level-System

Hamiltonian for Qubits ˆ H0 = ω0 |↑ ↑| assigns energy ω0 to spin-up-state and 0 to spin-down

slide-26
SLIDE 26

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

2-Level-System

Hamiltonian for Qubits ˆ H0 = ω0 |↑ ↑| assigns energy ω0 to spin-up-state and 0 to spin-down solution: a(t) = e−iω0t b(t) = 0

  • Larmor precession
slide-27
SLIDE 27

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-28
SLIDE 28

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Description of EM-Field

Hamiltonian with electromagnetic field ˆ H = ˆ H0 + ˆ HI(t) ˆ HI(t) = f (t) |↑ ↓| + f ∗(t) |↓ ↑| with f (t) = Qe−iǫt

slide-29
SLIDE 29

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Description of EM-Field

Hamiltonian with electromagnetic field ˆ H = ˆ H0 + ˆ HI(t) ˆ HI(t) = f (t) |↑ ↓| + f ∗(t) |↓ ↑| with f (t) = Qe−iǫt Matrix representation in the {|↑ , |↓}-basis: H(t) =

  • ω0

Q · e−iǫt Q · eiǫt

slide-30
SLIDE 30

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Solution to time dependent Hamiltonian

Plug into Schr¨

  • dinger equation for state
  • a(t), b(t)

T:

slide-31
SLIDE 31

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Solution to time dependent Hamiltonian

Plug into Schr¨

  • dinger equation for state
  • a(t), b(t)

T: i d dt a(t) = ω0 + f (t) · b(t) i d dt b(t) = f ∗(t) · a(t)

slide-32
SLIDE 32

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Solution to time dependent Hamiltonian

Plug into Schr¨

  • dinger equation for state
  • a(t), b(t)

T: i d dt a(t) = ω0 + f (t) · b(t) i d dt b(t) = f ∗(t) · a(t) Solution considering EM-field with |φ(t = 0) = |↑ a(t) = e− i

2 (ǫ+ω0)t ·

  • cos(βt) + i α

β sin(βt)

  • b(t) = −Q · i

β e−i ∆

2 t sin(βt)

slide-33
SLIDE 33

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate

slide-34
SLIDE 34

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate

Resonant case: ǫ = ω0 → ∆ = α = 0 and β = Q.

slide-35
SLIDE 35

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate

Resonant case: ǫ = ω0 → ∆ = α = 0 and β = Q. a(t) = e−iω0t cos(Qt) b(t) = −i sin(Qt)

slide-36
SLIDE 36

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate

Resonant case: ǫ = ω0 → ∆ = α = 0 and β = Q. a(t) = e−iω0t cos(Qt) b(t) = −i sin(Qt) Choose interaction time τ =

π 2Q → a(τ) = 0, b(τ) = −i

slide-37
SLIDE 37

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate

Resonant case: ǫ = ω0 → ∆ = α = 0 and β = Q. a(t) = e−iω0t cos(Qt) b(t) = −i sin(Qt) Choose interaction time τ =

π 2Q → a(τ) = 0, b(τ) = −i

|a| : 1 → 0 |b| : 0 → 1

  • =

⇒ |↑ → |↓ (NOT-Operation)

slide-38
SLIDE 38

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-39
SLIDE 39

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

slide-40
SLIDE 40

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

Definition ˆ ρ :=

n

  • i=1

pi |φi φi| with

n

  • i=1

pi = 1

slide-41
SLIDE 41

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

Definition ˆ ρ :=

n

  • i=1

pi |φi φi| with

n

  • i=1

pi = 1 Distinguish between mixed / pure state.

slide-42
SLIDE 42

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

Definition ˆ ρ :=

n

  • i=1

pi |φi φi| with

n

  • i=1

pi = 1 Distinguish between mixed / pure state. ˆ ρ describes pure state ⇐ ⇒ ˆ ρ2 = ˆ ρ ⇐ ⇒ Tr(ˆ ρ2) = 1

slide-43
SLIDE 43

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

Definition ˆ ρ :=

n

  • i=1

pi |φi φi| with

n

  • i=1

pi = 1 Distinguish between mixed / pure state. ˆ ρ describes pure state ⇐ ⇒ ˆ ρ2 = ˆ ρ ⇐ ⇒ Tr(ˆ ρ2) = 1 von Neumann-equation: d dt ˆ ρ(t) = −i[ ˆ H(t),ˆ ρ(t)]

