Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Solution to time dependent Hamiltonian � � T : Plug into Schr¨ odinger equation for state a ( t ) , b ( t ) i d d t a ( t ) = ω 0 + f ( t ) · b ( t ) i d d t b ( t ) = f ∗ ( t ) · a ( t ) Solution considering EM-field with | φ ( t = 0) � = |↑� cos( β t ) + i α a ( t ) = e − i 2 ( ǫ + ω 0 ) t · � � β sin( β t ) b ( t ) = − Q · i 2 t sin( β t ) β e − i ∆
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate Resonant case: ǫ = ω 0 → ∆ = α = 0 and β = Q .
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate Resonant case: ǫ = ω 0 → ∆ = α = 0 and β = Q . a ( t ) = e − i ω 0 t cos( Qt ) b ( t ) = − i sin( Qt )
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate Resonant case: ǫ = ω 0 → ∆ = α = 0 and β = Q . a ( t ) = e − i ω 0 t cos( Qt ) b ( t ) = − i sin( Qt ) Choose interaction time τ = 2 Q → a ( τ ) = 0, b ( τ ) = − i π
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate Resonant case: ǫ = ω 0 → ∆ = α = 0 and β = Q . a ( t ) = e − i ω 0 t cos( Qt ) b ( t ) = − i sin( Qt ) Choose interaction time τ = 2 Q → a ( τ ) = 0, b ( τ ) = − i π � | a | : 1 → 0 = ⇒ |↑� → |↓� | b | : 0 → 1 (NOT-Operation)
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
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix Definition n n � � ρ := ˆ p i | φ i � � φ i | with p i = 1 i =1 i =1
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix Definition n n � � ρ := ˆ p i | φ i � � φ i | with p i = 1 i =1 i =1 Distinguish between mixed / pure state.
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix Definition n n � � ρ := ˆ p i | φ i � � φ i | with p i = 1 i =1 i =1 Distinguish between mixed / pure state. ρ 2 = ˆ ρ 2 ) = 1 ρ describes pure state ⇐ ˆ ⇒ ˆ ρ ⇐ ⇒ Tr(ˆ
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix Definition n n � � ρ := ˆ p i | φ i � � φ i | with p i = 1 i =1 i =1 Distinguish between mixed / pure state. ρ 2 = ˆ ρ 2 ) = 1 ρ describes pure state ⇐ ˆ ⇒ ˆ ρ ⇐ ⇒ Tr(ˆ von Neumann-equation: d ρ ( t ) = − i [ ˆ d t ˆ H ( t ) , ˆ ρ ( t )]
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix Definition n n � � ρ := ˆ p i | φ i � � φ i | with p i = 1 i =1 i =1 Distinguish between mixed / pure state. ρ 2 = ˆ ρ 2 ) = 1 ρ describes pure state ⇐ ˆ ⇒ ˆ ρ ⇐ ⇒ Tr(ˆ von Neumann-equation: d ρ ( t ) = − i [ ˆ d t ˆ H ( t ) , ˆ ρ ( t )] ρ ( t 0 ) ˆ ρ ( t ) = ˆ U † ( t , t 0 ) unitary time evolution: ˆ U ( t , t 0 )ˆ
✶ Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix for spin-1/2-systems
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix for spin-1/2-systems Bloch-sphere representation ρ = 1 � � ✶ + � P · � ˆ σ ˆ 2 P = ( P x , P y , P z ): expectation value of the spin ˆ � � S = ( ˆ S x , ˆ S y , ˆ S z ) − → Polarization
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix for spin-1/2-systems Bloch-sphere representation ρ = 1 � � ✶ + � P · � ˆ σ ˆ 2 P = ( P x , P y , P z ): expectation value of the spin ˆ � � S = ( ˆ S x , ˆ S y , ˆ S z ) − → Polarization ρ describes a pure state if and only if | � Note: ˆ P | = 1
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix for spin-1/2-systems Bloch-sphere representation ρ = 1 � � ✶ + � P · � ˆ σ ˆ 2 P = ( P x , P y , P z ): expectation value of the spin ˆ � S = ( ˆ � S x , ˆ S y , ˆ S z ) − → Polarization ρ describes a pure state if and only if | � Note: ˆ P | = 1 � aa ∗ � ab ∗ ρ = a ∗ b bb ∗
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Density Matrix for spin-1/2-systems Bloch-sphere representation ρ = 1 � � ✶ + � P · � ˆ σ ˆ 2 P = ( P x , P y , P z ): expectation value of the spin ˆ � � S = ( ˆ S x , ˆ S y , ˆ S z ) − → Polarization ρ describes a pure state if and only if | � Note: ˆ P | = 1 � 1 + P z � aa ∗ � � ab ∗ = 1 P x − iP y ρ = a ∗ b bb ∗ P x + iP y 1 − P z 2
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
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence A state is called decoherent, if ”its interference is supressed” a . a Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information . 10th Anniversary Edition. Cambridge University Press, 2010. isbn : 978-1-107-00217-3.