slide-44
SLIDE 44

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix

Definition ˆ ρ :=

n

  • i=1

pi |φi φi| with

n

  • i=1

pi = 1 Distinguish between mixed / pure state. ˆ ρ describes pure state ⇐ ⇒ ˆ ρ2 = ˆ ρ ⇐ ⇒ Tr(ˆ ρ2) = 1 von Neumann-equation: d dt ˆ ρ(t) = −i[ ˆ H(t),ˆ ρ(t)] unitary time evolution: ˆ ρ(t) = ˆ U(t, t0)ˆ ρ(t0) ˆ U†(t, t0)

slide-45
SLIDE 45

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix for spin-1/2-systems

✶

slide-46
SLIDE 46

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix for spin-1/2-systems

Bloch-sphere representation ˆ ρ = 1 2

  • ✶ +

P · ˆ σ

  • P = (Px, Py, Pz): expectation value of the spin ˆ
  • S = ( ˆ

Sx, ˆ Sy, ˆ Sz) − → Polarization

slide-47
SLIDE 47

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix for spin-1/2-systems

Bloch-sphere representation ˆ ρ = 1 2

  • ✶ +

P · ˆ σ

  • P = (Px, Py, Pz): expectation value of the spin ˆ
  • S = ( ˆ

Sx, ˆ Sy, ˆ Sz) − → Polarization Note: ˆ ρ describes a pure state if and only if | P| = 1

slide-48
SLIDE 48

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix for spin-1/2-systems

Bloch-sphere representation ˆ ρ = 1 2

  • ✶ +

P · ˆ σ

  • P = (Px, Py, Pz): expectation value of the spin ˆ
  • S = ( ˆ

Sx, ˆ Sy, ˆ Sz) − → Polarization Note: ˆ ρ describes a pure state if and only if | P| = 1 ρ = aa∗ ab∗ a∗b bb∗

slide-49
SLIDE 49

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Density Matrix for spin-1/2-systems

Bloch-sphere representation ˆ ρ = 1 2

  • ✶ +

P · ˆ σ

  • P = (Px, Py, Pz): expectation value of the spin ˆ
  • S = ( ˆ

Sx, ˆ Sy, ˆ Sz) − → Polarization Note: ˆ ρ describes a pure state if and only if | P| = 1 ρ = aa∗ ab∗ a∗b bb∗

  • = 1

2 1 + Pz Px − iPy Px + iPy 1 − Pz

slide-50
SLIDE 50

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-51
SLIDE 51

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence

slide-52
SLIDE 52

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence

A state is called decoherent, if ”its interference is supressed”a.

aMichael A Nielsen and Isaac L Chuang. Quantum Computation and

Quantum Information. 10th Anniversary Edition. Cambridge University Press,

  • 2010. isbn: 978-1-107-00217-3.
slide-53
SLIDE 53

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence

A state is called decoherent, if ”its interference is supressed”a.

aMichael A Nielsen and Isaac L Chuang. Quantum Computation and

Quantum Information. 10th Anniversary Edition. Cambridge University Press,

  • 2010. isbn: 978-1-107-00217-3.

ˆ A := ψ| ˆ A |ψ = (a∗ ↑| + b∗ ↓|) ˆ A(a |↑ + b |↓) = aa∗A11 + bb∗A22 + b∗aA21 + ba∗A12

  • interference term
slide-54
SLIDE 54

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence

A state is called decoherent, if ”its interference is supressed”a.

aMichael A Nielsen and Isaac L Chuang. Quantum Computation and

Quantum Information. 10th Anniversary Edition. Cambridge University Press,

  • 2010. isbn: 978-1-107-00217-3.

ˆ A := ψ| ˆ A |ψ = (a∗ ↑| + b∗ ↓|) ˆ A(a |↑ + b |↓) = aa∗A11 + bb∗A22 + b∗aA21 + ba∗A12

  • interference term

For the density operator: off-diagonal elements vanish

slide-55
SLIDE 55

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Decoherence

A state is called decoherent, if ”its interference is supressed”a.

aMichael A Nielsen and Isaac L Chuang. Quantum Computation and

Quantum Information. 10th Anniversary Edition. Cambridge University Press,

  • 2010. isbn: 978-1-107-00217-3.

ˆ A := ψ| ˆ A |ψ = (a∗ ↑| + b∗ ↓|) ˆ A(a |↑ + b |↓) = aa∗A11 + bb∗A22 + b∗aA21 + ba∗A12

  • interference term

For the density operator: off-diagonal elements vanish For the Bloch-sphere: | P| decreases

slide-56
SLIDE 56

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

✶

slide-57
SLIDE 57

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system.

✶

slide-58
SLIDE 58

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects:

✶

slide-59
SLIDE 59

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession)

✶

slide-60
SLIDE 60

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.)

✶

slide-61
SLIDE 61

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.) Statistical physics: there are no isolated systems → assume (random) interactions of the system with the environment

✶

slide-62
SLIDE 62

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.) Statistical physics: there are no isolated systems → assume (random) interactions of the system with the environment Tr

  • ˆ

ρ2(t)

  • ✶
slide-63
SLIDE 63

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.) Statistical physics: there are no isolated systems → assume (random) interactions of the system with the environment Tr

  • ˆ

ρ2(t)

  • = Tr

ˆ U(t) ˆ ρ0 ˆ U†(t) ˆ U(t)

  • =✶

ˆ ρ0 ˆ U†(t)

slide-64
SLIDE 64

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.) Statistical physics: there are no isolated systems → assume (random) interactions of the system with the environment Tr

  • ˆ

ρ2(t)

  • = Tr

ˆ U(t) ˆ ρ0 ˆ U†(t) ˆ U(t)

  • =✶

ˆ ρ0 ˆ U†(t)

  • = Tr(ˆ

ρ2

0)

slide-65
SLIDE 65

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The problems so far

Von Neumann-equation describes isolated system. Does not produce certain effects: Dissipation of energy (Larmor precession) Change in entropy (| P| = const.) Statistical physics: there are no isolated systems → assume (random) interactions of the system with the environment Tr

  • ˆ

ρ2(t)

  • = Tr

ˆ U(t) ˆ ρ0 ˆ U†(t) ˆ U(t)

  • =✶

ˆ ρ0 ˆ U†(t)

  • = Tr(ˆ

ρ2

0)

→ pure state remains pure for all times

slide-66
SLIDE 66

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-67
SLIDE 67

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

slide-68
SLIDE 68

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

slide-69
SLIDE 69

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

Assume: t = 0: ˆ ˜ ρ = ˆ ρ ⊗ ˆ ρ(E)

slide-70
SLIDE 70

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

Assume: t = 0: ˆ ˜ ρ = ˆ ρ ⊗ ˆ ρ(E) Reduced density matrix: ˆ ρ(t) = TrENV ˆ ˜ ρ

slide-71
SLIDE 71

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

Assume: t = 0: ˆ ˜ ρ = ˆ ρ ⊗ ˆ ρ(E) Reduced density matrix: ˆ ρ(t) = TrENV ˆ ˜ ρ Define the Dynamical map: V (t) : ˆ ρ(0) → ˆ ρ(t)

slide-72
SLIDE 72

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Open system

Assume: t = 0: ˆ ˜ ρ = ˆ ρ ⊗ ˆ ρ(E) Reduced density matrix: ˆ ρ(t) = TrENV ˆ ˜ ρ Define the Dynamical map: V (t) : ˆ ρ(0) → ˆ ρ(t) Lindblad has showna that under certain assumptions, such time evolution can be written as a quantum mechanical master equation preserving the properties of ˆ ρ (positiveness, convexity).

aGoran Lindblad. “On the generators of quantum dynamical semigroups”.

In: Communications in Mathematical Physics 48 (1976), pp. 119–130.

slide-73
SLIDE 73

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Lindblad equation d dt ˆ ρ(t) = −i[ ˆ H(t),ˆ ρ(t)] +

  • µ>0
  • ˆ

Lµˆ ρ(t)ˆ L†

µ − 1

2 ˆ L†

µˆ

Lµ, ˆ ρ(t)

  • = −i[ ˆ

H(t),ˆ ρ(t)] + D[ˆ ρ(t)] =: L[ˆ ρ(t)]

slide-74
SLIDE 74

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Lindblad equation d dt ˆ ρ(t) = −i[ ˆ H(t),ˆ ρ(t)] +

  • µ>0
  • ˆ

Lµˆ ρ(t)ˆ L†

µ − 1

2 ˆ L†

µˆ

Lµ, ˆ ρ(t)

  • = −i[ ˆ

H(t),ˆ ρ(t)] + D[ˆ ρ(t)] =: L[ˆ ρ(t)] Form: Lindblad-operator = unitary evolution + dissipation

slide-75
SLIDE 75

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

The Lindblad equation d dt ˆ ρ(t) = −i[ ˆ H(t),ˆ ρ(t)] +

  • µ>0
  • ˆ

Lµˆ ρ(t)ˆ L†

µ − 1

2 ˆ L†

µˆ

Lµ, ˆ ρ(t)

  • = −i[ ˆ

H(t),ˆ ρ(t)] + D[ˆ ρ(t)] =: L[ˆ ρ(t)] Form: Lindblad-operator = unitary evolution + dissipation ˆ Lµ ... Lindblad operators

slide-76
SLIDE 76

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

slide-77
SLIDE 77

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ?