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence A state is called decoherent, if ”its interference is supressed” a . a Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information . 10th Anniversary Edition. Cambridge University Press, 2010. isbn : 978-1-107-00217-3. A | ψ � = ( a ∗ �↑| + b ∗ �↓| ) ˆ � ˆ A � := � ψ | ˆ A ( a |↑� + b |↓� ) = aa ∗ A 11 + bb ∗ A 22 + b ∗ aA 21 + ba ∗ A 12 � �� � interference term
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence A state is called decoherent, if ”its interference is supressed” a . a Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information . 10th Anniversary Edition. Cambridge University Press, 2010. isbn : 978-1-107-00217-3. A | ψ � = ( a ∗ �↑| + b ∗ �↓| ) ˆ � ˆ A � := � ψ | ˆ A ( a |↑� + b |↓� ) = aa ∗ A 11 + bb ∗ A 22 + b ∗ aA 21 + ba ∗ A 12 � �� � interference term For the density operator: off-diagonal elements vanish
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Decoherence A state is called decoherent, if ”its interference is supressed” a . a Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information . 10th Anniversary Edition. Cambridge University Press, 2010. isbn : 978-1-107-00217-3. A | ψ � = ( a ∗ �↑| + b ∗ �↓| ) ˆ � ˆ A � := � ψ | ˆ A ( a |↑� + b |↓� ) = aa ∗ A 11 + bb ∗ A 22 + b ∗ aA 21 + ba ∗ A 12 � �� � interference term For the density operator: off-diagonal elements vanish For the Bloch-sphere: | � P | decreases
✶ Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The problems so far
✶ Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The problems so far Von Neumann-equation describes isolated system.
✶ 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:
✶ 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)
✶ 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.)
✶ 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
✶ 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 � � ρ 2 ( t ) Tr ˆ
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 � ˆ � � � ρ 0 ˆ U † ( t ) ˆ ρ 0 ˆ ρ 2 ( t ) U † ( t ) Tr ˆ = Tr U ( t ) ˆ U ( t ) ˆ � �� � = ✶
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 � ˆ � � � ρ 0 ˆ U † ( t ) ˆ ρ 0 ˆ ρ 2 ( t ) U † ( t ) ρ 2 Tr ˆ = Tr U ( t ) ˆ U ( t ) ˆ = Tr(ˆ 0 ) � �� � = ✶
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 � ˆ � � � ρ 0 ˆ U † ( t ) ˆ ρ 0 ˆ ρ 2 ( t ) U † ( t ) ρ 2 Tr ˆ = Tr U ( t ) ˆ U ( t ) ˆ = Tr(ˆ 0 ) � �� � = ✶ → pure state remains pure for all times
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
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system Assume: t = 0: ˆ ρ ( E ) ρ = ˆ ˜ ρ ⊗ ˆ
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system Assume: t = 0: ˆ ρ ( E ) ρ = ˆ ˜ ρ ⊗ ˆ Reduced density matrix: ρ ( t ) = Tr ENV ˆ ˆ ρ ˜
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system Assume: t = 0: ˆ ρ ( E ) ρ = ˆ ˜ ρ ⊗ ˆ Reduced density matrix: ρ ( t ) = Tr ENV ˆ ˆ ρ ˜ Define the Dynamical map: V ( t ) : ˆ ρ (0) → ˆ ρ ( t )
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Open system Assume: t = 0: ˆ ρ ( E ) ρ = ˆ ˜ ρ ⊗ ˆ Reduced density matrix: ρ ( t ) = Tr ENV ˆ ˆ ρ ˜ Define the Dynamical map: V ( t ) : ˆ ρ (0) → ˆ ρ ( t ) Lindblad has shown a that under certain assumptions, such time evolution can be written as a quantum mechanical master equation preserving the properties of ˆ ρ (positiveness, convexity). a Goran Lindblad. “On the generators of quantum dynamical semigroups”. In: Communications in Mathematical Physics 48 (1976), pp. 119–130.
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Lindblad equation � �� d µ − 1 � � ˆ ρ ( t ) = − i [ ˆ ˆ ρ ( t )ˆ L † L † µ ˆ d t ˆ H ( t ) , ˆ ρ ( t )] + L µ ˆ L µ , ˆ ρ ( t ) 2 µ> 0 = − i [ ˆ H ( t ) , ˆ ρ ( t )] + D [ˆ ρ ( t )] =: L [ˆ ρ ( t )]
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Lindblad equation � �� d µ − 1 � � ˆ ρ ( t ) = − i [ ˆ ˆ ρ ( t )ˆ L † L † µ ˆ d t ˆ H ( t ) , ˆ ρ ( t )] + L µ ˆ L µ , ˆ ρ ( t ) 2 µ> 0 = − i [ ˆ H ( t ) , ˆ ρ ( t )] + D [ˆ ρ ( t )] =: L [ˆ ρ ( t )] Form: Lindblad-operator = unitary evolution + dissipation
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate The Lindblad equation � �� d µ − 1 � � ˆ ρ ( t ) = − i [ ˆ ˆ ρ ( t )ˆ L † L † µ ˆ d t ˆ H ( t ) , ˆ ρ ( t )] + L µ ˆ L µ , ˆ ρ ( t ) 2 µ> 0 = − i [ ˆ H ( t ) , ˆ ρ ( t )] + D [ˆ ρ ( t )] =: L [ˆ ρ ( t )] Form: Lindblad-operator = unitary evolution + dissipation ˆ L µ ... Lindblad operators
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ?
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ? ˆ σ − L µ ∼ ˆ
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ? ˆ σ − L µ ∼ ˆ produce amplitude damping
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ? ˆ σ − L µ ∼ ˆ produce amplitude damping ˆ L µ ∼ ˆ σ z
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ? ˆ σ − L µ ∼ ˆ produce amplitude damping ˆ L µ ∼ ˆ σ z produce phase damping
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Lindblad operators What is the meaning of the Lindblad operators ˆ L µ ? ˆ σ − L µ ∼ ˆ produce amplitude damping ˆ L µ ∼ ˆ σ z produce phase damping Note: one can consider more than one Lindblad operator
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
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Amplitude damped Qubit
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Amplitude damped Qubit L µ = √ γ ˆ Choose ˆ σ −
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Amplitude damped Qubit L µ = √ γ ˆ Choose ˆ σ − ˙ P z = − γ (1 + P z ) P x = − ω 0 P y − γ ˙ 2 P x P y = ω 0 P x − γ ˙ 2 P y
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Amplitude damped Qubit L µ = √ γ ˆ Choose ˆ σ − ˙ P z = − γ (1 + P z ) P x = − ω 0 P y − γ ˙ 2 P x P y = ω 0 P x − γ ˙ 2 P y � � e − γ 2 t P x ( t ) = − y 0 sin( ω 0 t ) + x 0 cos( ω 0 t ) � � e − γ 2 t P y ( t ) = y 0 cos( ω 0 t ) + x 0 sin( ω 0 t ) P z ( t ) = − 1 + ( z 0 + 1) e − γ t − → amplitude damping occurs ( P z ( t → ∞ ) = − 1)
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate Amplitude damped Qubit x ( t ) P 1 y ( t ) P z ( t ) P | � P ( t ) | 0.5 x(t) / y(t) / z(t) 0 -0.5 -1 0 2 4 6 8 10 12 14 t Figure : Plot of the time evolution obtained before. γ = 0 . 05, ω 0 = 1. At √ 3 · (1 , 1 , 1) T t = 0, the system was prepared with a polarization 1 /
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
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel L µ = √ γ ˆ Again, use ˆ σ − .