slide-78
SLIDE 78

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ? ˆ Lµ ∼ ˆ σ−

slide-79
SLIDE 79

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ? ˆ Lµ ∼ ˆ σ− produce amplitude damping

slide-80
SLIDE 80

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ? ˆ Lµ ∼ ˆ σ− produce amplitude damping ˆ Lµ ∼ ˆ σz

slide-81
SLIDE 81

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ? ˆ Lµ ∼ ˆ σ− produce amplitude damping ˆ Lµ ∼ ˆ σz produce phase damping

slide-82
SLIDE 82

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Lindblad operators

What is the meaning of the Lindblad operators ˆ Lµ? ˆ Lµ ∼ ˆ σ− produce amplitude damping ˆ Lµ ∼ ˆ σz produce phase damping Note: one can consider more than one Lindblad operator

slide-83
SLIDE 83

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-84
SLIDE 84

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Amplitude damped Qubit

slide-85
SLIDE 85

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Amplitude damped Qubit

Choose ˆ Lµ = √γˆ σ−

slide-86
SLIDE 86

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Amplitude damped Qubit

Choose ˆ Lµ = √γˆ σ− ˙ Pz = −γ(1 + Pz) ˙ Px = −ω0Py−γ 2Px ˙ Py = ω0Px−γ 2Py

slide-87
SLIDE 87

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Amplitude damped Qubit

Choose ˆ Lµ = √γˆ σ− ˙ Pz = −γ(1 + Pz) ˙ Px = −ω0Py−γ 2Px ˙ Py = ω0Px−γ 2Py Px(t) =

  • − y0 sin(ω0t) + x0 cos(ω0t)
  • e− γ

2 t

Py(t) =

  • y0 cos(ω0t) + x0 sin(ω0t)
  • e− γ

2 t

Pz(t) = −1 + (z0 + 1)e−γt − → amplitude damping occurs (Pz(t → ∞) = −1)

slide-88
SLIDE 88

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Amplitude damped Qubit

  • 1
  • 0.5

0.5 1 2 4 6 8 10 12 14 x(t) / y(t) / z(t) t P

x(t)

P

y(t)

P

z(t)

| P(t)|

Figure : Plot of the time evolution obtained before. γ = 0.05, ω0 = 1. At t = 0, the system was prepared with a polarization 1/ √ 3 · (1, 1, 1)T

slide-89
SLIDE 89

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-90
SLIDE 90

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

slide-91
SLIDE 91

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

Again, use ˆ Lµ = √γˆ σ−.

slide-92
SLIDE 92

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

Again, use ˆ Lµ = √γˆ σ−. Full Hamiltonian ˆ H = ˆ H0 + ˆ HI(t), plug into Lindblad equation

slide-93
SLIDE 93

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

Again, use ˆ Lµ = √γˆ σ−. Full Hamiltonian ˆ H = ˆ H0 + ˆ HI(t), plug into Lindblad equation d dt Px = 2QPz sin(ǫt) − ω0Py −γ 2Px d dt Py = −2QPz cos(ǫt) + ω0Px −γ 2Py d dt Pz = 2Q

  • Py cos(ǫt) + Px cos(ǫt)
  • −γ(1 + Pz)
slide-94
SLIDE 94

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

Again, use ˆ Lµ = √γˆ σ−. Full Hamiltonian ˆ H = ˆ H0 + ˆ HI(t), plug into Lindblad equation d dt Px = 2QPz sin(ǫt) − ω0Py −γ 2Px d dt Py = −2QPz cos(ǫt) + ω0Px −γ 2Py d dt Pz = 2Q

  • Py cos(ǫt) + Px cos(ǫt)
  • −γ(1 + Pz)

→ numerical solution

slide-95
SLIDE 95

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Amplitude Damping Channel

  • 1
  • 0.5

0.5 1 0.5 1 1.5 2 2.5 3 x(t) / y(t) / z(t) t P

x(t)

P

y(t)

P

z(t)

| P(t)| [1.546,-0.945]

Figure : Time evolution of a NOT-Gate with ˆ Lµ = √γˆ σ− (amplitude damping). ω0 = ǫ = 1 (resonance), Q = 1, γ = 0.05.

slide-96
SLIDE 96

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

slide-97
SLIDE 97

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

Use different Lindblad operator: ˆ Lµ = √ λˆ σz

slide-98
SLIDE 98

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

Use different Lindblad operator: ˆ Lµ = √ λˆ σz d dt Px = 2QPz sin(ǫt) − ω0y −2λx d dt Py = −2QPz cos(ǫt) + ω0x −2λy d dt Pz = 2Q

  • Py cos(ǫt) + x cos(ǫt)
slide-99
SLIDE 99

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

Use different Lindblad operator: ˆ Lµ = √ λˆ σz d dt Px = 2QPz sin(ǫt) − ω0y −2λx d dt Py = −2QPz cos(ǫt) + ω0x −2λy d dt Pz = 2Q

  • Py cos(ǫt) + x cos(ǫt)
  • Difference to amplitude damping: no (direct) damping of Pz
  • ccurs
slide-100
SLIDE 100

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

Use different Lindblad operator: ˆ Lµ = √ λˆ σz d dt Px = 2QPz sin(ǫt) − ω0y −2λx d dt Py = −2QPz cos(ǫt) + ω0x −2λy d dt Pz = 2Q

  • Py cos(ǫt) + x cos(ǫt)
  • Difference to amplitude damping: no (direct) damping of Pz
  • ccurs −

→ Phase Damping Channel

slide-101
SLIDE 101

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

NOT-Gate with Phase Damping Channel

  • 1
  • 0.5

0.5 1 0.5 1 1.5 2 2.5 3 x(t) / y(t) / z(t) t P

x(t)

P

y(t)

P

z(t)

| P(t)| [1.571,-0.924]

Figure : Time evolution of a qubit considering a noisy phase damping

  • channel. ω0 = ǫ = 1 (resonance), Q = 1, λ = 0.05.
slide-102
SLIDE 102

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

1 A Simple Model for NOT-Gate

Introduction to Qubits Interaction with Electromagnetic Field (NOT-Gate)

2 Decoherence & Dissipation

The Density Matrix Effect of decoherence

3 Treatment as an Open Quantum System

The Lindblad equation Qubit in presence of dissipation

4 NOT-Gate in Presence of Dissipation

Quantum Channels Entropy change

slide-103
SLIDE 103

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

✶

slide-104
SLIDE 104

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

From probability theory: S = −

i pi · ln(pi)

✶

slide-105
SLIDE 105

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

From probability theory: S = −

i pi · ln(pi)

− → measure information ✶

slide-106
SLIDE 106

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

From probability theory: S = −

i pi · ln(pi)

− → measure information von Neumann-entropy S = −Tr

  • ˆ

ρ · ln(ˆ ρ)

  • ✶
slide-107
SLIDE 107

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

From probability theory: S = −

i pi · ln(pi)

− → measure information von Neumann-entropy S = −Tr

  • ˆ

ρ · ln(ˆ ρ)

  • Note: S = 0 if and only if ˆ

ρ describes a pure state ✶

slide-108
SLIDE 108

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy in Quantum Systems

From probability theory: S = −

i pi · ln(pi)

− → measure information von Neumann-entropy S = −Tr

  • ˆ

ρ · ln(ˆ ρ)

  • Note: S = 0 if and only if ˆ

ρ describes a pure state maximally entangled state: ˆ ρ = 1

2✶ −

→ S = ln(2)

slide-109
SLIDE 109

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Long time evolution, phase damped

  • 1
  • 0.5

0.5 1 20 40 60 80 100 x(t) / y(t) / z(t) t P

x(t)

P

y(t)

P

z(t)

| P(t)|

Figure : NOT-Gate under consideration of a phase damping (ˆ Lµ = √ λˆ σz). Again, ω0 = ǫ = 1 (resonance), Q = 1, λ = 0.05

slide-110
SLIDE 110

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Long time evolution, amplitude damped

  • 1
  • 0.5

0.5 1 20 40 60 80 100 x(t) / y(t) / z(t) t P

x(t)

P

y(t)

P

z(t)

| P(t)|

Figure : NOT-Gate under consideration of an amplitude damping. Again, ω0 = ǫ = 1 (resonance), Q = 1, λ = 0.05

slide-111
SLIDE 111

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy change

0.1 0.2 0.3 0.4 0.5 0.6 0.7 50 100 150 200 S(t) t

Figure : Long time evolution of a NOT-operation under consideration of an amplitude damping. Again, ω0 = ǫ = 1 (resonance), Q = 1, λ = 0.05.

slide-112
SLIDE 112

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Entropy of amplitude damped qubit

0.1 0.2 0.3 0.4 0.5 0.6 0.7 50 100 150 200 S(t) t

Figure : Change in the von Neumann-entropy of a simple qubit (no EM interaction) considering amplitude damping as discussed before. Parameters are Q = 1, γ = 0.05.

slide-113
SLIDE 113

Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate

Thank you for your attention.