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel L µ = √ γ ˆ Again, use ˆ σ − . Full Hamiltonian ˆ H = ˆ H 0 + ˆ H I ( t ), plug into Lindblad equation
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel L µ = √ γ ˆ Again, use ˆ σ − . Full Hamiltonian ˆ H = ˆ H 0 + ˆ H I ( t ), plug into Lindblad equation d t P x = 2 QP z sin( ǫ t ) − ω 0 P y − γ d 2 P x d t P y = − 2 QP z cos( ǫ t ) + ω 0 P x − γ d 2 P y d � � d t P z = 2 Q P y cos( ǫ t ) + P x cos( ǫ t ) − γ (1 + P z )
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel L µ = √ γ ˆ Again, use ˆ σ − . Full Hamiltonian ˆ H = ˆ H 0 + ˆ H I ( t ), plug into Lindblad equation d t P x = 2 QP z sin( ǫ t ) − ω 0 P y − γ d 2 P x d t P y = − 2 QP z cos( ǫ t ) + ω 0 P x − γ d 2 P y d � � d t P z = 2 Q P y cos( ǫ t ) + P x cos( ǫ t ) − γ (1 + P z ) → numerical solution
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Amplitude Damping Channel x ( t ) P 1 y ( t ) P z ( t ) P | � P ( t ) | [1.546,-0.945] 0.5 x(t) / y(t) / z(t) 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 t L µ = √ γ ˆ σ − (amplitude Figure : Time evolution of a NOT-Gate with ˆ damping). ω 0 = ǫ = 1 (resonance), Q = 1, γ = 0 . 05.
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Phase Damping Channel
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Phase Damping Channel √ Use different Lindblad operator: ˆ L µ = λ ˆ σ z
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Phase Damping Channel √ Use different Lindblad operator: ˆ L µ = λ ˆ σ z d d t P x = 2 QP z sin( ǫ t ) − ω 0 y − 2 λ x d d t P y = − 2 QP z cos( ǫ t ) + ω 0 x − 2 λ y d � � d t P z = 2 Q P y cos( ǫ t ) + x cos( ǫ t )
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Phase Damping Channel √ Use different Lindblad operator: ˆ L µ = λ ˆ σ z d d t P x = 2 QP z sin( ǫ t ) − ω 0 y − 2 λ x d d t P y = − 2 QP z cos( ǫ t ) + ω 0 x − 2 λ y d � � d t P z = 2 Q P y cos( ǫ t ) + x cos( ǫ t ) Difference to amplitude damping: no (direct) damping of P z occurs
Simple NOT-Gate Decoherence & Dissipation Open Quantum System Decoherent NOT-Gate NOT-Gate with Phase Damping Channel √ Use different Lindblad operator: ˆ L µ = λ ˆ σ z d d t P x = 2 QP z sin( ǫ t ) − ω 0 y − 2 λ x d d t P y = − 2 QP z cos( ǫ t ) + ω 0 x − 2 λ y d � � d t P z = 2 Q P y cos( ǫ t ) + x cos( ǫ t ) Difference to amplitude damping: no (direct) damping of P z occurs − → Phase Damping Channel
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